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1686 2371

This is to certify that the
dissertation entitled

Engineering Applications of Microbial Chemotaxis

presented by

Mark Thomas Widman

has been accepted towards fulfillment
of the requirements for

 

 

Ph.D. degree in Chemical Engineering

K 7715M Zb/fl/Oék

 

Major professor

S/HK/C/7

Date

 

MSU is an A/lirnmtiw- At'ttnra/It'qiml Opportunity Institution 0-12771

 
 
   
    

 
 

LIBHMY
Mlchlgan State
Unlversity

PLACE IN RETURN BOX
to remove this checkout from your record.
TO AVOID FINES return on or before date due.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1/98 cJCIRC/DateDue.p65—p.14

 

 

ENGINEERING APPLICATIONS OF MICROBIAL CHEMOTAXIS
By

Mark Thomas Widman

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemical Engineering

1997

 

 

ENG

Chemo‘
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chemotaxis to
to develop or r
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ABSTRACT

ENGINEERING APPLICATIONS OF MICROBIAL CHEMOTAXIS
By

Mark Thomas Widman

Chemotaxis is the ability of an organism to move in response to gradients of
chemicals called chemoattractants. Many species of bacteria are known to exhibit
chemotaxis to a wide variety of chemoattractants. The first objective of this research was
to develop or refine tools and methods to study the chemotactic response of bacteria. The
tools and methods include the diffusion gradient chamber (DGC) and its associated units,
the laser diffraction capillary assay (LDCA), and microelectrodes and microbiosensors.
Several mathematical models of the DGC system were written that allowed predictions of
experimental outcomes to be made without running the actual experiments. The modeling
predictions were verified by comparison to experimental data. Methods to analyze the
modeling simulations were developed to further enhance the understanding of the
chemotactic response. These analysis methods included bacterial flux and receptor
saturation calculations.

The second objective of the research was to identify applications or systems in
which chemotaxis was beneficial, and then to develop ways to engineer those systems to
fully take advantage of the benefits offered through the chemotactic response. The three
applications that were selected were microbial competition, selection of mutants, and in

situ bioremediation. Chemotaxis was predicted by the mathematical model to impart a

 

 

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competitive ac'
preliminan' W
testunse wt
(alt. In situ
themotactit t
{initiation of
telngused tor
chemotaxis to
response of P

KC and ESt‘iit’

competitive advantage to a bacterial population in certain non-mixed environments, and
preliminary work was done to experimentally validate the predictions. The chemotactic
response was used as a selection agent in the isolation of desirable mutants of Escherichia
coli. In situ mutagenesis was performed on feedback inhibited E. coli, and the
chemotactic response was used to separate feedback resistant mutants from the
population of inhibited bacteria. Pseudomonas stutzeri strain KC, a bacterium currently
being used to remediate a carbon tetrachloride contaminated aquifer, was found to exhibit
chemotaxis to chemicals present in the aquifer. Experimental protocols to study the
response of Pseudomonas KC around objects, and competition between Pseudomonas

KC and Escherichia coli were developed.

 

 

 

The at
tori reported
performed in;
tort on the
responsible to

1 \soul
with me on 2
those researc
Marshall Bre(

Dr. .\1
{hate reache
DGC and me
microorganisr
ADI algorithr
and Faith Pep

IOWe
thank my wif
has Worked e:

her devotion r

ACKNOWLEDGMENTS

The efforts of many people have been instrumental in helping me to complete the
work reported in this thesis. I am grateful for the help of a number of undergraduates who
performed many of the experiments. Laura Booms and Petty Setiawan both did good
work on the Pseudomonas KC swarm plate experiments. Sebastian Schmidt was
responsible for making the Laser Diffraction Capillary Assay a reality.

I would like to thank my fellow graduate students, Tyler Ames, who collaborated

with me on a project that impacted both of our research programs, and Mark Mikola,

 

whose research on mutant selection led to another use for chemotaxis. Both of them and

Marshall Bredwell provided me with many insights into my research.

 

Dr. Mark Worden has mentored me through my four years, and with his guidance
I have reached the goals that I set for myself. Dave Emerson taught me how to use the
DGC, and more importantly, gave me a good introduction to handling and thinking about
microorganisms, from a non-engineering viewpoint. Chichia Chiu supplied me with the
ADI algorithm used to solve some of the math models. Candy McMaster, Julie Caywood
and Faith Peterson in the graduate office have also been invaluable resources and friends.

I owe a lot to my Mom and Dad, for everything. Most importantly, I would like to
thank my wife, Shelly. She has put up with me all through my graduate school years, and
has worked extremely hard so that I could stay in school. I hope to finally be able to repay

her devotion to me by proving to her that all this was really worth it.

iv

 

LIST OF TAIE
LIST OF FIG
LIST OF NO.‘
1. L\TRODL"

3. OBJECT 1\
3.3 DevelO'
3.3 Appliez

3. CHESIOT:

3.1 The tilt
3.3 Experir
4. DEVELOP

4.1 Improv
4.1.1 1m.
4.1.2 Qu
4.1.3 Me
4.1.4 Fag
4.1.5 Re
4.1.6 Mi

4.2 Derek)
4.2.1 B0
4.22 lni
4.2.3 Ex
4.2.4 C0
4.2.5 Mr
4.2.6 Re
4.2.7 Ef:
4.2.8 CC
4.2.9 A5

4-3Ana1yg
43.1 Fit
43-2 Re
4-3.4 CI.

4'4 Variati
'5 Laserc

5' MICROBI

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................ vii
LIST OF FIGURES .......................................................................................................... viii
LIST OF NOMENCLATURE ........................................................................................... xi
1. INTRODUCTION ............................................................................................................ 1
2. OBJECTIVES AND SIGNIFICANCE ............................................................................ 3
2.1 Deve10pment of tools ................................................................................................. 3
2.2 Applications for chemotaxis ...................................................................................... 4

3. CI-IEMOTAXIS BACKGROUND .................................................................................. 6
3.1 The chemotactic mechanism ...................................................................................... 6
3.2 Experimental systems for measuring chemotaxis ...................................................... 7

4. DEVELOPMENT OF TOOLS ...................................................................................... 11
4.1 Improvements to the DGC method .......................................................................... 11
4.1.1 Image acquisition system .................................................................................. 11
4.1.2 Quantification of cells by grayscale analysis .................................................... 12
4.1.3 Method to measure membrane mass transfer coefficient .................................. 15
4.1.4 Faster approach to steady-state gradients .......................................................... 17
4.1.5 Recycle of source and sink flasks ..................................................................... 20
4.1.6 Microsensors and microbiosensors ................................................................... 20

4.2 Development of model ............................................................................................. 24
4.2.1 Boundary conditions ......................................................................................... 28
4.2.2 Initial conditions ............................................................................................... 30
4.2.3 Experiments used to validate model ................................................................. 30
4.2.4 Computer simulations ....................................................................................... 33
4.2.5 Modeling parameters ......................................................................................... 33
4.2.6 Results of validation simulations ...................................................................... 36
4.2.7 Effects of grid-spacing on simulations .............................................................. 40
4.2.8 Concentration profiles of chemoattractants and nutrient .................................. 40
4.2.9 Assumptions and simplifications in the mathematical model ........................... 47

4.3 Analysis of modeling results .................................................................................... 50
4.3.1 Flux calculations ............................................................................................... 50
4.3.2 Receptor saturation ........................................................................................... 55
4.3.3 Pattern dynamics ............................................................................................... 57
4.3.4 Chemotactic wave in response to oxygen ......................................................... 58

4.4 Variations on original model ................................................................................... 5 8
4.5 Laser diffraction capillary assay .............................................................................. 61
5. MICROBIAL COMPETITION ..................................................................................... 66

 

 

 

5.1 Compe‘
5.3 Compe'
5.3 Modeli.
5.4 Analys:
5.4.1 Th1
5.4.3 Sat

5 5 Compe
5.5.1 E11
5.5.3 \‘a
5.5.3 \‘a
5.5.4 To
5.5.5 \'a
5.5.6 \‘a
5.5.7 Va
56 Future
6. SELECTK

6.1 Expert
6.3Mode1:

BIOREME

7.1 Experi
7.1.1 Sv
7.1.3 D(
7.1.3 M.

7-3 Future

8. SL'MMAF
APPENDIX
APPENDIX
APPENDIX
APPENDIX
APPENDIX
APPENDIX
APPENDIX
LIST OF RE

 

 

 

5.1 Competition background .......................................................................................... 66
5.2 Competition mathematical model ............................................................................ 69
5.3 Modeling parameters ............................................................................................... 71
5.4 Analysis of competition simulations ........................................................................ 73
5.4.1 The dynamic competition factor ....................................................................... 73
5.4.2 The average concentration curve ....................................................................... 75
5.4.3 Saturation of chemoreceptors ............................................................................ 77
5.5 Competition simulations .......................................................................................... 77
5.5.1 Effect of modeling parameters on competition results ..................................... 77
5.5.2 Variation of ans ................................................................................................ 78
5.5.3 Variation of vas .................................................................................................. 82
5.5.4 Variation of KDAS .............................................................................................. 84
5.5.5 Variation of chemoattractant concentration ...................................................... 87
5.5.6 Variation of inoculation conditions ................................................................... 90
5.5.7 Variation of nutrient initial conditions .............................................................. 92
5.6 Future work on competition ..................................................................................... 95
6. SELECTION OF MUTANTS ........................................................................................ 99
6.1 Experimental work ................................................................................................... 99
6.2 Modeling selection of mutants ............................................................................... 100
7. BIOREMEDIATION ................................................................................................... 1 13
7.1 Experimental results .............................................................................................. 115
7.1.1 Swarm plates ................................................................................................... 115
7.1.2 DGC experiments ............................................................................................ 119
7.1.3 Movement of wave around obstacles .............................................................. 123
7 .2 Future work on bioremediation .............................................................................. 126
8. SUMMARY ................................................................................................................. 129
APPENDIX A Miscellaneous chemotaxis experiments .................................................. 131
APPENDDI B The Alternating-Direction Implicit Method ............................................ 135
APPENDIX C Instructions for FORTRAN model .......................................................... 137
APPENDIX D The FORTRAN model ............................................................................ 141
APPENDIX E Input files for model. ................................................................................ 184
APPENDIX F Matlab program files ................................................................................ 187
APPENDIX G Dimensionless model .............................................................................. 196
LIST OF REFERENCES ................................................................................................. 199
vi

 

Table 1.Dilut
Table 3. Para
Table 3. Matt
Table 4. Base
Table 5. Base

Table 6. Para

 

LIST OF TABLES

Table 1. Dilution factor and OD for grayscale calibration ................................................. 13

Table 2. Parameter values for validation of model ............................................................ 36

Table 3. Maximum predicted cell fluxes for Run 1 at 20 hours. ....................................... 51

Table 4. Base case parameters for competition simulations. ............................................. 72

Table 5. Base case initial conditions for competition simulations ..................................... 87

Table 6. Parameter values for inhibition model ............................................................... 106
vii

 

 

 

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Figure 1. Rar.
Figure 3. Rut
Figure 3. Dia
Figure 4. .\'or
Figure 5. Cal
Figure 6. Det
Figure 7. Th1
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Figure 9. Th1
figure 10. E,
Figure 11. L‘
Fitrue 12. st
Figure 13. 31
Figure 14. 81
Figure 15. c
Figure 16. T
Figure 17 It
Figure 18.0
Figure 19. G

Figure 20. C

LIST OF FIGURES

Figure 1. Random walk ........................................................................................................ 6
Figure 2. Runs and tumbles in the presence of a gradient. .................................................. 7
Figure 3. Diagram of the diffusion gradient chamber. ......................................................... 9
Figure 4. Normalized grayscale vs. optical density for E. coli. ......................................... 14
Figure 5. Calibration curve for E. coli in DGC .................................................................. 15
Figure 6. Determination of the membrane mass transfer coefficient. ................................ 16
Figure 7. Three-slab pouring method. ................................................................................ 19
Figure 8. Simulation of approach to steady-state. .............................................................. 19
Figure 9. Three dimensional glucose microsensor calibration plane. ................................ 22
Figure 10. E. coli forming ring in response to glucose. ..................................................... 23

Figure 11. Use of microsensors, microbiosensors, and image analysis, DGC modeling...24

Figure 12. Schematic diagram of DGC, showing modeling coordinates. .......................... 32
Figure 13. Simulation compared to experiment for 1.0 mM aspartate source ................... 38
Figure 14. Simulation compared to experiment for 0.1 mM aspartate source. .................. 39
Figure 15. Cell concentration profiles across the two central axes of the DGC. ............... 40
Figure 16. Top: Typical cell profile. Bottom: nutrient profile. .......................................... 42
Figure 17 Top: Typical S profile. Bottom: Q profile. ........................................................ 43
Figure 18. Oxygen gradients in the DGC arena ................................................................. 44
Figure 19. Glucose gradients at several positions in the DGC .......................................... 46
Figure 20. Comparison of model with and without hyperbolic tangent term. ................... 49
viii

 

 

 

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Figure 31. F1
Figure 33. F1
Figure 33. F1
Figure 34. It
Figure 35 To
Figure 36. 52
Figure 37. F1
Figure 38. 11
Figure 39. C-
Figure 30. I
Figure 31 V;
Figure 31 0
Figure 33_ v
Figure 34. V
Figure 35: v
Figure 36. p.
Figure 37, p
Figure 38. D
Figure 39. C
Figure 40. C

Figure 41_ C

Figure 21. Fluxes due to response to S chemoattractant. ................................................... 52

Figure 22. Fluxes due to response to Q chemoattractant. .................................................. 53
Figure 23. Fluxes due to random motility. ......................................................................... 54
Figure 24. Total mass fluxes. ............................................................................................. 54
Figure 25 Top: 438 for Run 1. Bottom: dig for Run 2. Note different vertical scale ............. 56
Figure 26. Sample simulations from second-generation model. ........................................ 61
Figure 27. Filling the capillary in the LDCA ..................................................................... 63
Figure 28. The random motility coefficient of Pseudomas KC. ........................................ 65
Figure 29. Coordinate axes used for competition analysis. ............................................... 74
Figure 30. Typical simulation, illustrating average concentration curve calculation. ....... 76
Figure 31 Variation of XOAS- ............................................................................................... 79
Figure 32. Other results of variation of 960/43 ...................................................................... 80
Figure 33. Variation of vAs. ............................................................................................... 83
Figure 34. Variation of KDAS. Time=30 h for all figures. .. ................................................ 85
Figure 35: Variation of S source concentration. Time=28 h for all figures. ...................... 89
Figure 36. P0p. A inoculated farther from S source than pop. B. ...................................... 90
Figure 37. P0p. A inoculated to the side and away from S source ..................................... 91
Figure 38. Dynamic competition factors for two inoculation patterns. ............................. 92
Figure 39. Competition for nutrient diffusing in at x = 5 cm. ........................................... 94
Figure 40. Competition between Pseudomonas KC and E. coli in swarm plates. ............. 97
Figure 41. Colony counts in competition experiment. ....................................................... 98
ix

 

 

 

 

 

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Figure 54. C
Figure 55. Sr
Figure 56. Sr
Figure 57. P:
Figure 58. C

Figure 59. T

Figure 42. Mutant selection by chemotactic response (source at top of pictures). .......... 100

Figure 43. E. coli responding to chemoattractant gradient. ............................................. 104
Figure 44. E. coli responding to inhibitor gradient and chemoattractant gradient. .......... 107
Figure 45 . Simulation of experiment shown in Figure 42. .............................................. 109
Figure 46. Mutant appears far back in population. .......................................................... 111
Figure 47. Mutant now chemotactic to S. ....................................................................... .112
Figure 48. Chemotactic rings of Pseudomonas KC at varying nitrate levels. ................. 116
Figure 49. Nitrate concentration = 50 mg/l, varying acetate ............................................ 117
Figure 50. Nitrate concentration = 500 mg/l, varying acetate. ......................................... 118
Figure 51. Pseudomonas KC responding to glucose (left) and aspartate(right). ............. 119
Figure 52. Pseudomonas KC responding to acetate and nitrate in a DGC. ..................... 122
Figure 53. Movement of wave around obstacles in swarm plates. .................................. 123
Figure 54. Chemotaxis of Pseudomonas KC around an object. ...................................... 125
Figure 55. Set-up to make glass—bead wall in DGC. ........................................................ 127
Figure 56. Swarm plates with various chemoattractants .................................................. 132
Figure 57. Pattern formation in the DGC ......................................................................... 133
Figure 58. Chemotaxis through a hollow object. ............................................................. 134
Figure 59. The alternating-direction implicit ethod. ........................................................ 135
x

 

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flux

fluX

total

inhil

LIST OF NOMENCLATURE

nutrient saturation constant

chemoattractant Q saturation constant
chemoattractant S saturation constant

diffusion coefficient of H

diffusion coefficient of Q

diffusion coefficient of S

average maximum flux

glucose concentration

nutrient density

cell flux

cell receptor saturation constant for chemoattractant Q
cell receptor saturation constant for chemoattractant S
inhibition constant

mass transfer coefficient for Q

mass transfer coefficient for S

flux of cells due to chemotaxis to S

flux of cells due to chemotaxis to Q

flux of cells due to random motility

total number of receptors for a chemoattractant

inhibitor P density

xi

 

 

 

chem

nucrc

chern

thne

ceH d

' . rolur

niaxi
maxi

speci

chen
perc
yield
cent
ungl
chen
Chen
rand
rand
Cher

dyn:

ena

chemoattractant Q density

microbiosensor reading

chemoattractant S density

time

cell density

volume of DGC arena

maximum growth rate on H

maximum specific growth rate modified for inhibition
specific chemoattractant consumption coefficient for Q
specific chemoattractant consumption coefficient for S
chemotactic velocity

percent oxygen saturation

yield coefficient for cells growing on H

cell tumbling frequency

single cell swimming speed

chemotactic sensitivity coefficient

chemotactic sensitivity coefficient modified for inhibition

random motility coefficient
random motility coefficient modified for inhibition
chemotactic response factor for chemoattractant S

dynamic competition factor

xii

 

 

 

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1. TNTRC

Non-r
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and modeli

1. INTRODUCTION

Non-uniform distributions of chemical compounds, pH, dissolved oxygen, light
energy, temperature and other factors are common in many microbial ecosystems.
Microorganisms able to position themselves optimally with respect to gradients may have
1 competitive advantage over other organisms. The ability of an organism to move in
response to a gradient of a chemical species is known as chemotaxis. Many bacteria have
men shown to exhibit chemotaxis (Adler, 1972; Berg and Tedesco, 1975; Harwood et al.,
1990; Kato et al., 1990; Macnab, 1987; Yamamoto and Irnae, 1993).

There were two main objectives for this research. The first objective was to
levelop or refine tools and methods to study the chemotactic response of bacteria. The
0013 that were used include the diffusion gradient chamber (DGC), several versions of
nathematical models of the DGC system, the laser diffraction capillary assay (LDCA),
1nd microelectrodes and microbiosensors.

The second objective was to identify applications or systems in which .it might be
>ossible for chemotaxis to be of a beneficial nature, and then to deveIOp ways to engineer
hose systems so as to realize the full benefits offered by the chemotactic response. The
hree applications that were studied were microbial competition, selection of mutants, and
n situ bioremediation.

Section 2 explores the two main objectives in more detail, and describes their
ignificance to the scientific community. Section 3 gives background on the experimental

.nd modeling methods employed by others to study chemotaxis. The results of this

 

 

 

 

 

research 12101e
methods thatr
57 describe I
and the work
srrnrnan 01 1h

Include
hell in the ma
Trier are incl
chemotactic 11
outlines the alt
equations. Apr
Which is prese
program. The I
the quantities p

the model is de

 

2

search project are presented in the next four sections. Section 4 examines the tools and
:thods that were developed to study chemotaxis over the course of the project. Sections
1 describe the applications that were identified for improvement through chemotaxis,
i the work that was done to engineer the systems to make use of chemotaxis. A
nmary of the work, and the conclusions that were obtained are given in Section 8.

Included in Appendix A are results of some interesting experiments that did not fit
11 in the main text of the thesis, or have not been fully analyzed for their importance.
ey are included to show the range of growth patterns that can arise due to the
:motactic response, including rings and geometric patterns. Appendix B briefly
:lines the altemating-direction implicit (ADI) algorithm for solving partial differential
Iations. Appendix C gives instructions for the FORTRAN ADI model, one version of
ich is presented in Appendix D. Appendix E gives the input files for the FORTRAN
gram. The Matlab files used to produce many of the graphs and to calculate some of
quantities presented in the thesis are given in Appendix F. A dimensionless version of

model is developed in Appendix G.

 

 

 

 

2. OBJEC'

3.1 DeveIOj

The f1
modeling met
The diffusion
used in this re
Emerson et at
SIstem. Cons
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responding 1.
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inhibitor grai
quantitative 1r
without actuaj

T0 pn
andmicroorg;
Ihesfiltaramet
the Iager diffr;
OilherandOm

OBJECTIVES AND SIGNIFICANCE

. Development of tools

The first objective of this research was to improve existing experimental and
deling methods for studying chemotaxis, or to develop new methods, as necessary.
e diffusion gradient chamber (Emerson et al., 1994) was the main experimental tool
:d in this research. The DGC was well established and tested for chemotaxis studies by
Ierson et al., but some further enhancements to the method were made to optimize the
:tem. Considerable research was devoted to developing mathematical models of
ferent Operating modes of the DGC system. These modes included one cell population
.ponding to one chemoattractant; two cell populations responding to multiple
:moattractants; and two cell populations responding to two chemoattractants and an
Libitor gradient. These models were experimentally validated and then used for
mtitative interpretation Of experimental results, and for exploring chemotactic behavior
:hout actually running time-consuming experiments.

TO produce useful simulations that accurately modeled real chemical gradients
I microorganisms, parameter values had to be Obtained. Several methods of measuring
36 parameters were developed as part Of the research effort. These methods included
laser diffraction capillary assay (Schmidt et al., 1997) which allowed for measurement
he random motility coefficient, and microelectrodes and microbiosensors (Peteu et al.,

5), Which enabled high—resolution measurements Of gradients of oxygen and glucose.

 

 

 

 

 

2.2 Applie

The St
chemotactic n
has the first s
of bacreria c
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Applications for chemotaxis

The second Objective Of the research was to identify applications in which the
totactic response could be exploited to benefit the application. Microbial competition
he first such application selected. Many studies have examined how two populations
acteria compete in well-mixed environments, but much less is known about
petition in spatially structured systems, where chemical gradients can occur. It is
tthesized that chemotaxis can impart a competitive advantage by allowing cells to
:ion themselves in conditions optimal for their survival. The mathematical model of
)GC provided a means to explore this hypothesis. Preliminary work on experimental
rods to validate the model's predictions was also initiated.

Another application benefitted by the chemotactic response was the selection of
[III strains of bacteria that exhibited a desired trait. Specifically, chemotaxis was used
raw mutant strains of Escherichia coli into a region of the DGC having a higher
entration of an inhibitor than their non-mutagenized progenitors could tolerate. This
Od of selection by chemotaxis resulted in an improved strain of E. coli being
ned for the industrial fermentation of a valuable product.

A third application for which chemotaxis was studied was in situ bioremediation.
)acterium Pseudomonas KC, which is able to degrade carbon tetrachloride without
Icing harmful by-products, was chosen as the model organism. Pseudomonas KC is

My being used in Schoolcraft, MI, to bioremediate a contaminated aquifer. The

 

 

 

 

 

chemotactic 1

medium on m

 

otactic behavior of Pseudomonas KC was studied, and the effects of a porous

um on motility and transport were observed.

 

 

 

3. CHEMI

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the Presence

 

 

IEMOTAXIS BACKGROUND

he chemotactic mechanism

otility and chemotactic response Of the enteric bacteria Escherichia coli and
rella typhimurium have been studied in detail (Berg and Brown, 1972; Macnab,
When these peritrichously flagellated bacteria spin their flagella counterclockwise,
gella move in a synchronized bundle, causing the cell to swim forward, or run.
the cell reverses the spin of the flagella, the bundle separates, causing the cell to
and randomly reorient itself before running in a new direction. The overall effect
series of runs followed by tumbles is referred to as a random walk, as shown in

1.

Figure 1. Random walk

.vironment with no chemoattractants, the run length is independent of direction. In

ence of a chemoattractant gradient, the cells monitor the time rate of change of

 

 

 

occupancy t
concentratior
extended run
frequency ap
result is a bin

in Figure 3.

3'2 EXperi

EXper
assay (Adler,

1972; NOSSal

(SFDCI (Fort

7

upancy of receptors for the chemoattractant during a run. If an increasing
centration gradient is encountered, the tumbling frequency decreases, causing an
ended run length. If a decreasing concentration gradient is encountered, the tumbling

uency apparently remains at the basal level (Berg, 1988; Macnab, 1987). The net

 

tlt is a biased random walk toward the higher chemoattractant concentration, as shown

igure 2.

    

 

Low I I . I : High;

Concentration

Figure 2. Runs and tumbles in the presence of a gradient.

Experimental systems for measuring chemotaxis

Experimental systems used to study microbial chemotaxis include the capillary
(Adler, 1972; Nikita er al., 1992; Staffeld et al., 1987), the motility plate (Adler,
; Nossal, 1972; Wolfe and Berg, 1989), and the Stopped Flow Diffusion Chamber

C) (Ford, 1992; Ford et al., 1990). In the capillary assay, the open end of a capillary

 

 

 

 

 

filled with bt
for a fixed at
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thiferent co
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:d with buffer solution containing a chemoattractant is inserted into a cell suspension

a fixed amount of time. The capillary is then removed, and the number of cells that

_.>

fired the capillary are counted. By comparing this number for capillaries having
Lrent concentrations of chemoattractant, chemotaxis can be quantified. A
ematical model of the capillary assay has been developed (Rivero-Hudec and
enburger, 1986) that allows calculation of two important modeling parameters, the
om motility coefficient and the chemotactic sensitivity coefficient.

A motility plate consists of a petri dish containing a semi-solid agar medium
ng a uniform distribution of a consumable chemoattractant (Adler, 1972). The center
le plate is inoculated with cells. As they grow and consume the chemoattractant, a
ient is formed that induces migration of the cells outward from the inoculation zone.
motility plate is a simple system to use, but, because the gradient is formed by
tlar metabolism, the gradient is difficult to control and quantify. Also, the
iotactic response to non—metabolizable chemoattractants cannot be studied in a
city plate.

The SFDC is a rectangular chamber into which two impinging streams Of medium

atroduced. One stream contains the chemoattractant, and the other contains the cells

terest. When the flows are suddenly stopped, a step—change in chemoattractant is

 

gximated where the two streams intersect. Diffusion of the chemoattractant creates

(ft 11

<2

:itially steep gradient that decays over time. The chemotactic migration is measured

h
ght diffraction. A mathematical model of the system has been developed to calculate

 

 

 

 

the chemoatt
cells.

.\'one
tell characte
hate develop

microbial che

previOUSIy PU

 

9
chemoattractant gradients and to analyze the resulting chemotactic mi gration Of the
s.
1 None of the experimental systems described above are convenient for establishing

1 characterized, steady-state gradients or multiple gradients in multiple directions. We

3 developed the Diffusion Gradient Chamber (DGC), shown in Figure 3, for studying

 

lrobial chemotaxis under such conditions. Details of the DGC system have been

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Reservoir
j n 11 l n,
e _________________ I e 4
ET Arena ("m a
2 er— ‘5: ......
E _ __'. _ gm ES
- . _________________ , ’t
e 2 t 9 Gasket
E1 1 U El Semi— ermeoble
/ \ mem rane
Inlet Outlet

Figure 3. Diagram of the diffusion gradient chamber.

ously published (Emerson et al., 1994), so only a brief description is given here. The

consists of a square arena (5 cm x 5 cm x 1 cm) bounded by a reservoir on each

 

 

 

 

«in?

side. Each re
byan imperr
arena and res
gel through
chemoattract
concentratior
concentratior
across the gr
then the char
monitored lit
The 1
31111111316 gm
Chamber. GT
inoculation,
Simultaneous
Tilt DGC ha
microsengOr
inOCUIEITIOn
i“Oculation i
maintain an .

gaseous reag

 

10

side. Each reservoir is separated from the arena either by a semi-permeable membrane, or
by an impermeable silicone sheet, depending on whether substrate diffusion between the
arena and reservoir is desired. The arena is filled with medium containing dilute agarose
gel through which the cells can readily swim. Different concentrations of the
chemoattractant(s) are maintained in the reservoirs. Diffusion from the higher
concentration reservoir (the source) through the membrane and across the gel to the lower
concentration reservoir (the sink) results in the development of a continuous gradient
across the gel. The gradient is allowed to establish for a specified amount of time, and
:hen the chamber is inoculated. Growth and movement of the microbial populations are
nonitored from above by light diffraction.

The DGC has several advantages over other methods of studying chemotaxis.
vIultiple gradients in different directions may be established simultaneously in the
:hamber. Gradients may be allowed to approach a linear steady-state profile before
noculation, or transient gradients can be used. These gradients can be initiated
imultaneously, or in a staggered fashion to simulate pulsed influxes of chemoattractants.
The DGC has been designed with a removable lid so that samples may be withdrawn or
microsensor readings taken without sacrificing the experiment. Many different
moculation protocols may be used, including uniform inoculation across the gel,
loculation in a line, or at a point. Gas ports in the chamber provide the possibility to
laintain an inert gas headspace above the arena for anaerobic experiments, or to supply

aseous reagents to the microbes.

 

 

 

 

4. DEVEI
4.1 lmpro

4.1.1 Image
In prr
the DGC wet
was laboriot
course of [In
were detelo;
The f
rcx ccnc
The Pulnix c
‘Qllanta, Mc
Written by Se
Chemotactic
Smdiertarbeit
PICiures at pp
4‘0 for Wind
Two
ca[Horas (C01

other to a P 0‘

 

DEVELOPMENT OF TOOLS

Improvements to the DGC method

1 Image acquisition system

In previous work done in the DGC system (Emerson et al., 1994), photographs of
DGC were taken manually with a 35 mm camera. This method gave good results, but
laborious and did not lend itself well to computerized image analysis. Over the
.‘se of the work presented in this thesis, several automated image analysis systems
:developed.

The first computer-based image analysis system was built using a PULNiX TM-
1 CCD-camera (PULNiX America Inc., Sunnyvale, CA) mounted above the DGC.
Pulnix camera was attached to a PC through a WinVisionPro video capture board
tnta, Mountain View, CA). A time-lapse capture program called AutOCap was
en by Sebastian Schmidt ("Development of Novel Methods to Measure Random and
notactic Microbial Motility at the Community Level", Sebastian Schmidt,
ienarbeit, Michigan State University, 1995) and used to automatically capture
res at pre-set intervals. Captured images were converted into RAW format in Matlab
)r Windows, and then analyzed.

Two other image analysis systems were installed based on Color QuickCam
ras (Connectix, San Mateo, CA). One Quickcam was connected to a PC, and the
to a Power Macintosh 7200/90. The software included with the Quickcams allowed

11

 

 

 

for timed-in
PL'LViX sys

afreetrare ir

4.1.2 Quan

To q
curve relatir.
has grown
glycerol for
were taken
madmac
acquired. Tl
(lithe cultur
bar. Anothe
medium, 0p
added [0 1111
IHOCulated, '

The
1116 1:100,()(
average but

12
ned-image capture. Images from these units, and, later in the research, from the
liX system, were analyzed using NIH Image (http://rsb.info.nih.gov/NIH—I1VIAGE/),

ware image analysis program.

Quantification of cells by grayscale analysis

To quantify the number of cells in a DGC through image analysis, a calibration
relating cell number to image grayscale was developed. Escherichia coli HCB 33
;rown in M63 medium (described in more detail in Section 4.2.3) with 5 mM
01 for 30 hours at 30°C. Replicate measures of the optical density of the culture
taken at a wavelength of 630 nm. Forty milliliters of the cell culture were then
to a clean DGC containing no agar, and three consecutive images of the DGC were
ed. The DGC was then emptied and cleaned for the next reading. A 0.1 ml aliquot
culture was placed in triplicate on LB agar plates and spread uniformly with a glass
nother volume of the cell culture was then diluted by a factor of 1:1.5 with M63
m, optical density measurements were taken, 40 ml of the diluted solution was
to the DGC, three more images were acquired, and three more LB plates were
Ited. This process was repeated seven times at the dilutions shown in Table 1.

The plates from each dilution were incubated at 30°C for 18 hours. At this time,
00,000 dilution plates had the most suitable number of colonies to count. The
a number Of colonies in each plate was 448, with a standard deviation of 36

s. The original culture therefore contained approximately 4.48><108 cells/ml.

 

Table 1. Dilution factor and OD for grayscale calibration

13

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dilution Optical Density
(ml cells/ml total) (7t=630 nm)

no dilution 0.762 0.789
1:1.5 0.559 0.577
1:2 0.453 0.424
1:3 0.327 0.299
1:4 0.322 0.270
1:5 0.191 0.192
1:10 0.082 0.066
1:15 0.054 0.041
1:20 0.025 0.028
1:100 -- --
1:1000 -- --
1:10,000 -- --
1:100,000 -- --

 

vlatlab 4.0 for Windows was used to analyze the images. The image was

:d to RAW format with Image Alchemy, and then read into Matlab as an image

 

 

 

 

mefleme
imgfima
dilution. Fin:
tononnahze
meraged at

ihmmdm

Tm

DGC.Thn
wfihiyhr.
[his (IISSen

mmmmw

 

14
. The average grayscale value of each picture was calculated in Matlab. The three
.ges for each dilution were then averaged together to give one grand average value per
ttion. Finally, each grand average was divided by the maximum of the grand averages
tormalize the values into a range from 0 to 1. The Optical density readings were also
raged at each dilution. The relationship between optical density and grayscale is

strated in Figure 4.

 

 

Normalized grayscale

 

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6
Optical Density

 

Figure 4. Normalized grayscale vs. Optical density for E. coli.

The normalized grayscale was next plotted against the number of cells in the
C. This yielded a linear calibration, as shown in Figure 5. Other work in collaboration
1 Tyler Ames, a Ph.D. candidate in Dr. Worden's research group, but not included in

dissertation, has also shown that soybean plant cell concentration has a linear

tionship with grayscale in the DGC (Ames, 1997).

 

 

4.1.3 Metht

An e
the membra
following Se
arena by a s
Was iTXtd in
[01116 grour
harden, The
take Place in
ahOFiZOrltal

A su

3000 “1L of

 

 

 

 

 

 

15

1
9 0.8 ~-
8
9
<6 -_
a 0.6
.0
a
g 0.4 ~-
E
o
2 0.2

0 l l l 7

O 1 2 3 4 5

# Of cells x 10'8

Figure 5. Calibration curve for E. coli in DGC.

3 Method to measure membrane mass transfer coefficient

An experiment was designed to measure the mass transfer coefficient, ks, across
membrane. This parameter is important in the modeling work introduced in the
wing sections. A DGC system was prepared having one reservoir separated from the
a by a semi-permeable membrane, and the other three sealed. After the membrane
fixed in the chamber, the chamber was stood on one side with the membrane parallel
1e ground. 400 11L Of 0.3% agar were spread onto the membrane and allowed to
en. The agar layer was applied to duplicate any fouling of the membrane that could
place in a DGC experiment. After the agar layer solidified, the DGC was returned to
rizontal position, and filled with 45 ml of RO water.

A sucrose solution was pumped through the reservoir from a flask containing

lmL Of 30 g/L sucrose. The solution was recycled back into the 3000 mL flask. A stir

 

 

 

bar was plac

model for thi

where S is Il
A is the are.
concentratio

5:0 and line

Sugar cone.

measuremer

7’

_‘
ix.)

In (Ssource/(Ssourco-S))
O _(D
51>. or)

0%

16
;ar was placed in the DGC, and the DGC sat on a magnetic stir plate. The mathematical
odel for this system is
l

V flicks/us —S) (1)

arena source

Where S is the concentration of sugar in the DGC arena, Varena is the volume of the arena,
x is the area of the membrane (3.06 cmz) available for mass transfer, and Ssource is the

oncentration in the source flask. Equation ( 1) can be solved with the initial condition

=0 and linearized as

 

—S V

source arena

2
1n[ Ssource J: kSA t ( )
S .

ugar concentrations were measured over time by HPLC analysis. The results of

leasurements for both sucrose and glucose are shown in Figure 6.

 

 

 

  
  
 

 

 

 

 

 

 

 

Sucrose Glucose
1.6 1.6
53 1‘2 k=o.28 chh :3 1-2 k=0.31 cm/h
25 08 - £3: 03 l
E
3 R2 = 0.9988 3 R2 = 0.9977
‘5? 0-4i 9’; 0.4 l
.5 2
0 I l —*r 0 cr 1; l l
O 20 4O 6O 80 O 20 4O 60 80
time(hr) time(hr)

Figure 6. Determination of the membrane mass transfer coefficient.

 

 

 

4.1.4 Pastel

In St
conventiona'
arena. \l'hel
process beg
mathematicz

state diffusit

Where 3 is 1

Equation l 3

Where \ 13

boundary CO

17

f4 Faster approach to steady-state gradients

!

. In some instances, steady-state gradients were desirable in the DGC. In the

Eventional Operating mode, gel without the gradient chemical was poured into the

na. When the gradient chemical was introduced into one reservoir, the diffusion

 

Fess began, and eventually a linear steady-state gradient would be established. A
hematical model of the system was developed to describe this process. The unsteady-

e diffusion equation, for the case with no cells present, is given by

35
——- = D V25
8: 5

(3)

:re S is the gradient chemical and D3 is the diffusion coefficient. In two-dimensions,

.ation ( 3) can be rewritten as

825 825 (4)
a = . 55*?

;re y is the direction parallel to the gradient and z is the vertical direction. The

dary conditions at the membranes are given by
I

 

( 5)
’35 = 1(5— (SIFO " SSink)
ay D.
he sink reservoir (position y=0 cm) and
( 6)

(3‘ —k
$16 = —D-: (S|y=5 — SSource)

 

 

 

hhwm
assumed to

boundary CO

The
equation iii
eoui'entiona
steady-state.
state solutic
solution at 1
within 95‘}
necessary tc

A ne
achieve the
Work refere
schematical
COttcentratic
Wired and
“0 gradient

50% 0f the

18
' the source reservoir (position y=5 cm). The source and sink flask concentrations were

;umed to be constant throughout the experiment. For all other boundaries, the no-flux

 

undary condition was applied

l
sts = o (7)

The Matlab Partial Differential Equation Toolbox was used to solve the diffusion
1ation with a finite element method. The predicted time to reach steady—state for the
aventional method was 33 days, using sucrose as the model chemical. To define

ady-state, the elliptical equation was solved by the tool box to give the actual steady-

 

te solution. Then the parabolic equation was solved, and compared to the elliptical
ution at several time points. The first time point at which the parabolic solution was

bin 95% of the elliptic solution at every node in the DGC was defined as the time

 

:essary to reach steady-state.

A new method of pouring gel slabs was developed to reduce the time necessary to
.ieve the steady-state gradient. This method was developed as part of the plant cell
rk referenced in Section 4.1.2. The new method of pouring the slabs is shown

ematically in Figure 7. The DGC was tilted at an angle and Layer 1, containing a

 

centration of the gradient chemical equal to that used in the source reservoir, was

Ulmlll

fitted and allowed to solidify. The DGC was returned to level, and Layer 2, containing

h‘Illl

“gradient chemical, was poured and allowed to solidify. Finally, Layer 3, containing

6 of the gradient chemical concentration used in Layer 1, was poured.

 

 

The Matlab

distance frorr

 

19
_U‘r—— 50m —————>l

 

 

0.4 cm ayer nocu um T
T Layer 2 2.2 cm
2 Layer 1

 

 

 

L,

Figure 7. Three-slab pouring method.

Matlab simulation of this system, depicting the concentration as a function of

tnce from the sink reservoir, is shown in Figure 8 for three time-points.

'20
10

   

0
Sucrose (g/l) 5 hr

 

012345

Position (cm)

Figure 8. Simulation of approach to steady-state.

:ime necessary to reach steady—state for the three—slab pouring method is predicted to

1 hours, which is approximately 20 times less than the conventional pouring method.

 

4.15 Recy

Ano
flasks. In t
through the
method wet
the flasks a
flask m
concentratit
lift/week
result is spt
{185k conce

mode.

“5 Micr

Mic.
Worden 10 1
01.. 1996a;
microelecm
"HCFOdectn
conslimes g

miceoelecm

20

5 Recycle of source and sink flasks

Another in the operating method was the use of recycle in the source and sink
ks. In the original DGC method, fluid is pumped out of the source or sink flask,
>ugh the correSponding reservoir on the DGC, and then to a waste container. This
thod worked well, but the time an experiment could run was limited by the volume of
flasks and the pump speed. The idea of recycling medium from the reservoir to the
;k was tested. Sucrose was the gradient chemical, at an initial source flask
reentration of 20 g/L. In six different experiments, only 0.26i0.13 g/L-week (or
%/week) were lost from the flasks due to transfer across the membrane. Although this
ult is specific to the plant cell system and to sucrose, it suggests that in general, the

3k concentrations will not change significantly if the system is operated in recycle

de.

.6 Microsensors and microbiosensors

Microsensors were developed by David Emerson, Serban Peteu, and Mark
rden to measure oxygen, glucose and other chemicals (Peteu et al., 1996; Emerson et
1996a; Emerson et al., 1996b). Briefly, the microsensors are Clark-type oxygen
roelectrodes. The addition of an enzyme, such as glucose oxidase, to the tip of the
oelectrode produces a microbiosensor. The enzyme catalyzes a reaction that
umes glucose and oxygen. The rate of disappearance of oxygen is measured by the

>electrode, and allows for the glucose concentration to be measured.

 

 

A P<
for the glue
linked to th
hmmm
function of

lit the data I

Where R is
percent oxy
least-square
Parameters

calibration l

 

21
A portion of this research project was devoted to deve10ping a calibration method
the glucose microbiosensors. Because the reaction at the tip of the microbiosensor is
ked to'the oxygen concentration, calibrations at varying oxygen concentrations had to
performed. A group of microbiosensor readings, in picoamps, was obtained as a
action of percent oxygen saturation and glucose concentration. The simplest model to

the data to is a linear plane, whose equation is given by

R=aG+bX+d (8)

here R is the microbiosensor reading, G is the glucose concentration, and X is the
:rcent oxygen saturation. The Solver tool in Microsoft Excel 7.0 was used to find the
ast-squares best fit between the experimental points and the model, by varying the
trameters a, b, and d simultaneously. Four isoclines of the resulting three-dimensional

libration plane are shown in Figure 9.

 

 

An
microbiose;
dmuhaneOI
medlUm W;
for m0f€ dE

bacteria hat

22

 

 

 

400
00 mM glucose 0
A136 mM glucose
2 __ x476 mM glucose
9; 300 [1816 mMglucose
8’
1.; Symbol: experiment 0 -
9 200 “c Line = best fit plane X
8
8
3 ‘ II
a 100 —— . A "
° A El
0 l + l #1 r
O 20 40 60 80 100

% 02 saturation

Figure 9. Three dimensional glucose microsensor calibration plane.

An example of an experiment in which the oxygen microsensor, the glucose
icrobiosensor, and the image analysis system for cell grayscale were used
nultaneously is shown in Figure 11. A swarm plate containing 1 mM glucose in M63
:dium was inoculated with 10 {AL actively growing E. coli HCB 33 (see Section 4.2.3

f more details on medium and strain). After approximately 24 hours of growth, a ring of

:teria had formed, as shown in Figure 10.

 

 

 

klicrosensc
An image V
11 Shows

chemotactit

 

23

 

Figure 10. E. coli forming ring in response to glucose.

Microsensor and microbiosensor readings were taken across the ring at a depth of 1 mm.
An image was recorded of the swarm plate with the Pulnix image analysis system. Figure
11 shows the combined results of the sensor readings and the image analysis. The

chemotactic wave (see Section 4.2) is visible where the gradients have the steepest slopes.

 

Figure l

4.2 Den

The
Coupled CC

or Carbon 3

Where Ju i
lllllfiem (1‘
review, St

migration

24

 

 

 

 

 

 

 

1200 30
_ c‘a" ”70
100° ° n “ ,. Edge of wave
A t‘ _ 60 %
O
E 800 -— .. ii a
"’ - 50 9
_o' .. <5
C H o .0
8 600 " __ 40 g
‘1’ N
8 n __ 30 O
.3 400 .- +Glucose g
‘9 +oxygen .. ' Inoculation__ 20 o\..
200 __ —Cell Profile 90"“
- 10
U
0 . . “_-..___: : : :: : : , , 0
-15 -1o -5 0 5 10 15 20

Distance (mm)

Figure 11. Use of microsensors, microbiosensors, and image analysis, DGC modeling

4.2 Development of model

 

The mathematical model of the Diffusion Gradient Chamber system consists of
coupled conservation equations for the microbes, the chemoattractant(s), and the nutrient

or carbon source. The cell balance equation may be written as

a (9)
§=—V-Ju +f(H)u

where Ju is the cell flux, 11 is the cell density, and f(H) is a function for cell growth on a
nutrient (H). Many forms of the flux equations for chemotaxis have been suggested (for a
review, see Ford, 1992). A commonly used constitutive equation to describe cell

migration at the population level is that derived by Keller and Segel (1971):

 

there it is
model pres

chemotactir

there \"',,5

chemotactir

The
the RTBL r
W has be
IItodel for
assumes th;
to the Spa

(Flymjer, e.

 

25

J, =—uVu+Vuu (10)

where u is the random motility coefficient and Vu is the chemotactic velocity. For the

model presented here, the Keller-Segel equation has been modified by adding a second

chemotactic flux, to give

Ju =—,uVu+VuSu+V,Qu (11)

l

where Vus is the chemotactic velocity in response to a chemoattractant S, and VUQ is the

chemotactic velocity to a chemoattractant Q. Combining Equations (9) and ( l 1) gives

au
8:

(12)

 

=uV2u—V-(V.su)—V-(V.Qu)+f<H>u

The chemotactic velocity equation prOposed by Rivero, et al. (1989) (known as
the RTBL model) is strictly applicable only to one-dimensional movement of the bacteria,
but has been shown to give good agreement with a more rigorous three-dimensional
.nodel for a range of parameter values (Frymier, et al., 1992). The RTBL model also
assumes that the temporal gradient of chemoattractant has a small contribution compared
:0 the spatial gradient, and evidence to support this assumption has been provided

:Frymier, et al., 1994). The RTBL chemotaxis term is given by

(13)
VS =vtanh w—MVS}
“ (K0, +5)

 

 

 

where t) is
total numbr
for the rec

velocity car

where 105.

terms of llll

A similar e

Where X00
dissoeiatio
Chemotacti
from man
that both c

assay

 

26
here 1) is the swimming speed of a single cell, 0 is the tumbling frequency, NTS is the
ital number of receptors for the chemoattractant S, and KDS is the dissociation constant
tr the receptor-S complex. For shallow gradients, the modified RTBL chemotactic

:locity can be approximated by

KDS VS (14)
(Km + Sf

 

Vus = 105

here X08: the chemotactic sensitivity coefficient to the attractant S, can be expressed in

arms of individual cell parameters by

X05 =002Nrs (15)

similar equation can be written for the chemoattractant Q as

K (16)
DQ 2 VQ
KDQ+Q)

 

VuQ = log (

here XOQ is the chemotactic sensitivity coefficient to the attractant Q and KDQ is the _
ssociation constant for the receptor-Q complex. Rivero et al. (1989) showed that the
remotactic sensitivity coefficient and the random motility coefficient can be calculated
om individual cell parameters, and Rivero-Hudec and Lauffenburger (1986) showed
at both can be measured experimentally through population assays such as the capillary

say.

 

 

 

The
and (16) w

yield

6+“er
dt
where v is
half-saturat
chemotactit
better thar
reproducet
achieved g
coli to simt

The
Monod—[yp
Chemoattra
“Puke. Th
nutrient (l

chemoattra

The

 

27
The cell balance used for the DGC system is derived by combining Equations (14)

d (16) with the random motility flux term and a term for growth (on substrate H) to

:1d

9a K K vH (17)
_= V2 _ 0 —L— _ . DQ
9t ’1 u ZOSV [[(Kos + S)2 }VS] XOQV [[(KDQ +Q)2 }VQ]+ C0 + H u

tere v is the maximum specific growth rate of the cells growing on H, and C0 is the

 

 

lf-saturation constant for growth on H. The assumption of simple additivity of the

:motactic responses to multiple attractant gradients has been shown to be as good or

 

:ter than more complex interaction models, but lacks the ability to completely
iI‘OdllCC experimental observations (Strauss, et al., 1995). Boon and Herpigny (1986)
rieved good agreement with experimental results when modeling the response of E.

7i to simultaneous gradients of glucose and oxygen using this assumption.

 

The cell growth term and the chemoattractant consumption terms are modeled as
mod-type (Bailey and Ollis, 1986) saturation processes. Growth of cells due to
:moattractant uptake is assumed to be negligible compared to growth due to nutrient
.ake. This assumption is reasonable for many experimental systems because the

rient (usually glycerol) is present at a much higher concentration than the

mo attractant .

The two chemoattractant balances are given by

1255 u (18)

as 2
———= VS-
31‘ D5 CS+S

 

 

and

where D; a
chemoattra
consumptit

The

Where DH ;
0n H.

In 1
diffusion c
aniougly

4-2-1 Bou

A 1'
illustrate tl
2'5 cm, am

on all sou,

 

28

vQQ (19>

 

where D5 and DQ are the chemoattractant diffusion coefficients; vs and VQ are the specific
chemoattractant consumption coefficients; and Cs and CQ are the saturation constants for
consumption of S and Q, respectively.

The nutrient balance is given by

a—H-=DHV2H—
3t

vH L (20)
C0 + H YH

 

where DH is the nutrient diffusion coefficient, and YB is the yield coefficient for growth

 

on H.

In both of the chemoattractant balances and in the nutrient balance, the respective
diffusion coefficients are assumed to be constant and the medium isotropic. We have
previously confirmed that substrate diffusion into the DGC is accurately described by the

model (Emerson 81‘ al., 1994).

1.2.1 Boundary conditions

A two-dimensional schematic representation of the DGC is shown in Figure 12 to
,llustrate the geometric parameters used in the mathematical model. The dimension R is
2.5 cm, and r is 2 cm. For the cell balance, a zero total flux boundary condition is applied

in all boundaries (Q) of the chamber. This boundary condition, Shown below, states that

 

the overall

[02610 on 1

Net
boundaries
diffuse ac

transfer. T]

where Sm
resfirt'oirs,
Penneable

Ou

Walls of [h

confllilOn

 

29
3 overall flux, given by the sum of the random motility and chemotaxis fluxes, is equal

zero on the boundaries.

2 KDS KDQ _ (21)
{#V u ‘Zosv'[[(1{m +S)2 lVSl—ZOQV-lllKDQ +Q)2 }VQlln - O

Neumann boundary conditions are used for the chemoattractant S on the

 

 

undaries where the reservoir is open to the arena. The chemoattractant is assumed to
fuse across the semi—permeable membrane, which imposes a resistance to mass

nsfer. This boundary condition is written as

( 22)

 

 

 

.ere Sres,S and Stem are the concentrations of the chemoattractant in the south and north

ervoirs, respectively, and ks is the mass transfer coefficient for S across the semi-

meable membrane.

Outside the reservoir openings, there is no flux of chemoattractant across the

118 of the chamber:

35 (23)

85 __ :0
832%

3y

=0 and

 

 

1‘0

:he reservoir is sealed with a non—permeable membrane, then the no flux boundary

tdition would apply across the entire side. The boundary conditions for Q and H are

 

 

 

similar ma
thesis. the
initially pr:

C 01le Ilil at.

4.2.2 Init
h
inoculated

shape of ti

Where up, i:
mdw'we
exPerimen
lhal X':0 3

Th
initial cop
Concentrat

XWME(

413E“

Ex

Sirepmmy‘

3O
irnilar mathematically to those of S. For all simulations presented in this section of the
hesis, the concentration of Q in each Open reservoir was equal to the concentration of Q
nitially present in the arena, and the concentration of H in each reservoir was equal to the

oncentration of H initially in the arena.

..2.2 Initial conditions

In the experiments used to validate the model, the center of the chamber was
aoculated with a micropipette. An exponential function was used to approximate the

hape of the injected cell peak:

 

u (24)

11(X’, I): 0
y exp(w1/x’2 + y’z)

 

rhere uo is the initial concentration of cells and w is a peak width factor. The values of uo

nd w were calculated to yield the same number of cells in the peak as were added

 

xperimentally (~4x107 cells, Emerson et al., 1994). The variables x' and y' are defined so
rat x'=0 and y'=0 occur at the center of the arena.

The initial condition for the chemoattractant S is S(x,y)=0 for all x and y. The
titial condition for H is that H(x,y)=Ho for all x and y, where H0 is the initial
)ncentration of H in the arena. The initial condition on Q is that Q(x,y)=Q0 for all x and

. where Q0 is the initial concentration of Q in the arena.

2.3 Experiments used to validate model
Experiments in the DGC were carried out as described in Emerson et al. (1994). A

reptomycin resistant strain of E. coli HCB 33 (=RP437) that is Wild tYP€ for motility

 

and chemoraxis w
This medium was
methionine. and t]
Tne glycerol ser
chemoattractant i
and either 0.1 m!
1163 medium on
initially containe.
in LB broth.

In thesee
north the sink. 1

pm pore-size p0]

31

chemotaxis was used. A mineral salts medium (M63) was used for all experiments.
3 medium was supplemented with 2 mM glycerol, the amino acids histidine, leucine,
hionine, and threonine required for this auxotrophic strain, and 125 ug/l streptomycin.
2 glycerol served as the carbon and energy source for growth, but it is not a
moattractant for E. coli. The source reservoir contained supplemented M63 medium
either 0.1 mM or 1 mM aspartate, while the sink reservoir contained supplemented
3 medium only. The medium in the arena was stabilized with 0.15% agarose and
ially contained no aspartate. The E. coli were grown overnight with aeration at 30°C
.B broth.

In these experiments, the south reservoir (see Figure 12) was the source, and the

th the sink. These source and sink reservoirs were separated from the arena by 0.05

pore-size polycarbonate filter membranes (Poretics Corp., Liverrnore, CA).

v

 

 

 

 

Figure

The cast
silastic sheeting
autoclaved at 12
in the flasks as I
sterile filters. Th
through a sterile

Both the
reservoirs at 2.5

Operated for a

32

 

North Reservoir

 

 

 

 

 

 

 

 

 

 

 

 

'= rn
O

a 8
a, H
a) 30
a: (D
01 H—R———>I g
‘8 ' 2
' O
E . ='

I d

v “‘f r .

X South Reservoir

 

 

 

Figure 12. Schematic diagram of DGC, showing modeling coordinates.

The cast and west reservoirs were sealed off from the arena with non-permeable
silastic sheeting (Dow Corning, Inc., Midland, MI). The source and sink liquids were
autoclaved at 121°C to sterilize, and allowed to cool. To prevent a vacuum from forming
in the flasks as liquid was pumped out, the flasks were vented to the atmOSphere through
sterile filters. The gas port to the DGC (see Figure 3) was also vented to the atmosphere
through a sterile filter, to allow the headspace to be replenished with fresh air.

Both the source and sink solutions were pumped through their respective
reservoirs at 2.5 nil/hr. To allow the aspartate gradient to partially form, the system was

Operated for a given time before inoculation. The amount of time the gradient was

 

 

allowed 10 form t
the arena of the D

the cells et'enl." [l

4.2.4 Computer

The math:
algorithm (Carnal
difference equati
equation. The fit
only in the y-dire
ADI method is a
discretization err
More details on t
FORTRAN 77 at
Q. and H were sr
Was added, in or
from the FORTE

MATLAB progr:

4.2-5 MOdeling

In order
determined inde

laser diffraction

33
allowed to form before inoculation was varied between experiments. The center point in
the arena of the DGC was inoculated with 10 u] of E. coli using a micropipette to disperse

the cells evenly throughout the depth of the agarose.

4.2.4 Computer simulations

The mathematical model was solved with an Alternating Direction Implicit (ADI)
algorithm (Carnahan et al., 1969; Chapra and Canale, 1988). The ADI method uses two
difference equations to solve each two-dimensional unsteady—state partial differential
equation. The first difference equation is implicit only in the x—direction and the second
only in the y-direction. The equations are solved in succession at time steps of At/2. The
ADI method is an unconditionally stable method with which convergence occurs with a
discretization error of the order [(At)2+(Ax)2]. For the model presented here, Ax=Ay.
More details on the ADI method can be found in Appendix B. The program is written in
FORTRAN 77 and executed in the UNIX operating system. The balance equations for S,
Q, and H were solved by the model for a specified amount of time before the cell balance
was added, in order to simulate the gradient initiation time before inoculation. Output
from the FORTRAN program was imported to MATLAB version 4.0 for Windows. The

MATLAB program was used for image analysis and for graphical output.

4.2.5 Modeling parameters
In order to validate the mathematical model, all of the parameter values were

determined independently. The random motility coefficient, u, was measured using a

laser diffraction capillary assay (LDCA) deve10ped in our laboratory (Schmidt et al.,

 

 

1997). The LDC.
movement in car
and modeling. th
indicate that the
range of 0.15 to
the experimenter.
expenments. The
assumed to be ir
rate. in. was de

battenal pOPUl'di?

recalculate the g
The diift

correlation (Will

and glycerol. re:

5111 by a eorrel
(ire
en. 1984). A

“‘1 .
tens growing on

Values 1'
dissociation "a
‘ . «Mil

11011: ‘I T
- til‘ i '
pk liICIdl'

34
1997). The LDCA consists of a capillary tube through which a laser is directed. Cell
movement in varying agar concentrations can be observed, and through image analysis
and modeling, the random motility coefficient can be determined. LDCA experiments
indicate that the random motility decreases linearly with agar concentration over the
range of 0.15 to 0.30% agar. The random motility coefficient usedin the simulations is
the experimentally measured value for the agar concentration used in the DGC
experiments. The LDCA is described in more detail in Section 4.5. For this model, it is
assumed to be independent of the chemoattractant gradient. The maximum cell growth
rate, vH, was determined from the experimentally measured doubling time, rd, of the

bacterial population using the relationship

-1132 (25>

v
H
t .1

to Calculate the growth rate.

The diffusion coefficient for aspartate (D3) was calculated by‘the Wilke-Chang
correlation (Wilke and Chang, 1955). The diffusion coefficients for Do and DH (oxygen
and glycerol, respectively) at 25°C were obtained from. the literature, and adjusted to
30°C by a correlation for the temperature dependence of diffusion coefficients (Perry and
Green, 1984). An approximate value for Ya was calculated by an electron balance for
cells growing on glycerot.

Values for the chemotactic sensitivity coefficient for aspartate, x05, and the
dissociation constant for; the receptor-attractant complex for aspartate, K03, were taken

Irom the iiterature. ‘s’alues tor the chemotactic sensitivity to various chemoattractants

 

 

have been PUbli
published by M3‘

The value
range of those Ct
specific chemoat
the maximum ce
csdmated by adj
DGC iEmerson.
glycerol). 8 (as;
is given by kr-DC
0f the membrane
and Chang. 195
predicted outcor

coefficient. Valu

35
e been published (Ford, 1992). Values for the dissociation constant have been
tlished by Macnab (1987).

The values forthe saturation constants CH, Cs, and CQ were chosen to be in the
ge of those commonly reported for E. coli (Bailey and Ollis, 1986). The maximum
cific chemoattractant consumption coefficient (vs) was taken to be close in value to
maximum cell growth rate on H. The mass transfer coefficients kH, kg, and kQ were
mated by adjusting the k value for glucose, determined in previous work with the
-C (Emerson, et al., 1994) by the ratio of the molecular weight of glucose to H
ICCI'OI), S (aspartate), or Q (oxygen). The mass transfer coefficient across a membrane
;iven by k=Deff/l, where Deff is the effective diffusion coefficient, and l is the thickness
he membrane (Cussler, 1984). Deg varies with the molar volume, V, to the -0.5 (Wilke
, Chang, 1955) to ~06 power (Sitaraman et al., 1963). In results not shown, the
:licted outcomes were relatively insensitive to small variations in the mass transfer

fficient. Values for parameters used in the modeling simulations are given in Table 2.

 

 

 

416 Results 0

The cell
those Observed
“’38 introduced
chamber at 21 cc

Center of the ch.

Thfi ext

Figllre 13 The

36

Table 2. Parameter values for validation of model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Value

ll 0.0010 cm2 h"
m 0.080 cm2 h"

m 0.085 cm2 11'1

C0 4.08x10'6 gH cm?
CQ 6.70x10’8 gQ cm'3
Cs 5.50x10'rgscmj’r
DH 0.01 cm2 11'1

DQ 0.09 cm2 h"1

Ds 0.033 cm2 h'1
KDQ 3.30x10‘5 gQ cm'
KDS 2.00x10'6 gs cm‘3
v 0.35 h"

VQ 0.02 gQ gu"l h'1

vS 0.60 gs g," h“
YH 0.50 gu/gH

 

 

 

 

 

 

6 Results of validation simulations

The cell growth and migration patterns predicted by the model were compared to

;se observed experimentally under two sets of conditions. In Run 1, 1.0 mM aspartate

,-

MM

3 introduced into the south reservoir. Glycerol was initially present throughout the

) ‘lll‘

Timber at a concentration of 5 mM. After the gradient was established for 6 hours, the

iter of the chamber was inoculated with E. coli.

The experimental photos and modeling results at four time points are shown in

Lure 13. The results of the computer simulation are shown in the left-hand column and

 

photographs of tl
modeling simulal
observed experirr
rate of cells. chz
developed and in
increased with ti
approached the t
source slowed 11
source. The thii
chemoattractant
In Run
lOlHLVI aspanat
in the left-hand
exPeriment, the
RU“ 1. chemota.
“'3th were evit
shapes of the 1
nglb pattern

aPproached the

37
tographs of the experiment in the right-hand column. In both the photos and the
fdeling simulations, lighter areas correspond to higher cell density. Several trends
:erved experimentally were reproduced by the model. The first trend was the band or

ire of cells, characterized by a locally high cell concentration near the pattern edge, that

:eloped and migrated toward the chemoattractant source. The height of the wave peak

 

[eased with time. The second trend was that the velocity of the wave decreased as it
roached the chemoattractant source. Because the edge of the pattern closest to the
rec slowed more than the rest, the pattern tended to broaden as it approached the
rce. The third trend was i the less prominent wave that migrated away from the
moattractant source reservoir.

In Run 2, the chemoattractant concentration in the source (south) reservoir
lmM aSpartate) was one tenth that in Run 1. Figure 14 shows the computer simulation
;he left-hand column and experimental photographs in the right-hand column. In this

befiment, the chamber was inoculated 6.5 hours after the gradient was initiated. As in

n l, chemotactic migration toward the aspartate source and formation of a chemotactic
l

&

five were evident in both the experiment and the model predictions. However, the

T es of the patterns were significantly different in the two runs. In particular, the

\‘\] “

*wth pattern for Run 2 was more elongated and exhibited less flattening as it

iroached the source.

 

 

 

 

Figure 'l

38

. ma‘wmmmmwwmmm‘mmmwx

 

 

Figure 13. Simulation compared to experiment for 1.0 mM aspartate source

 

39

 

Figure 14. Simulation compared to experiment for 0.1 mM aspartate source.

 

 

42.7 Effects of

The simu
equally spaced n:
of the interval si
and Run 1 was s
two central axes
on the predicted

did show a mark

tally .‘l [(D ‘ (u/rc‘)

( ‘0" do-

I:igure 15

4'2'8 Content
The ce‘
COUCemraUOH ]

All fOur gram

40

Effects of grid-spacing on simulations

The simulations shown in Figure 13 and Figure 14 used a grid consisting of 81
ly spaced nodes in each direction, for a total of 812 (6561) points. To test the effects
: interval size on the predicted profiles, the grid was reduced to 512 (2601) points,
Lun 1 was simulated again. Figure 15, which gives the cell concentrations along the
entral axes of the DGC, shows that the increase from 512 to 812 had almost no effect
c predicted profiles. Other simulations (not shown) at grid-spacings of 252 and 152

row a marked difference from the results at 512 intervals.

 

 

 

 
 

 

 

 

 

  

 

 

 

 

 

 

 

2 _ O 51 intervals 7 T .
81 intervals 0 51 intervals
6 _ 2': 6 ._ -—81 inten/als
U
39 5 ~-
2 T 3 4 --
N
5‘ ._
8 “r .8 3
Q
:2 23
a 3
4 U 1 __
0 I r . -::==:::=::n O :3
0 1 2 3 4 5 0 1 2 3 4 5

 

x-dimension (cm) y-dimenslon (cm)

Figure 15. Cell concentration profiles across the two central axes of the DGC.

Concentration profiles of chemoattractants and nutrient

The cell patterns developed in response to the underlying, time-dependent
:ntration profiles of the chemoattractants and nutrient. Examples of these latter
es are shown, along with the corresponding cell profile, in Figure 16 and Figure 17.

our graphs have the concentration, g/cm3, on the vertical axis, and the spatial

 

4.2.7 Effects of

The simu
equally spaced n
of the interval sf
and Run 1 was 5
mo central axes
on the predicted

did Show a marl

‘ (ti/1T)

os~

H density x I"

1

0.4 —

v
V

Figure l

4'2'8 COncen

The c
concentration
profiles are sf

All fOUr grill

40

l.2.7 Effects of grid-spacing on simulations
The simulations shown in Figure 13 and Figure 14 used a grid consisting of 81
:qually spaced nodes in each direction, for a total of 812 (6561) points. To test the effects
>f the interval size on the predicted profiles, the grid was reduced to 512 (2601) points,

md Run 1 was simulated again. Figure 15, which gives the cell concentrations along the

 

.wo central axes of the DGC, shows that the increase from 512 to 812 had almost no effect
)n the predicted profiles. Other simulations (not shown) at grid-spacings of 252 and 152

lid show a marked difference from the results at 512 intervals.

 

 

 

 
 

 

 

 

  

 

 

 

 

 

 

 

 

2 " o 51 intervals 7 "
. O 51 intervals

A _81 Intervals "‘ 6 __ ——-81 intervals
8 1.6 “ 8
5° 3 5 —~
2 1.2 -- 2 4 _-
’i N
3* ,3‘ __
'g 0.8 g 3
“g 1: 2 a-
E 04 E!
U ‘ u 1--

O , l o ,____"_"__,__ ---

0 1 2 3 4 5 0 1 2 3 4 5

x-dimenslon (cm) y-dimension (cm)

Figure 15. Cell concentration profiles across the two central axes of the DGC.

4.2.8 Concentration profiles of chemoattractants and nutrient

The cell patterns developed in response to the underlying, time-dependent
:oncentration profiles of the chemoattractants and nutrient. Examples of these latter
profiles are shown, along with the corresponding cell profile, in Figure 16 and Figure 17.

All four graphs have the concentration, g/cm3, on the vertical axis, and the spatial

 

 

 

position. in cm.

graphing prograi

 

41
sition, in cm, on the horizontal axes. Note that the jagged edges are artifacts of the

aphing program.

 

 

 

'F‘

42

 

Figure 16. Top: Typical cell profile. Bottom: nutrient profile.

43

“was;
‘ “was“
‘smm‘“““
“‘ ‘\“\‘
\
o

\‘ .‘¢

‘ .
“\‘3 ‘

“t\“““

‘ “\““‘

"“‘.“

.o a,“

,a

 

Figure
17 To '
p. T ‘
ypical S profile. Bottom
: Q profi
1e.

 

 

Experimental VI
Figure 18 (’Wid
oxygen profiles

away from the 2

Oxygen Concentration (p M)

T0 val
measured in a
the Same as .
compared to ;

[CSUhS 0f the

44

imental verification of these profiles was more challenging. However, as shown in
3 18 (W idman et al., 1997), microelectrodes have been used to confirm that sharp
:n profiles exist across both the cell from moving toward the aspartate source and

from the aspartate source.

250

 

 

 

 

 

lnsrde cell population pl A a .. -o
I
.d A

I
a . r
V I
C I
'2 ID '
e 150 "" I l
5 55' .
8 .
o A
o 100 -~ I /
c l
a, I
a) ’ 5
Q 9 /
o I A

50 « (I ’
‘ I
4'- -A- 4 ’ Outside cell population
0 r r i i .
-6 -4 -2 o 2 4 6

Distance from Wave Front (mm)

Figure 18. Oxygen gradients in the DGC arena

To validate the model's predictions of chemical gradients, glucose gradients were
.1de in a miniature DGC using microbiosensors. The miniature DGC is fashioned
me as a regular DGC, but has an arena length and width of only 3.0 cm, as
ared to 5.0 cm for the regular DGC. Figure 19 (Widman et al., 1997) shows the

s of these measurements for a single time point, along four different lines in the

 

 

miniature DGC.
openings not C(

model accounts

45
;e DGC. The variations in the profiles for the different lines arise from reservoir
:3 not completely spanning the width of the DGC, as shown in Figure 12. The

‘zcounts for this geometry.

 

 

 

257.

n
U a I I n.-
I a I

:2...- . 0.500 OIOU-a-mv

‘ A i

u... ”w
I)» 1 v 1

:21. -UCOU avg-003.0

 

46

 

Line 1: 1.5 mm from top

 

 

 

 

 

in 5 10 15 20 25
I Position [mm]

Line 2: 7.5 mm from top

2500

 

2000-

1500--

1000--

500--

Glucose Conc. firM]

 

 

 

 

Position [mm]

 

Line 3: 12.5 mm from top

 

 

    

 

 

L r r a 4'
o 5 1o 15 20 25
Position [mm]

 

Line 4: 17.5 mm from top

2500

 

2000‘-

1500-~

1000-

500~~

 

Glucose Conc. [ng]

 

 

 

Position [mm]

 

Line 2

Line 1

 

 

 

 

 

 

 

 

 

Glucose Source

-—-~-~-—-~44-----~ m

 

 

 

 

 

 

\\ Line 3

 

 

Line 4 /
4'

 

 

 

Figure 19. Glucose gradients at several positions in the DGC

 

 

 

 

 

 

4.2.9 Assumpt

The mor
and boundary c
between the er
assumptions USI

In the l
depend upon b
neglect these c
random motilit
to chemotaxis.
relatively inser

Anothe
bacteria are n
Equation ( 13))
assay (Ford e1
C0ncentrations
Chemoattractar
prevents the 11
Cells. In Our r
the Simlllatior
speed of 7.92

gradient 51ml:

47

ptions and simplifications in the mathematical model

'3 model was able to reproduce several experimental trends for different initial '
ary conditions using the same set of parameters. The reasonable agreement

1e experimental and modeling results provides support for the simplifying

 

: 3 used. These assumptions are discussed below in more detail.

he RTBL model of chemotaxis, the random motility coefficient is shown to
;on both temporal and spatial chemoattractant gradients. We have chosen to
ese dependencies and assume constant 11. The magnitude of the flux due to
)tility is predicted to be several orders of magnitude smaller than the fluxes due
lXiS. In results not shown, the predicted growth patterns have been found to be
nsensitive to it. Therefore an exact calculation of u is not deemed necessary.
other assumption is that the gradients of chemoattractant encountered by the
re relatively shallow (i.e. Equation (14) is a reasonable approximation to
13)). This assumption has been found to be adequate for modeling the capillary
‘d et al., 1990; Rivero and Lauffenburger, 1986). However, the locally high
ions of cells in the chemotactic waves are associated with significant
ictant gradients. The use of the hyperbolic tangent term in Equation (13)
1e predicted chemotactic velocity from exceeding the swimming speed of the
hr modeling simulations, the maximum predicted velocity at any time during

.tion is only 0.154 cm/h (0.43 uni/s), much less than a typical cell swimming

 

:92 cm/h (22 urn/s) (Frymier, et al., 1994). To further validate the shallow-

 

,implification, velocity profiles from the hyperbolic tangent model and the

 

 

 

 

shallow gradien
gradients of S 1
(Ford and laufl
swimming spec
the number of i
used in the sim
sensitivity of 6.
in Figure 20 to
introduced by

encountered in

48
ow gradient model were calculated by Equations (13) and (14), respectively, for the
ients of S predicted by the model. Typical values for the individual cell parameters

:1 and Lauffenburger, 1990) for E. coli were chosen for use in Equation (13). The cell

 

.nning speed, 1), was taken to be 30 urn/s, and the tumbling frequency multiplied by

pumber of receptors, ONT, was 75 s. The value for KD, 2.0x10'6, was the same as that
:in the simulations. Equation (15) was used to calculate a value for the chemotactic
tivity of 6.74x10'4 cm2/s from the single cell parameters. The comparisons are shown
gure 20 for the first and last time points in Runs 1 and 2. There is virtually no error

duced by substituting Equation (14) for Equation (13) for any of the gradients

untered in either run.

 

 

 

(1‘
(3}).
. (.‘i
((7:
U (o :2

Eu:

-G\

Eer-U
«undo

« >«.00_0> 0.

0

0. w

~I>t

Figure

 

 

49

7.0E-05 -- Run 1

 

—Tanh model
6.0E-05 -l 0 Shallow gradient model

 

 

 

5.0E-05 -

l

4.0E-05 -

l

3.0E-05 -

2.0E-05 -

 

Chemotactic velocity to S [cm/s]

1.0E-05 -

 

ctctcttccg, .‘
0.0E+00 Jagauuuuu 'c.

i 1
-1 .0E-05

 

 

y-Positlon [cm]

    

 

7.0E-05 -—
Run 2
6.0E—05 -—
5.0E-05 _. 0 Shallow gradient model

 

 

 

4.0E-05 -

3.0E-05 a

2.0E-05 -

Chemotactic velocity to S [cm/s]

1.0E-05 —

 

0.0E+00 ‘ " . l

 

-1 .OE-05
y-Posltlon [cm]

Figure 20. Comparison of model with and without hyperbolic tangent term.

 

 

 

A third
constant throu
uniformly dist
gradients to f
significant ve
rnicroelectrode

partially respo:
4.3 Analysi

4.3.1 Flux ca

The m
from chemota

0f fluxes due t

Where J W is th
[0 S'gradients
magnllllde am
20 hr and are
auow COITesI
ConesPonds n

 

50
A third assumption is that the concentration of all components in the model is
instant throughout the depth (the z-direction) of the gel. Oxygen, which is initially
1iformly dispersed in the gel, is quickly consumed by the cells, causing vertical
"adients to form. In experiments (Emerson, unpublished data), we have measured
gnificant vertical oxygen gradients in the zone of cell growth with oxygen
icroelectrodes. The variation in oxygen concentration with depth is thought to be

utially responsible for the deviations between the experimental and predicted profiles.

.3 Analysis of modeling results

3.1 Flux calculations

The model was used to calculate the direction and magnitude of cell fluxes arising
om chemotaxis and random motility in the DGC. Equation (11) was rewritten in terms

i'fluxes due to single driving forces as

Ju : Jim + JuS + JuQ (26)

here J W is the flux due to random motility, J uS is the flux due to chemotaxis in response

S-gradients, and J llQ is the flux due to chemotaxis in response to Q-gradients. The
agnitude and direction of each flux term were calculated individually for Run 1 at time
) hr and are presented at each node point in Figure 21—Figure 24. The length of each

row corresponds to the magnitude of the flux, and the direction of the arrow

>rresponds to the direction of the flux. The tail of the arrow 1s located at the pomt where

e flux is calculated. (Note that in Figure 21-Figure 24 only the top 20% of the fluxes are

 

 

 

shown to reduc
the lighter (wh
illustrated. Th
chemoattractan
arrows onto th
profile. and th
Figure 33 sho'
obtained by ad

type of cell tlu

 

51
vn to reduce the number of arrows and to improve clarity.) In each of these figures,
ighter (whiter) shading corresponds to higher concentration of the component being
trated. The top graph in Figure 21 overlays the J uS flux arrows onto the
moattractant S gradient profile, and the lower graph in Figure 21 overlays the J us flux
ws onto the cell profile. The top graph in Figure 22 shows J uQ overlayed onto the Q
ile, and the bottom graph in Figure 22 shows J uQ overlayed onto the cell profile.
re 23 shows Juu, the flux due to random motility. Figure 24 shows the total flux,
ined by adding the three flux components together. The maximum magnitude of each

of cell flux is given in Table 3.

 

Table 3. Maximum predicted cell fluxes for Run 1 at 20 hours.

 

 

 

 

 

 

 

Flux component Maximum magnitude of flux
(x 105 gcells/cmz‘h)
Jus 8.05
J uQ 3.83
J m, 0.0000131
J u 1 1.4

 

 

 

 

 

 

52

 

5
2
FE: conical»

x—position [mm]

 

5O

5
2
7:5 coEonl>

x-position [mm]

Figure 21. Fluxes due to response to S chemoattractant.

 
 
 

y—position [mm]

25
x—position [mm]

50

y-position [mm]
N
01

 

25
x—position [mm]

Figure 22. Fluxes due to response to Q chemoattractant.

 

y—position [mm]
N
01

25
x—positlon [mm]

Figure 23. Fluxes due to random motility.

y—position [mm]
l\)
U!

25
x—position [mm]

Figure 24. Total mass fluxes.

 

 

 

432 Receptl

The (16
be calculated 1
05. as the aver
Sdivided by t

TECCplOTS WCTI

The value of
response is c
zero. the chei
factor has be
13 and Figu
Calculation (

of the maxirr

55

3.2 Receptor saturation

The degree to which receptor saturation suppresses the chemotactic response can
: calculated from the simulation results. We define a global chemotactic response factor,
;, as the average cell flux (along a line of constant x) in response to the chemoattractant
divided by the average cell flux (along the same constant x line) that would result if the

:ceptors were completely free of any S. This definition is expressed mathematically as

 

 

 

 

2R 2R 2 ‘2 (27)
KB, 35 R s as
{[9503 (K03 +S)2 3y “idy lid); iii“- KDS] 3y uddy
5 = 2R 2R : 2R
1 8S 3S
———u d d —u d
{[9605 KDSBy :iy ii y {[3}; iy

he value of ¢s will vary between 0 and 1. If tbs is close to one, then the chemotactic
spouse is close to the maximum it can attain for the given gradient. As (1)3 approaches
arc, the chemotactic response to the gradient of S diminishes. The chemotactic response
lctor has been calculated for Run 1 and Run 2 at each of the time points shown in Figure
3 and Figure 14. Figure 25 shows (pg plotted against the x-position for both runs.
alculation of (pg was limited to areas where the cell concentration was greater than 0.1%

Fthe maximum cell concentration occurring at that time point.

 

 

 

0.2

4’s

Figure 2

56

 

 

 

 

 

0.20

' ' ' 11 hr
—0— 17 hr
5 —20 hr

0 N I ‘
l \‘ ,I \ —O—23 hr

V
B B

(D
e 0.10 -

‘_
‘_

 

 

 

0.00 ‘r

 

.1-

 

0.90

 

 

I - - - 9hr
—0—15hr

—18 hr

—O--24 hr

I
0.80 -— '

0.70 ~—

 

 

 

 

0.60 --

 

0.50 -_

<l>s

0.40 -+-

0.30 --

 

0.20 *-

O.10 ~F

 

 

 

0.00

—h—
‘1—
.-

x-position [cm]

Figure 25 Top: $3 for Run 1. Bottom: (pg for Run 2. Note different vertical scale.

 

4.3.3 Pattern

Twoh
first hypothCS
reducing the c
takes longer f
near the sourc

The cl
first hypothes
03 was deterr
point A at tin
of the patterr
values vary 1
lesser degree
the maximun
Run l.

The 2
While the 24
occur, the y:
in the center.

Suggest that ;

57

4.3.3 Pattern dynamics

Two hypotheses could explain the flattening of the pattern seen in Figure 13. The
first hypothesis is that the cell receptors become saturated for the chemoattractant S,
reducing the cell’s chemotactic response to gradients in S. The second hypothesis is that it
takes longer for the cells to consume the higher concentrations of chemoattractant found
near the source.

The chemotactic response factor values (Figure 25) provide strong support for the
first hypothesis. In Run 1, (p3 values lie between 0.02 and 0.18 at all time points for which
(1)3 was determined. The values are the lowest in the middle of the pattern (indicated by
point A at time 17 hr), where the flattening is the most evident, and highest near the edges
of the pattern (indicated by the two points labeled B at time 17 hr). In Run 2, the tbs
values vary between 0.25 and 0.9, indicating that chemotaxis is suppressed to a much
lesser degree by receptor saturation than in Run 1. This result would be expected since
the maximum chemoattractant concentration in Run 2 is approximately one tenth that in
Run 1.

The 23 hr (pg curve for for Run 1 is flat from approximately 1.7 cm to 3.2 cm,
while the 24 hr (1)3 curve for Run 2 has a much rounder appearance. For broadening to
occur, the y-component of the wave’s velocity at the sides of the pattern must exceed that
in the center. The shapes of the $3 curves are consistent with such a velocity gradient, and

Suggest that receptor saturation is at least partially responsible for the pattern broadening.

 

 

 

4.3.4 Chemo

Mover
source reservt
was not reprt
that this beha
second chemt
second chem
coli (Adler. 1
significant or
away from th
moving away

gradient.

44 Variat

The
exllellments
Compounds,
however’ an
ComPlelt m0

First
COmpetition

modificatim

58

4.3.4 Chemotactic wave in response to oxygen

Movement of a portion of the chemotactic wave away from the chemoattractant
source reservoir (i.e. toward the north) was an unexpected experimental result. This effect
was not reproduced in the simulations with a single chemoattractant. We hypothesized
that this behavior could arise in response to gradients formed by the consumption of a
second chemoattractant that is uniformly distributed throughout the chamber initially. The
second chemoattractant has been identified as oxygen, a known chemoattractant for E.
coli (Adler, 1972). The microelectrode measurements shown in Figure 18 indicate that
significant oxygen gradients coincide with the cell wave fronts moving both toward and
away from the applied chemoattractant gradient. Model predictions indicate that the wave
moving away from the aspartate source tracks an oxygen gradient rather than an aspartate

gradient.

4.4 Variations on original model

The original model, presented in Section 4.2, gave good agreement with
experiments for one bacterial population responding to gradients of two chemical
compounds, and growing on a third. The resulting simulations were not perfect matches,
however, and experimental data, such as the oxygen gradients, suggested that a more
complex model was needed to match the experimental results more closely.

First, a second bacterial balance was added to the model. This balance allowed
competition or other multi-population phenomena to be studied. More details on this

modification to the model are given in Sections 5 and 6.

 

 

 

 

In addi
second-genera
and the nutrie
speaking, this
Q were oxyge
oxygen. If (
improvement
and 01115. 19E
takes into 3C1
period of it
chemoattract;
growth. The
motility. for.

All it
cells to grol
comPound u
coli, or acetz

The :

du.

we

Where Vei l

popuiaUOn‘

 

59

 

In addition to the second cell balance, several improvements were built into this
second-generation model. These included making consumption of the chemoattractant S
and the nutrient H dependent on the concentration of chemoattractant, Q. Biologically
speaking, this can be interpreted as modeling Q as the electron acceptor in the system. If
Q were oxygen, then the cell's metabolism is aerobic, with electrons being donated to
oxygen. If Q were nitrate, anaerobic denitrification could be modeled. Another

improvement was the addition of death terms and endogenous metabolism terms (Bailey

 

and Ollis, 1986). The death terms model the death of cells, while endogenous metabolism

1

:takes into account the loss of cell mass due to internal consumption, such as during a

 

period of low nutrient availability. Cell maintenance terms were added to the
chemoattractant and nutrient balances to allow for consumption for uses other than

growth. The cell maintenance terms could include nutrient consumed for energy for

 

motility, for example.

An important modification to the second-generation model was the ability of the
cells to grow on the chemoattractant S. This gives the model the ability to model a
compound which is both a chemoattractant and a growth nutrient, such as glucose for E.
coli, or acetate for Pseudomonas KC (see Section 7).

The second-generation model for cell population i (i=a or b) is given by

8L1,
797:“:‘V2 up _V.(ch)is _V.(V It)“, +

 

1121‘ i

. . 2
Q vth + vlSS __ veg u,- __ d Ll ( 8)
C,,, + H C“ + S

where vet is the endogenous metabolism term and dth; is the death term for the ith

population. The balance for the consumable compound j (i=8 or H) is now given by

where vpm is

term is given

1111616 ij IS 1

term. for the 1

(9Q _
E _
u‘here Yiq is
endogenous 1

Two
ShOWn in p
metabolism 1
to Obtain ac C

60

if 2 1' Q (29)
— V
31—11) j 2["”'C+JC +Qu']

 

where Vijm is the total consumption coefficient for the ith population growing on j. This

term is given by

v.. ( 30)
ijm — Y ij

'1

where Yij is the yield coefficient for cells growing on j and cmij is the cell maintenance

term, for the consumption of j. The balance for the compound Q is given by

 

 

 

mg

a Q _ rhm H vism S vet Q ( 31)
BI—qu—VQ 21“th C,,,+H+Y C+S+YiC,q+Q“‘]

where Yiq is the yield coefficient for the consumption of Q by the ith population used for
endogenous metabolism.

Two time-points from a sample simulation using the second-generation model are
shown in Figure 26. Values for unknown parameters, such as the endogeneous
metabolism terms and the consumption terms were estimated. More work will be needed
to obtain accurate values for these parameters, but the model has been shown to produce

results that resemble the experimental data.

 

 

 

4.5 Laser

The 1
this research
Student, Seb;
Will be presl

1997.

Moti
COtifficiem :
moving thro
Therefom, 3

allows the r:

61

 

Figure 26. Sample simulations from second-generation model.

4.5 Laser diffraction capillary assay

The laser diffraction capillary assay was conceived and developed as a portion of
this research project, but the majority of the work was done by an Aachen exchange
student, Sebastian Schmidt. For this reason, only a brief summary of the LDCA method
will be presented within the body of this work. For further details, see Schmidt et al.,
1997.

Motility parameters necessary for the modeling work, such as the random motility
coefficient and the chemotactic sensitivity coefficient, were not known for bacteria
moving through a semi—solid medium, such as the dilute agar gel in the arena of the DGC.
Therefore, a method to independently measure these parameters was needed. The LDCA

allows the random motility coefficient to be measured at various agar gel concentrations,

 

 

mdmmmm
coefficients It

Thel
mounted on 1
was achieved
uupenuonsi
mwmmnh
alle/Ne lase

experiment.

62
and preliminary experiments indicate that it will also allow chemotactic sensitivity
coefficients to be measured as a function of agar concentration.

The LDCA consists of a 1.6 mm inside diameter glass capillary, which can be
mounted on a microscope stage. A step change in cell concentration inside the capillary
was achieved by successively inserting the tip of the capillary into each of the two agar
suspensions in the culture tubes (Figure 27). The capillaries were filled over a total length
of 30 m, then mounted on the microscope stage. The end of the capillary opposite from
a He/Ne laser was sealed with silicone grease to avoid unwanted convection during the

experiment.

 

 

 

The laser w;
Was mounte
15 minutes,
mElintained ;

The
Latlffenburg

dimenSlOna]

63

 

 

x=-L ' x=0 =L

= Agar suspension with no cells

- = Agar suspension with cells

Figure 27. Filling the capillary in the LDCA

  

The laser was shone axially through the capillary. A CCD camera, attached to a computer,
was mounted on the optical lens of the microscope. Images were captured at 1.5 minutes,

15 minutes, and then at 30 minute intervals. A typical run lasted 3 hours. The LDCA was

maintained at room temperature for all runs.

The mathematical model for the LDCA is similar to that used by Ford and

Lauffenburger (1992) for the SFDC, with the chemotaxis terms omitted. The one-

dimensional cell balance is given by

 

 

 

 

The analytica

using the b0t

The concem

This SiOpe c
The model i
m01ility We
a function (
Points, The

confidence

64

a_B_ 323 (32)
at flaxz

 

The analytical solution to this model is

B 1 .. (33>
——2-erfc Tilt

 

using the boundary conditions

B(x,0)=1for—L<x<0 (34)
B(x,0)=0for0<x<L (35)
83 (36)

dB
x(-le)—3;(L,t)—0

The concentration gradient at x=0 is given by differentiating Equation ( 33) to yield

33 _ —0.5 (37)

75777317

 

This slope can be experimentally measured at each time—point from the captured images.
The model is fit to the slopes by adjusting the random motility coefficient, 11. The random
motility coefficient for the bacteria Pseudomonas stutzeri KC, measured by the LDCA as
a function of agar concentration, is shown in Figure 28. The open circles show the data
points. The solid line is the best fit of the model, and the dashed lines represent the 95%

confidence interval for the fit. The random motility coefficient for a zero agar

 

 

concentration
order of mag
cmzls). The 11
displayed a ‘

concentratior

65
concentration, estimated by extrapolation, is 2.0)(10'6 cmzls. This value is of the same

order of magnitude as that reported by Segel et al. (1977) for P. fluorenscens (6><10'6
cm2/s). The important finding of the LDCA work was that the random motility coefficient

displayed a linear relationship with agar concentration, at least in the region of agar

concentration tested.

 

25

11 x107 (cmzls)

 

 

 

 

0 h r
0.1 0.2 0.3

% Agar Concentration

Figure 28. The random motility coefficient of Pseudomas KC.

 

 

 

5. MICRC

5.1 Compl

Cherr
to gradients 1
119761 Stated
Since most e
sole criterion

Thee
utilizing mat
microbial pt
confined do)
motility Cou
01- (1982') w
Of chemotax
growth rates
bacterial p0)
(Lauffenbm
populations
Sufficiently
had a highe

geometry ‘

5. MICROBIAL COMPETITION

5.1 Competition background

Chemotactic microorganisms able to position themselves optimally with respect
to gradients may have a competitive advantage over other organisms. Chet and Mitchell
(1976) stated that chemotaxis "presumably gives. . .microorganisms a selective advantage.
Since most ecosystems are not fully mixed, enzyme kinetics cannot be postulated as the
sole criterion for the competitive advantage of one microorganism over another."

The effects of chemotaxis on competition have been studied on a theoretical basis,
utilizing mathematical models to predict possible outcomes of competition between two
microbial populations. Lauffenburger et al. (1981) showed that in a one-dimensional,
confined domain, with a substrate diffusing in from one side, increasing levels of random
motility could be detrimental to the survival of a bacterial pOpulation. Lauffenburger et
al. (1982) went on to show that if the substrate was also a chemoattractant, a certain level
of chemotaxis could impart an advantage to a population, canceling the effects of lower
growth rates or higher random motilities. Both of these studies involved only a single
bacterial population. Lauffenburger and Calcagno (1983), building on the previous work
(Lauffenburger et al., 1981; Lauffenburger et al., 1982) modeled two randomly motile
populations simultaneously. They found that if a slower growing population also had
sufficiently lower random motility, it could out-compete a faster growing population that
had a higher random motility. Kelly et al. (1988), using a model systemiwith the same

geometry as the Lauffenburger papers, modeled two chemotactic pOpulations
66

 

simultaneous]
above which
immotile pop
The n
steady-state :
first populati
would survil
authors poin
order of a ye
occur over 1
motility ant
environmen
Rela

from an ex]
51min of Pr
CUlture, bur
(1985) stud
Strain of 11.
Chemotacti
Similar gro
Otl

Organisms

67
simultaneously. Their model predicted that there was a minimum level of chemotaxis
above which a population would have a competitive advantage over a non-chemotactic,
immotile population.

The model of Kelly et al., which included a death term and therefore allowed for
steady-state solutions, predicted that at steady-state, three conditions could arise: (1) the
first pOpulation would survive and the second would disappear, (2) the second population
would survive, while the first disappeared, or (3) both populations would coexist. The
authors pointed out that the time to attain steady-state predicted by their model is on the
order of a year, and in real systems, it is likely that variations in chemical gradients would
occur over that time period. They concluded that the ability to "consider the effects of
motility and chemotaxis on population growth and competition in rapidly changing
environments is clear".

Relatively few studies have addressed the effects of chemotaxis on competition
from an experimental vieWpoint. Pilgram and Williams (1976) found that a chemotactic
strain of Proteus mirabilis outgrew a non-chemotactic but still motile strain in stationary
culture, but that the two strains grew equally well in mixed’culture. Kennedy and Lawless
(1985) studied a motile, chemotactic strain of Pseudomonas fluorescens and a immotile
strain of the same species, and found that in unmixed aerobic and anaerobic soils, the
chemotactic strain survived significantly better than the non-chemotactic strain. Both had
similar growth characteristics in mixed culture.

Other work has focused on the interaction of two or more populations of

Organisms growing in the presence of spatially varying environments. Caldwell and

 

 

Hirsch (1972
gradients int
layer. and grl
study did no
could have l
the local con
Tilm.
structured he
dispersal rat
movement c
studied how
competition
In t1
Spatially 51p.
Called the d
the format
Microorgan
movement
mathematic
balance, an
Widman et

(1) the $111):

68
Hirsch (1972) studied the growth of microorganisms in two-dimensional concentration
gradients in their steady-state diffusion plate. The organisms were immobilized in an agar
layer, and growth was quantified as a function of position in the gradients. Although this
study did not address competition directly, the researchers did find that the organisms
could have vastly different growth characteristics, depending upon their adaptability to
the local concentration conditions.

Tilman (1994), taking a broad ecological perspective of competition in spatially
structured habitats, noted that coexistence of species could occur if a species with a high
dispersal rate could move into a zone not inhabited by a superior competitor with less
movement capability. Holmes et al. (1994), again from an ecologically based viewpoint,
studied how partial differential equations could be used to study, among other things,
competition, dispersal, and dispersal-mediated coexistence in spatially structured systems.

In this study, a system of partial differential equations was used to describe a
spatially structured system in which two bacterial populations were growing. The system,
called the diffusion gradient chamber (DGC), has been established as a tool that allows
the formation of two-dimensional chemical gradients (Emerson et al., 1994).
Microorganisms inoculated into the chamber can be observed, and their growth and
movement properties recorded. Widman et al., (1997) (see Section 4.2), developed a
mathematical model of the DGC that included two chemoattractant balances, a nutrient
balance, and a cell balance. In addition to modeling a different physical geometry, the
Widman et al. model offered three features not found in the Kelly et al., (1988) model:

(1) the substrate does not necessarily have to be a chemoattractant, and vice versa; (2) the

 

 

model equati
transient m0
analyzed wit
extended to

competitions

5.2 Comp

The
system. intr

balance. Th-

Where, for i
is the nutri
Coefficient
the receptc
KDiQ is the
gTOWth rat.

Th.

the second

69
model equations are solved in two dimensions, as opposed to one dimension; and (3) the
transient model can be solved over relatively short time periods, and the data can be
analyzed without requiring a steady-state solution. In this study, the model of the DGC is
extended to include a second cell balance, which allows predictions of the outcomes of

competitions between two microbial populations.

5.2 Competition mathematical model

The system of coupled partial differential equations that describes the DGC
system, introduced in Widman et al. (1997), was extended to include a second cell

balance. The cell balances for Populations A and B are given by

 

38
W ( >

au. K . KDiQ
—'=fl.-V2u,-_Z.~ V __‘L’ uiVS —Zi V _— uin + ”i
at OS [[(Kots +S)2 OQ (KDrQ +Q)2 C1” + H

where, for i=A or B, u, is the cell concentration, S and Q are the two chemoattractants, H

is the nutrient, u, is the random motility coefficient, X015 is the chemotactic sensitivity

 

coefficient to the attractant S for the ith population, KDiS is the dissociation constant for

the receptor-S complex, inQ is the chemotactic sensitivity coefficient to the attractant Q,
KDiQ is the dissociation constant for the receptor-S complex, V,“ is the maximum specific

growth rate, and CiH is the half-saturation constant for growth on H.
The chemoattractant and nutrient balances were changed to reflect the addition of

the second cell population. The chemoattractant balances are given by

where, for j:
chemoattract
chemoattract
chemoattract
As (1
more of the
chemoattrac
the reservoir
Which is a cl

The

Where DH i
glOWIh of tl

The
batted in
balance are
nutrient are

“11 popula

 

where, for j=S or Q, Dj is the chemoattractant diffusion coefficient, Vij is the specific

chemoattractant consumption coefficient for the ith population consuming the jth
chemoattractant, and Cij is the saturation constants for ith population consuming the jth
chemoattractant.

As described in Section 4.2.3, the chemoattractant S is introduced from one or
more of the DGC reservoirs, and forms diffusion gradient across the gel in the arena. The
chemoattractant Q is initially present at a constant concentration throughout the gel and
the reservoirs. In the experiments used to validate the model, Q was most likely oxygen,
which is a chemoattractant for many bacteria.

The nutrient balance becomes

flzDHVZH—X

B VrHH _L_‘i_ (40)
at i=A

C... + H Y...
where DH is the nutrient diffusion coefficient, and YiH is the yield coefficient for the
growth of the ith population on H.

The assumptions made in the original, single population model are given and
justified in Section 4.2.9. The new assumptions introduced by including the second cell
balance are that the effects of the two populations on the chemoattractants and the
nutrient are additive (Lauffenburger and Calcagno, 1983; Kelly et al., 1988); that neither

cell population preferentially consumes one chemoattractant or nutrient over the other;

 

and that the:

predation or

5.3 Mode?

.Vlanj
parameters 1
correlations.
model was i

In th

71
and that there are no direct interaction effects between the two populations, such as

predation or parasitism.

5.3 Modeling parameters

Many of the parameters used in the model are given in Section 4.2.5. The
parameters were measured independently, obtained from the literature, or calculated from
correlations. All the parameters had reasonable values for a population of E. coli, and the
model was verified by comparing its simulations to experiments with E. coli.

In this study, the base set of parameters for each population is given in Table 4.

 

Frt
their effec
Where the
parameter

72

Table 4. Base case parameters for competition simulations.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Value

(i = a or b)

u, 0.0010 cm2 h'1
XOAQ 0.085 cm2 h“
XOBQ 0

XOAs 0.020 cm?1
Xoas 0

Cir; 4.08x10’6 gH cm'3
CiQ 6.70x10'8 gQ cm’r
Cis 5.50x10'5 gs cm‘f
DH 0.01 c? h'1

DQ 0.033 cmTh’l

D5 0.033 cmThT
KDtQ 3.301.105 g., cm'3
Kors 2.00x10'vgs cm'3
vAH 0.35 h'1

VBH 0.5 h'I

VAQ 0'02 8Q gu'l h-r
vBQ 0

MS 0.60 gs a.‘1 h'1
vBs 0

YiH 0.50 gu/gH

 

 

 

 

From this base set, individual parameters for Population A were varied to test
their effects on the outcomes of competition simulations. In many mathematical models,
Where the effects of parameters are unknown, a dimensionless model enables many
parameter effects to be studied at one time. In the case of this particular model, however,

a dimensionless model did not decrease the number of parameters, and was therefore not

 

useful for pa
the attractant
the specific
chosen as 11
inoculation j
simulations.

while that it
rate for P0p
(Hansen ant
including 0
that it is onl

In 2
inoculation
The base 0

Chamber, a

54 Anal

5.4.1 The
Ty
populallor

Populatior

[631111 is

73
useful for parametric studies (see Appendix G). The chemotactic sensitivity coefficient to

the attractant S (xOAs), the dissociation constant for the receptor-S complex (KDAS), and
the specific chemoattractant consumption coefficient for the attractant S (vAs) were

chosen as the variables to be manipulated. The initial amount of H present and the
inoculation pattern of the two populations were also varied in some simulations. For all

simulations, the maximum specific growth rate of P0pulation B (ttB) was set to 0.5 h'l,

while that for Population A was left at the base case value of 0.35 h]. The larger growth
rate for Population B would give it a competitive advantage in a well-mixed environment
(Hansen and Hubbell, 1980). Population B does not interact with either chemoattractant,
including chemotaxis or consumption. In other words, the base case for Population B is
that it is only able to move by random motility, and it consumes only H.

In all simulations, the gradient was allowed to initialize for 6 hours before
inoculation. After inoculation, the duration of the simulated experiment was 30 hours.
The base case initial condition for the cells is that they are inoculated in the center of the

chamber, at an equal concentration, as described in Widman et al. (1997).

5.4 Analysis of competition simulations

5.4.1 The dynamic competition factor

Typically, the result of a competition experiment between two microbial
pOpulations in well-mixed culture is quantified by the death or disappearance of one
population and the continued survival of the other. This type of steady-state competition

result is not readily amenable to the transient DGC model. Therefore, a dynamic

 

 

competition

masses of th

where V is ‘
the model.
dimensional

depth of the

 

W1.

Value of W

a Value of

74
competition factor (tlt), was defined to give a time-variant measure of the ratio of the

masses of the populations; ‘1’ is defined by

”I“ (x, y,z,l‘)dV (41)

1”“) = yyyu,(x,y,z,.)av
V

where V is the total volume of the gel, and ua(x,y,z,t) and ub(x,y,z,t) are calculated from
the model. The coordinate axes are defined in Figure 29. It is assumed that the two-
dimensional concentration profiles predicted by the model are constant throughout the
depth of the gel (the z-direction).

Source side

 

X Sink side

Figure 29. Coordinate axes used for competition analysis.

When \p = 1, both populations have the same total mass present in the chamber. A
value of tyr < 1, indicates that Population B has a higher total mass than Population A, and

a value of \p > 1 indicates that Population B has a lower total mass than Population A.

 

 

Therefore, 1
instantaneou
time. and vit

Anot
derived fror

masses. and

lftp‘>0,tl
B.lftp<1

although its

5.4.2 The

Ant
P0pulation
average cc
X-directior

COllcentrat

75
Therefore, for the purposes of this study, Population A will be said to have an

instantaneous competitive advantage over Population B at a given time if \p > 1 at that

time, and vice versa.
Another measure of the competitive relationship of the two populations can be

derived from Ill. The local competition rate, ty‘, is the rate of change of the ratio of the

masses, and can be found by

, d '1’ (42)

If 111' > 0, then the mass of Population A is increasing relative to the mass of Population
B. If t)! < 1, but 111' > 1, Population A may still have the overall competitive advantage,

although its total mass has not yet become greater than the total mass of B.

5.4.2 The average concentration curve

 

Another useful approach to studying the spatial arrangement of the two competing
pOpulations was the development of the average concentration curve. This curve plots the
average concentration of the population along each line perpendicular to the gradient (the
x-direction) vs. the position parallel to the gradient (the y-direction). The average

concentration is calculated (for Population A, for example) by

x=L n n . (43)
j uA(x, y)|yAyAzdx Ema, y)|yAyAzAx,. 2...,(1, y)) Y ,
i=1 : i=1 '

HA(Y) = x=0

 

AxAyAz z nAxAyAz n

where L is t1
and Ay are .

the gel. Figt

point on the

image.

 

TOT any 1
ConCentra

the Simul:

76
where L is the width of the chamber, n is the number of grid points in the x-direction, Ax
and Ay are determined by the grid spacing and are always equal, and Az is the depth of

the gel. Figure 30 shows a typical competition simulation, and one line used to calculate a

point on the total mass curve. The chemoattractant S is diffusing in from the top of the

image.

15 hours 20 hours 25 hours

    

Line of constant y
along which one
point of average
concentration
curve calculated

 

Figure 30. Typical simulation, illustrating average concentration curve calculation.

It should be noted that the average concentration curve can be related to the

dynamic competition factor as

jit‘A(y,t)dy (44)

WU): j'rZB(y,t)dy

for any time point. By studying the dynamic competition factor and the average

concentration curve, some measure of the spatially developing competitive outcomes of

the simulations can be found.

 

5.4.3 Satur-

The
utilized to e:
quantity use
fora given 1

the equality

Where Fm,

of constant

5-5 Com

Ma
W33 also 1
”We. and
The Matl.

appTOle;

55.1 an

In

in Table 4

77

5.4.3 Saturation of chemoreceptors

The global chemotactic response factor, 03, defined in Section 4.3.2, can be
utilized to explain some of the responses that occur in the following simulations. Another
quantity used to analyze some of the simulation results was the maximum flux attainable
for a given S gradient. This term is the integrand of the denominator of the second term of

the equality in Equation ( 27), and can be defined as

10.45 é‘gd (45)
y
KDAS 8y x L

W

O

F,,,,.(x) =

where Fmax(x) is the average maximum flux attainable for a given S gradient, along a line

of constant x.

5.5 Competition simulations

Matlab v. 4.2c1 was used to visualize the output of the FORTRAN model. Matlab
was also used to solve for the dynamic competition factor, the average concentration
curve, and the global chemotactic response factor (see Appendix F for the Matlab code).

The Matlab function "trapz.m", which utilizes the trapezoidal method, was used to

approximate the solution to the integrals.

5.5.1 Effect of modeling parameters on competition results
In the following sets of simulations, the parameters for Population B are as shown

in Table 4, The indicated parameter for Population A is varied in each set of simulations.

 

 

 

 

 

 

Range findi
showed a gt
range to prt
concentratit
hours after

prior to cel
number is 1
al.. 1997).

simulation

5.5.2 Var:

Tilt
0.01-0.071
points in
Correspont
gray. Pop
Chemoattr
the moon}
dynamicc

trend.

 

78
Range finding runs (results not shown) were performed to find a parameter range that
showed a good variation in results. The parameter of interest was then varied within this
range to produce the desired simulation sets. Both populations were inoculated in equal
concentrations in the center of the chamber. The simulated experimental time was 30
hours after inoculation of the cells. The chemoattractant gradient was initiated for 6 hours
prior to cell inoculation. The grid-spacing used for all simulations was 412 points. This
number is less than the 512 points used in the study that validated the model (W idman er
al., 1997), but was found to be sufficiently accurate and allowed for the fastest possible

simulation times.

5.5.2 Variation of ans

The chemotactic sensitivity factor for Population A, X0As. was varied between
0.01-0.07 cmzlh to assess its affect on the outcome of simulated competitions. Three time
points in this simulation for X0AS=O~O7 are shown in Figure 30. The lighter areas
correspond to higher cell densities. Both Populations A and B are shown in shades of
gray. Population A is the one which moves out in a chemotactic wave toward the
chemoattractant source, while P0pulation B remains in the center of the chamber, around
the inoculation point. Figure 31A shows the dynamic competition factor. In many of the

dynamic competition graphs shown, the initial 10 hour period shows a similar decreasing

trend.

 

 

 

 

 

Thi
center of t]
Figure 31E
graph and
the positic
Chamber 1
reached a.
limb Popu
C0“Central
so they W
hour time
have Th.

the Chemt

79

 

 

A XOAs=0.07

—U— ___ .
:XOAs=0.05 1-4 -- chs 0 O1

 

XOA5=0'03
......... xOAs=0.05

   

XaAs=0-03 o XOAs=0-07

 

 

 

 

 

”o

,_

X

<

‘5

C

.2

A "' (3‘
5 08- g g o...
9‘ 5 a.

0.6 3 V 0.6

o

04 3.
° 3’ 0.4
0'2 ----I a 02
II... > '

'I'llll... x0A8=O 01 <
o r 4 o

0 10 20 30 40
Time (h) y-Position (cm)

Figure 31 Variation of XOAS-

This trend occurs because significant concentrations of S do not diffuse into the
center of the chamber until approximately 10 hours after inoculation (results not shown).
Figure 31B shows the average concentration curves for four values of X0As, at 25 h. In this
graph and all subsequent average concentration curve figures, the attractant source is at
the position y=5 cm. For higher levels of XOAs. Population A encountered the wall of the
chamber before 30 hours. In the graphs of Figure 31, only times before the wall was
reached are shown in the graphs. Figure 32A shows the average concentration curve for
both Populations A and B, for the case where dis=0-03 cmzlh, at 25 hours. The average
concentration curves for Population B in most of the other simulations were very similar,
so they will not be shown in future graphs. Figure 32B shows the density maps of the 25
hour time-point of Populations A and B for XoAs=0~03 cmZ/h, illustrating the chemotactic

wave. The wave is the high concentration band of cells of Population A moving toward

the chemoattractant source.

  
 
    
 

 

 

 

o
o .
l 2*? .4.__.444U ,

Q
N

3
' 3 cm )
Cell concentration x10 (9!
_O

.,}U, A

0

Note that
the highes
This does

that it is a

T1

the cell p.
Figure 31
grOWth a.
I)OIJUIatir
(To/13201
highest 1.

Popuiaill

8O

 

 

 

   

 

0.5

61‘

E A

° l

B) 0.4 T PopulationA
v o
P) :l

a Population B . ,

E 3‘ l

t: 0.3

.9 s

i-l l

E s‘

E 0.2 ‘y _‘

0 E

o

8

o 0.1 ‘1

T.»

o n

o ..... y .....
0 1 2 3 4 5

y-Position (cm)

Figure 32. Other results of variation of XOAS

Note that in this figure, and all figures of the density map type, the cell population with
the highest concentration at any point in the chamber is the one that appears in the graph.
This does not necessarily mean that the other population does not exist at this point, only
that it is at a lower concentration.

The chemotactic sensitivity coefficient is a measure of the magnitude with which
the cell population migrates chemotactically in response to a chemoattractant gradient. In
Figure 31A it is apparent that below a certain X0 value (in this case, X0AS=O~01 cmZ/h), the
growth advantage of Population B is so large that, in the time-frame of the experiment,
Population A is overgrown by B, as indicated by 111' < O at all times. At higher levels
(X0A520.03 cmz/h), the sign of 111' becomes positive after about 10 hours, and at the
highest levels of XOAs. 11! reaches a value greater than one, indicating that the total mass of

Population A has become greater than the total mass of Population B. For this set of

 

conditions, .
total mass 0
The
the chemot
average cor
side of the
Population
peak of P0
two populz
of higher l
with B.
Th1
sensitivity
This indie
Chiming c
increasing
Cells mov
cells may
and there
CaITying .
adVantagr

and the n'

8 1
conditions, chemotaxis can be said to impart a competitive advantage, as measured by the
total mass of microorganisms present, to Population A.
The average concentration curves of Figure 31B give another view of the effect of

the chemotactic sensitivity. As X0As increases, two things occur. First, the peak of the

average concentration curve shifts toward the attractant source, which is located at the
side of the chamber corresponding to y=5 cm. The average concentration curve for
Population B remains close to the center of the chamber, as illustrated in Figure 32A. The
peak of Population A moving further from the center indicates greater separation of the
two populations. By moving away from Population B, Population A is able to enter areas
of higher nutrient (H) concentration, that can be consumed without having to compete
with B.

The second trend that can be observed in Figure 31B is that as the chemotactic
sensitivity increases, the maximum height of the average concentration curve increases.
This indicates that not only is the total mass of Population A increasing, but also that the
carrying capacity of the chemotactic wave (Widman er al., 1997) of Population A is
increasing. The chemotactic wave is better illustrated in Figure 32B. The bright band of
cells moving toward the chemoattractant source is the chemotactic wave. This wave of
cells may allow for faster consumption of the chemoattractant, creating sharper gradients,
and therefore allowing for faster chemotaxis toward the source. By increasing the
calTying capacity of the wave, the cells may be gaining an even larger competitive
advantage by being able to consume larger concentrations of both the chemoattractant S

and the nutrient H as the wave travels through the region.

 

 

 

5.5.3 Vari:

The
coefficient.
dynamic cc
cm) profile
33C shows

33D show

82

5.5.3 Variation of vas

The second parameter varied was the specific chemoattractant consumption
coefficient, vAs. It was varied between 0 and 1.0 gsguA'lh'l. Figure 33A shows the
dynamic competition factor for five values of vAs. Figure 33B shows a centerline (x=2.5
cm) profile of the chemoattractant S gradient at 30 hours for each value of vAs. Figure

33C shows a density map profile for both populations with vAs=l .0 gsguA'lh'1 and Figure

 

33D shows the profiles for vAs=O gsguA‘lh'l, both at 30 hours.

 

 

0.1 °

 

011 the d
Flgm‘e ’

Other w.

83

 

 

 

 

 

1 1.2
0.9 — A B
0,8 - VAs=1-0 1
A
0.7 -- ”E 0.8
0.6 -- 0
.7: VAS=0.6 @
V 0.5 -- 0.6
9‘ ‘b
0.4 -' v- .
VAs=0.3 X 0.4 lncreastng v.15
0'3 " VAs=0.06 ‘0
0.2 -- VAS=0 0.2
0.1 ‘r
0 ‘fi 1 l . 0
0 10 20 30 40 0 1 2 3 4 5
Time (h) y-Position (cm)

Population A

Population B

 

Population A

\

Population B

Figure 33. Variation of vAs.

Increasing the specific chemoattractant consumption coefficient, vAs, had an effect

on the dynamic competition factor curves similar to that of increasing XOAs, as shown in

Figure 33A. The reasons for the increased competitive advantage are not the same,

however. Figure 33B shows the S concentration profiles for the position x=2.5 cm, or, in

Other words, down the center of the DGC. As vAs is increased, a given mass of cells can

 

 

consume :
encountere
response,
vAg:0, me
positive n
were grea
This can;
of S, the
When v,
subseque:
33C, Whe
and the ‘
consume

(X0A3=0.(

5.5.4 v1

COmplex

343 she

Show '1]

 

84
consume a larger quantity of S, and therefore increase the gradient of S that is
encountered by the wave of A. This increase in slope enables a faster chemotactic
response, and therefore a more rapid separation of P0pulations A and B. Even with

vA5=O, meaning no consumption of the chemoattractant S, the value of w' does become

positive near the end of the simulation (Figure 33A). If the dimensions of the chamber
were greater, it is likely that w' would become positive, even with no consumption of S.
This can again be explained by the carrying capacity of the wave. With no consumption
of S, the gradient is relatively shallow, and the carrying capacity of the wave is low.
When vAs is high, the gradients are sharper, which increases chemotaxis, and
subsequently increases the carrying capacity of the wave. This is illustrated in Figure
33C, where VAS is high and the wave is very apparent, and in Figure 33D, where vAszO,
and the wave is just beginning to appear at 30 hours. The ability of Population A to
consume S gives it a greater competitive advantage, but at this level of chemotaxis

(X0A3=O.02), the ability to consume S is not necessary for A to compete successfully.

5.5.4 Variation of KDAS

The third parameter varied was the dissociation constant for the receptor-S
complex, KDAS. The parameter range was 1><10'7 - 8><10'6 g/cm3. Figure 34A and Figure
34B show the w(t) values for five values of KDAS. Note that the curve for KDAS=1><10'6 is

shown in both figures, for comparison. The arrows Show the direction of increasing KDAS.

 

b-.. N.25 «:9 now x =5“.

«asuxmv new x COSQbCOOCOO Onwfigw><

 

‘P (t)

Fun-x X 103 (90A ct“.2 hd)

Average concentration x 10’ (gems)

1
A
0.8 u-
0.6 ‘ -
F KDAs=1x10 6
0'4 -_ KDAhs=5X‘|0'7
0.2 _l Kms=mo"
0 i—‘ l r I
0 10 20 30 40

0.8 ‘-

0.6 <

0.4

0.2 ‘-

O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:3 05-. {Xx

l.

 

 

 

)

 

x-Position (cm) x-Position (cm)

 

 

 

 

y-Position (cm)

Figure 34. Variation of KDAS. Time=30 h for all figures.

\<
J ‘x «‘2!
0.1 )(xxxxxfl xxx X

 

 

Xxxxxxx

 

 

In ?
competitiv
1x106, inc
reasons fo
average rr
legend in
the global
maximum
Consequc
when S :
points it
decrease
gradient.
value fc
Optimur.
gradient
the ave
Value 0‘

Values

interim

86

In Figure 34A, for KDAS < 1x106, increasing KDAS made Population A more
competitive, as indicated by larger w values at the later time points. However, above
1x10'6, increasing KDAS caused \y to decrease at later times, as shown in Figure 34B. The
reasons for these trends are shown in Figure 34C and Figure 34D. Figure 34C shows the
average maximum attainable fluxes for the corresponding S-gradient, at 30 hours. The
legend in Figure 34C applies to Figure 34D and Figure 34E, as well. Figure 34D shows
the global chemotactic response factor for the five parameter values, also at 30 hours. The
maximum mass flux is inversely proportional to KDAS, as shown in Equation (45).
Consequently, at the lowest values of KDAS, Fmax reaches its highest levels. However,
when S >> KDAS, (e.g. for the lowest values of KDAS), the value of (1)3 approaches 0 at all
points in the chamber. As KDAS increases, the level of saturation of the receptors
decreases, and therefore the population responds with a higher chemotactic flux to the S-
gradient. The trade—off between the opposing trends of Fmax and (pg results in an optimum
value for KDAS of about 1x10‘6 g/cm3, where flux is high and saturation is low. This
optimum KDAs value is most likely dependent on the concentration and SIOpe of the
gradient. Evidence for an optimum KDAS value is also given in Figure 34E, which shows
the average concentration profiles for Population A for a low, intermediate, and high
value of KDAS at 30 hours. Although the peaks of the curves for the intermediate and high
values appear at about the same y-position, the carrying capacity of the wave for the

intermediate value is clearly greatest.

 

 

 

 

SSJS‘Var

In
vahd.Th
meES
coefficien

values, wf

l
1.32x10
Concenu
Figure 3
source
Concent

10 and i

87
5.5.5 Variation of chemoattractant concentration
In this group of simulations, the initial conditions for the chemoattractant S were
varied. The base case set of initial conditions for all of the chemical compounds is given
in Table 5. For all simulations, the east and west reservoirs were sealed (the mass transfer
coefficient was set to 0). The parameters for Population A were set to the base case

values, with the exception of XOAs, which had a value of 0.05 cmZ/h.

Table 5. Base case initial conditions for competition simulations.

 

 

 

 

 

 

Compound and location Value (g/cm3)
S in arena, south reservoir 0

S in north reservoir 1.32><10"4

Q in arena, N and S reservoirs 1.32><10'5

H in arena, N and S reservoirs 4.6x10’4
P0pulation A, Population B 5x10'6

 

 

 

 

The concentration of S in the north reservoir was varied from 1.32><10‘6 to
1.32><10'2 g/cm3. Figure 35A and Figure 35B show the dynamic competition factor for six
concentrations of S in the source reservoir. Like the results shown in Figure 34A and B,
Figure 35A and B also show a reversal in the \[l trend above a certain chemoattractant
source concentration. For values of S S 1.32x10’6, w' is always negative. As the
concentration is increased, the Sign of w‘ changes after a certain amount of time (between

10 and 20 hours) in each simulation. Above a certain concentration, however, the u! trend

 

 

 

 

 

begins to «
the trade-c

Fig
concentra‘
mmgnm
time-pom
35C and i

is that as

concentr:
Opposing
simulatir
conditio:
for three
indicate.
growth 1

Wall of

Caflying

88
begins to decrease with increasing concentration. This effect can again be explained by
the trade-off between the maximum flux and saturation of receptors.

Figure 35C and Figure 35D show the maximum obtainable flux for the six
concentrations. Note that the y-axis has a different scale in the two figures. To avoid
using simulations where the cells had encountered the wall of the chamber, the 28 hour
time-point was used for all graphs that show a trend at a single time. The graphs in Figure
35C and Figure 35D have been split because the scales are so different. The overall trend
is that as the concentration of S increases, so does the maximum attainable flux.

Figure 35E shows the tbs curves for the four lowest concentrations. At the higher
concentrations, the (1)5 values were extremely close to O at every position. The same
opposing trends that were observed in the response to KDAS are apparent in this set of
simulations. The level of S that minimizes saturation and maximizes flux imparts the best
conditions for survival to Population A. In Figure 35F, the average concentration curves
for three concentrations are shown. At the lowest concentration, a wave does not form, as
indicated by the lack of a local maximum in cell concentration near the source side of the
growth region. At the intermediate concentration, a strong wave has traveled nearly to the
wall of the chamber. At the highest concentration, a wave is apparent, but has a low

carrying capacity and has not moved as far from the inoculation point.

 

 

Ar: N.Ee .39 an... x .35".

 

89

 

 

wt)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o 10 20 30 40
Time (h)
7
C 1.32005
9 6 " .5"
'2 Z:
‘3' 5 - ‘t‘
E l e
o 4 o
‘5
a, i 3
v v
‘2: 3 " "b
1- 1-
X 2 .. X
2 l a
IL 1 T “-
o . .
o 1 2 a 4 5 o 1 2 3 4
x-Position (cm) x-Position (cm)
1 1.6
1.3200“ F
V 1.4-4
08- E 2 l
' i x ....
c
.9
0.6 i 6.6x10'° ‘5 1.
b 6“
9- i\/ 3 0 0'8“
0.4 i 1.32x10‘5 5 32
O 0.6 --
o
a:
0.2 “ g o_4 ._
1.32x10“ 5 0.2g
0 4—— r a 4. r
o 1 2 3 4 5 o . . if .
0 1 2 3 4 5

X-POSitiOfl (cm) y-Position (cm)

Figure 35: Variation of S source concentration. Time=28 h for all figures.

 

 

5.5.6 Var

A
microbial
S-source
Altemati‘

chemoatt

 

(101)) in
A was
chemor
reSpon
enter 2

Compe

90
5.5.6 Variation of inoculation conditions
A large variety of inoculation patterns can be envisioned that could influence the
microbial competition. For example, one population could be inoculated further from the
S-source than the other, possibly disadvantaging the population further from the S-source.

Alternatively, two different inoculation points equidistant from the source of the

chemoattractant S could be used.

   

Figure 36. Pop. A inoculated farther from S source than pop. B.

In Figure 36, a the first approach was tested. The source of S is from the north
(top) in each image. The cell properties are the same as used in Section 5.5.5. Population
A was inoculated further from the S—source, but directly in line with Population B. The
chemotactic wave in Population A still was able to form, although it may be more in
response to the chemoattractant Q than to S. As the S gradient develops over time, it will
enter areas of the DGC where Population A is present, and Population A may gain the

competitive advantage.

 

 

i
discusse
moved t
Populati

l
in Figu
POpulat
(approx

indicati

POpulat
COmpet
WCH-es

test if

91

 

Figure 37. Pop. A inoculated to the side and away from S source.

Figure 37 shows the results for a combination of the two inoculation classes
discussed. Population B has been moved closer to the S-source, while Population A was
moved to one side, away from Population B. Again the chemotactic wave is apparent, and
Population A is able to move into a larger area of the chamber than Population B.

Figure 38, which shows the dynamic competition factor curves for the simulations
in Figure 36 and Figure 37, indicates that in both figures, the S-gradient is giving
Population A some advantage, because it is only after S reaches Population A
(approximately 10 hours), that the dynamic competition factor begins to increase (rp'>0),
indicating that Population A is gaining mass at a faster rate than B.

It is apparent from these few test situations that the inoculation pattern of the two
pOpulations can play an important role in determining which population will have the
competitive advantage. Another situatiOn that could be tested is to have one population

well—established in the DGC, and introducing the other at a much lower concentration, to

test if it can successfully "invade" and be coexistent with the initial population. The

 

results of

the time 1

5.5.7 i

compo
being

Popula
nutrier
hl'potl

given

 

92
results of all of these variation of inoculation point simulations are highly dependent upon

the time point being studied.

 

 

 

 

1

0.3 -—

0-6 " Figure 36
A ,o
4-0 ,1
V _

0.4 -- -

0.2-- t

Figure 37
0 l l i r r
o 5 10 15 20 25 30
Time (h)

Figure 38. Dynamic competition factors for two inoculation patterns.

5.5.7 Variation of nutrient initial conditions

Another hypothesis that was explored with the model was that if the nutrient
compound was diffusing in from one reservoir with the chemoattractant, as opposed to
being uniformly distributed through the gel, then the chemotactic response of a

pOpulation might give it the competitive advantage by moving that population toward the

nutrient, while the other was left behind in a low nutrient environment. To test this

hypothesis, a simulation was performed where the two populations had the parameters

given in Table 4, except XOAs=O.05 cmZ/h, VAH=O-5 hJa and VBH=O-7 114' The nutrient was

 

 

 

present i
was pre:
same as
simulati

39C sho

93
present in the north reservoir (x = 5 cm) at a concentration of 8.6><10'4 g/cm3, and no H
was present in the gel initially. The chemoattractant and cell initial conditions were the
same as given in Table 5. Figure 39A shows the dynamic competition factor for this
simulation, and Figure 39B shows the average concentration curves at time = 30 h. Figure

39C shows the centerline profiles (y=2.5 cm) of the H and S compounds.

 

 

 

3
i-i )< 1 O3 gg/ksrvw

HCVer
Populz

393. ti

0

O

94

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.5 6‘ 0.8
E PopulationA
3 T a 0‘7 T ----- PopulatlonB
L “i: 0.5-i
2.5 — F
x
L g 0.5 «i
A 2- 3
=3 S 0.4--
"‘ — a
15 i 8 0.3i
8
1 0
o 0.21.
8’
0.5 a I- 0.1 .
l s 1 ‘
<
o + + : 4| + 0 ~41; .
0 5 10 15 20 25 30 0 1 2 3 4 5
Time (h) Position (cm)
1 0.12
0.8 L - - H profile
" S profile _ 009
(V) (‘0
E
8 0.6 T 5
CD C)
no _ 0.06 ”O
1- 1—
x 0.4 "L' x
I U)
_ 0.03
0.2 i—
I”
0 d I. _____*--‘---"T- O

 

 

 

 

Position (cm)

Figure 39. Competition for nutrient diffusing in at x = 5 cm.

Figure 39A illustrates that even though Population B has a growth advantage, it
never obtains a higher mass than Population A. This is explained by the fact that
POpulation B never experiences a high concentration of the nutrient. As shown in Figure

39B, the chemotactic response of Population A has allowed it to move toward the source

of nutrie
complete
mass of
likely is
consume

1
pronoun
was a f:
exist. cl
examplt

discuss

5.6 F‘

methor
should
single

Cheme

and th
Studie

hOWe‘

95
of nutrient and chemoattractant. Figure 39C illustrates . that the nutrient has been
completely consumed by P0pulation A, and is not available to Population B. The small
mass of Population B that appears close to the source of S and H in Figure 39B most
likely is the product of a small amount of growth that occurred before Population A fully
consumed the H that had diffused to the center of the chamber.

In this simulation, the chemotactic response of Population A gave it an even more
pronounced advantage in that u: never had a value less than 1, even though Population B
was a faster grower. In a system where gradients of both chemoattractants and nutrients
exist, chemotaxis may be extremely important in determining the competitive winner. An
example of a real system where these conditions might occur is bioremediation, as

discussed in Section 7.

5.6 Future work on competition

The next step needed in the competition studies is to develop experimental
methods to corroborate the modeling predictions. The focus of the experimental work
should be on finding mutant bacteria that would differ from the main strain only in a
single transport property, such as random motility, or response to a single
chemoattractant.

Many transport mutants of E. coli have been previously isolated and cataloged,
and the genes involved in the specific mutations are well defined. Emerson et al. (1994)
studied the growth of such E. coli mutants in the DGC. At the time of this writing,

however, Pseudomonas KC (see Section 7) was the focus of this portion of the research

 

 

 

 

 

project. l
efforts tc
working
mutants

desired 1

possess
mutant 2
after the
marker

competf

Work) 1
Escherr
more (
mediur
asparta
grown.
Optical

l:igure

96
project. Well-characterized mutants of Pseudomonas KC are not currently available, but
efforts to develop them are underway in Dr. Craig Criddle's laboratory. Petty Setiawan,
working on a project that was part of this thesis work, screened 120 Pseudomonas KC
mutants for their motith and chemotaxis parameters. None were found that have the
desired prOperties that would make them useful in the competition experiments.

One way to perform these experiments would be to select transport mutants which
possess a selection marker, such as antibiotic resistance or fluorescence. A mixture of the
mutant and non-mutant strains could then be inoculated in a swarm plate or the DGC, and
after the growth and motility patterns had formed, samples could be taken. The selection
marker would allow each strain to be counted individually, and the outcome of the
competition could be quantified.

A preliminary competition experiment has been performed (Petty Setiawan's
work) to begin to study the effects of chemotaxis on competitions. In this experiment,
Escherichia coli was competed against Pseudomona stutzeri strain KC (see Section 7 for
more details on Pseudomonas KC). Swarm plates were poured that contained M9
medium at pH 7.5. The plates contained 2 mM glycerol as the nutrient source and 0.1 mM
aSpartate as the chemoattractant. Two plates were inoculated with 20 ill of either actively
growing Pseudomonas KC or E. coli culture that had been adjusted to have the same
Optical densities. The plates were incubated at 27 °C for 48 hours. These plates, shown in

Figure 40, allowed the response of the individual populations to be observed.

 

 

Fi

density
for 48
sample
diluted
Were i1

KC C0

97

 

    

Mixture of two populations

Figure 40. Competition between Pseudomonas KC and E. coli in swarm plates.

In a third swarm plate, a 20 ul aliquot of an equal mixture, based on optical
density, of the two populations was inoculated in a swarm plate and incubated at 27 °C
for 48 hours. An image of this swarm plate is also given in Figure 40. After 48 hours,
samples were taken at 5 locations in the competition swarm plate. The samples were
diluted by a factor of l:10,000 and plated on nutrient agar plates. The nutrient agar plates
were incubated for two days at 27 °C. Individual colonies of E. coli and Pseudomonas

KC could be differentiated by their unique morphologies, and counted. The results are

 

 

shown it

plate.

results
The P5
the €te

illiO arr

gradiei
(glycei

Was in

98

shown in Figure 41. The origin of the x-axis corresponds to the center point of the swarm

plate.

onn

 

Concentration x 104 (cells/ml)

 

 

 

 

 

Position (cm)

 

Figure 41. Colony counts in competition experiment.

The E. coli remained primarily in the center of the swarm plate. Although the
results are not symmetrical, the Pseudomonas KC have moved outward from the center.
The Pseudomonas KC may be at a competitive advantage in this case if the resources in
the center of the plate are fully consumed. The Pseudomonas KC would be able to move
into areas at the edge of the growth front where resources are still plentiful.

More work needs to be done with experiments such as this one. For instance,
gradient measurements would be useful in determining if resources such as the nutrient
(glycerol) were still available in the center of the plate. Also, in this experiment, no effort

was made to control the relative growth rates of the two populations.

 

6.5fldl

a M. S.
were pe
Mutaget
Chambt

Bioclzei

DAHP
metabo
feedbar
tested 1
had in
anaIOg
away f
lid Wa:
for 10
that w
The cl

glucos

6. SELECTION OF MUTANTS

6.1 Experimental work

This portion of the research was completed as a collaboration with Mark Mikola,
a M. S. candidate in Dr. Worden's research group. The experimental portions of the work
were performed entirely by Mr. Mikola, and are detailed in Mikola (1996) and in "In-situ
Mutagenesis and Chemotactic Selection of Microorganisms in a Diffusion Gradient
Chamber" by M. R. Mikola, M. T. Widman, and R. M. Worden, submitted to Applied
Biochemistry and Biotechnology.

To summarize, it was desired to obtain a strain of E. coli whose isozyme of
DAHP synthase, designated AroF, was not inhibited by L-tyrosine, a product in the
metabolic pathway. To achieve this goal, a population of E. coli that experienced
feedback resistance from tyrosine was inoculated into a DGC. The E. coli had previously
tested positive for chemotaxis toward glucose in a separate DGC experiment. The DGC
had in its source flask 125 ttM m-fluorotyrosine (m-FT), a non-metabolizable tyrosine
analogue, and 5 mM glucose. The bacteria only grew in the portion of the DGC farthest
away from the source of m-FT (see Figure 42A). After the initial growth period, the DGC
lid was removed, and the DGC was exposed to ultraviolet radiation at a distance of 60 cm
for 10 seconds. The lid was replaced and the incubation continued. Blooms of mutants
that were less inhibited by the m-FT appeared after the mutation process (Figure 42B).
The chemotactic response to the glucose gradient drew the mutants toward the source of

glucose and m-FT, as seen in Figure 42C.
99

-—

 

 

 

 

Fig

6.2

modifie
cell ba'
Also. a

by but]

Where
consu

satur;

100

 

Figure 42. Mutant selection by chemotactic response (source at top of pictures).

6.2 Modeling selection of mutants

The mathematical model of the DGC system presented in Section 4.2 was
modified and used to simulate strain selection in the DGC. For this application, a second
cell balance was added to account for both the original (A) and the mutant (B) strains.
Also, a balance was added to account for diffusion and cellular uptake of the inhibitor (P)

by both strains:

:92_
8t _

vaFP vbPP ( 46)

a ”[7
Cap + P C“. + P

 

 

DPVZP—

where Dp is the diffusion coefficient of the inhibitor; Vap and pr are the specific
consumption coefficients for Populations A and B consuming P; and Cap and Cbp are the

saturation constants for consumption of P by Populations A and B, respectively.

chemoat
growth 1
terms cl

Ollis, 15

where
respont
sensitii

pOpula

by

Where
uninh

motili

101
The chemotactic sensitivity coefficients for both populations responding to the
chemoattractants S and Q, the random motility coefficients, and the maximum specific
growth rates were modified to incorporate the inhibition effects. The form of the modified
terms chosen is analogous to noncompetitive inhibition in enzyme kinetics (Bailey and

Ollis, 1986). The modified chemotactic sensitivity is given by

 

, _ x... (47)

 

where x'Oij is the inhibited chemotactic sensitivity for the ith population (i=a or b)

 

responding to the jth chemoattractant (j=S or Q), XOij is the uninhibited chemotactic
sensitivity, and Kfijx is the inhibition constant for the chemotactic sensitivity of the ith
population to the th chemoattractant. The modified random motility coefficient is given
by

I “i ( 48)

 

 

where u'i is the inhibited random motility COCfflClCI'lt for the 1t population, it, is the
uninhibited random motility coefficient, and K1,l1 is the inhibition constant for the random

motility of the ith population. The inhibited maximum specific growth rate is given by

 

where v
H, till 1
specific

cell bal

where

includt

Where
the yi
(Widi
Equat

trend:

funct

Calcu

 

I vii-l ( 49)

 

 

where v'iH is the inhibited maximum specific growth rate for the ith population growing on
H, win is the maximum specific growth rate, and KEV is the inhibition constant for the
specific growth rate of the ith population. In terms of the variables defined above, the two

cell balance equations are given by

( 50)

 

where u, is the ith cell population (either a or b). The nutrient balance was also modified to

include the inhibited maximum specific growth rates:

8H
———=D VZH—
at ”

_aiaiga._____véaH _u_,_ (51)
C.H+H Yay CbH+H YbH

where DH is the diffusion coefficient for H, Cm is the half-saturation constant, and YiH is
the yield coefficient. Values for the modeling constants were taken from the literature
(Widman et al., 1997) insofar as possible. Values for the inhibition constants included in
Equations ( 47)-( 49) were chosen that gave reasonable agreement with the experimental

trends.

Figure 43 shows a time sequence of contour plots depicting cell concentration as a

function of position. The contour lines show constant concentration isoclines, as

calculated by a built-in Matlab program, The chemoattractant (glucose) gradient is

 

 

indkxne
Thereis
thetop

observe

 

103
indicated by gray shading, with lighter gray indicating a higher glucose concentration.
There is no m-FT in this simulation. The source reservoir is on the side correSponding to
the top of each figure. Bias of the cell’s migration in this direction is comparable to that

observed experimentally.

 

 

 

104

Increasing Time

 

 

 

Figure 43. E. coli responding to chemoattractant gradient.

 

 

shown
evohni
(nuytt
unhcat
43.is
inhhnt
DCKl
paant
Popul
bound
The -
cheni
secon
the g
reser
and .
inoct

POpU

105

The ability of the mathematical model to reproduce the experimental trends
shown in Figure 42 was evaluated. Figure 44 shows the predicted growth-pattern
evolution in which gradients of both a chemoattractant and an inhibitor are applied, and
only the population sensitive to the inhibitor is present. In this figure, the gray shading
indicates the inhibitor gradient. The effect of chemotaxis, which was significant in Figure
43, is overwhelmed by the effect of the inhibitor. As the concentration gradient of the
inhibitor becomes established, cell growth occurs predominantly in the sink end of the
DGC. This trend was observed experimentally in Figure 42. The base case set of

parameters for all of the simulations shown is given in Table 6. In these simulations,

 

Population A was the inhibited population, while Population B was the mutant. The
boundary and initial conditions are similar to those presented in Sections 4.2.1 and 4.2.2.
The inhibitor concentration (P) in the source reservoir was 0.0132 g/cm3. The
chemoattractant concentration (S) in the source reservoir was 0.000132 g/cm3. The
second chemoattractant concentration (Q) in the source and sink reservoirs and initially in
the gel was 0.0000132 g/cm3. The nutrient concentration (H) in the source and sink
reservoirs and initially in the gel was 0.000046 g/cm3. The value of uao was 3><10‘6 g/cm3,
and ubO was 3><10'7 g/cm3. The gradients were allowed to initiate for 12 h before
inoculation of Population A. Population B appeared in the simulation 15 h after

Population A. The simulation continued for 25 h after the appearance of Population B.

The
shown in I
evolution in
only the p0j
indicates [iii
43, is oven
inhibitor be
DGC. This
parameters
Population
boundary;
The inhib
chemoattri
second Chi
the gel w
reservoirs
and ub0
inoculatic

POPUlatio

105
The ability of the mathematical model to reproduce the experimental trends
shown in Figure 42 was evaluated. Figure 44 shows the predicted growth—pattern

evolution in which gradients of both a chemoattractant and an inhibitor are applied, and

 

only the population sensitive to the inhibitor is present. In this figure, the gray shading
indicates the inhibitor gradient. The effect of chemotaxis, which was significant in Figure
43, is overwhelmed by the effect of the inhibitor. As the concentration gradient of the
inhibitor becomes established, cell growth occurs predominantly in the sink end of the
DGC. This trend was observed experimentally in Figure 42. The base case set of

parameters for all of the simulations shown is given in Table 6. In these simulations,

 

Population A was the inhibited population, while Population B was the mutant. The
boundary and initial conditions are similar to those presented in Sections 4.2.1 and 4.2.2.

The inhibitor concentration (P) in the source reservoir was 0.0132 g/cm3. The

 

chemoattractant concentration (8) in the source reservoir was 0.000132 g/cm3. The
second chemoattractant concentration (Q) in the source and sink reservoirs and initially in
the gel was 0.0000132 g/cm3. The nutrient concentration (H) in the source and sink
reservoirs and initially in the gel was 0.000046 g/cm3. The value of uao was 3><10'6 g/cm3,
and ubO was 3x107 g/crn3. The gradients were allowed to initiate for 12 h before
inoculation of Population A. Population B appeared in the simulation 15 h after

Population A. The simulation continued for 25 h after the appearance of Population B.

106

Table 6. Parameter values for inhibition model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Value

(i=a or b)

p, 0.010 cm2 h'l
XOiQ 0.08 chh"

m, 0.08 cmth

CiH 4.08x10'6 gH cm'3 i
CiQ 6.70x10'8 gQ crn‘3
CS 5.50x10'5 gs crn‘3
DH 0.01 or? hj

DQ 0.033 crthr
DS 0.033 cmfh’l

Dp 0.033 cmfh'l
KDiQ 3.30x10'5EQ crn'3
KDis 2.00x10'fgs crn'3
v,“ 0.35 h~I

ViQ 0-02 gQ gr:1 h-1
vis 0.60 gs g,,'1 h’1
YiH 0‘50 gu/gH

Klajx 1.0x10'6 g cm‘T
K1,, 10x10T g crnT
Km 1.0x10'6 g cm:

 

 

 

 

Figui

107

 

Increasing Time

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A

Figure 44. E. coli responding to inhibitor gradient and chemoattractant gradient.

 

Figt
B) that is it
the inhibitr
significant
3 lower cor
effect of th
moves prt
predomina
experimen

Th1
favorable
that lose I
chemoattr:
method sl
inhibtion i
the trends

be elucida

 

108

Figure 45 is a simulation of a mutation event giving rise to a mutant (Population
B) that is insensitive to the inhibitor. As in Figure 44, Population A, which is sensitive to
the inhibitor, grows preferentially near the sink. Population B takes longer to appear in
significant concentration, because this population was initiated after Population A, and at
a lower concentration than Population A, to simulate the UV mutagenesis. The beneficial
effect of the chemoattractant in separating the two populations is evident, as Population B
moves preferentially toward the source reservoir, while Population A remains
predominantly near the sink reservoir. These trends are similar to those observed
experimentally in Figure 42.

The model allowed the effectiveness of the method to be explored under less
favorable conditions than were experienced in the experimental work, such as mutants
that lose their chemotactic re3ponse, or that appear in areas of the DGC far from the
chemoattractant source. The model predicts that even in these worse case scenarios, the
method should still allow the desired mutants to be selected. The noncompetitive
inhibtion form used in the modeling (Equation ( 47), for example) adequately reproduced
the trends of the experiment, but other forms may be more physically realistic, and may

be elucidated through further experiments and modeling.

 

 

 

109

 

 

Increasing Time

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\\.=.
A

 

Figure 45. Simulation of experiment shown in Figure 42.

The r
asimulation
of Populatio
two populat
predominan
to Q. T0 es
repeated wf
chemotaxis

potent the t

1 10
The model can also be used to test other situations. For example, Figure 46 shows
a simulation of the case where the mutation occurred in the middle of the growth pattern
of Population A, and the mutant (Population B) was not chemotactic to S. In this case, the
two populations were still able to be separated, but in this case the separation is based

predominantly on the influence of the inhibitor gradient and spreading due to chemotaxis

"

to Q. To estimate the benefit obtained by chemotaxis toward S, this simulation was then

 

repeated with chemotaxis to S reinstated. The results, shown in Figure 47, indicate that
chemotaxis toward S does further enhance the rate of separation. Presumably, the more

potent the chemoattractant (i.e., the higher the x'mj value) the greater the enhancement.

 

 

111

 

Increasing Time

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 46. Mutant appears far back in population.

 

112

 

 

 

Increasing Time

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 47. Mutant now chemotactic to S.

7. BIORI
The
environmen
e101,. 1993:
references,
or consider
might sugg
that severa
through so:
Rec
porous me
Walk mod
the ratio (
porous mt
column ti.
both the r
replaced
eCOiOgica
roots by j

and Chen-

7. BIOREMEDIATION

The transport of bacteria through porous media, specifically soil or aquifer
environments, has been studied in both experimental and modeling systems (Abu~Ash0ur,
et al., 1993; Sarkar, et al., 1994a; Sarkar et al., 1994b, for example). In a number of these
references, the chemotactic movement of the bacteria has been ignored (Tan, et al., 1994)
or considered too complex to include, even though results have been recorded which
might suggest chemotaxis is important. For example, Abu-Ashour et al. (1993) reported
that several field and laboratory experiments found that the average velocity of bacteria
through soil was faster than the velocity of tracers and of the groundwater.

Recently, a few studies have addressed the role motility plays in transport through
porous media. Duffy et al. (1995) attempted to adapt the homogeneous media random
walk model to a porous media. They utilized a tortuosity factor that was proportional to
the ratio of the random motility in bulk liquid to the effective random motility in the
porous media. Their model gave good qualitative agreement with an experimental sand
column through which Pseudomonas putida swam. Barton and Ford (1997) showed that
both the random motility coefficient and the chemotactic sensitivity coefficient could. be
replaced with effective values that incorporate the effect of the porous media. An
ecological system in which chemotaxis through soil is important is the colonization of
roots by Rhizobium meliloti. Soby and Bergman (1983) demonstrated that active motility

and chemotaxis were necessary for efficient spreading of the bacteria through soil.

113

 

 

 

The:
an in situ
phenanthren
the areas of
They blame
sands. reter
contaminar
presented 2
model inc
experimen

PSt
tetrachlori
chloroforr
other mic
microbe '
€Xperime
1996). 1
bioaugm.
Schoolcr
as an elt
aCccptor

COmami:

114

The effective transport of bacteria through the soil is an important requirement for
an in situ bioremediation project. Devare and Alexander (1995) found that a
phenanthrene-metabolizing Pseudomonas sp. could adequately remove phenanthrene in
the areas of a soil column in which the bacteria was inoculated, but not in other areas.
They blamed this result on bacterial retention by the clay soil, and noted that in aquifer
sands, retention was much less. Their conclusion was that transport of bacteria to the
contaminant was very important to the success of bioremediation. Bosma et al. (1988)
presented a model for bacteria consuming xenobiotic chemicals in a soil column. Their
model incorporated chemotactic movement, and its results compared well with
experimental data.

Pseudomonas stutzeri KC is a denitrifying bacterium able to degrade carbon
tetrachloride (CT) into carbon dioxide and nonvolatile products without the production of
chloroform (Criddle et al., 1990; Dybas et al., 1995; Mayotte et al., 1996). In contrast,
other microbes typically convert CT to chloroform. The outstanding potential of this
microbe for bioremediation of CT spills has been demonstrated both in shaker-flask
experiments (Tatara et al., 1993), and a model aquifer system (Witt 1994; Mayotte et al.,
1996). The Michigan Department of Natural Resources is sponsoring a major
bioaugmentation field experiment with this organism in a CT-contaminated aquifer near
Schoolcraft, Michigan. In this experiment, acetate is periodically injected into the ground
as an electron donor for the reaction, and the naturally occurring nitrate is the electron
acceptor. The Pseudomonas KC is introduced into the aquifer soil, and the flow of

contaminated water is directed through a "biofence" of the bacteria.

 

 

Chet
in the labo
packed witl
equal to tha
indicated ti
front was n

15 cm/day.

7.1 Expt

7.1.1 Swa

Stu
KC t0 var
were prep;
adjusted tr
SOlutionsr
into petri
Pseudmm
exlitifimer
SCientific

Fi
levels of:

nitrate c0

1 15
Chemotactic movement of Pseudomonas KC has been observed in sand columns
in the laboratory (personal communication with Mike Witt). The sand column was
packed with aquifer sand, and supplemented with acetate. The nitrate concentration was
equal to that observed in the Schoolcraft aquifer. Preliminary experimental measurements
indicated that the chemotactic movement was in response to nitrate gradients. The cell

front was moving at a velocity approximately 8 cm/day faster than the water velocity of

15 cm/day.

7 .1 Experimental results

7 .1.1 Swarm plates

Studies were initiated to characterize the chemotactic response of Pseudomonas
KC to various possible chemoattractants. Solutions of Medium D (Tatara, et al., 1993)
were prepared with varying levels of acetate and/or nitrate. The pH of the medium was
adjusted to 8.2. Swarm plates (Petty Setiawan's work) were made using the Medium D
solutions supplemented with 0.25% high strength agar gel. The gel mixture was poured

into petri dishes and allowed to solidify. A 20 til aliquot of actively growing

Pseudomonas KC culture was inoculated in the center of the plate. For the anaerobic
experiments, the plates were incubated in a GasPak 150TM Anaerobic System (VWR
Scientific). For aerobic experiments, the plates were incubated on the lab bench.

Figure 48 shows the chemotactic ring patterns formed in response to varying
levels of nitrate, at a constant acetate concentration of 1 g/l. In the image on the left, the

nitrate concentration was 1 g/l, and on the right, the nitrate concentration was 2.5 g/l. A11

 

 

 

 

swarm plate
to l g nitre
Pseudomom
acetate to 11
image of F
occurred. C
inner ringt
consumptit

may be du-

 

experirr

the left

1 l6
swarm plate images were taken 48 hours after inoculation. The mass ratio of l g acetate
to 1 g nitrate is the balanced ratio for consumption of the two chemical species by
Pseudomonas KC (personal communication with Dr. Craig Criddle). When the ratio of
acetate to nitrate was equal to one, only a single ring appeared, as shown in the left-hand
image of Figure 48. In the right-hand image, when nitrate is in excess, a double ring
occurred. One hypothesis is that the outer ring is due to consumption of acetate, and the
inner ring due to the use of the remaining nitrate, or an intermediate such as nitrite, for the
consumption of another carbon source, such as endogenous metabolism. The outer ring

may be due to acetate, nitrate, or a combination of both.

 

Figure 48. Chemotactic rings of Pseudomonas KC at varying nitrate levels.

In the next experiments, the acetate concentration was varied. Figure 49 shows
experiments at a constant nitrate level of 50 mg/l, and acetate varying from 100 mg/l on

the left to 1000 mg/l on the right. In the low acetate image, there is no noticeable

 

movement i
miniml let
considerablt
Figure 48. I
hand image
image 100(
the many '
appeared v
bubbles m

denitriftca‘

 

117
movement from the inoculation condition. Both growth and motility seem to be at a
minimal level. In the higher acetate image, however, the high density ring indicates
considerable growth, but the diameter of the ring indicates that motility is slower than in
Figure 48. In the next set of experiments, the nitrate level was set to 500 mg/l. In the left-
hand image in Figure 50, the acetate concentration was 100 mg/l, and in the right-hand
image 1000 mg/l. At 100 mg/l acetate, only a single ring forms. In the 1000 mg/l image,
the many bright white dots are bubbles that appeared in the gel. The bubbles only
appeared when the acetate to nitrate ratio was greater than one. We hypothesize that the
bubbles may contain nitrogen that forms when excess acetate is present to complete the

denitrification reaction shown in Equation ( 52)

 

Figure 49. Nitrate concentration = 50 mg/l, varying acetate.

 

Pseudomc

acetate is
acceptor t

nitrogen,

mEdium.
found ti
r2iiiging
fOrrned.

at the h]

Pseudo

118

 

Figure 50. Nitrate concentration = 500 mg/l, varying acetate.

Pseudomonas KC is known to preferentially use nitrate before consuming nitrite. When
acetate is in excess, however, the Pseudomonas may utilize the nitrite as an electron
acceptor to further consume the remaining acetate. This would result in the formation of

nitrogen, and possibly the bubbles.

No; —> No; —> N2 (52)

In aerobic experiments with the Pseudomonas KC, no nitrate was added to the
medium. Instead, oxygen was used as the electron acceptor. In results not shown, it was
found that no ring formed at an acetate concentration of 100 mg/l. For concentrations
ranging between 1000 mg/l and 1750 mg/l, a ring of approximately 4.7 cm in diameter
formed. At a concentration of 2500 mg/l, no ring was observed. These results suggest that
at the higher concentrations of acetate, the chemotactic receptors may be saturated.

Additional experiments were performed to test the aerobic response of

Pseudomonas KC to attractants other than acetate. Medium M9 (Maniatis et al., 1982),

 

adjusted to
medium we
more cells
experiment
supplemen

carbon sou

  

119
adjusted to pH 7 .5, was used for the medium. In the left-hand image in Figure 52, the M9
medium was supplemented with 1 mM glucose. A distinct chemotactic ring formed, but
more cells seemed to remain in the center of the plate than in the acetate/nitrate
experiments shown above. In the right-hand image of Figure 52, the medium was

supplemented with 0.1 mM aspartate as the chemoattractant, and 2 mM glycerol as the

carbon source. In this instance, a fairly wide, diffuse ring was observed.

 

Figure 51. Pseudomonas KC responding to glucose (left) and aspartate(right).

7.1.2 DGC experiments

Experiments were performed in the DGC to test the response of the Pseudomonas
KC to applied gradients of acetate and nitrate. Medium D-, containing 3.5 g/l K2HP04,
1.24 g/l KH2P04, and 1.0 g/l (NH4)2SO4 was used as the minimal medium for the

experiment. The pH of the medium was adjusted to 8.1 with KOH. After autoclaving, 2

 

mid of l M
to test the g
experiments
comparable
contained 1
minimize t
chemotacti
degradatio:
An

nitrate ant
addition tt
sink flask
and 0.25‘.
C011 grow
toward ti
nearest t
alilieared
Source 1
Samples

the DG(

151118

120

m1/1 of 1 M MgSO4 was added to the medium. Shake flask experiments were performed
to test the growth characteristics of Pseudomonas KC in this medium. The shake flask
experiments showed that the maximum growth, as measured by optical density, was
comparable to growth in the complete Medium D recipe. The Medium D- recipe
contained fewer components, which was useful not only for simplicity, but also to
minimize the number of consumable chemical components that could possibly elicit a
chemotactic response. The effects of using Medium D- on carbon tetrachloride
degradation were not studied as part of this work.

An initial DGC experiment was performed to test the chemotactic response to
nitrate and acetate in Medium D- (results not shown). The source flask contained, in
addition to 800 ml of Medium D—, 5 mM sodium acetate and 5 mM sodium nitrate. The
sink flask contained 800 ml of Medium D—. The arena of the DGC contained Medium D-
and 0.25% high-strength agar gel. After 28 hours of growth, a pattern similar to the E.
coli growth patterns shown in Figure 13, had developed, with a definite bias in movement
toward the acetate/nitrate source. A 0.5 ml sample was removed from the gel at the point
nearest to the source reservoir. A similar sample was removed from the gel at what
appeared to be the center of the chemotactic wave moving toward the acetate/nitrate
source reservoir. The samples were injected into shake flasks and incubated. These
samples were taken in an effort to isolate bacteria that were optimized for chemotaxis in
the DGC environment.

A second DGC experiment was performed with the same parameters. This time, a

15 til aliquot of the cell culture taken from the spot nearest the wall in the previous

 

 

 

 

 

experiment
22 hours b:
Figure 52.

inoculation

121
experiment was used to inoculate the chamber. The gradient was allowed to initialize for
22 hours before inoculating the bacteria. The results of this experiment are shown in
Figure 52. The times shown in the corners of the images correspond to the time after

inoculation of the bacteria. The acetate/nitrate source is at the bottom in each image.

 

 

 

122

 

Figure 52. Pseudomonas KC responding to acetate and nitrate in a DGC.

 

 

7.1.3 Mo‘

A
otherwise
objects. Tr
experimet
sodium at
KOH. Ti
connector
solidify,
Figure 53
across th
ends oft

the wave

 

123

7 .1.3 Movement of wave around obstacles

A first logical step in understanding the effect of solids on chemotaxis in an
otherwise homogenous medium would be to study a medium which has one or a few
objects. To test the response of the chemotactic wave to obstacles in its path, swarm plate
experiments were performed (Laura Booms' work). Medium D- supplemented with 5 g/L
sodium acetate, 3 g/L sodium nitrate, and 0.25% agar was adjusted to a pH of 8.2 with
KOH. The plates were poured, and before the agar solidified, two sterilized tubing
connectors with sealed ends objects were placed in the gel. The gel was allowed to
solidify, and then 20 111 of Pseudomonas KC culture were inoculated in the center. In
Figure 53A, taken 26 hours after inoculation, the wave has moved approximately halfway
across the tubing connectors. In Figure 53B (42 hours), the wave has moved beyond the
ends of the connectors. The objects did not appear to perceptibly inhibit the movement of

the wave.

 

Figure 53. Movement of wave around obstacles in swarm plates.

To
performed
diameter 1
the path 0

P5
and the st
in a micrc

15 pl 0ft
the gradit

A
The time
the botto
in Figure
experimt

will not

124

To test how a large object would affect the chemotactic wave, an experiment was
performed under the following conditions in a DGC. A cut-off glass test tube 1 cm in
diameter was inserted into the gel, close to the source reservoir, to serve as an object in
the path of the chemotactic wave.

Pseudomonas KC were grown for 3 days in a 100 ml flask containing Medium D-
and the same ratio of acetate and nitrate as in the source flask. 6 m1 of culture were spun
in a microcentrifuge, and the resulting pellets were resuspended in 0.5 ml of fresh media.

15 1.11 of the resuspended culture were inoculated into the center of the arena 5 hours after

the gradient was initiated.

A time-sequence of image captures from the experiment is shown in Figure 54.
The time increases from left to right, and t0p to bottom. The source reservoir is located at
the bottom in each picture. The edge of the wave just encountered the tube in Figure 54A.
In Figure 54D the cells have passed the object and have rejoined on the other side. This
experiment indicates that, at least in this very simple single-object situation, an obstacle

will not catastrophically disrupt the chemotactic movement of the cell population.

 

 

125

 

 

 

Figure 54. Chemotaxis of Pseudomonas KC around an object.

 

7.2 Fut

Ft
hypothesi
focus on
a wall of
DGC wa
cm from
into the
allowed
solidifie
Experirr
designer
glass be

Structur

could b.

7.2 Future work on bioremediation

Further experimental and modeling work will be necessary to prove the
hypothesis that chemotaxis can enhance a bioremediation effort. First, studies should
focus on the effects of a porous medium on the chemotactic response. A method to create
a wall of glass beads, sand, or soil in the DGC has been developed (see Figure 55). A
DGC was modified to allow two specially cut microscope slides to be placed 1 cm and 2
cm from one wall. Glass beads were placed between the slides, and the gel was poured
into the chamber. Small gaps between the slides and the bottom cover of the DGC
allowed the gel to penetrate under the slides and up into the beads. After the gel
solidified, the slides were carefully removed, and the three gel regions combined.
Experiments to test how the bacteria are transported through the porous wall should be
designed and performed with and without chemoattractant gradients present. Initially,
glass beads of known diameter could be used to give a system with well—defined pore
structure. Next, a more environmentally realistic wall material, such as aquifer sands,

could be used.

126

v

 

 

 

Chemo
source

contami
had a it
used to
model 1
model

Worder

pOI‘OUS

Carbon

127

Microscope slides

 

Chemoattractant F3611 1 '
source Inocu atron
pornt

 

 

 

 

 

 

Figure 55. Set-up to make glass-bead wall in DGC.

A version of the mathematical model was developed that included a balance for a
contaminant. The balance was similar to the nutrient and chemoattractant balances, but
had a term that accounted for disappearance of the contaminant. This model could be
used to predict the enhanced contaminant degradation when chemotaxis occurs. The
model was never fully tested and debugged, and suitable parameters for the contaminant
model were not found. A new cellular dynamics model has been proposed by Drs.
Worden and Lastoskie which would allow for predictions of the behavior of cells in a
porous medium.

For the Pseudomonas KC system specifically, a method to quantify the amount of

Carbon tetrachloride in a DGC will need to be developed. A control experiment with no

applied c?
degradatic
movemen'
compare 1
growth st
independe
Or
bioremed
cells witl
injection
possible
uncontan
pulse che
how the
(Section
maximiz
0Plimize
zones. 1
bioreme

money (

128
applied chemoattractant gradient should be performed to measure the amount of
degradation of carbon tetrachloride that takes place in the absence of chemotactic
movement. Then an experiment with an applied chemoattractant gradient should be run to
compare the degradation rate to the control experiment. Factors such as the amount of
growth should be accounted for in the analysis, so that the effect of chemotaxis can be
independently evaluated.

Other experiments to discover ways in which chemotaxis could improve a
bioremediation effort could focus on the concept of controlling the dissemination of the
cells with chemoattractant gradients. One hypothesis is that by carefully placing the
injection wells through which the chemoattractant is introduced into the soil, it may be
possible to direct the spreading of the bacteria so as to minimize the losses into
uncontaminated zones. A relatively simple experiment to test this hypothesis would be to
pulse chemoattractant into different reservoirs of the DGC at timed intervals, and observe
how the cell population responded. In the section of this thesis on microbial competition
(Section 5), the model predicted that optimal chemoattractant concentrations exist that
maximize the chemotactic wave properties, such as the carrying capacity, and could
optimize the delivery protocols to obtain the best transport of cells to the contaminated
zones. The mathematical model should prove to be an extremely useful tool to explore
bioremediation, and, through modeling simulations, considerable experimental time and

money can be saved.

 

 

8. SUM

Tl
engineeri
response.
nricrobio
diffusion
chemoat'
mathem:
having I
predictit

involve:

respons
mathen
the pre
advant
technic
compe
eXploi
could

inhibi

8. SUMMARY

The focus of this research was on microbial chemotaxis, and how it could benefit
engineering applications. Tools were developed to aid in the study of the chemotactic
response. Experimental tools included the diffusion gradient chamber, microsensors and
microbiosensors, and the laser diffusion capillary assay. A mathematical model of the
diffusion gradient chamber was also developed that allowed two cell balances, two
chemoattractant compounds, and a nutrient compound to be modeled simultaneously. The
mathematical model allowed predictions of chemotactic responses to be made without
having to run time consuming experiments. The model was validated by comparing its
predictions to experimental data from DGC experiments. The validation experiments
involved Escherichia coli responding to gradients of aspartate.

The applications that were identified that could benefit from the chemotactic
response were microbial competition, selection of mutants, and bioremediation. The
mathematical model was used to explore the effects of several modeling parameters on
the predicted competitive outcomes. Chemotaxis was shown to give a competitive
advantage to a bacterial population in certain cases in non-mixed environments. Several
techniques to analyze the modeling predictions were developed, including the dynamic
competition factor. For the selection of mutants t0pic, the chemotactic response was
exploited to draw mutants of a bacterial species into an area in which the parent strain
could not grow well. Again, the mathematical model was used to more fully explore the

inhibition and chemotaxis interaction. The model predicted that the selection method

129

 

 

 

 

 

should wr
bioremed
mmdm

obstacles
hmMm
were per
benveen

asanaqt

130
should work well in various scenarios. Finally, the response of an important bacterium for
bioremediation, Pseudomonas stutzeri strain KC, was studied. Pseudomonas KC was
found to be chemotactic to nitrate and acetate. The reaction of the chemotactic wave to
obstacles also studied as a first step in examining chemotaxis through porous media.

Preliminary competition experiments between Pseudomonas KC and Escherichia coli

 

were performed. Further experiments could lead to a better understanding of the interplay
between chemotaxis and competition in environments important to bioremediation, such

as an aquifer.

 

 

 

 

 

 

 

APPENDICES

 

 

 

 

APPENDIX A

 

 

APPEl‘

Miscella

1‘
section 1

chemotz

organis:
require:
aliquot
56A, g
gel. Th
be attr
oxyger
and 1

define
aspan
multi}
how
the of
after

resea:

APPENDIX A

Miscellaneous chemotaxis experiments

Appendix A includes selected experimental results that do not directly fit into a
section of the thesis, but that are interesting or informative in the context of studying the
chemotactic response.

, Figure 56 shows three swarm plate experiments. In each of these experiments, the
organism used was E. Coli HCB 33. M63 minimal medium, supplemented with the
required amino acids and streptomycin (see Section 4.2.3), was the base medium. A 10 111
aliquot of actively growing culture was inoculated into the center of each plate. In Figure
56A, glucose was initially present at a constant concentration of 3 mM throughout the
gel. The image was taken 20 hours after inoculation. The multiple waves or bands could
be attributed to chemotaxis to other chemoattractants, one of which would likely be
oxygen. Figure 56B, also 20 hours after inoculation, shows the response to 1 mM glucose
and 1 mM aspartate. In this case, a very uniform single wave formed, that is sharply
defined on the outer edge. Since at least two known chemoattractants (glucose and
aspartate) were present, and possibly a third with oxygen, it might be assumed that
multiple bands would occur. This did not occur experimentally in this particular case,
however. It is possible that the consumption of the two chemoattractants was similar, and
the chemotactic waves overlapped closely. The image in Figure 56C was taken 42 hours
after inoculation. This swarm plate contained 1 mM glucose and 3 mM glycerol. Some

research has reported that glycerol can be an inhibitor of chemotaxis (see Zhulin et al.,

131

 

 

 

 

 

1997 f0

conditio

 

Patten

format
1995;

Symmt
0r S. 1

substa

and g
Strept.

inocu

M63.

132

1997 for a review), which might explain the slower swarming under the glycerol

conditions.

   

Figure 56. Swarm plates with various chemoattractants.

Pattern formation

An interesting phenomenon that is thought to be related to chemotaxis is the
formation of patterns in bacterial populations (Agladze et al., 1993; Woodward et al.,
1995; Budrene and Berg, 1995). In these studies, swarm plates were used to study the
symmetrical patterns, sometimes described as looking like snowflakes, formed by E. coli
or S. typhimurium. The current hypothesis is that the bacteria exude a chemoattractant
substance, which causes conglomeration of bacteria in patterns.

A DGC experiment was set-up to test the response of E. coli HCB 33 to aspartate
and glucose. M63 minimal medium was used with the required amino acids and
streptomycin. The source concentrations were 3 mM aspartate and 5 mM glucose. The
inoculation culture was grown in a shake flask with 10 mM glucose and supplemented

M63. The gradients were allowed to initiate for 5 1/2 hours prior to inoculation. A 10 ml

 

aliquot
the cent
until 1t
DGC,

concen'

hours)

 

produ
HOWe
the c.

T688211

MOW

the C

 

133
aliquot of cells that had been growing in the shake flask for 22 hours was inoculated in
the center of the chamber. Because of a problem with the camera, no images were taken
until 16 hours after inoculation. At that time, a very unique pattern had formed in the
DGC, as shown in Figure 57A. This pattern includes spots indicating local high

concentrations of E. coli. The pattern continued to evolve, as shown in Figure 57B (18

hours) and Figure 57C (20 hours). Attempts to reproduce these results failed.

 

Figure 57. Pattern formation in the DGC.

A version of the mathematical was developed to incorporate a chemoattractant
produced by the bacteria. This model was tested, and appeared to be working properly.
However, no patterns were observed in the simulations. Work may be necessary to find
the correct parameter combinations that allow for pattern formation, if this line of
research is continued.

Movement around objects

A group of swarm plate experiments was performed to analyze the movement of

the chemotactic wave of Pseudomonas KC around objects. The experimental conditions

 

 

 

are des
object
channe
around
was [11:
the cor
cells vs
gel sur
hypotl
apprm
be har
in ca;

capillz

134
are described in Section 7.1.3. In Figure 58, three objects were placed in the gel. The
object in the lower right—hand section of the gel was a tubing connector with an open
channel through it. It appeared that the cells swam more quickly through the opening than
around the object. Two hypotheses were formed to explain this phenomenon. The first
was that the tubing connector was full of water remaining from the autoclave cycle. When
the connector was immersed in the gel, the gel may not have displaced the water. The
cells would then encounter less resistance to swimming in the water as compared to the
gel surrounding the connector, and could move through the connector faster. The second
hypothesis was that, in the narrow confines of the connector (the channel is
approximately 0.4 mm in diameter), the cells tumbling would be confined, and it would
be harder for the cells to change swimming direction. Evidence of this has been observed
in capillary assays (Liu and Papadopoulos, 1995; Liu et al., 1996), although their

capillaries had much smaller diameters.

 

Figure 58. Chemotaxis through a hollow object.

 

APPENDIX B

 

APPEl

The All

and fro
tndiagc

time st:

APPENDIX B

The Alternating-Direction Implicit Method

This discussion and the figures have been adapted from Chapra and Canale (1988)
and from Carnahan et al. ( 1969). The altemating—direction implicit (ADD scheme uses
tridiagonal matrices to solve parabolic equations in two or more spatial dimensions. Each

time step is broken into two steps, as shown in .

 

o Explicit

 

o Implicit

 

 

 

 

 

 

 

 

 

 

First half-step Second half-step

Figure 59. The alternating-direction implicit ethod.

The first step to solve the simple two-dimensional diffusion equation given in Equation (

4) is written

135

 

 

 

 

such tl
implici

Equatiu

where
tridiag

apprO)

so tha

explic

Again

136

—1.j 1,j+l .,j—1

At / 2 ’ 5 (M2 + (Ay)2

 

 

SSW — Silij D 51:14 ‘25:]. + S," Sim/2 - 25:?” +S."+l/2] (53)

such that the approximation of 828/ax2 is explicit, while the approximation of 828/8y2 is
implicit. For the model presented in Section 4.2, the grid was square, and thus Ax=Ay, so

Equation ( 53) may be rearranged as

— 1W2 + 2(1+ ”5,33%”? — 154“” = 15:11.]. + 2(1— A)S,'jj + 215,1, , ( 54)

uj-l l,j+l

where X=DSAU(Ax)2. When Equation ( 54) is written for each point on the grid, a

tridiagonal set of simultaneous equations results. For the second half step, Equation ( 4) is

 

 

approximated by
Stiff _ Si’fil/z _ Sir-ii; _ 25:31 + Sin—iij + Still/12 — 25:?” + 5:;32 ( 55)
At / 2 5 (my (Ayf

so that now the approximation of azs/axz is implicit, and the approximation of 828/8y2 is

explicit. Equation ( 55) can be rearranged to yield

— 15:17}, + 2(1+ MS“ — 15,17} , = 2.53”“ + 2(1— ”5,7?” + 25,737,112 ( 56)

ivj [hi—1

Again, when Equation ( 56) is written for the entire grid, a tridiagonal matrix results.

 

 

 

 

 

APPENDIX C

 

 

APPE

Instrur

Th
(ua ant
chamo
that th<
0 Th
0 Fil

OEa

Onaf

diffus

Spaci
the g
grow
eXpe

outp

 

APPENDIX C

Instructions for FORTRAN model

The program currently solves five coupled PDE'S. These include two cell balances
(ua and ub), two chemoattractants (s and q) and a growth nutrient (h). The sides of the
chamber are identified by compass directions (n,s,e,w). For the version of the program
that these instructions pertain to, the chemoattractant q should be thought of as oxygen.
0 The unit basis for all variables is mass=grams, length=centimeters, time=hours.
0 Filenames are italicized, while variable names are in "quotes".

0 Each input file is formatted, so be sure to keep numbers in the proper positions.

On a Sun or HP:
1. Modify the param file.

This file contains the parameters of the model, including chemotactic sensitivities,
diffusion coefficients, mass transport coefficients, etc.

The param file also contains information about the simulation run-time and grid
Spacing. The value of "m" sets the grid size (m x m matrix). "Timeinit" sets the time that
the gradients establish before cell inoculation. "Timeinoc" sets the time that the cells
grow in the chamber. "Timeinit" plus "timeinoc" gives the total simulated time of the

experiment. Param also contains instructions for the simulation that specify a movie

output or single graph output.

137

_'

 

 

chemt
begin:
exam]
gel is
begin:
conce
initial
and "1

cell in

"Num
four g
"time:
Para"
not pr
4. n

S

“36d.

138

A list of variables contained in the param file and their definition is included at the
end of this section.

2. Modify the init file.

This file sets the initial conditions, including initial cell concentration,
chemoattractant boundary conditions, and nutrient boundary conditions. The parameters
beginning with r (rsn, for example) correspond to the reservoir concentrations. In the
example, rsn is reservoir, s-chemoattractant, north side. The initial concentration in the
gel is also set in init. For example, "stini" sets the concentration of s in the gel at the
beginning of the simulation. The nutrient concentration is usually set equal to the nutrient
concentration in the four reservoirs. For a chemoattractant that will make a gradient, the
initial concentration in the gel is usually 0. The cell concentration is initialized by "uaint"
and "ubini". Note that these are not the initial concentrations, but are a parameter for the
cell initialization subroutine.

3. Modify the "time" file.

"Numti" tells the program how many matrices to print. For example, for "numti" =04,
four graphs would be produced at times specified by "timel

"time2", etc. The times are time after inoculation. Note that the variable "timeinoc" in
param specifies how long the simulation will run, so any time longer than "timeinoc" will
not print a graph.

4. If necessary, modify the FORTRAN program itself.

Subroutine uaO sets the initial cell distribution. Currently, an exponential peak is

used.

 

 

Tl
both f
be cha
differe

Ti

0

bound

comp‘

Toru

word

Dati

The
Calle

Whh

 

139

The subroutines f and ifs set the nutrient and chemoattractant uptake functions. Note:
both f and fs must have identical functionalities. If one is changed, the other must also
be changed. Both functions correspond to the same term in the equations, but are slightly
different because of the finite differencing algorithm.

The subroutine chemofl sets the functionality of the chemotaxis term.

Other subroutines set the windows of the reservoirs, calculate velocities, set
boundary conditions, and calculate certain other values.
5. To run the program, a Sun computer must be used. If an HP is the current
computer, open a window to a Sun.
To run the program in batch mode, type:

batch
An at> prompt should appear. Type the following commands (..J = return, underlined
words should be replaced with appropriate name, and bold indicates pressing control and
D at the same time):

at> f77 _tllegamej .J

at> a.out .J

at> mailx -s "subject" mail-address J

at> ctrl-D

The simulation is complete when mail bearing the "subject" is received. At this time, files
called lcella.m, lcellb.m, latts.m, lattq.m, and lnut.m should be in the directory from

which the program ran. These files are formatted for Matlab.

 

 

 

 

 

Matlal

1.A

2.A

3. Ty
exam
4. I
For 6
shoul

5. Tv

will
Higl
give
Mat

help

140

Matlab instructions:
1. At a Sun or HP, start Matlab by typing:

matlab
2. At the Matlab prompt, switch to the directory containing the simulation files:

cd path\directog
3. Type the name of the file you wish to work with, without the m extension. For
example, to look at the cell Population a matrix, type lcella
4. To check if the matrices have been properly loaded, type who
For example, if "numti" = 4, and if individual graphs were specified, then there
should be 4 matrices named ual, ua2, ua3, and ua4.
5. Two types of graphs are useful for viewing:
Three dimensional plots are created with the surf command. surf(ual) will yield a 3-D
plot, with Matlab default view angle and axis.
Overhead view type plots are created with the appmult command.

appmult(ual,3,'pcolor')
will give a good overhead plot. The value 3 can be changed to give better or worse plots.
Higher numbers give smoother graphs, but take longer to view and print. Lower numbers

give coarser plots, but are faster.

Matlab has a fairly good help feature. Typing help will list general help topics, or type

help command to get specific help.

 

 

 

 

 

APPENDIX D

 

APP:

Thel

canb

Popu

APPENDIX D

The FORTRAN model

This is the FORTRAN model used to solve the competition model. This model
can be reduced to the one-cell population model simply by setting the initial condition for

Population B to zero in the initialization file (see Appendix E).

implicit double precision (a—h,o-z)

C
parameter (nmax=100,kmax=100)

C
dimension x(0:nmax),y(0:nmax)
dimension ual(O:nmax,0:nmax),ua2(0:nmax,0:nmax)
dimension ub1(0:nmax,0:nmax),ub2(0:nmax,0:nmax)
dimension H1(O:nmax,0:nmax),H2(O:nmax,O:nmax)
dimension S 1(O:nmax,0:nmax),82(0:nmax,0:nmax)
dimension q1(O:nmax,0:nmax),q2(0:nmax,0:nmax)
dimension aa(0:nmax),bb(0:nmax),cc(0:nmax)
dimension funua(O:nmax),funs(0:nmax),funh(0:nmax)
dimension rhsua(0:nmax),rhss(0:nmax),rhsh(0:nmax)
dimension funub(0:nmax)
dimension rhsq(0:nmax),funq(0:nmax),rhsub(0:nmax)
dimension uua(0:nmax),us(0:nmax),uh(0:nmax),uq(0:nmax),uub(0:nmax)
dimension mcount3(0:kmax), time(Ozkmax)
dimension velm(0:500),vell(0:500)
dimension distm(0:500),distl(0:500),timm(0:500),timl(0:500)

open(16,file='lcella.m')
open(32,file='lcellb.m’)
open( 17,file='latts.m')
open(23 ,file='1attq.m')
open(18,file='lnut.m')
open( 1 9,fi1e='time')
open(24,file=‘init')
open(22,file=‘param')
open(30,file='distmax')
open(3 1 ,file='distlim')

141

 

 

142

read(22,101)m
read(22, 103)timinit
read(22,103)timinoc
read(22, 102)numcel
read(22,102)numvel
read(22,102)kstep
read(22,100)dmin
read(22,102)movc
read(22,100)R
read(22, 100)ayoff
read(22, 100)axoff
read(22, 100)byoff
read(22, 100)bxoff
read(22, 100)awid
read(22, 100)bwid
read(22,100)Dh
read(22, 100)thn
read(22, 100)ths
read(22, 100)the
read(22, 100)thw
read(22, 100)Ds
read(22, 100)tsn
read(22, 100)tss
read(22, 100)tse
read(22, 100)tsw
read(22,100)Dacs
read(22, 100)Dbcs
read(22, 100)DKas
read(22, 100)DKbs
read(22, 100)cas
read(22, 100)vas
read(22, 100)Yas
read(22, 100)cbs
read(22,100)vbs
read(22,100)Ybs
read(22, 100)Dq
read(22, 100)tqn
read(22, 100)th
read(22, 100)tqe
read(22, 100)tqw
read(22,100)Dacq
read(22, 100)Dbcq
read(22, 100)DKaq

 

 

143

read(22, 100)DKbq
read(22, 100)caq
read(22, 100)vaq
read(22, 100)Yaq
read(22, 100)cbq
read(22, 100)qu
read(22, 100)qu
read(22,100)Dua
read(22,100)va
read(22,100)ca
read(22, 100)Ya
read(22,100)Dub
read(22, 100)vb
read(22, 100)cb
read(22,100)Yb
read(24, 100)uaini
read(24,100)ubini
read(24, 100)rsn
read(24,100)rss
read(24, 100)rse
read(24,100)rsw
read(24, 100)rqn
read(24, 100)th
read(24, 100)rqe
read(24,100)rqw
read(24, 100)rhn
read(24, 100)rhs
read(24, 100)rhe
read(24, 100)rhw
read(24, 100)stini
read(24, 100)qtini
read(24, 100)htini
100 format(6x,dl3.10)
101 format(4x,i4)
102 format(7x,i7)
103 format(8x,d13.10)
if(movc.eq.2)then
write(l6,*)'ua=['
write(32,*)'ub=['
write(17,*)'s=['
write(23,*)'q=['
write(18,*)'h=['
endif

 

C

144

hr=2.0d0*R/m
ht=0.15625d0*hr*hr
mcount1=timinit/ht+1
mcount2=timinocfht+1
write(*,*)'hr =',hr
write(*,*)'ht =',ht
write(*,*)'mcount1=',mcount1
write(*,*)'mcount2=',mcount2

read(19,201)numti
do 555 i=1,numti
read(19,202)time(i)
mcount3(i)=time(i)/ht
time(i)=mcount3(i)*ht
write(*,*)'Real time',i,'=',time(i)
555 continue
201 forrnat(6x,i2)
202 format(6x,d6.2)
C Calculate values of hr and ht
C
C set up the initial gradient of attractant and nutrient
C
C set up the initial chemoattractant concentration in chamber
do 10 i=0,m
do 5 j=0,m
S l (i,j)=stini
q1(i,j)=qtini
H1(i,j)=htini
ua1(i,j)=0.0d0
ub1(i,j)=0.0d0
5 continue
10 continue
C
D0=Dh*(ht/2.0d0)/(hr**2)
D1=Ds*(ht/2.0d0)/(hr**2)
D2=Dua*(ht/2.0d0)/(hr**2)
D3=Dacs*(ht/2.0d0)/(hr**2)
D4=Dq*(ht/2.0d0)/(hr**2)
D5=Dacq*(ht/2.0d0)/(hr**2)
D6=Dub*(ht/2.0d0)/(hr**2)
D7=Dbcs*(ht/2.0d0)/(hr**2)
D8=Dbcq*(ht/2.0d0)/(hr**2)
C
C solving the ode

 

145

do 1000 k=1,mcount1

C
C take care of boundary points:
C
i=0
do 20 i=0,m

t=tr(i,hr,tss,Ds)
call rhbound(m,S1(i,0),S1(i,1),rss,D1,hr,t,rhss(i))
funs(i)=1.0d0
t=tr(i,hr,tqs,Dq)
call rhbound(m,q1(i,0),q1(i,1),rqs,D4,hr,t,rhsq(i))
funq(i)=1 .0d0
t=tr(i,hr,ths,Dh)
call rhbound(m,H1(i,O),H1(i,1),rhs,D0,hr,t,rhsh(i))
funh(i)=1 .0d0
20 continue
C
call PDMTRIX(m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRIX(m,DO,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh)
do 25 i=0,m
82(i,0)=us(i)
q2(i.0>=uq<i)
h2(i,0)=uh(i)
25 continue
j=m
do 30 i=0,m
=tr(i,hr,tsn,Ds)
call rhbound(m,S l(i,m),S 1(i,m-1),rsn,Dl ,hr,t,rhss(i))
funs(i)=1 .0d0
t=tr(i,hr,tqn,Dq) -
call rhbound(m,q1(i,m),q1(i,m-1),rqn,D4,hr,t,rhsq(i))
funq(i)=1.0d0
t=tr(i,hr,thn,Dh)
call rhbound(m,H1(i,m),H1(i,m-1),rhn,D0,hr,t,rhsh(i))
funh(i)=1 .0d0
30 continue

C
call PDMTRIX(m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)

 

 

146

call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRIX(m,D0,funh,aa,bb,cc)
call T RISOLV(m,aa,bb,cc,rhsh,uh)
do 35 i=0,m

S2(i,m)=us(i)

q2(i,m)=uq(i)

h2(i,m)=uh(i)

35 continue

C take care of interior points:
C
do 50 j=1,m-1
t=tr(j ,hr,tsw,Ds)
call rh2(m,S l(0,j),S 1(0,j-1),S 1(0,j+1),rsw,D 1 ,hr,t,rhss(0))
funs(0)=1.0d0+D1*hr*tr(j,hr,tsw,Ds)
t=tr(i ,hr,tqw,Dq)
call rh2(m,q1(0,j),q1(0,j- 1),q1 (O,j+1),rqw,D4,hr,t,rhsq(0))
funq(0)= 1 .0d0+D4*hr*tr(j ,hr,tqw,Dq)
=tr(j ,hr,thw,Dh)
call rh2(m,h1(0,j),h1(0,j- 1),h1(0,j+1),rhw,D0,hr,t,rhsh(0))
funh(0)=1.0d0+D0*hr*tr(j,hr,thw,Dh)

 

=tr(j ,hr,tse,Ds)

call rh2(m,S1(m,j),S1(m,j—1),S1(m,j+1),rse,D1,hr,t,rhss(m))
funs(m)=1.0d0+D1 *hr*tr(j ,hr,tse,Ds)

t=tr(j ,hr,tqe,Dq)

call rh2(m,q] (m,j),q1(m,j-1),q1(m,j+1),rqe,D4,hr,t,rhsq(m))
funq(m)=1 .0d0+D4*hr*tr(j ,hr,tqe,Dq)

t=tr(j ,hr,the,Dh)

call rh2(m,h1(m,j),h1(m,j-1),h1(m,j+1),rhe,D0,hr,t,rhsh(m))
funh(m)=1 .0d0+D0*hr*tr(i ,hr,the,Dh)

do 40 i=1 ,m—l
rhss(i)=S1(i,j)+D1*(S1(i,j+1)-2.0d0*S1(i,j)+Sl(i,j-1))
funs(i)=1.0d0
rhsq(i)=q1(i,j)+D4*(q1(i,j+1)—2.0d0*q1(i,j)+q1(i,j-1))
funq(i)=1.0d0
rhsh(i)=hl(i,j)+D0*(h1(i,j+1)—2.0d0*h1(i,j)+h1(i,j-1))
funh(i)=1.0d0

40 continue

call PDMTRIX(m,D1,funs,aa,bb,cc)

call TRISOLV(m,aa,bb,cc,rhss,us)

call PDMTRIX(m,D4,funq,aa,bb,cc)

 

147

call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRIX(m,D0,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh)
do 45 i=0,m

S2(i,j)=us(i)

h2(i,j)=uh(i)

45 continue
50 continue
C
C implicit in y direction:
C
do 55 i=0,m
do 56 j=0,m

S 1 (i,j)=S2(i,j)
q1(1,j)=q2(i,j)
h1(i,j)=h2(i,j)

56 continue

55 continue

C

C take care of boundary points:

C

 

i=0
do 60 j=0,m
t=tr(j ,hr,tsw,Ds)
call rhbound(m,S 1(0,j),S 1(1,j),rsw,D1,hr,t,rhss(j))
funs(0)=1.0d0
t=tr(j ,hr,tqw,Dq)
call rhbound(m,ql(0,i),q1(1,i),rqw,D4,hr,t,rhsqG))
funq(0)=1.0d0
t=tr(j ,hr,thw,Dh)
call rhbound(m,Hl(O,j),H1(1 ,j),rhw,D0,hr,t,rhsh(i))
funh(0)=1.0d0
60 continue

call PDMTRD((m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRIX(m,D0,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh)
do 65 j=0,m

82(0,j)=us(i)

q2(0,j)=uq(i)

 

148

h2(0,j)=uh(i)

65 continue
C
i=m
do 70 j=0,m

t=tr(i ,hr,tse,Ds)

call rhbound(m,S 1(m,j),S 1(m-1,j),rse,D1,hr,t,rhss(j))
funs(0)=1.0d0

t=tr(i,hr,tqe,Dq)

call rhbound(m,q1(m,j),q1(m-1,j),rqe,D4,hr,t,rhsq(j))
funq(0)=1 .0d0

t=tr(i,hr,the,Dh)

call rhbound(m,H1(m,j),H1(m—1,j),rhe,D0,hr,t,rhsh(j))
funh(0)=1 .0d0

70 continue

call PDMTRIX(m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRD((m,D0,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh)
do 75 j=0,m

32(m,j)=US(i)

q2(m,j)=uq(i)

h2(m,j)=uh(i)

75 continue

C take care of interior points:
C
do 90 i: 1 ,m-l
C
t=tr(i,hr,tss,Ds)
call rh2(m,S 1(i,0),S1(i-1,0),S1(i+1,0),rss,D1,hr,t,rhss(0))
funs(0)=1.0d0+D1*hr*tr(i,hr,tss,Ds)
=tr(i,hr,tqs,Dq)
call rh2(m,q1(i,0),q1(i-1,0),q1(i+1,0),rqs,D4,hr,t,rhsq(0))
funq(0)= 1 .0d0+D4*hr* tr(i,hr,tqs,Dq)
t=tr(i,hr,ths,Dh)
call rh2(m,h1(i,0),h1(i-1,0),h1(i-1,0),rhs,D0,hr,t,rhsh(0))
funh(0)=1 .0d0+D0*hr*tr(i,hr,ths,Dh)

=tr(i,hr,tsn,Ds)
call rh2(m,S1(i,m),S1(i-1,m),S1(i+1,m),rsn,D1,hr,t,rhss(m))

 

 

149

funs(m)=] .0d0+D1*hr*tr(i,hr,tsn,Ds)

t=tr(i,hr,tqn,Dq)

call rh2(m,q1(i,m),q1(i-l,m),q1(i+1,m),rqn,D4,hr,t,rhsq(m))
funq(m)= 1 .0d0+D4*hr* tr(i,hr,tqn,Dq)

t=tr(i,hr,thn,Dh)

call rh2(m,h1(i,m),hl(i-1,m),h1(i-1,m),rhn,D0,hr,t,rhsh(m))
funh(m)=1 .0d0+D0*hr*tr(i,hr,thn,Dh)

 

C
do 80 j=1,m—1
rhss(i)=S 1(i,j)+D1*(S 1(i+1,j)-2.0d0*S l(i,j)+S 1(i-1,j))
funs(i)=1.0d0
rhsq(i)=q1(i,j)+D4*(q1(i+1,j)—2.0d0*q1(i,j)+q1(i-1,j))
funq(j)=1.0d0
rhsh(i)=h1(i,j)+D0*(h1(i+1,j)-2.0d0*h1(i,j)+h1(i-1,j))
funh(i)=1.0d0
80 continue
C
call PDMTRIX(m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRIX(m,D0,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh)
do 85 j=0,m
32(i,i)=US(i)
q2(i,i)=uq(i)
h2(i,j)=uh(i)
85 continue
90 continue
C
do 95 j=0,m
do 96 i=0,m
S 1 (i,j)=32(i,i)
C11 (i,j)=q2(i,j)
h1(i,j)=h2(i,j)
96 continue
95 continue
C
1000 continue
C
C set up the initial cell condition (inoculation)
C
C

axoff=axoff*hr

-‘

 

 

150

ayoff=ayoft*hr
bxoff=bxoff*hr
byoff=byoff*hr
do 1010 i=0,m
x(i)=hr*i-R
do 1005 j=0,m
yG)=hr*j-R
ual (i,j)=ua0(x(i) ,y(j),uaini,ayoff,axoff,awid)
ub1(i,j)=ub0(x(i),y(j),ubini,byoff,bxoff,bwid)
1005 continue
1010 continue
C
C solving the ode
C
kt0=- 1
kt20=- 1
mcount4= 1
1vel= 1
mvel=1
distm(0)=0.
distl(0)=0.
kvel= 1

(:****************************************************************

 

*******

do 2000 k=1,mcount2+1

C

C track the steps

C

kt1=klmcount2

C

C if(kt1.eq.1) then

C kt0=kt1

if(k.eq.mcount3(mcount4)) then
C
C print out the results
if(movc.eq. 1)then
if(mcount4.lt. 10)then
write(16,360)'ua',mcount4,' =
write(32,360)'ub',mcount4,‘ =
write(17,360)'s',mcount4,‘ = ['
write(23,360)'q',mcount4,' = ['
write(18,360)'h',mcount4,' = ['
360 format(a2,i1,a4)
elseif(mcount4.gt.9)then

[l
[I

 

 

151

write(16,36l)'ua',mcount4,' = ['
write(32,361)'ub',mcount4,’ = ['
write(17,361)'s',mcount4,' = ['
write(23,361)'q',mcount4,' = ['
write(18,361)'h',mcount4,' = ['
361 format(a2,i2,a4)
endif
do 1210 j=0,m

write(16,575)(ua1(i,j),i=0,m)

write(32,575)(ub 1 (i,j),i=0,m)

write(17,575)(S1(i,j),i=0,m)

write(23,575)(q1(i,j),i=0,m)

write(l 8,575)(h1(i,j),i=0,m)

12 10 continue

write(16,*)'];'

write(32,*)'];'

write(17,*)'];'

write(23,*)'];'

write(18,*)'];'

57 5 format(2x,101(e10.4,2x))
mcount4=mcount4+l

elseif(movc.eq.2)then
C
C print out the results

do 121 1 j=0,m
write(16,575)(ua1(i,j),i=0,m)
write(32,575)(ub1(i,j),i=0,m)
write( 17,57 5)(S 1(i,j),i=0,m)
write(23,575)(q1(i,j),i=0,m)
write(18,575)(h1(i,j),i=0,m)
121 1 continue

mcount4=mcount4+1

C
else
endif
endif
C

C kt2: index for status of the program
C
kt2=k/(mcount2/ 10)
if(kt2.gt.kt20) then

 

 

 

 

 

C

152

kt20=kt2
write(6,*) 'kt2=',kt2
else

endif

C implicit in x direction:

C

C take care of boundary points:

C
C

C

rhss(0)=s1(0,0)+D1*(S 1(0, 1)-S 1 (0,0))
funs(0)=l.0d0+(ht/2.0d0)*fs(cas,vas,s1(0,0))*ua1(0,0)/Yas
funs(O)=funs(0)+(ht/2.0d0)*fs(cbs,vbs,s1(0,0))*ub1(0,0)/Ybs
rhsq(0)=q1(0,0)+D4*(q1(0,1)-ql(0,0))

funq(0)=1 .0d0+(ht/2.0d0)*fs(caq,vaq,q1(0,0))*ua1(0,0)/Yaq
funq(0)=funq(0)+(ht/2.0d0)*fs(cbq,qu,q1(0,0))*ub1(0,0)/qu
rhsh(0)=h1(0,0)+D0*(h1(0,1)-h1(0,0))
funh(0)=1.0d0+(ht/2.0d0)*fs(ca,va,h1(0,0))*ua1(0,0)/Ya
funh(0)=funh(0)+(ht/2.0d0) *fs(cb,vb,h1 (0,0))*ub1 (0,0)/Yb
rhss(m)=S 1(m,0)+D1*(s 1(m, 1)-s 1(m,0))

funs(m)=1 .0d0+(ht/2.0d0)*fs(cas,vas,s1(m,0))*ua1(m,0)/Yas
funs(m)=funs(m)+(ht/2.0d0)*fs(cbs,vbs,s1(m,0))*ub1(m,0)/Ybs
rhsq(m)=q1 (m,0)+D4*(q1(m, 1)-q1(m,0))

funq(m)=1 .0d0+(ht/2.0d0)*fs(caq,vas,q1(m,0))*ua1(m,0)/Yaq
funq(m)=funq(m)+(ht/2.0d0)*fs(cbq,vbs,q1(m,0))*ub1(m,O)/qu
rhsh(m)=h1(m,0)+DO*(h1(m, 1)-h1(m,0))
funh(m)=1.0d0+(ht/2.0d0)*fs(ca,va,h1(m,0))*ua1(m,0)/Ya
funh(m)=funh(m)+(ht/2.0d0)*fs(cb,vb,h1(m,0))*ub1(m,0)/Yb

i=0

C Cell group a

a12=0.5d0*chemof1(DKas,S1(1,0))*ua1(1,0)
a12=a12+0.5d0*chemof1(DKas,S 1 (0,0))*ua1 (0,0)
b12=0.5d0*chemof1(DKas,S1(0,1))*ua1(0,1)
b12=b12+0.5d0*chemof1(DKas,S1(0,0))*ua1(0,0)
c12=0.5d0*chemof1(DKaq,q1(1,0))*ua1(1,0)
c12=c12+0.5d0*chemof1(DKaq,q1(0,0))*ua1(0,0)
d12=0.5d0*chemof1(DKaq,q1(0,1))*ua1(0, 1)
d12=d12+0.5d0*chemof1 (DKaq,q 1(0,0))*ua1(0,0)
rhsua(0)=ua1(0,0)+D2*(ua1(0,1)-ua1(0,0))
rhsua(0)=rhsua(0)+(ht/2.0d0)*f(ca,va,hl (0,0))*ua1(0,0)
rhsua(0)=rhsua(0)-D3*a12*(s1(1,0)-s1(0,0))
rhsua(0)=rhsua(0)—D3*b12*(S1(0,1)-s1(0,0))
rhsua(0)=rhsua(0)-D5 *c 12* (q 1 (1 ,O)-q1(0,0))

 

 

 

153

rhsua(0)=rhsua(0)-D5 *d12*(q1(0,1)-q1(0,0))
funua(0)=1.0d0
C
a21=0.5d0*chemof1(DKas,S 1(m,0))*ua1(m,0)
a21=a2l+0.5d0*chemof1(DKas,S1(m—1,0))*ua1(m-1,0)
b12=0.5d0*chemof1(DKas,S1(m,1))*ua1(m,l)
b12=b12+0.5d0*chemof1(DKas,S1(m,0))*ua1(m,0)
c2 I=0.5d0*chemof1(DKaq,q 1(m,0))*ua1 (m,0)
021=ch+0.5d0*chemof1(DKaq,ql(m—1,0))*ua1(m-1,0)
d12=O.5d0*chemof1(DKaq,q1(m,1))*ua1(m, 1)
d12=d12+0.5d0*chemof1(DKaq,q1(m,0))*ua1(m,0)
rhsua(m)=ual(m,0)+D2*(ua1(m, 1)-ua1 (m,0))
rhsua(m)=rhsua(m)+(ht/2.0d0)*f(ca,va,h1(m,0))*ua1(m,0)
rhsua(m)=rhsua(m)-D3 *a21*(s1(m-1,0)—sl(m,0))
rhsua(m)=rhsua(m)-D3 *b12*(sl(m,1)-s1(m,0))
rhsua(m)=rhsua(m)-D5*c2 l *(q1(m-1 ,0)~q1(m,0))
rhsua(m)=rhsua(m)—D5 *d12*(q1(m, 1)—q1(m,0))
funua(m)= 1 .0d0
C Cell group b
a12=0.5d0*chemof1(DKbs,S1(1,0))*ub1(1,0)
a12=312+0.5d0*chemof1(DKbs,S1(0,0))*ub1(0,0)
b12=0.5d0*chemof1(DKbs,S 1(0,1))*ub1(0, 1)
b 12=b12+0.5d0*chemof1(DKbs,S 1(0,0))*ub1(0,0)
c12=0.5d0*chemof1(DKbq,q1(1,0))*ub1(1,0)
cl2=c12+O.5d0*chemof1(DKbq,ql(0,0))*ub1(0,0)
d12=0.5d0*chemof1(DKbq,ql(0,1))*ub1(0, 1)
d12=d12+0.5d0*chemof1(DKbq,q1(0,0))*ub1(0,0)
rhsub(0)=ub1(0,0)+D6*(ub1(0,1)-ub1(0,0))
rhsub(0)=rhsub(0)+(ht/2.0d0)*f(cb,vb,h1(0,0))*ub1(0,0)
rhsub(0)=rhsub(0)—~D7*a12*(s1(1,0)—s1(0,0))
rhsub(0)=rhsub(0)—D7 *b 1 2* (S 1(0,1)~sl(0,0))
rhsub(0)=rhsub(O)-D8 *c 12*(q1( 1 ,0)-q1(0,0))
rhsub(0)=rhsub(0)—D8 *d12*(q1(0,1)-q1(0,0))
funub(0)=1 .0d0

 

 

C
a21=0.5d0*chemof1(DKbs,S1(m,0))*ub1(m,0)
a21=a21+0.5d0*chemof1(DKbs,S1(m-1,0))*ub1(m— 1,0)
b12=0.5d0*chemof1(DKbs,S1(m,1))*ub1(m,1)
b12=b12+0.5d0*chemof1(DKbs,S1(m,0))*ub1(m,0)
c21=0.5d0*chemof1(DKbq,q1(m,0))*ub1 (m,0)
c21=c2l+0.5d0*chemof1(DKbq,q1(m-1,0))*ub1(m~1,0)
d12=0.5d0*chemof1(DKbq,q1(m,1))*ub1(m,1)
d12=d12+0.5d0*chemof1(DKbq,q1(m,0))*ub1(m,0)
rhsub(m)=ub1(m,0)+D6*(ub1(m,1)-ub1(m,0))

 

 

154

rhsub(m)=rhsub(m)+(ht/2.0d0)*f(cb,vb,h1(m,0))*ub1(m,0)
rhsub(m)=rhsub(m)—D7*a21*(s1(m-1,0)-s1(m,0))
rhsub(m)=rhsub(m)-D7*b12*(s l(m,1)-sl(m,0))
rhsub(m)=rhsub(m)-D8*021*(ql(m—1,0)-q1(m,0))
rhsub(m)=rhsub(m)-D8*d12*(q 1 (m, 1)-q1 (m,0))
funub(m)=1.0d0
C
do 1020 i=1 ,m-l
C
t=tr(i,hr,tss,Ds)
call rhbound(m,S1(i,0),S1(i,1),rss,D1,hr,t,rhss(i))
funs(i)=1 .0d0+(ht/2.0d0)*fs(cas,vas,s1(i,0))*ua1(i,0)/Yas
funs(i)=funs(i)+(ht/2.0d0)*fs(cbs,vbs,s1(i,0))*ub1(i,0)/Ybs
t=tr(i,hr,tqs,Dq)
call rhbound(m,q 1 (i,0),q 1 (i, 1),rqs,D4,hr,t,rhsq(i))
funq(i)=1.0d0+(ht/2.0d0)*fs(caq,vaq,q1(i,0))*ua1(i,0)/Yaq
funq(i)=funq(i)+(ht/2.0d0)*fs(cbq,qu,q1(i,0))*ub1(i,0)leq
t=tr(i,hr,ths,Dh)
call rhbound(m,H1(i,0),H1(i, 1),rhs,D0,hr,t,rhsh(i))
funh(i)= 1 .0d0+(ht/2.0d0)*fs(ca,va,h1(i,0))*ua1(i,0)lYa
funh(i)=funh(i)+(ht/2.0d0)*fs(cb,vb,h1(i,0))*ub1(i,O)/Yb
C Cell group a
a12=0.5d0*chemof1(DKas,S1(i+1,0))*ua1(i+1,0)
a12=a12+0.5d0*chemof1(DKas,S1(i,0))*ua1(i,0)
a21=0.5d0*chemof1(DKas,S1(i,0))*ua1(i,0)
a21=a2l+0.5d0*chemof1(DKas,S1(i-1,0))*ua1(i-1,0)
b1 2=0.5d0*chemof1(DKas,S 1 (i, 1 ))*ua1 (i, 1)
b12=b12+0.5d0*chemof1(DKas,S1(i,0))*ua1(i,0)
c12=0.5d0*chemof1(DKaq,q1(i+1,0))*ua1(i+1,0)
c12=c12+0.5d0*chemof1(DKaq,ql(i,0))*ua1(i,0)
ch=0.5d0*chemof1(DKaq,q1(i,0))*ua1(i,0)
c21=c2l+0.5d0*chemof1(DKaq,q1(i~1,0))*ua1(i—l ,0)
d 1 2=0.5d0*chemof1 (DKaq,q l (i, 1 ))*ua1 (i, 1)
d12=d12+0.5d0*chemof1 (DKaq,q1(i,0))*ua1 (i,0)
rhsua(i)=ua1 (i,0)+D2*(ua1 (i, 1)-ua1(i,0))
rhsua(i)=rhsua(i)+(ht/2.0d0)*f(ca,va,h l(i,0))*ua1(i,0)
rhsua(i)=rhsua(i)—D3 *a12*(s 1(i+1 ,0)-s 1(i,0))
rhsua(i)=rhsua(i)-D3 *a21*(s1(i-1,0)-sl(i,0))
rhsua(i)=rhsua(i)-D3*b12*(S 1 (i, 1)-S 1(i,0))
rhsua(i)=rhsua(i)-D5 *012*(ql(i+1,0)-q1(i,0))
rhsua(i)=rhsua(i)-D5 *c2 1 *(q1(i-1,0)-q1(i,0))
rhsua(i)=rhsua(i)—D5 *d12*(q1(i,1)-q1(i,0))
funua(i)=1.0d0
C Cell group b

 

 

 

 

 

 

 

 

 

155

a12=0.5d0*chemofl (DKbs,S 1(i+1,0))*ub1(i+1,0)
a12=al2+0.5d0*chemof1(DKbs,S1(i,0))*ub1(i,0)
a21=0.5d0*chemof1(DKbs,S1(i,0))*ub1(i,0)
a21=a21+0.5d0*chemof1(DKbs,S1(i-1,0))*ub1(i-1,0)
b 1 2=0.5d0*chemof1 (DKbs,S 1(i, 1))*ub l (i, 1)
b12=b12+0.5d0*chem0f1(DKbs,S l(i,0))*ub1(i,0)
c12=0.5d0*chemof1(DKbq,q1(i+] ,O))*ub1(i+1,0)
c12=c12+0.5d0*chemofl(DKbq,q1(i,0))*ub1(i,0)
c21=0.5d0*chemof1(DKbq,q1(i,0))*ub1(i,0)
c2 l=c2 1+0.5d0*chemof1(DKbq,q1(i-1 ,0))*ub 1(i- 1,0) ‘
d 1 2=0.5d0*chemof1 (DKbq,q 1 (i, 1))*ub 1 (i, 1)
d12=d12+0.5d0*chemofl(DKbq,ql(i,0))*ub1(i,0)
rhsub(i)=ub1(i,0)+D6*(ub1(i,1)-ub1(i,0))
rhsub(i)=rhsub(i)+(ht/2.0d0)*f(cb,vb,h1(i,0))*ub1(i,0)
rhsub(i)=rhsub(i)—D7 *a12*(s 1(i+1 ,0)-SI (i,0))
rhsub(i)=rhsub(i)-D7*a21*(sl(i-1,0)-s1(i,0))

. rhsub(i)=rhsub(i)—D7*b12*(S1(i,1)—S1(i,0))
rhsub(i)=rhsub(i)-D8 *c 12*(q1(i+1 ,0)—q1(i,0))
rhsub(i)=rhsub(i)—D8 *021 *(q1(i-1,0)-q1(i,0))

rhsub(i)=rhsub(i)—D8*d12*(q1(i,1)-q1(i,0))
funub(i)=1 .0d0
1020 continue
C

 

 

call PDMTRIX(m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRIX(m,D0,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh)
call PDMTRIX(m,D2,funua,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsua,uua)
call PDMTRD((m,D6,funub,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsub,uub)
do 1025 i=0,m

82(i,0)=us(i)

q2(i.0)=uq<i>

h2(i,0)=uh(i)

ua2(i,0)=uua(i)

ub2(i,0)=uub(i)

1025 continue
C
i=m
C

rhss(0)=S 1(0,m)+Dl*(sl (0,m-1)-s 1(0,m))

 

 

 

 

156

funs(0)=1.0d0+(ht/2.0d0)*fs(cas,vas,s1(0,m))*ua1(0,m)/Yas
funs(0)=funs(0)+(ht/2.0d0)*fs(cbs,vbs,s1(0,m))*ub1(0,m)/Ybs
rhsq(0)=ql(0,m)+D4*(q1(0,m-1)-q1(0,m))

funq(0)=1 .0d0+(ht/2.0d0)*fs(caq,vaq,q1(0,m))*ual(0,m)/Yaq
funq(0)=funq(0)+(ht/2.0d0)*fs(cbq,qu,q1(0,m))*ubl(0,m)/qu
rhsh(0)=h1(0,m)+D0*(h1(0,m-1)-h1(0,m))

funh(0)=1 .0d0+(ht/2.0d0)*fs(ca,va,h1 (0,m))*ua1 (0,m)/Y a
funh(0)=funh(0)+(ht/2.0d0)*fs(cb,vb,h1(0,m))*ub1(0,m)/Yb
rhss(m)=S1(m,m)+D1*(s1(m,m-1)-sl(m,m))
funs(m)=1.0d0+(ht/2.0d0)*fs(cas,vas,s1(m,m))*ua1(m,m)/Yas
funs(m)=funs(m)+(ht/2.0d0)*fs(cbs,vbs,s1(m,m))*ub1(m,m)/Ybs
rhsq(m)=q1(m,m)+D4*(q1(m,m- 1)-q1(m,m))
funq(m)=1.0d0+(ht/2.0d0)*fs(caq,vaq,q1(m,m))*ua1(m,m)/Yaq
funq(m)=funq(m)+(ht/2.0d0)*fs(cbq,qu,q1(m,m))*ub1(m,m)/qu
rhsh(m)=h 1(m,m)+D0*(h1(m,m— 1 )—h1(m,m))
funh(m)=1.0d0+(ht/2.0d0)*fs(ca,va,h1(m,m))*ua1(m,m)/Ya
funh(m)=funh(m)+(ht/2.0d0)*fs(cb,vb,h1(m,m))*ub1 (m,m)/Yb

C Cell group a

00

a12=0.5d0*chemof1(DKas,S1(1,m))*ua1(1,m)
a12=a12+0.5d0*chemof1(DKas,S1(0,m))*ua1(0,m)
b21=0.5d0*chemof1(DKas,S1(0,m))*ua1(0,m)
b21=b21+0.5d0*chemof1(DKas,S1(0,m-1))*ua1(0,m-1)
c12=0.5d0*chemof1(DKaq,q1(1,m))*ua1(1,m)
c12=c12+0.5d0*chemof1(DKaq,q1(0,m))*ua1(0,m)
d21=O.5d0*chemofl(DKaq,q1(0,m))*ual(0,m)
d21=d21+0.5d0*chemof1(DKaq,q1(0,rn-1))*ua1(0,m-1)
rhsua(O)=ua1(0,m)+D2*(ua1(0,m—1)-ua1(0,m))
rhsua(O)=rhsua(0)+(ht/2.0d0)*f(ca,va,h1(0,m))*ua1 (0,m)
rhsua(O)=rhsua(0)-D3 *a12*(sl (1 ,m)-s 1 (0,m))
rhsua(O)=rhsua(0)-D3 *b21*(S 1(0,m—1)-s 1(0,m))
rhsua(O)=rhsua(0)-D5*c12*(q1(1,m)-q1(0,m))
rhsua(O)=rhsua(0)—D5*d21*(q1(0,m-1)-q1(0,m))
funua(0)=1.0d0

a21=0.5d0*chemof1(DKas,S1(m,m))*ua1(m,m)
a21=a21+0.5d0*chemof1(DKas,S 1(m—1 ,m))*ua1(m—1,m)
b2l=0.5d0*chemof1(DKas,S 1(m,m))*ua1(m,m)
b21=b21+0.5d0*chemof1(DKas,S 1(m,m—1))*ua1(m,m— 1)
021:0.5d0*chemof1 (DKaq,q 1 (m,m))*ua1(m,m)
c2l=c21+0.5d0*chemof1(DKaq,q1(m-1,m))*ua1(m—1,m)
d21=0.5d0*chemof1(DKaq,ql (m,m))*ual (m,m)
d21=d21+0.5d0*chemof1 (DKaq,q1(m,m—1))*ua1(m,m— 1)
rhsua(m)=ua](m,m)+D2*(ua1(m,m—1)-ua1(m,m))

 

 

157

rhsua(m)=rhsua(m)+(ht/2.0dO)*f(ca,va,h1(m,m))*ua1(m,m)
rhsua(m)=rhsua(m)-D3 *a21 *(s l (m— 1 ,m)-s 1 (m,m))
rhsua(m)=rhsua(m)-D3 *b2 1 *(S l(m,m- 1 )-s 1 (m,m))
rhsua(m)=rhsua(m)—D5*c21*(q1(m-1,m)-q1(m,m))
rhsua(m)=rhsua(m)-D5 *d2 1 *(q 1(m,m— 1)-q1 (m,m))
funua(m)=1 .0d0
C

C Cell group b
a12=0.5d0*chemof1(DKbs,S1(1,m))*ub1(1,m)

a12=a12+O.5d0*chemof1(DKbs,S1(0,m))*ub1(0,m)
b21=0.5d0*chemof1(DKbs,S1(0,m))*ub1(0,m)
b21=b21+0.5d0*chemof1(DKbs,S 1(0,m-1))*ub1(0,m- 1)
c12=0.5d0*chemof1(DKbq,q1(1,m))*ubl(l,m)
c12=c12+0.5d0*chemof1(DKbq,q1(0,m))*ub1(0,m)
d21=0.5d0*chemof1(DKbq,q1(0,m))*ub1(0,m)
d2 l=d21+0.5d0*chemof1(DKbq,q1(0,m-1))*ubl(0,m- 1)
rhsub(0)=ub1(0,m)+D6*(ub 1(0,m- 1)-ub1(0,m))
rhsub(0)=rhsub(0)+(ht/2.0d0)*f(cb,vb,h1(0,m))*ub1(0,m)
rhsub(0)=rhsub(0)-D7*a12*(s1(1,m)—s 1(0,m))
rhsub(0)=rhsub(0)—D7*b2 1 * (S 1(0,m—1)—sl(0,m))
rhsub(0)=rhsub(O)-D8*c12*(q](1,m)-q1(0,m))
rhsub(0)=rhsub(0)-D8 *d21*(q1(0,m-1)-q1(0,m))
funub(0)=1 .Od0

00

a21=0.5d0*chemof1(DKbs,S1(m,m))*ub1(m,m)
a21:321+0.5d0*chemof1(DKbs,S1(m—1,m))*ub1(m—1,m)
b21=0.5d0*chemof1(DKbs,S1(m,m))*ub1(m,m)
b21=b21+0.5d0*chemof1(DKbs,S l(m,m-1))*ub1(m,m-1)
c21=0.5d0*chemof1(DKbq,q1(m,m))*ub1(m,m)
021=c21+0.5d0*chemof1(DKbq,q1(m-1,m))*ub1(m-1,m)
d21=0.5d0*chemof1(DKbq,q1(m,m))*ub1(m,m)
d21=d21+O.5d0*chemof1(DKbq,q1(m,m-1))*ub1(m,m— 1)
rhsub(m)=ubl (m,m)+D6*(ub1(m,m- 1)—ub1(m,m))
rhsub(m)=rhsub(m)+(ht/2.0dO)*f(cb,vb,h1(m,m))*ub1(m,m)
rhsub(m)=rhsub(m)—D7*a21*(s 1(m-1,m)—sl(m,m))
rhsub(m)=rhsub(m)-D7*b21 *( S 1(m,m- 1)—s 1 (m,m))
rhsub(m)=rhsub(m)-D8*c21 * (q 1 (m— 1 ,m)-q 1 (m,m))
rhsub(m)=rhsub(m)—D8*d21*(ql(m,m—1)-q1(m,m))

funub(m)=l .0d0
do 1030 i=1,m—1

t=tr(i,hr,tsn,Ds)

 

 

 

 

158

call rhbound(m,S 1(i,m),S 1(i,m-1),rsn,D1,hr,t,rhss(i))
funs(i)=1 .0d0+(ht/2.0d0)*fs(cas,vas,s1(i,m))*ua1(i,m)/Y as
funs(i)=funs(i)+(ht/2.0d0)*fs(cbs,vbs,s1(i,m))*ubl(i,m)/Ybs
t=tr(i,hr,tqn,Dq)
call rhbound(m,q1(i,m),q1(i,m-1),rqn,D4,hr,t,rhsq(i))
funq(i)=1.0d0+(ht/2.0d0)*fs(caq,vaq,q1(i,m))*ua1(i,m)/Yaq
funq(i)=funq(i)+(ht/2.0d0)*fs(cbq,qu,q1(i,m))*ub 1(i,m)leq
t=tr(i,hr,thn,Dh)
call rhbound(m,H1(i,m),H l (i,m- 1),rhn,D0,hr,t,rhsh(i))
funh(i)=1.0d0+(ht/2.0d0)*fs(ca,va,h1(i,m))*ua1(i,m)/Ya
funh(i)=funh(i)+(ht/2.0d0)*fs(cb,vb,h1(i,m))*ubl (i,m)/Yb
C Cell group a
a12=0.5d0*chemof1(DKas,S1(i+1,m))*ua1(i+l,m)
a12=a12+0.5d0*chemof1(DKas,S 1(i,m))*ua1(i,m)
a21=0.5d0*chemof1(DKas,S1(i,m))*ua1(i,m)
a21=a21+0.5d0*chemof1(DKas,S1(i-1,m))*ua1(i—1,m)
b21=0.5d0*chemof1(DKas,S1(i,m))*ua1(i,m)
b21=b21+0.5d0*chemofl(DKas,S1(i,m-1))*ual(i,m-1)
c12=0.5d0*chemof1(DKaq,q1 (i+1,m))*ua1(i+l ,m)
c12=c12+0.5d0*chemof1(DKaq,q1(i,m))*ua1(i,m)
021=0.5d0*chemof1(DKaq,q1(i,m))*ua1(i,m)
021:021+0.5d0*chemof1(DKaq,q](i-1,m))*ual(i-1,m)
d21=0.5d0*chemof1(DKaq,ql(i,m))*ua1(i,m)
d21=d21+0.5d0*chemof1(DKaq,q1(i,m—1))*ua1(i,m-1)
rhsua(i)=ua](i,m)+D2*(ua1(i,m-l)-ua1(i,m))
rhsua(i)=rhsua(i)+(ht/2.0dO)*f(ca,va,hl (i,m))*ua1(i,m)
rhsua(i)=rhsua(i)-D3 *a12*(sl(i+1,m)—sl(i,m))
rhsua(i)=rhsua(i)-D3 *a21*(sl(i-1,m)-sl(i,m))
rhsua(i)=rhsua(i)-D3 *b21*(S 1(i,m- 1)-S 1(i,m))
rhsua(i)=rhsua(i)-D5*cl2*(q1(i+1,m)-ql(i,m))
rhsua(i)=rhsua(i)-D5 *021 *(q1(i-1 ,m)-q1(i,m))
rhsua(i)=rhsua(i)-D5*d21*(q1(i,m-1)-q1(i,m))
funua(i)=1 .0d0

C Cell group b
a12=0.5d0*chemof1(DKbs,S1(i+1,m))*ub1(i+1,m)

a12=a12+0.5d0*chemof1(DKbs,S1(i,m))*ub1(i,m)
a21=0.5d0*chemof1(DKbs,S 1(i,m))*ubl(i,m)
a21=a21+0.5d0*chemofl(DKbs,S1(i-l,m))*ub1(i—1,m)
b21=0.5d0*chemof1(DKbs,S 1(i,m))*ub1 (i,m)
b21=b2l+0.5d0*chemof1(DKbs,S1(i,m-1))*ub1(i,m-1)
c l2=0.5d0*chemofl(DKbq,q1(i+1,m))*ub1(i+1,m)
c12=c12+0.5d0*chemof1(DKbq,q1(i,m))*ub1(i,m)
021=0.5d0*chemof1(DKbq,q1(i,m))*ub1(i,m)
c21:021+0.5d0*chemof1(DKbq,ql(i-1,m))*ub1(i-1,m)

 

 

 

 

C
1030
C

1035

159

d21=0.5d0*chemof1(DKbq,q1(i,m))*ub1(i,m)
d21=d21+0.5d0*chemof1(DKbq,q1(i,m-1))*ub1(i,m-l)
rhsub(i)=ub1(i,m)+D6*(ub] (i,m-1)-ub1(i,m))
rhsub(i)=rhsub(i)+(ht/2.0d0)*f(cb,vb,h1(i,m))*ub1(i,m)
rhsub(i)=rhsub(i)-D7*a12 *(sl (i+1,m)-s 1(i,m))
rhsub(i)=rhsub(i)—D7*a21*(sl(i-1,m)-sl(i,m))
rhsub(i)=rhsub(i)-D7*b21*(S1(i,m—1)-Sl(i,m))
rhsub(i)=rhsub(i)-D8*c12*(q1(i+1 ,m)-q1(i,m))
rhsub(i)=rhsub(i)-D8*021 *(q1(i-1 ,m)-q1(i,m))
rhsub(i)=rhsub(i)-D8*d21*(q1(i,m—1)-q1(i,m))
funub(i)=1.0d0

conunue

call PDMTRIX(m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRIX(m,D0,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh)
call PDMTRD((m,D2,funua,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsua,uua)
call PDMTRIX(m,D6,funub,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsub,uub)
do 1035 i=0,m .

S2(i,m)=us(i)

q2(i,m)=uq(i)

h2(i,m)=uh(i)

ua2(i,m)=uua(i)

ub2(i,m)=uub(i)
continue

C take care of interior points:

C

C

do 1050 j=1,m~1

t=tr(j ,hr,tsw,Ds)

call rh2(m,Sl(0,j),S1(0,j—1),S1(0,j+1),rsw,D1,hr,t,rhss(0))
funs(0)=1.0d0+(ht/2.0d0)*fs(cas,vas,s1(0,j))*ua1(0,j)/Yas
funs(O)=funs(0)+(ht/2.0d0)*fs(cbs,vbs,s1(0,j))*ub1(0,j)/Ybs
funs(0)=funs(0)+D1 *hr*tr(i,hr,tsts)

t=tr(j ,hr,tqw,Dq)

call rh2(m,q1(0,j),q1(0,j—1),q1(0,j+1),rqw,D4,hr,t,rhsq(0))
funq(0)=1.0d0+(ht/2.0d0)*fs(caq,vaq,q1(0,j))*ua1(0,j)/Yaq

 

 

C

 

 

160

funq(0)=funq(0)+(ht/2.0d0)*fs(cbq,qu,q1(0,j))*ub1(0,j)/qu
funq(0)=funq(0)+D4 *hr*tr(j ,hr,tqw,Dq)

t=tr(j ,hr,thw,Dh)

call rh2(m,h1(0,j),h1(0,j-1),h1(0,j+1),rhw,D0,hr,t,rhsh(0))
funh(0)=1.0d0+(ht/2.0d0)*fs(ca,va,h1(0,j))*ua1(0,j)/Ya
funh(0)=funh(0)+(ht/2.0d0)*fs(cb,vb,h1(0,j))*ub1(0,j)/Yb
funh(0)=funh(0)+D0*hr*tr(j ,hr,thw,Dh)

t=tr(j ,hr,tse,Ds)

call rh2(m,S1(m,j),Sl(m,j-1),S1(m,j+1),rse,D1,hr,t,rhss(m))
funs(m)=1.0d0+(ht/2.0d0)*fs(cas,vas,s1(m,j))*ua1(m,j)/Yas
funs(m)=funs(m)+(ht/2.0d0)*fs(cbs,vbs,s1(m,j))*ub1(m,j)les
funs(m)=funs(m)+D1 *hr*tr(j ,hr,tsw,Ds)

t=tr(j ,hr,tqe,Dq)

'call rh2(m,q1 (m,j),q 1(m,j-1),q 1(m,j+1 ),rqe,D4,hr,t,rhsq(m))
funq(m)=1.0d0+(ht/2.0d0)*fs(caq,vaq,q1(m,j))*ua1(m,j)/Yaq
funq(m)=funq(m)+(ht/2.0d0)*fs(cbq,qu,q1(m,j))*ubl(m,j)/qu
funq(m)=funq(m)+D4*hr*tr(j,hr,tqw,Dq)

t=tr(i ,hr,the,Dh)

call rh2(m,h1(m,j),h1(m,j-1),h1(m,j+1),rhe,D0,hr,t,rhsh(m))
funh(m)=1.0d0+(ht/2.0d0)*fs(ca,va,h1(m,j))*ua1(m,j)/Ya
funh(m)=funh(m)+(ht/2.0d0)*fs(cb,vb,h1(m,j))*ub1(m,j)/Yb
funh(m)=funh(m)+D0*hr*tr(j ,hr,thw,Dh)

i=0

C Cell group a

a12=0.5d0*chemof1(DKas,S1(i+1,j))*ua1(i+1,j)
312=a12+0.5d0*chemof1(DKas,S l(i,j))*ua1(i,j)
b12=0.5d0*chemof1(DKas,S1(i,j+1))*ua1(i,j+1)
b12=b12+0.5d0*chemof1(DKas,S1(i,j))*ual(i,j)
b21=0.5d0*chemof1(DKas,S1(i,j))*ua1(i,j)
b21=b21+0.5d0*chemof1(DKas,S1(i,j-1))*ua1(i,j-l)
c12=0.5d0*chemof1(DKaq,q1(i+1,j))*ua1(i+1,j)
c12=c12+0.5d0*chemof1(DKaq,q1(i,j))*ua1(i,j)
d12=0.5d0*chemof1(DKaq,q1(i,j+1))*ua1(i,j+1)
d12=d12+0.5d0*chemof1(DKaq,q1(i,j))*ual(i,j)
d21=0.5d0*chemof1(DKaq,q1(i,j))*ua1(i,j)
d21=d21+0.5d0*chemof1(DKaq,q1(i,j-1))*ua1(i,j- 1)
rhsua(O)=ua1(0,j)+D2*(ua1(0,j+1)-ua1(0,j))
+D2*(ua1(0,j-1)-ua1(0,j))
rhsua(O)=rhsua(0)+(ht/2.0d0)*f(ca,va,h1(0,j))*ua1(0,j)
rhsua(O)=rhsua(0)—D3*a12*(s1(1,j)-s1(0,j))
rhsua(O)=rhsua(0)-D3*b12*(sl(0,j+1)-s1(0,j))
rhsua(O)=rhsua(0)-D3*b21*(s1(0,j—1)—s1(0,j))
rhsua(O)=rhsua(0)-D5 *c 12*(q 1 (1 ,j)-q1(0,j))

 

 

 

C

161

rhsua(O)=rhsua(0)-D5*d12*(ql(0,J+1)-q1(0,j))
rhsua(O)=rhsua(0)-D5 *d2] *(q1(0,j-1)-q1(0,j))
funua(0)=1.0d0

i=m

a21=0.5d0*chemofl(DKas,Sl(i,j))*ua1(i,j)
a21=a21+0.5d0*chemof1(DKas,S1(i-l,j))*ua1(i-1,j)
b12=0.5d0*chemof1(DKas,S1(i,j+1))*ua1(i,j+1)
b12=b12+0.5d0*chemof1(DKas,S1(i,j))*ua1(i,j)
b21=0.5d0*chemof1(DKas,S1(i,j))*ua1(i,j)
b21=b21+0.5d0*chemof1(DKas,S1(i,j-l))*ua1(i,j-1)
c21=0.5d0*chemof1(DKaq,q1(i,j))*ua1(i,j)
c21=c21+0.5d0*chemof1(DKaq,ql(i-1,j))*ual(i-1,j)
d12=0.5d0*chemof1(DKaq,q l (i,j+1))*ua1 (i,j+ 1)
d12=d12+0.5d0*chemof 1 (DKaq,q] (i, j))*ual (i,j)
d21=O.5d0*chemof1(DKaq,q1(i,j))*ua1(i,j)
d21=d21+0.5d0*chemof1(DKaq,q1(i,j-1))*ua1(i,j- 1)
rhsua(m)=ua1(m,j)+D2*(ua1(m,j+1)-ua1(m,j))
+D2*(ua1(m,j-1)—ua1(m,j))
rhsua(m)=rhsua(m)+(ht/2.0d0)*f(ca,va,hl(m,j))*ua1(m,j)
rhsua(m)=rhsua(m)-D3*a21*(sl(m—1,j)-s1(m,j))
rhsua(m)=rhsua(m)—D3*b12*(s1(m,j+1)-s1(m,j))
rhsua(m)=rhsua(m)-D3 *b21 *(s 1 (m,j- 1)-s 1(m,j))
rhsua(m)=rhsua(m)-D5*c2l*(q1(m-1,l)‘q1(m,l))
rhsua(m)=rhsua(m)—D5*d12*(q1(m,j+1)-q1(m,j))
rhsua(m)=rhsua(m)—D5*d21*(q1(m,j-l)-q1(m,j))
funua(i)=1.0d0

C Cell group b

a12=0.5d0*chemofl(DKbs,S1(i+l,j))*ubl(i+1,j)
a12=a12+0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)
b12=0.5d0*chemof1(DKbs,S1(i,j+1))*ub1(i,j+1)
b12=b12+0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)
b21=0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)
b21=b21+0.5d0*chemof1(DKbs,S1(i,j-1))*ub1(i,j-1)
c12=0.5d0*chemof1(DKbq,q1(i+1,j))*ub1(i+1,j)
c12=c12+0.Sd0*chemof1(DKbq,q1(i,j))*ub1(i,j)
d12=0.5d0*chemof1(DKbq,q1(i,j+1))*ub1(i,j+1)
d12=d12+0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)
d21=0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)
d21=d21+0.5d0*chemof1(DKbq,ql(i,j-1))*ub1(i,j-1)
rhsub(0)=ub1(0,j)+D6*(ub1(0,j+1)-ub1(0,j))
+D6*(ub1(0,j-1)-ub1(0,j))

 

 

 

162

rhsub(0)=rhsub(0)+(ht/2.0d0)*f(Cb,Vb,hl (0,j))*ub1 (OJ)
rhsub(0)=rhsub(0)-D7*a12*(s1(1,j)—sl(O,J))
rhsub(0)=rhsub(0)-D7 *b 1 2*(s 1 (0, j +1 )-s 1 (0, j ))
rhsub(0)=rhsub(0)—D7*b21 *(sl(0,j-1)-s 1(0,j))
rhsub(0)=rhsub(0)-D8 *cl2*(q1(1,j)-q1(0,j))
rhsub(0)=rhsub(O)-D8*d 12 * (q 1 (0,j+ 1 )-q 1 (0, j))
rhsub(0)=rhsub(0)-D8 *d2] *(q 1 (O,j-1)-q1 (0,j))
funub(0)=1 .OdO

i=m

a21=0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)
a21=a21+0.5d0*chemof1(DKbs,S1(i-1,j))*ub1 (i- 1 ,j)
b12=0.5d0*chemof1(DKbs,S1(i,j+1))*ub1(i,j+l)
b12=b12+0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)
b21=0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)
b21=b21+0.5d0*chemof1(DKbs,S1(i,j-1))*ub1(i,j-l)
c21=0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)
c21=ch+0.5d0*chemof1(DKbq,q1(i-1,j))*ub1(i-1,j)
d12=0.5d0*chemof1(DKbq,q1(i,j+1))*ub1(i,j+1)
d12=d12+0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)
d21=0.5d0*chemof1(DKbq,q1(i,j))*ubl(i,j)
d21=d21+0.5d0*chemof1(DKbq,q1(i,j-1))*ub1(i,j-1)
rhsub(m)=ub1(m,j)+D6*(ubl(m,j+l)-ub1(m,j))
+D6*(ub1(m,j-1)—ub1(m,j))
rhsub(m)=rhsub(m)+(ht/2.0d0)*f(cb,vb,h1(m,j))*ub 1 (m, j)
rhsub(m)=rhsub(m)-D7*a21*(sl(m-1,j)-s l(m,j))
rhsub(m)=rhsub(m)—D7*b12*(s 1(m,j+1)—s1(m,j))
rhsub(m)=rhsub(m)-D7*b21*(s1(m,j-1)—sl(m,j))
rhsub(m)=rhsub(m)—D8*c21*(q1(m—1,j)-q1(m,1))
rhsub(m)=rhsub(m)—D8*d12*(q1(m,j+1)-q1(m,j))
rhsub(m)=rhsub(m)-D8*d21*(q1(m,j-1)-q1(m,j))
funub(i)=1.0d0

 

do 1040 i=1,m—1

rhss(i)=S1(i,j)+D1*(S1(i,j+1)-2.0d0*Sl(i,j)+S1(i,j—1))
funs(i)=1.0d0+(ht/2.0d0)*fs(cas,vas,s1(i,j))*ual(i,j)/Yas
funs(i)=funs(i)+(ht/2.0d0)*fs(cbs,vbs,s1(i,j))*ub1(i,j)/Ybs
rhsq(i)=q1(i,j)+D4*(q1(i,j+1)~2.0d0*q1(i,j)+q1(i,j-1))
funq(i)=1.0d0+(ht/2.0d0)*fs(caq,vaq,q1(i,j))*ua1(i,j)/Yaq
funq(i)=funq(i)+(ht/2.0d0)*fs(cbq,qu,q1(i,j))*ub1(i,j)/qu
rhsh(i)=h1(i,j)+D0*(h1(i,j+1)-2.0d0*h1(i,j)+h1(i,j-1))
funh(i)=1.0d0+(ht/2.0d0)*fs(ca,va,h1(i,j))*ual(i,j)/Ya

163

funh(i)=funh(i)+(ht/2.0d0)*fs(cb,vb,h1(i,j))*ubl (i,j)/Yb
C Cell group a
a12=0.5d0*chemofl(DKas,S1(i+1,j))*ua1(i+1,j)
a12=a12+0.5d0*chemof1(DKas,S1(i,j))*ua1(i,j)
a21=0.5d0*chemof1(DKas,S 1(i,j))*ua1(i,j)
a21=a21+0.5d0*chemof1(DKas,S1(i-1,j))*ua1(i~1,j)
b12=0.5d0*chemof1(DKas,S1(i,j+1))*ua1(i,j+1)
b12=b12+0.5d0*chemof1(DKas,S1(i,j))*ua1(i,j)
b21=0.5d0*chemof1(DKas,S1(i,j))*ua1(i,j)
b21=b21+0.5d0*chemof1(DKas,S1(i,j-1))*ua1(i,j—l)
c12=0.5d0*chemof1(DKaq,q1(i+1 ,j))*ua1(i+1,j)
c12=ch+0.5d0*chemof1(DKaq,q1(i,j))*ua1(i,j)
c2 l=0.5d0*chemof1(DKaq,q1(i,j))*ua1(i,j)
c21=c21+0.5d0*chemofl(DKaq,q1(i-1,j))*ua1(i-1,j)
d12=0.5d0*chemof1(DKaq,ql(i,j+1))*ua1(i,j+1)
d12=d12+0.5d0*chemof1(DKaq,q1(i,j))*ua1(i,j)
d21=0.5d0*chemof1 (DKaq,q1(i,j))*ua1(i,j)
d21=d21+0.5d0*chemof1(DKaq,q1(i,j-1))*ua1(i,j- 1)
rhsua(i)=ua1(i,j)+D2*(ua1(i,j+1)—ua1(i,j))
* +D2*(ua1(i,j-1)-ua1(i,j))
rhsua(i)=rhsua(i)+(ht/2.0d0)*f(ca,va,h 1 (i,j))*ua1(i,j)
rhsua(i)=rhsua(i)-D3*a12*(sl(i+1,j)-s1(i,j))
rhsua(i)=rhsua(i)-D3*a21*(sl(i-1,j)-sl(i,j))
rhsua(i)=rhsua(i)-D3*b 12*(s 1 (i,j+1)—s l (i,j))
rhsua(i)=rhsua(i)-D3*b21 *(s 1(i,j-1)—sl(i,j))
rhsua(i)=rhsua(i)-D5*c12*(q1(i+1,j)—ql(i,j))
rhsua(i)=rhsua(i)-D5*021*(q1(i— 1 ,j)-q1(i,j))
rhsua(i)=rhsua(i)-D5*d12*(q1(i,j+1)~q1(i,j))
rhsua(i)=rhsua(i)-D5*d2l *(ql(i,j-1)-q1(i,j))
funua(i)=1 .0d0
C Cell group b
a12=0.5d0*chemofl(DKbs,S1(i+1,j))*ub1(i+1,j)
a12=a12+0.5d0*chemofl(DKbs,S1(i,j))*ub1(i,j)
a21=0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)
a21=a21+0.5d0*chemof1(DKbs,S1(i-1,j))*ub1(i-1,j)
b12=0.5d0*chemof1(DKbs,S1(i,j+1))*ub1(i,j+1)
b12=b12+0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)
b21=0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)
b21=b21+0.5d0*chemofl(DKbs,S 1(i,j-1))*ub1(i,j— 1)
ch=0.5dO*chemof1(DKbq,q1(i+1,j))*ub1(i+1,j)
c12=c12+0.5d0*chemof1(DKbq,q1(i,j))*ub](i,j)
021=0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)
c21=c21+0.5d0*chemof1(DKbq,q1(i-1,j))*ub1(i~1,j)
d12=0.5d0*chemofl(DKbq,q1(i,j+1))*ub1(i,j+1)

 

 

164

d12=d12+0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)
d21=0.5d0*chemof1(DKbq,q1(i,j))*ubl (i,j)
d21=d21+0.5d0*chemof1(DKbq,q1(i,j-1))*ub1(i,j-1)
rhsub(i)=ub1(i,j)+D6*(ub1(i,j+1)-ub1(i,j))
* +D6*(ub1(i,j-l)-ub1(i,j))
rhsub(i)=rhsub(i)+(ht/2.0d0)*f(cb,vb,h I (i,j)) *ub 1 (i, j)
rhsub(i)=rhsub(i)-D7 *a12*(s 1(i+1,j)-s 1(i,j))
rhsub(i)=rhsub(i)-D7*a21*(sl(i-1,j)-sl(i,j))
rhsub(i)=rhsub(i)—D7*bl 2* (s 1 (i, j+ 1 )-s 1 (i,j))
rhsub(i)=rhsub(i)—D7*b2 1 *(s 1 (i, j — 1 )-s 1 (i,j))
rhsub(i)=rhsub(i)-D8*c12*(q1(i+1,j)-q1(i,j))
rhsub(i)=rhsub(i)-D8 *c21*(q1(i-1,j)-q1(i,j))
rhsub(i)=rhsub(i)-D8*d12*(q1(i,j+1)-q1(i,j))
rhsub(i)=rhsub(i)-D8*d21*(q1(i,j-1)—q1(i,j))
funub(i)=1 .0d0
C
1040 continue
C
call PDMTRIX(m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRD((m,D0,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh)
call PDMTRIX(m,D2,funua,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsua,uua)
call PDMTRIX(m,D6,funub,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsub,uub)
do 1045 i=0,m
82(i,j )=us(i)
q2(i,j)=uq(i)
h2(i,j)=uh(i)
ua2(i,j)=uua(i)
ub2(i,j)=uub(i)
1045 continue
1050 confinue
C
C implicit in y direction:
C
do 1055 i=0,m
do 1056 j=0,m
s 1(i,j>=82(i.j>
q1(1.1)=q2<i.j)
h1(i,j)=h2(i,j)

 

 

165

ual(i,j)=ua2(i,j)
ub1(i,j)=ub2(i,j)
1056 continue
1055 continue

C
C take care of boundary points:
C
i=0
C

rhss(0)=S 1(0,0)+D1*(S 1(1,0)-S 1(0,0))
rhss(0)=rhss(O)-(ht/2.0d0)*f(cas,vas,s1(0,0))*ua1(0,0)/Yas
rhss(0)=rhss(0)-(ht/2.0d0)*f(cbs,vbs,s1(0,0))*ub1(0,0)/Ybs

funs(0)=1 .0d0

rhsq(O)=q1(0,0)+D4*(q1(1,0)-q1(0,0))
rhsq(0)=rhsq(0)-(ht/2.0dO)*f(caq,vaq,q1(0,0))*ua1(0,0)/Yaq
rhsq(0)=rhsq(0)—(ht/2.0d0)*f(cbq,qu,q1(0,0))*ub1(0,0)/qu

funq(0)=1 .0d0

rhsh(0)=h1(0,0)+D0*(h1(1,0)-h1(0,0))
rhsh(0)=rhsh(0)-(ht/2.0dO)*f(ca,va,h1(0,0))*ua1(0,0)/Ya
rhsh(0)=rhsh(0)-(ht/2.0d0)*f(cb,vb,h1(0,0))*ub1(0,0)/Yb

funh(0)=1 .0d0

rhss(m)=S1(0,m)+D1*(S1(1,m)-Sl(0,m))
rhss(m)=rhss(m)—(ht/2.0d0)*f(cas,vas,s1(0,m))*ua1(0,m)/Yas
rhss(m)=rhss(m)-(ht/2.0d0)*f(cbs,vbs,s 1(0,m))*ub1(0,m)/Ybs

funs(m)=1.0d0

rhsq(m)=q](0,m)+D4*(q1(1,m)-q1(0,m))
rhsq(m)=rhsq(m)-(ht/2.0d0)*f(caq,vaq,q1(0,m))*ua1(0,m)/Yaq
rhsq(m)=rhsq(m)-(ht/2.0d0)*f(cbq,qu,q1(0,m))*ub 1(0,m)/qu

funq(m)=1 .OdO

rhsh(m)=h1(0,m)+D0*(h1(1,m)-h1(0,m))
rhsh(m)=rhsh(m)—(ht/2.0dO)*f(ca,va,h1(0,m))*ual (0,m)/Ya
rhsh(m)=rhsh(m)-(ht/2.0d0)*f(cb,vb,h1(0,m))*ub1(0,m)/Yb

funh(m)=1 .0d0

C Cell group a
a12=0.5d0*chemofl(DKas,S 1(1,0))*ua1 (1,0)
a12=a12+0.5d0*chemof1(DKas,S1(0,0))*ua1(0,0)
b12=0.5d0*chemof1(DKas,S 1(0,1))*ua1(0, 1)
b12=b12+0.5d0*chemof1(DKas,S1(0,0))*ua1(0,0)
c12=0.5d0*chemof1(DKaq,q1(1,0))*ua1(1,0)
c12=c12+0.5d0*chemof1(DKaq,q1(0,0))*ua1(0,0)
d12=0.5d0*chemof1(DKaq,q1(0,1))*ua1(0, 1)
d12=d12+0.5d0*chemof1(DKaq,q1(0,0))*ua1(0,0)
rhsua(O)=ua1(0,0)+D2*(ua1(1,0)-ua1(0,0))
rhsua(O)=rhsua(0)+(ht/2.0dO)*f(ca,va,h1(0,0))*ua1 (0,0)

 

 

166

rhsua(O)=rhsua(0) -D3 *a12* (s 1(1 ,0)-s1(0,0))

rhsua(O)=rhsua(0)-D3 *b 1 2* (S 1(0, 1)-s 1 (0,0))

rhsua(O)=rhsua(0)-D5*c12*(q1(l,0)-q1(0,0))

rhsua(O)=rhsua(0)-D5 *d12* (q 1 (0, 1)-q1(0,0))
funua(0)=1 .0d0

a12=0.5d0*chemofl(DKas,S1(1,m))*ua1(1,m)
a12=a12+0.5d0*chemof1(DKas,S l(0,m))*ua1(0,m)
b21=0.5d0*chemof1(DKas,S1(0,m-1))*ua1(0,m-1)
b21=b21+0.5d0*chemof1(DKas,S1(0,m))*ua1(0,m)
c 12=0.5d0*chemof1(DKaq,ql (1 ,m))*ual(1,m)
012=c12+0.5d0*chemof1(DKaq,q1(0,m))*ua1(0,m)
d21=0.5d0*chemof1(DKaq,ql (0,m— 1))*ua1(0,m- 1)
d21=d21+0.5d0*chemof1(DKaq,q1(0,m))*ua1(0,m)
rhsua(m)=ua1(0,m)+D2*(ua1 (1 ,m)-ua1(0,m))
rhsua(m)=rhsua(m)+(ht/2.0d0)*f(ca,va,h1(0,m))*ual(0,m)
rhsua(m)=rhsua(m)-D3 *a12*(s 1 (1,m)-sl(0,m))
rhsua(m)=rhsua(m)-D3*b2 1 *(s 1 (0,m—1)—s 1 (0,m))
rhsua(m)=rhsua(m)—D5*c12*(q1(1,m)-q1(0,m))
rhsua(m)=rhsua(m)-D5*d21*(ql(0,m-l)—q1(0,m))
funua(m)=1.0d0
C
C Cell group b
a12=0.5d0*chemof1 (DKbs,S 1(1,0))*ub 1 (1 ,0)
a12=a12+0.5d0*chemof1(DKbs,S1(0,0))*ub1(0,0)
b12=0.5d0*chemof1(DKbs,S1(0,1))*ub1(0,1)
b12=b12+0.5d0*chemof1(DKbs,S 1(0,0))*ub1(0,0)
c12=0.5d0*chemof1(DKbq,q1(1,0))*ub1(1,0)
c12=c12+0.5d0*chemof1(DKbq,q1(0,0))*ub1(0,0)
d12=0.5d0*chemof1(DKbq,q1(0, l))*ub1(0,1)
d12=d12+0.5d0*chemof1(DKbq,q1(0,0))*ub1(0,0)
rhsub(0)=ub1(0,0)+D6*(ub1(1,0)-ub1(0,0))
rhsub(0)=rhsub(0)+(ht/2.0d0)*f(cb,vb,h1(0,0))*ub1(0,0)
rhsub(0)=rhsub(0)-D7*a12*(s1(1,0)-sl(0,0))
rhsub(0)=rhsub(0)-D7*b12*(S1(0,1)-s1(0,0))
rhsub(0)=rhsub(0)-D8 *c 12*(q1(1,0)-q1(0,0))
rhsub(0)=rhsub(0)-D8*d12*(q1(0,1)-q1(0,0))
funub(0)=1.0d0

a12=0.5d0*chemofl(DKbs,S l(1,m))*ub1(1,m)
312=a12+0.5d0*chemof1(DKbs,S1(0,m))*ub1(0,m)
b21=0.5d0*chemof1(DKbs,S1(0,m—1))*ub1(0,m-1)
b21=b21+0.5d0*chemof1(DKbs,S1(0,m))*ub1(0,m)
c 12=0.5d0*chemof1(DKbq,ql(1,m))*ub1(1,m)

 

167

c12=c12+0.5d0*chemof1(DKbq,q1(O,m))*ub1(0,m)
d21=0.5d0*chemof1(DKbq,q1(0,m—1))*ub1(0,m- 1)
d21=d21+0.5d0*chemof1(DKbq,q1(0,m))*ub1(0,m)
rhsub(m)=ub 1 (0,m)+D6*(ub 1 (1 ,m)-ub 1 (0,m))
rhsub(m)=rhsub(m)+(ht/2.0d0)*f(cb,vb,h1(0,m))*ub1(0,m)
rhsub(m)=rhsub(m)—D7*a12*(s 1 (1 ,m)-s 1 (0,m))
rhsub(m)=rhsub(m)-D7*b21 *(s 1(0,m-1)—s 1 (0,m))
rhsub(m)=rhsub(m)—D8 *c 12*(q1( 1 ,m)-q1(0,m))
rhsub(m)=rhsub(m)-D8 *d21*(q1(0,m-1)-q1(0,m))
funub (m)=1 .0d0

do 1060 j=1,m-1

t=tr(j ,hr,tsw,Ds)
call rhbound(m,S1(0,j),Sl(1,j),rsw,D1,hr,t,rhss(j))
rhss(j)=rhss(j)-(ht/2.0d0)*f(cas,vas,s 1 (0,j))*ua1(0,j )/Y as
rhss(j)=rhss(j)-(ht/2.0d0)*f(cbs,vbs,s l(0,j))*ub 1 (0,j )/Ybs
funs(0)=1.0d0
t=tr(j ,hr,tqw,Dq)
call rhbound(m,q](0,j),q1(1,j),rqw,D4,hr,t,rhsqG))
rhsq(j)=rhsq(j)-(ht/2.0d0)*f(caq,vaq,q1(0,j))*ua1(0,j)lYaq
rhsq(j)=rhsq(j)-(ht/2.0d0)*f(cbq,qu,q1(0,j))*ub1 (0,j)leq
funq(0)=1 .OdO
t=tr(j ,hr,thw,Dh)
call rhbound(m,h1(0,j),h1(1,j),rhw,D0,hr,t,rhsh(i))
rhsh(i)=rhsh(j)—(ht/2.0d0)*f(ca,va,h1(0,j))*ua1(0,j)/Ya
rhsh(i)=rhsh(j)-(ht/2.0d0)*f(cb,vb,h1(0,j))*ub1(0,j)/Yb
funh(0)=1.0d0

C Cell group a
a12=0.5d0*chemofl(DKas,S1(1,j))*ua1(l,j)

a12=a12+0.5d0*chemof1(DKas,S1(0,j))*ua1(0,j)
b12=0.5d0*chemof1(DKas,S1(0,j+1))*ua1(0,j+1)
b12=b12+0.5d0*chemof1(DKas,S1(0,j))*ua1(0,j)
b21=0.5d0*chemof1(DKas,S1(0,j-1))*ua1(0,j-1)
b21=b21+0.5d0*chemof1 (DKas,S 1 (0,j))*ua1(0,j)
c12=0.5d0*chemof1(DKaq,q1(1,j))*ua1(1 ,j)
cl2=c12+0.5d0*chemof1(DKaq,q1(0,j))*ua1(0,j)
d12=0.5d0*chemof1(DKaq,ql(0,j+1))*ua1(0,j+1)
d12=d12+0.5d0*chemof1 (DKaq,q1(0,j))*ua1(0,j)
d21=0.5d0*chemof1(DKaq,q1(0,j-1))*ua1(0,j-1)
d21=d21+0.5d0*chemof1(DKaq,q1(0,j))*ua1(0,j)
rhsua(i)=ua1(0,j)+D2*(ua1(1,j)—ua1(0,j))
rhsua(i)=rhsua(i)+(ht/2.0d0)*f(ca,va,h1(0,j))*ua1(0,j)
rhsua(j)=rhsua(j)-D3*a12*(s1 (1 ,j)—sl(0,j))

 

 

168

rhsua(i)=rhsua(i)-D3*b12*(S1(0,j+1)-sl(0,j))
rhsua(i)=rhsua(i)-D3*b21*(S1(0,j-1)-sl(0,j))
rhsua(i)=rhsua(i)-D5 *012*(q1(1,j)-q1(0,j))
rhsua(i)=rhsua(i)-D5*d12*(q1(0,j+1)-q1(0,l))
rhsua(i)=rhsua(j)-D5*d21*(q 1 (0,j-1)-q1 (0,j))
funua(i)=1.0d0
C
C Cell group b
a12=0.5d0*chemofl(DKbs,S 1(1,j))*ub1(1,j)
a12=a12+0.5d0*chemof1(DKbs,S 1 (0,j))*ub1(0,j)
b12=0.5d0*chemof1(DKbs,S1(0,j+1))*ub1(0,j+1)
b12=b12+0.5d0*chemof1(DKbs,S l(0,j))*ub1(0,j)
b21=0.5d0*chemof1(DKbs,S1(0,j-1))*ub1(0,j-1)
b21=b21+O.5d0*chemof1(DKbs,Sl(0,j))*ub1(0,j)
012:0.5d0*chemof1(DKbq,q1(1,j))*ub1(1,j)
c12=c12+0.5d0*chemof1(DKbq,q1(0,j))*ub1(0,j)
d12=0.5d0*chemof1(DKbq,q1(0,j+1))*Ub1(0,l+1)
d12=d12+0.5d0*chemof1(DKbq,ql (0,j))*ub1(0,j)
d21=0.5d0*chemof1(DKbq,q1(0,j-1))*ub1(0,j-1)
d21=d21+0.5d0*chemof1(DKbq,q1(0,j))*ub1(0,j)
rhsub(i)=ub1(0,j)+D6*(ub1(1,j)-ub1(0,j))
rhsub(i)=rhsub(j)+(ht/2.0d0)*f(cb,vb,h1(0,j))*ub1(0,j)
rhsub(i)=rhsub(i)—D7*a12*(s1(1,j)-81(0,j))
rhsub(i)=rhsub(j)-D7*b12*(S 1(0,j+1)-s 1 (0,j))
rhsub(i)=rhsub(j)-D7*b21*(S1(0,j-1)—sl(0,j))
rhsub(i)=rhsub(j)-D8*c12*(q1(1,j)—q1(0,j))
rhsub(i)=rhsub(j)-D8*d12*(q1(0,j+1)—q1(0,j))
rhsub(i)=rhsub(j)-D8*d21*(ql(0,j-1)—q1(0,j))
funub(j)=1.0d0
1060 continue
C

 

 

call PDMTRIX(m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRD((m,D0,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh)
call PDMTRIX(m,D2,funua,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsua,uua)
call PDMTRIX(m,D6,funub,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsub,uub)
do 1065 j=0,m
82(Oaj)=u8(i)
q2(0,j)=uq(i)

1065
C

C

169

h2<0,j>=uho>

ua2(0,j)=uua(j)

ub2(0,j)=uub(j)
continue

i=m

rhss(0)=Sl(m,0)+D1*(S 1(m—1,0)-S 1(m,0))
rhss(0)=rhss(0)-(ht/2.0d0)*f(cas,vas,s 1(m,0))*ua1 (m,0)/Y as
rhss(0)=rhss(0)-(ht/2.0d0)*f(cbs,vbs,s1(m,0))*ub1(m,O)/Ybs

funs(0)=1 .0d0

rhsq(0)=q1(m,0)+D4*(q1(m-1,0)—q1(m,0))
rhsq(0)=rhsq(0)-(ht/2.0d0)*f(caq,vaq,q1(m,0))*ua1(m,0)/Yaq
rhsq(0)=rhsq(0)—(ht/2.0d0)*f(cbq,qu,q1(m,0))*ub1(m,0)leq

funq(0)=1 .OdO

rhsh(0)=h1(m,0)+DO*(h1(m-1,0)-h1(m,0))
rhsh(0)=rhsh(0)—(ht/2.0d0)*f(ca,va,h1(m,0))*ua1(m,0)/Ya
rhsh(0)=rhsh(0)-(ht/2.0d0)*f(cb,vb,h1(m,0))*ub1(m,0)/Yb

funh(0)=1 .0d0

rhss(m)=S 1(m,m)+D1 *(S 1(m-1,m)—S 1(m,m))
rhss(m)=rhss(m)~(ht/2.0d0)*f(cas,vas,s 1 (m,m))*ual (m,m)/Y as
rhss(m)=rhss(m)—(ht/2.0d0)*f(cbs,vbs,s1(m,m))*ub1(m,m)/Ybs

funs(m)=1.0d0

rhsq(m)=q1(m,m)+D4*(ql(m-1,m)-q1(m,m))
rhsq(m)=rhsq(m)-(ht/2.0d0)*f(caq,vaq,q1(m,m))*ua1(m,m)/Yaq
rhsq(m)=rhsq(m)-(ht/2.0d0)*f(cbq,qu,q1(m,m))*ub1(m,m)/qu

funq(m)=1 .0d0

rhsh(m)=hl(m,m)+D0*(h1(m-1,m)—h1(m,m))
rhsh(m)=rhsh(m)—(ht/2.0d0)*f(ca,va,h1(m,m))*ual (m,m)/Ya
rhsh(m)=rhsh(m)-(ht/2.0d0)*f(cb,vb,hl (m,m))*ub l(m,m)/Yb

funh(m)=1 .0d0

C Cell group a

a21=0.5d0*chemof1(DKas,S1(m-1,0))*ua1(m-1,0)
a21=a2l+0.5d0*chemof1(DKas,S 1(m,0))*ual(m,0)
b12=0.5d0*chemof1(DKas,S1(m,1))*ua1(m,1)
b12=bl2+0.5d0*chemof1(DKas,S 1(m,0))*ua1(m,0)
021:0.5d0*chemof1(DKaq,q1(m—1,0))*ua1(m—1,0)
c21:021+0.5d0*chemof1(DKaq,q1(m,0))*ua1(m,0)
d12=0.5d0*chemof1(DKaq,q1 (m, 1))*ua1(m, 1)
d12=d12+0.5d0*chemof1(DKaq,q1(m,0))*ua1(m,0)
rhsua(O)=ua1(m,0)+D2*(ua1(m-1,0)-ua1(m,0))
rhsua(O)=rhsua(0)+(ht/2.0d0)*f(ca,va,h1(m,0))*ua1(m,0)
rhsua(O)=rhsua(0)—D3 *a21 *(s 1(m— 1 ,0)-sl(m,0))
rhsua(O)=rhsua(0)-D3 *b12*(S 1(m, 1)-s 1(m,0))

 

170

rhsua(O)=rhsua(0)-D5 *c21 *(q 1 (m- 1,0)-q1(m,0))
rhsua(O)=rhsua(0)-D5 *d 1 2* (q 1 (m, 1)-q1 (m,0))
funua(0)= 1 .OdO

a21=0.5d0*chemof1(DKas,S1(m-1,m))*ua1(m—1,m)
a21=a21+0.5d0*chemof1(DKas,S1(m,m))*ua1(m,m)
b21=0.5d0*chemofl (DKas,S 1(m,m— l ))*ual (m,m— 1)
b21=b21+0.5d0*chemof1(DKas,S 1(m,m))*ua1(m,m)
c2l=0.5d0*chemof1(DKaq,ql(m-1,m))*ual(m-1,m)
c21=c21+0.5d0*chemof1(DKaq,q1(m,m))*ual(m,m)
d21=0.5d0*chemof1(DKaq,q1(m,m-1))*ua1(m,m-1)
d21=d21+0.5d0*chemof1(DKaq,q1(m,m))*ua1(m,m)
rhsua(m)=ua1(m,m)+D2*(ua1(m-1,m)—ua1(m,m))
rhsua(m)=rhsua(m)+(ht/2.0d0)*f(ca,va,h1(m,m))*ua1(m,m)
rhsua(m)=rhsua(m)-D3 *a21 *(sl(m-1,m)-sl(m,m))
rhsua(m)=rhsua(m)-D3 *b21*(sl(m,m—1)—sl(m,m))
rhsua(m)=rhsua(m)-D5 *c21*(q1(m—1,m)-q1(m,m))
rhsua(m)=rhsua(m)—D5 *d21*(q1(m,m—l)—q1(m,m))
funua(m)=1.0d0
C
C Cell group b
a21=0.5d0*chemof1(DKbs,S 1(m-1,0))*ub1(m—1,0)
a21=a21+0.5d0*chemof1(DKbs,S 1(m,0))*ub1(m,0)
b 12=0.5d0*chemof1(DKbs,S 1(m, 1))*ub1(m, 1)
b12=b12+0.5d0*chemof1(DKbs,S1(m,0))*ub1(m,0)
021=0.5d0*chemof1(DKbq,ql(m-1,0))*ub1(m—1,0)
021:021+0.5d0*chemof1(DKbq,q1(m,0))*ub1(m,0)
d12=0.5d0*chemof1(DKbq,q1(m, 1))*ub1(m,1)
d12=d12+0.5d0*chemof1(DKbq,ql(m,0))*ub1(m,0)
rhsub(0)=ub1(m,0)+D6*(ub](m—1,0)-ub1(m,0))
rhsub(0)=rhsub(0)+(ht/2.0d0)*f(cb,vb,h1(m,0))*ub1(m,0)
rhsub(0)=rhsub(0)-D7*a21 *(s 1(m—l ,0)-s 1(m,0))
rhsub(0)=rhsub(0)-D7*b12*(S 1(m, 1)-s 1(m,0))
rhsub(0)=rhsub(0)-D8*c21*(ql(m—1,0)-q1(m,0))
rhsub(0)=rhsub(0)-D8 *d12*(q1(m,1)-q1(m,0))
funub(0)=-1.0d0

a21=0.5d0*chemof1(DKbs,S1(m-1,m))*ub1(m-1,m)
a21=a21+0.5d0*chemof1(DKbs,S1(m,m))*ubl(m,m)
b21 =0.5d0*chemof1(DKbs,S 1(m,m— 1))*ub1(m,m- 1)
b21=b21+0.5d0*chemof1(DKbs,S l(m,m))*ub1(m,m)
c21=0.5d0*chemof1(DKbq,q1(m-1,m))*ubl(m—1,m)

 

 

171

c21=c2l+0.5d0*chemof1(DKbq,q1(m,m))*ub1(m,m)
d21=0.5d0*chemof1(DKbq,q1(m,m-1))*ub1(m,m—1)
d21=d21+0.5d0*chemof1(DKbq,ql(m,m))*ub1(m,m)
rhsub(m)=ub1(m,m)+D6*(ub1(m-1 ,m)-ub1(m,m))
rhsub(m)=rhsub(m)+(ht/2.0d0)*f(cb,vb,h1(m,m))*ub 1(m,m)
rhsub(m)=rhsub(m)—D7*a21*(s1(m-1,m)-s1(m,m))
rhsub(m)=rhsub(m)-D7*b21 *(sl(m,m—1)-sl(m,m))
rhsub(m)=rhsub(m)-D8*c21*(ql(m—1,m)-q1(m,m))
rhsub(m)=rhsub(m)-D8 *d21*(q1(m,m- 1)-q 1 (m,m))
funub(m)=1.0d0

do 1070 j=1,m-1

t=tr(i ,hr,tse,Ds)
call rhbound(m,S 1 (m,j),S 1(m-1,j),rse,D1,hr,t,rhss(i))
rhss(i)=rhss(j)-(ht/2.0d0)*f(cas,vas,s1(m,j))*ua1(m,j)/Y as
rhss(i)=rhss(i)-(ht/2.0d0)*f(cbs,vbs,s1(m,j))*ub1(m,j)/Ybs
funs(0)=1 .0d0
t=tr(j ,hr,tqe,Dq)
call rhbound(m,ql(m,j),ql(m—1,j),rqe,D4,hr,t,rhsq(i))
rhsq(j)=rhsqG)-(ht/2.0d0)*f(caq,vaq,q1(m,j))*ua1(m,j)/Yaq
rhsq(j)=rhsq(j)-(ht/2.0d0)*f(cbq,qu,q1(m,j))*ub1(m,j)/qu
funq(0)=1 .0d0
t=tr(j ,hr,the,Dh)
call rhbound(m,hl(m,j),h1(m—1,j),rhe,D0,hr,t,rhsh(j))
rhsh(j)=rhsh(j)-(ht/2.0d0)*f(ca,va,h1(m,j))*ua1(m,j)/Ya
rhsh(j)=rhsh(j)-(ht/2.0d0)*f(cb,vb,h1(m,j))*ub1(m,j)/Yb
funh(0)=1 .0d0

C Cell group a
a21=0.5d0*chemof1(DKas,S1(m-1,j))*ua1(m—1,j)

a21=a21+0.5d0*chemof1(DKas,S1(m,j))*ua1(m,j)
b12=0.5d0*chemof1(DKas,S1(m,j+1))*ual(m,j+1)
b12=b12+0.5d0*chemof1(DKas,S1(m,j))*ua1(m,j)
b21=0.5d0*chemof 1(DKas,S 1(m,j- 1))*ua1(m,j- 1)
b21=b21+O.5d0*chemof1(DKas,S1(m,j))*ua1(m,j)
c21=0.5d0*chemof1(DKaq,q1(m-1,j))*ua1(m-1,1)
c21=c2l+0.5d0*chemof1(DKaq,ql (m,j))*ua1(m,j)
d12:0.5d0*chemof1(DKaq,ql(m,j+1))*ua1(m,j+1)
d12=d12+0.5d0*chemof1(DKaq,ql (m,j))*ua1(m,j)
d21=0.5d0*chemof1(DKaq,q1(m,j-1))*ua1(m,j-1)
d21=d21+0.5d0*chemof1(DKaq,ql(m,j))*ua1(m,j)
rhsua(i)=ua](m,j)+D2*(ua1(m-1,j)—ua1(m,j))
rhsua(i)=rhsua(j )+(ht/2.0d0)*f(ca,va,h1(m,j))*ua1(m,j)
rhsua(i)=rhsua(j)-D3*a21 *(sl(m—1,j)-sl(m,j))

 

 

172

rhsua(i)=rhsua(j)-D3*b12*(S1(m,j+1)-s1(m,j))
rhsuafi)=rhsua(j)-D3*b21*(S1(m,j-1)-sl(m,j))
rhsua(i)=rhsuaG)-D5*c21*(ql(m—1,j)-q1(m,j))
rhsua(i)=rhsua(i)-D5*d12*(q1(m,j+1)-q1(m,j))
rhsua(i)=rhsua(j)-D5*d21*(ql(m,j-1)-ql(m,j))
funua(i)=1 .0d0
C
C Cell group b
a21=0.5d0*chemof1(DKbs,Sl(m—1,j))*ub1(m-1,j)
a21=a21+0.5d0*chemof1(DKbs,S1(m,j))*ub1(m,j)
b12=0.5d0*chemof1(DKbs,S l(m,j+1))*ub1(m,j+1)
b12=b12+0.5d0*chemof1(DKbs,S1(m,j))*ub1(m,j)
b21=0.5d0*chemofl(DKbs,S 1(m,j-1))*ub 1(m,j-1)
b21=b21+0.5d0*chemof1(DKbs,S1(m,j))*ub1(m,j)
c21=0.5d0*chemof1(DKbq,ql(m-1 ,j))*ub1(m-1,j)
c21=c21+0.5d0*chemof1(DKbq,q1(m,j))*ubl (m,j)
d12=0.5d0*chemof1(DKbq,q1(m,j+1))*ub1(m,l+1)
d12=d12+0.5d0*chemof1(DKbq,q1 (m,j))*ub 1(m,j)
d21=0.5d0*chemof1(DKbq,q1(m,j-1))*ub1(m,j-1)
d21=d2 1+0.5d0*chemof1 (DKbq,q1(m,j ))*ub1(m,j)
rhsub(i)=ub1(m,j)+D6*(ub1(m-1,j)—ub1(m,j))
rhsub(i)=rhsub(j )+(ht/2.0d0)*f(cb,vb,h1 (m,j)) *ub 1 (m, j)
rhsub(j )=rhsub(j)-D7 *a2 1 *(s 1 (m— 1 ,j)-s 1 (m,j ))
rhsub(j)=rhsub(j)-D7*b12*(S 1(m,j+1)-s 1 (m,j))
rhsub(i)=rhsub(i)-D7*b21*(S1(m,j-1)-sl(m,j))
rhsub(j)=rhsub(j)-D8*c21*(ql(m-l,j)-q1(m,j))
rhsub(i)=rhsub(i)—D8*d12*(q1(m,j+1)—q1(m,j))
rhsub(i)=rhsub(j)-D8*d21*(ql(m,j-l)-q1(m,j))
funub(i)=1 .0d0
1070 continue
C

 

 

 

call PDMTRIX(m,D1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D4,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRD((m,DO,funh,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsh,uh) '
call PDMTRlX(m,D2,funua,aa,bb,cc)
call TRIS OLV(m,aa,bb,cc,rhsua,uua)
call PDMTRIX(m,D6,funub,aa,bb,cc)
call TRISOLV(m,aa‘,bb,cc,rhsub,uub)
do 1075 j=0,m
sztm.j)=us0)
q2(m,j)=uq(i)

173

h2(m,j)=uh(i)

ua2(m,j)=uua(j)

ub2(m,j)=uub0')
1075 continue

 

C take care of interior points:
C
do 1090 i=1,m- 1
C
t=tr(i,hr,tss,Ds)
call rh2(m,S1(i,0),Sl(i—1,0),S1(i+1,0),rss,D1,hr,t,rhss(0))
rhss(O)=rhss(0)-(ht/2.0d0)*f(cas,vas,s1(i,0))*ua1(i,0)/Yas
rhss(0)=rhss(0)-(ht/2.0d0)*f(cbs,vbs,s1(i,0))*ub1 (i,0)/Ybs
funs(0)=l .0d0+D1*hr*tr(i,hr,tss,Ds)
t=tr(i,hr,tqs,Dq)
call rh2(m,q1(i,0),q1(i-1,0),q1(i+1,0),rqs,D4,hr,t,rhsq(0))
rhsq(0)=rhsq(O)-(ht/2.0d0)*f(caq,vaq,q1(i,0))*ua1(i,0)/Yaq
rhsq(0)=rhsq(0)-(ht/2.0d0)*f(cbq,qu,q1(i,0))*ub1(i,0)/qu
funq(0)=1.0d0+D4*hr*tr(i,hr,tqs,Dq)
t=tr(i,hr,ths,Dh)
call rh2(m,h1(i,0),h1(i-1,0),h1(i+1,0),rhs,D0,hr,t,rhsh(0))
rhsh(0)=rhsh(0)-(ht/2.0d0)*f(ca,va,h1(i,0))*ua1(i,0)/Ya
rhsh(0)=rhsh(0)-(ht/2.0dO)*f(cb,vb,h1(i,0))*ub1(i,0)/Yb
funh(0)=1 .0d0+D0*hr*tr(i,hr,ths,Dh)
t=tr(i,hr,tsn,Ds)
call rh2(m,S1(i,m),S1(i-1,m),S1(i+1,m),rsn,D1,hr,t,rhss(m))
rhss(m)=rhss(m)-(ht/2.0dO)*f(cas,vas,s1(i,m))*ua1(i,m)/Yas
rhss(m)=rhss(m)-(ht/2.0d0)*f(cbs,vbs,s1(i,m))*ub1(i,m)/Ybs
funs(m)=1.0d0+D1*hr*tr(i,hr,tsn,Ds)
t=tr(i,hr,tqn,Dq)
call rh2(m,ql(i,m),q1(i-1,m),q1(i+1,m),rqn,D4,hr,t,rhsq(m))
rhsq(m)=rhsq(m)—(ht/2.0d0)*f(caq,vaq,q1(i,m))*ua1(i,m)/Yaq
rhsq(m)=rhsq(m)-(ht/2.0d0)*f(cbq,qu,q1(i,m))*ub1(i,m)/qu
funq(m)=1 .0d0+D4*hr*tr(i,hr,tqn,Dq)
t=tr(i,hr,thn,Dh)
call rh2(m,h1(i,m),h1(i-1,m),hl(i+1,m),rhn,D0,hr,t,rhsh(m))
rhsh(m)=rhsh(m)—(ht/2.0dO)*f(ca,va,hl(i,m))*ua1(i,m)/Ya
rhsh(m)=rhsh(m)—(ht/2.0d0)*f(cb,vb,h1 (i,m))*ub1(i,m)/Yb
funh(m)=1 .0d0+D0*hr*tr(i,hr,thn,Dh)

 

C
i=0
C Cell group a
a1 2=0.5d0*chemof1(DKas,S 1(i+1 ,0))*ua1(i+1,0)
a12=a12+0.5d0*chemof1(DKas,S1(i,0))*ua1(i,0)

('30

 

 

174

a21=0.5d0*chemof1(DKas,S1(i,0))*ua1(i,0)
a21=a21+0.5d0*chemof1(DKas,S 1(i-1,0))*ua1(i- 1,0)
b12=0.5d0*chemof1(DKas,S 1(i,1))*ua1 (i, l)
b12=b12+0.5d0*chemof1(DKas,S1(i,0))*ual(i,0)
c12=0.5d0*chemof1(DKaq,q1(i+1,0))*ua1(i+1,0)

c 12=c12+0.5d0*chemof1(DKaq,q1(i,0))*ua1(i,0)
c21=0.5d0*chemof1(DKaq,q1(i,0))*ua1(i,0)

021=c21+0.5d0*chemof1(DKaq,q1(i-1 ,0))*ua1(i- 1 ,0)

d12=0.5d0*chemof1(DKaq,q](i,1))*ua1(i,1)

d12=d12+0.5d0*chemof1(DKaq,q1(i,0))*ua1(i,0) .

rhsua(O)=ua1(i,0)+D2*(ua1(i+1,0)-ua1(i,0))
* +D2*(ual (i-1,0)-ua1(i,0))
rhsua(O)=rhsua(0)+(ht/2.0d0)*f(ca,va,h1(i,0))*ua1(i,0)
rhsua(O)=rhsua(0)-D3 *a12*(s 1(i+1,0)-s 1(i,0))
rhsua(O)=rhsua(0)-D3*a21*(sl(i-1,0)-sl(i,0))

' rhsua(O)=rhsua(0)-D3*b12*(S1(i,1)-sl(i,0))
rhsua(O)=rhsua(0)-D5*c12*(q1(i+1,0)-q1(i,0))
rhsua(O)=rhsua(O)-D5*021*(q1(i-1,0)-q1(i,0))
rhsua(O)=rhsua(0)—D5 *d12*(q1(i,1)-q1(i,0))

funua(0)=l .0d0

 

 

i=m

 

a12=0.5d0*chemofl(DKas,S l(i+1,m))*ua1(i+1,m)
a12=a12+0.5d0*chemof1(DKas,S1(i,m))*ua1(i,m)
a21=0.5d0*chemof1(DKas,S1(i,m))*ua1(i,m)
a21=a21+0.5d0*chemof1(DKas,S 1(i-1,m))*ua1(i-1,m)
b21=0.5d0*chemofl(DKas,S1(i,m))*ua1(i,m)
b21=b21+0.5d0*chemof1(DKas,S1(i,m-1))*ua1(i,m-1)
c12=0.5d0*chemof1(DKaq,q1(i+1,m))*ua1(i+1,m)
c12=c12+0.5d0*chemof1(DKaq,q1(i,m))*ua1(i,m)
c21=0.5d0*chemof1(DKaq,ql(i,m))*ual(i,m)
c2l=c21+0.5d0*chemof1(DKaq,ql(i-1 ,m))*ual(i-1 ,m)
d21=0.5d0*chemof1(DKaq,ql (i,m))*ua1(i,m)
d21=d2l+0.5d0*chemof1(DKaq,q1(i,m-1))*ua1(i,m-1)
rhsua(m)=ua1(i,m)+D2*(ua1(i+1,m)-ual(i,m))
* +D2*(ua1(i-1,m)-ual(i,m))
rhsua(m)=rhsua(m)+(ht/2.0d0)*f(ca,va,h1(i,m))*ua1(i,m)
rhsua(m)=rhsua(m)—D3 *al2*(s1(i+1,m)-Sl(1,m))
rhsua(m)=rhsua(m)-D3*a21*(sl(i-1,m)-Sl(i,m))
rhsua(m)=rhsua(m)-D3*b21*(S 1 (i,m-1)-s 1(i,m))
rhsua(m)=rhsua(m)-D5*c12*(q1(i+1,m)-q1(i,m))
rhsua(m)=rhsua(m)-D5*021*(ql(i-1,m)-q1(i,m))

C

175

rhsua(m)=rhsua(m)—D5*d21*(ql(i,m—1)-ql(i,m))

funua(m)= l .0d0

C Cell group b

00

a12=0.5d0*chemofl(DKbs,S1(i+1,0))*ub1(i+l,0)
a12=312+0.5d0*chemof1(DKbs,Sl(i,0))*ub1(i,0)
a21=0.5d0*chemof1(DKbs,S1(i,0))*ub1(1,0)
a21=a21+0.5d0*chemof1(DKbs,S 1(1- 1 ,0))*ub1(i-1,0)
b12=0.5d0*chemof1(DKbs,S1(i,1))*ub1(1, 1)
b12=b12+0.5d0*chemof1(DKbs,S1(1,0))*ub1(i,0)
c12=0.5d0*chemof1(DKbq,ql(i+1,0))*ub1(i+1,0)
c12=c12+0.5d0*chemof1(DKbq,q1(i,0))*ubl(i,0)
c21=0.5d0*chemof1(DKbq,q1(1,0))*ub1 (1,0)
c21=c21+0.5d0*chemof1(DKbq,q1(i-1,0))*ub1(i—1,0)
d12=0.5d0*chemof1(DKbq,ql(i, 1))*ub1(i, 1)
d12=d12+0.5d0*chemof1(DKbq,q1(i,0))*ub 1 (1,0)

rhsub(0)=ub1 (i,0)+D6*(ub 1(i+ 1 ,0)-uh 1(i,0))

+D6*(ub1(1—1,0)-ub1(i,0))
rhsub(0)=rhsub(0)+(ht/2.0d0)*f(cb,vb,h1 (i,0))*ub1 (1,0)

rhsub(0)=rhsub(0)—D7*a12*(s1(i+1,0)-sl(i,0))
rhsub(0)=rhsub(0)—D7*a21 *(sl(i-1,0)—sl (1,0))
rhsub(0)=rhsub(0)—D7*b12*(S l (i,1)-sl(i,0))
rhsub(0)=rhsub(0)-D8*c12*(q1(i+1,0)—q1(i,0))
rhsub(0)=rhsub(0)—D8*c21*(ql(i-1,0)—q1(i,0))
rhsub(0)=rhsub(0)-D8*d12*(q1(i,1)—q1(i,0))

funub(0)=1.0d0

J=m

a12=0.5d0*chemof 1 (DKbs,S1(i+1,m))*ubl(i+1,m)
a12=a12+0.5d0*chemof1(DKbs,S1(i,m))*ub1(i,m)
a21=0.5d0*chemof1(DKbs,Sl(i,m))*ub1(i,m)
a21=a21+0.5d0*chem0f1(DKbs,S1(1-1,m))*ub1(i-1,m)
b21=0.5d0*chemof1(DKbs,Sl(i,m))*ub1(i,m)
b21=b2l+0.5d0*chem0f1(DKbs,S1(i,m-1))*ubl(i,m—1)
c12=0.5d0*chemof1(DKbq,q1(i+1,m))*ub1(i+1,m)
cl2=c12+0.5d0*chemof1(DKbq,q1(i,m))*ub1(i,rn)
c21=0.5d0*chemof1(DKbq,q1(i,m))*ubl(i,m)
c21=c21+0.5d0*chem0f1(DKbq,q1(i-1,m))*ub1(i—1,m)
d21=0.5d0*chem0f1(DKbq,q1(i,m))*ubl(i,m)
d21=d21+0.5d0*chern0f1 (DKbq,q 1 (i,m- 1 ))*ub l (i,m— 1)

rhsub(m)=ub1(i,m)+D6*(ub1(i+1,In)-ub1(i,rn))

+D6*(ub1(i-1,m)-ubl(i,m))

 

176

rhsub(m)=rhsub(m)+(ht/2.0d0)*f(cb,vb,h1(i,m))*ub 1 (i,m)
rhsub(m)=rhsub(m)-D7*a12*(s1(1+1 ,m)—s 1 (i,m))
rhsub(m)=rhsub(m)—D7* a21 *(s 1 (1-1 ,m)-s1(i.m))
rhsub(m)=rhsub(m)-D7 *b21 *(S 1(1,m- l)-S 1(1,m))
rhsub(m)=rhsub(m)—D8*c12*(q1(i+1,m)-q1(i,m))
rhsub(m)=rhsub(m)—D8*c2 1 * (q 1 (1— 1 ,m)-q 1(1,m))
rhsub(m)=rhsub(m)-D8 *d21 *(q 1 (i,m- 1)-q1(i,m))

funub(m)= 1 .0d0

do 1080 j=1,m-1

rhss(j)=Sl(1,j)+D1*(S1(1+l,j)-2.0d0*Sl(i,j)+S1(i-1,j))
rhss(j)=rhss(i)-(ht/2.0d0)*f(cas,vas,s 1 (i,j))*ual (1, j)/Y as
rhss(j)=rhss(j)-(ht/2.0d0)*f(cbs,vbs,s1(i,j))*ub1(i,j)/Ybs

funs(i)=1.0d0

rhsq(j)=q1(1,j)+D4*(q1(1+1 ,j)-2.0d0*q1 (i,j)+q 1(1-1 ,j))
rhsq(j)=rhsq(j)-(ht/2.0d0)*f(caq,vaq,q1(i,j))*ual(i,j)/Yaq
rhsq(j)=rhsq(j)—(ht/2.0d0)*f(cbq,qu,q1(i,j))*ub1(i,j)/qu

funq(i)=1.0d0

rhsh(i)=h1(1,j)+D0*(h1(1+1,j)-2.0d0*h1(i,j)+h1(i-1,j))
rhsh(i)=rhsh(j)-(ht/2.0d0)*f(ca,va,h1(i,j))*ual (i,j )/Y a
rhsh(i)=rhsh(j)-(ht/2.0d0)*f(cb,vb,h1(i,j))*ub1(i,j)/Yb

funh(i)= 1 .OdO

C Cell group a

al2=0.5d0*chemof1(DKas,S1(1+1,j))*ua1(i+1,j)
a12=a12+0.5d0*chemof1(DKas,S1(i,j))*ua1(1,j)
a21=0.5d0*chemof1(DKas,S 1(i,j))*ual(1,j)
a21=a21+0.5d0*chemof1(DKas,S1(1-1,j))*ua1(1—1,j)
b12=0.5d0*chemof1(DKas,S l (1,j+1))*ua1(1,j+1)
b12=b12+0.5d0*chemof1(DKas,S1(1,j))*ua1(1,j)
b21=0.5d0*chemofl(DKas,S1(i,j))*ua1(i,j)
b21=b21+0.5d0*chemof1(DKas,S 1(1,j- 1))*ua1(i,j-1)
cl2=0.5d0*chemofl(DKaq,ql(1+1,j))*ua1(i+1,j)
c 12=c12+0.5d0*chemof1(DKaq,ql(i,j))*ua1 (i,j)
c21=0.5d0*chemof1(DKaq,ql(i,j))*ua1(i,j)
c21=c21+0.5d0*chemof1(DKaq,ql(i-1,j))*ua1(i-1,j)
d12=0.5d0*chemof1(DKaq,q1(i,j+1))*ua1(i,j+1)
d12=d12+0.5d0*chemof1(DKaq,ql(1,j))*ua1(1,j)
d21=0.5d0*chemof1(DKaq,ql(i,j))*ual(i,j)
d21=d21+0.5d0*chemof1(DKaq,ql(1,j-1))*ua1(i,j—1)

rhsua(i)=ua1(i,j)+D2*(ua1(i+1,j)—ua1(i,j))

* +D2*(ua1(i-1,j)-ua1(1,j))
rhsua(i)=rhsua(j)+(ht/2.0d0)*f(ca,va,h1(i,j))*ua1(i,j)
rhsua(i)=rhsua(j)-D3*a12*(sl(1+1,j)-s1(i,j))

 

 

 

177

rhsua(i)=rhsua(i)-D3*a21*(s1(1- 1 ,j)—sl(i,j))
rhsua(i)=rhsua(i)-D3*b12*(S1(i,j+1)-S1(i,j»
rhsua(i)=rhsua(i)—D3 *b2 1 *(s 1 (i,j-1)-S 1 (i,j))
rhsua(i)=rhsua(j)-D5*c12*(q1(i+1,j)-q1(1,j))
rhsan')=rhsua(i)—D5*021*(q1 (1—1 ,j)—ql (i,j))
rhsua(i)=rhsua(j)—D5*d12*(q1(i,j+1)-(110,11)
rhsua(i)=rhsua(j)—D5*d21*(ql(1,j-l)-q1(i,j))
funua(i)=1 .OdO
C Cell group b

a12=0.5d0*chemofl(DKbs,S1(1+1,j))*ub1(i+1,j)

a12=a12+0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)

a21=0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)

a21=a21+0.5d0*chemof1(DKbs,S1(1—1,j))*ub1(i-1,j)

b12=0.5d0*chemof1(DKbs,S1(1,j+1))*ub1(1,j+1)

b12=b12+0.5d0*chemof1(DKbs,S1(i,j))*ub1(i,j)

b21=0.5d0*chemof 1(DKbs,S 1(i,j))*ub1(i,j)

b21=b21+0.5d0*chemof1(DKbs,S 1(1,j- 1))*ub1(i,j— 1)

c12=0.5d0*chemof1(DKbq,ql(1+1,j))*ub1(i+1,j) ’

c12=c12+0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)

021:0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)

c21:021+0.5d0*chemof1(DKbq,q1(1~1,j))*ub1(i-1,j)

d12=0.5d0*chemof1(DKbq,q1(i,j+1))*ub1(i,j+1)

d12=d12+0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)

d21=0.5d0*chemof1(DKbq,q1(i,j))*ub1(i,j)

d21=d2 1+0.5d0*chemof1(DKbq,q1(i,j-1))*ub1(i,j- 1)

rhsub(i)=ub1(i,j)+D6*(ub1(i+1,j)-ub1(i,j))
* +D6*(ub1(i-1,j)-ub1(i,j))
rhsub(i)=rhsub(j)+(ht/2.0d0)*f(cb,vb,h1(i,j))*ub1(i,j)
rhsub(i)=rhsub(i)-D7*a12*(s1(1+1,j)-sl(1,j))
rhsub(j)=rhsub(j)-D7*a21*(sl(i—1,j)-s1(i,j))
rhsub(i)=rhsub(i)-D7*b12*(s1(i,j+1)-sl(i,j))
rhsub(i)=rhsub(j)-D7 *b2 1 *(s 1(1,j- 1)—s1(i,j))
rhsub(i)=rhsub(j)-D8*c12*(q1(i+1,j)-q1(i,j))
rhsub(i)=rhsub(i)-D8*021*(ql(1-1,j)-q1(1,j))
rhsub(i)=rhsub(j)-D8*dl 2*(q1(i,j+1)-q1(i,j))
rhsub(i)=rhsub(j)~D8*d21*(q1(i,j-1)-q1(i,j))
funub(j)=1.0d0
1080 continue
C

 

 

 

call PDMTRIX(m,D 1,funs,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhss,us)
call PDMTRIX(m,D1,funq,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsq,uq)
call PDMTRIX(m,D0,funh,aa,bb,cc)

178

call TRIS OLV(m,aa,bb,cc,rhsh,uh)
call PDMTRD{(m,D2,funua,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsua,uua)
call PDMTRIX(m,D6,funub,aa,bb,cc)
call TRISOLV(m,aa,bb,cc,rhsub,uub)
do 1085 j=0,m
32(i,j)=usti)
q2(i.j)=uqti)
h2(i,j)=uh(i)
ua2(i,j)=uua(i)
ub2(i,j)=uub(i)
1085 continue
1090 continue

C
do 1095 j=0,m
do 1096 i=0,m
81(i.j)=sz<i.j>
q1(i,j)=q2(i,j)
h1(i,j)=h2(i,j)
ua1(i,j)=ua2(i,j)
ub1(1,j)=ub2(i,j)
1096 continue
1095 confinue
C
2000 continue
C

call veloc(mvel,lvel,distm,t1mm,distl,t1ml,velm,ve11)
write(30,*)'timemax,distmax,velm'
write(3 1,*)'timelim,distlim,vell'
kvel2=mcount2lnumvel
do 2060 i=1,mvel—1
write(30,*)timm(i),',',distm(i) ,',',ve1m(1)
2060 continue
do 2062 1: 1 ,lvel-l
write(31,*)timl(i),',',distl(i),',',vell(1)
2062 continue
if(movc.eq.2)then
write(16,*)'];'
write(l7,*)'];’
write(23,*)'] ;’
write(18,*)'];'
endif
stop
end

 

 

 

 

 

179

C
C
C The following four functions give the initial conditions
C
C
function ua0(x,y,ua1ni,yoff,xoff,aw1d)
implicit double precision (a-h,o-z)
301 format(6x,d10.9)
r=sqrt((x-xoff)**2+(y—yoff)**2)
if(r.1e.0.3d0) then
ua0=uaini/dexp(awid*r)
else
ua0=0.0d0
endif
return
end

function ub0(x,y,ubini,yoff,xoff,bwid)
implicit double precision (a—h,o-z)
301 format(6x,d10.9)
r=sqrt((x-xoff)**2+(y-yoff)**2)
if(r.le.0.3d0) then
ubO=ub1ni/dexp(bwid*r)
else
ub0=0.0d0
endif
return
end
C
C
C The function f(x) is the uptake of nutrient
C
function f(ca,v,x)
implicit double precision (a-h,o-z)
if(x.ge.0.0d0) then
f=v*x/(ca+x)
else
f=0.0d0
endif
return
end
C
C fs=f(x)/x
function fs(c,v,x)

 

 

180

implicit double precision (a-h,o-z)
if(x.ge.0.0d0) then
fs=v/(c+x)
else
fs=0.0d0
endif
return
end
C
C chemof1*chemof2: non-linear chemotaxis coefficient function
C chemofl is defined by Keller-Segel model
C
function chemof1(DK,x)
implicit double precision (a-h,o-z)
if(x.ge.0.0d0) then
chemof1=DK/(DK+x)**2
else
.chemof1=0.0d0
endif
return
end
C
C function diffu is defined by Monod's law
C
function diffu(v2,c2,eps,x)
implicit double precision (a—h,o-z)
if(x.ge.0.0d0) then
diffu=v2*x/(02+x)+eps
else
diffu=eps
endif
return
end
C tr(i): sets tr to 0 if outside window, or to desired value inside window
function tr(i,h,tk,dxy)
implicit double precision (a—h,o-z)
x=-2.5d0+1*h
y=abs(x)
if(y.le.2.0d0) then
tr=tk/dxy
else
tr=0.0d0
endif
return

 

 

 

181

end
C
C rhbound: subroutine to calculate boundary conditions at
C sink or reservoir.
C
subroutine rhbound(n,a,b,c,d,hr,t,rh)
implicit double precision (a-h,o—z)
C dimension a(0:n),b(0:n),rh(0:n)
rh=a+d*(b-a-hr*t*(a-c))
return
end
C rh2: subroutine to calculate boundary conditions at
C sink or reservoir.
C
subroutine rh2(n,a,b,e,c,d,hr,t,rh)
implicit double precision (a-h,o-z)
C dimension a(0:n),b(0:n),rh(0:n)
rh=a+d*(b-2*a+e+hr*t*c)
return
end
C
C maxcalc: finds maximum density on wave
C
subroutine maxcalc(ua1,k,m,ht,hr,umax,pmax,t)
implicit double precision (a—h,o-z)
dimension ua1(0: 100,0: 100)
mcent=m/2
1max=0
5 if(ua1(mcent,imax+1).ge.ua1(mcent,imax))then
imax=imax+1
goto 5
endif
umax=ua1 (mcent,imax)
pmax=2.5d0'-float(imax)*hr
C write(*,*)'pmax=',pmax
=k*ht
C write(*,*)'t=',t
return
end
C
C limcalc: finds first point where minimum cell density occurs
C
subroutine limcalc(ua1,k,m,ht,hr,dmin,ulim,plim,t)
implicit double precision (a-h,o-z) '

 

 

 

182

dimension ua1(0: 100,0: 100)
mcent=m/2
imax=0
5 if(ua1(mcent,imax+1).1t.dmin)then
imax=imax+1
goto 5
endif
ulim=ua1(mcent,imax)
plim=2.5d0-float(imax)*hr
C write(*,*)'plim=',plim
t=k*ht
C write(*,*)'t=',t
return
end

subroutine veloc(mvel,lve1,distm,timm,distl,timl,velm,vell)
implicit double precision (a—h,o-z)
dimension distm(0: 100),distl(0: 100)
dimension t1mm(0: 100),timl(0: 100)
dimension velm(0: 100),ve11(0: 100)
do 2089 i=2,mvel-1
velm(i—1)=(distm(i)-distm(i-1))/(timm(i)-timm(i- 1))
2089 continue
do 2099 i=2,lvel-1
vell(i-1)=(distl(i)-distl(i-1))/(timl(i)-timl(i- 1))
2099 continue
return
end
C
C SOLVES THE TRIDIAGONAL SYSTEM OF LINEAR EQUATIONS
C
SUBROUTINE TRIS OLV(N,A,B,C,FUN,X)
implicit double precision (a—h,o-z)
DIMENSION A(O:N),B(0:N),C(O:N),FUN(0:N),X(0:N)
DIMENSION ARFA(0:500),BATA(0:500),GAMA(O:500),GUN(0:500)

B(O)=0.0d0
C(N)=0.0d0

ARFA(O)=A(0)
GAMA(O)=C(0)
DO 4000 I=1,N
GAMA(I)=C(I)
BATA(D=B (D/ARFA(I- 1)

 

 

 

 

183

ARFA(I)=A(I)-BATA(I) *C(I— 1)
4000 CONTINUE
GUN(O)=FUN(0)
DO 4999 I=1,N
GUN(I)=FUN(I)-BATA(I)*GUN(I- 1)
4999 CONTINUE
X(N)=GUN(N)/ARFA(N)
DO 4888 I=1,N
X(N-I)=(GUN(N-I)—GAMA(N-I)*X(N-I+1))/ARFA(N-I)
4888 CONTINUE
C
RETURN
END
C
C SET UP THE TRIDIAGONAL MATRD( for constant coefficient terms
C
SUBROUTINE PDMTRIX(n,dxy,fu,aa,bb,cc)
implicit double precision (a-h,o-z)
DINIENSION fu(0:n),aa(0:n),bb(0:n),cc(0:n)
bb(0)=0.0
aa(0)=fu(0)+1 .0d0*dxy
cc(0)=-1 .0d0*dxy
bb(n)=-1 .0d0*dxy
aa(n)=fu(n)+1.0d0*dxy
cc(n)=0.0

do 5000 i=1,n-1
bb(i)=-dxy
cc(i)=-dxy
aa(i)=fu(i)+2.0d0*dxy
5000 continue

return
end

 

 

 

APPENDIX E

 

APPENDIX E

Input files for model

the
.. file 5615
I! am
. 1 E. The Par '
Sample in put files for the program given 1n Appendlx f1 5615 the initial
. Il'nit" ] 6
values of all parameters and the time for solution. The 1 cifiCS which
, ll ' 6
conditions for each population and chemical component The

tfine—points should be printed to the output files.
Sample param file:

In: 41

t1minit= 06.0d0
timin0c= 30.0d0
numcel: 2

numvel: 100

kstep: 20

dmin = 0.000001d0 .
move: 1 1=individual matrices, 2=m0V15
R =. 2.5d0

ayoff: 0.0d0

axoff: 0.0d0

byoff= 0.0d0

bxoff: 0.0d0

awid = 1.0d0

bwid = 1.0d0

Dh — 0.01d0

thn 0.155d0

ths 0.155d0

the 0.000d0

thw 0.000d0

Ds 0.033d0

tsn 0.1071d0

tss 0.1071d0

tse 0.000d0

tsw 0.000d0

Dacs = 0.03d0

Dbcs = 0.00d0

DKas = 0.00000200d0

184

 

185

DKbs = 0.00000200d0
cas = 0.0000550d0
vas = 0.600d0

Yas = 1.0000d0

cbs = 0.0000550d0

vbs = 0.000d0
Ybs = 1.0000d0
Dq = 0.033d0
tqn = 0.891d0
tqs = 0.891d0
tqe = 0.000d0
tqw = 0.000d0
Dacq= 0.080d0
Dbcq = 0.00d0

DKaq = 0.0000330d0
DKbq = 0.0000330d0
caq = 0.000000067d0
vaq = 0.02d0

Yaq = 1.0000d0

cbq = 0.000000067d0

qu = 0.0d0

qu = 1.0000d0
Dua = 0.00100d0
va = 0.35d0

ca = 0.00000408d0
Ya = 0.350d0
Dub = 0.0010d0
vb = 0.50d0

cb = 0.00000408d0
Yb = 0.50d0

Sample init file:

uaini: 0.000005d0
ubini= 0.000005 d0

rsn = 0.000132d0
rss = 0.00000d0

rse = 0.0000d0

rsw = 0.0000d0
rqn = 0.0000132d0
rqs = 0.0000132d0
rqe = 0.0000132d0
rqw = 0.0000132d0

 

 

 

186

rhn = 0.00046d0
rhs = 0.00046d0
the = 0.00046d0
rhw = 0.00046d0
stini= 0.00000d0
qtini: 0.0000132d0
htini= 0.00046d0

Sample time file:

numti=04

time1= 10.0d0
time2= 15.0d0
time3= 20.0d0
time4= 30.0d0

 

 

APPENDIX F

 

APPENDIX F

Matlab program files

The following Matlab program files were created for use with Matlab 4.2c for
Windows. No guarantees are made that they will work with more recent versions.
Commands enclosed in quotes should be entered into Matlab exactly as typed.

The file appmultm plots the cell matrix as a density plot. The simulation plots in
Figure 13 were created using appmultm, for example. The syntax is
"appmult(u,#,'pcolor')" where u is the name of the matrix to be plotted, and # is the
amount of interpolation; typically 2 or 3 works well. (Note: I did not write this file. It is
modified from the version found at the Mathworks site, www.mathworks.com.)

function hfi = app_int(x,y,z,s,dc)
% This function approximates interpolated shading by interpolating
% data and using flat shading. The inputs are the data that was used
% to generate the original object, a scale factor, and a string
% that contains the drawing command. The
% function returns a handle to the new
% object.
%
% Syntax 1:For just Z—Data (e.g., surf(z) => app_int(z,s,dc))
% z = zdata, s = scaling factor and dc = drawing command used to
% create the plot
%
% Syntax 2: For x,y,z Data ( e.g., surf(x,y,z) ) => app_int
% (x,y,z,s,dc)
%
% Example 1: [x,y,z] = peaks;surf(z);shading interp
% app_int(z,3,'surf) interpolate by a factor of 3
%
% Example 2: [x,y,z ] = peaks; surf(x,y,z); shading interp
% app_int(x,y,z,3,‘surf')interpolate by a factor of 3
if ( nargin == 3 )
do = z;

187

 

 

 

188

z = x;
s = y;
[m n] = size(z);
x = 1:n; y = (1:m)';
mscal = m*s;
nscal = n*s;
xi = linspace(1,n,nscal);
yi = linspace(1,m,mscal)';
21 = interp2(x,y,z,xi,yi);
% figure;
cmd = ['hfi=' dc '(xi,yi,zi);']; i
eval(cmd);
shading flat;
elseif ( nargin == 5 )
[m n] = size(z);
mscal = m*s;
nscal = n*s;
mint = min(min(x));
maxt = max(max(x));
xi = linspace(mint,maxt,mscal);
mint = min(min(y));
maxt = max(max(y));
yi = linspace(mint,maxt,nscal)';
zi = interp2(x,y,z,xi,yi);
figure;
cmd = ['hfi=' dc '(xi,yi,zi);'];
eval(cmd);
shading flat;
end
colormap(gray)
axis('square')

 

 

The program surftwom allows two cell matrices to be plotted on the same figure,
with different colormaps. Examples of this program are Figure 33C and Figure 33D. The
syntax is "surftwo(u1,u2)" where 111 and u2 are the two cell matrices.

%Program to plot two cell populations on one figure

%with different colormaps.

function[h]=surftwo(ua,ub)
h(1)=surf(ua),axis equal

189

hold on

h(2)=surf(ub);

hold off

m=64;

cmin=min(ua(: ));

cmax=max(ua(: ));

cl=min(m, round((m-1)*(ua- -cmin)/(cmax- -cmin))+1);
cmin=min(ub(: ));

cmax=max(ub(:));
c2=min(m,round((m-1)*(ub-cmin)/(cmax-cmin))+1)+64;
set(h(1),'cdata',cl);

set(h(2),'cdata',c2);

caxis([min(cl(:)) maX(c2(:))]);

view(2);

shading interp;

colormap([bone(64);copper(64)]);

 

The file centmassm calculates the average mass for two cell matrices. An
example is Figure 35F. The syntax is "[cma,cmb]=centmass(u1,u2)" where 111 and u2 are

the cell matrices. The returned variables cma and cmb may be plotted with "plot(cma),

 

hold, plot(cmb)".

function[cma, cmb]=centmass(ua, ub)

%calculate the total mass along line perpendicular to gradient,
%by summing across each line parallel to gradient.
[r,c]=size(ua);

centline=round(r/2);

for 1: 1 :r;

cma(i)=sum(ua(i,:))/(r- 1);

cmb(i)=sum(ub(i,:))/(r—1);

end '

The file competit.m computes the dynamic competition factor for two cell

populations. Examples are Figure 35A and Figure 35B. The syntax is "competit", where

190
the value for num (the number of timepoints) and the cell matrices, have already been
entered into the Matlab variable space. The returned variable rab is the dynamic
competition factor, and is the ratio of the total mass of Population A to the total mass of
Population B. A plot can be created using "plot(rab)".

%Computes total mass of a and b and ratio a/b for uai, ubi where i=1:num
%To use, first specify value for num, then type competit

clear ta,clear tb,clear rab

for i=1:num;

[ta(i)]=totmass(eva1( ['ua',num28tr(i)]), 1 .6);
[tb(i)]=totmass(eval(['ub’,num28tr(i)]),1.6);

rab(i)=ta(i)/tb(i);

end

The file totmass.m is a subroutine called by the program competit.m.

function[tmass]=totmass(u,ht)

% Calculates total mass under cell profile

% Assumes length and width of DGC = 5 cm 4

% ht is the height of the gel, usually 1.6 cm

% Find number of increments

[r,c]=size(u);

% Compute area under 2-D plot, multiply by l*w*h for mass
tmass=trapz(trapz(u))*5/(r-1)*5/(c-1)*ht;

The program totvelm calculates the mass fluxes due to chemotaxis to S,
chemotaxis to Q, random motility, and the total mass flux. Examples are Figure 21-Figure
24. The syntax is
"[uxs,uys,uxq,uyq,uxu,uyu,uxt,uyt,mag,dir,dxc,dyc,aoverh]=totvel(kds,xos,kdq,xoq,du,s,

q,u,minv)" where kds and kdq are the dissociation constants for the receptor—S and

 

191
receptor-Q complexes, respectively; xos and xoq are the chemotactic sensitivity
coefficients for S and Q; du is the random motility coefficient; 5, q, and u are the S-
chemoattractant, Q-chemoattractant, and cell matrices, respectively; and minv is a
minimum flux below which no values are reported. The returned variables uxs and uys
are the x and y components of the flux due to chemotaxis to S; uxq and uyq are the x and
y components of the flux due to chemotaxis to Q; uxu and uyu are the x and y
components of the flux due to random motility; and uxt and uyt are the x and y
components of the total mass flux. The fluxes can be plotted with the quiver command, as

illustrated near the end of the program.

function[uxs,uys,uxq,uyq,uxu,uyu,uxt,uyt,mag,dir,dxc,dyc,aoverh]=totvel(kds,xos
,kdq,xoq,du,s,q,u,minv)
%This program calculates the flux matrices for chemotaxis to S and Q,
%(uxs,uys,uxq,uyq), random motility (uxu,uyu), and the total flux (uxt,uyt).
[m,n]=size(u);
dir=zeros(m,n);
%Calls to the subroutine chemo, which is where intermediate fluxes actually
calculated.
[uxs,uys]=chemo(kds,xos,s,u,minv);
[uxq,uyq]=chemo(kdq,xoq,q,u,minv);
[uxu,uyu]=randmot(du,u,minv);
%Total flux found by summing components of flux.
uxt=uxs+uxq+uxu;
uyt=uys+uyq+uyu;
%Magnitude of maximum flux calculated.
mag=sqrt(uxt.*uxt+uyt.*uyt);
%Maximum flux magnitude found.
maxvel=max(max(abs(mag)));
%A minimum flux value to be graphed is calculated. Minv=5 works well. (80% of
fluxes graphed).
minvel=maxvel/minv;
%Loop to remove fluxes whose magnitudes are lower than minvel.
fori = lzm,
for j = 1:n;
if abs(mag(i,j)) < minvel
uxt(i,j)=0;

 

 

191

receptor-Q complexes, respectively; xos and xoq are the chemotactic sensitivity
coefficients for S and Q; du is the random motility coefficient; 3, q, and u are the S-
chemoattractant, Q-chemoattractant, and cell matrices, respectively; and minv is a
minimum flux below which no values are reported. The returned variables uxs and uys
are the x and y components of the flux due to chemotaxis to S; uxq and uyq are the x and
y components of the flux due to chemotaxis to Q; uxu and uyu are the x and y
components of the flux due to random motility; and uxt and uyt are the x and y
components of the total mass flux. The fluxes can be plotted with the quiver command, as
illustrated near the end of the program.

function[uxs,uys,uxq,uyq,uxu,uyu,uxt,uyt,mag,dir,dxc,dyc,aoverh]=totvel(kds,xos
,kdq,xoq,du,s,q,u,minv)
%This program calculates the flux matrices for chemotaxis to S and Q,
%(uxs,uys,uxq,uyq), random motility (uxu,uyu), and the total flux (uxt,uyt).
[m,n]=size(u);
dir=zeros(m,n);
%Calls to the subroutine chemo, which is where intermediate fluxes actually
calculated.
[uxs,uys]=chemo(kds,xos,s,u,minv);
[uxq,uyq]=chemo(kdq,xoq,q,u,minv);
[uxu,uyu]=randmot(du,u,minv);
%Total flux found by summing components of flux.
uxt=uxs+uxq+uxu;
uyt=uys+uyq+uyu;
%Magnitude of maximum flux calculated.
mag=sqrt(uxt.*uxt+uyt.*uyt);
%Maximum flux magnitude found.
maxvel=max(max(abs(mag)));
%A minimum flux value to be graphed is calculated. Minv=5 works well. (80% of
fluxes graphed).
minvel=maxvel/minv;
%Loop to remove fluxes whose magnitudes are lower than minvel.
for 1 = l:m,
for j = 1:n;
if abs(mag(i,j)) < minvel
uxt(i,j)=0;

 

 

192

uyt(i,j)=0;
end
end
end
%subplot(2,2,1),quiver(uxs,uys,3,'w'),title('chemo s'),axis square
%subplot(2,2,2),quiver(uxq,uyq,3,'w'),title('chemo q'),axis square
%subplot(2,2,3),quiver(uxu,uyu,3,'w'),t1t1e('randmot'),axis square
%subplot(2,2,4),quiver(uxt,uyt,3,‘w'),title('total flux'),axis square
dxc=zeros(m,n);
dyc=zeros(m,n);

for i = l:m,
for j = 1:n;
if mag(i,j)~=0
dxc(i,j)=uxt(i,j)/mag(i,j);
dycti,j)=uyt(i,j)/mag(i,j);
end
end
end

The file chemo.m is called as a subroutine 1n totvel.m.

function[ux,uy,satx,saty]=chemo(kd,xo,c,u,minv)

%calculate the partial derivatives of c in the x and y directions
[m,n]=size(c);

dx=5/m;

dy=5/n;

[px,py]=gradient(c,dx,dy);

%calculate the matrix kd/(kd+c)"2

for i = l:m,
for j = 1:n;
sat(i,j)=kd/(kd+c(i,j))"2;
end
end

%calculate the velocity coefficient matrix kd/(kd+c)"2 *px(or py)

%the .* operator causes only the corresponding two matrix positions to be
multiplied

%rather than real matrix multiplication

satx=xo*(sat.*px);

saty=xo*(sat.*py);

%clear all values of u below a minimum value

maxu=max(max(u));

 

193

minu=maxul 1000;

for i = l:m,
for j = 1:n;
if u(i,j) < minu
u(i,j)=0;
end
end
end

%multiplies the cell density times the velocity coefficient
ux=u.*satx;

uy=u.*saty;

mag=sqrt(ux.*ux+uy.*uy);
maxvel=max(max(abs(mag)));

minvel=maxve1/minv;

%maxvely=max(max(abs(uy)));
%minvely=maxvely/minv;

fori = l:m,
for j = 1:n;
if abs(mag(i,j)) < minvel
ux(i,j)=0;
uy(i,j)=0;
end
end
end

The file randmotm is called as a subroutine in the program totve1.m

function[ux,uy]=randmot(du,u,minv)
%calculate the partial derivatives of u in the x and y directions
[px,py]=gradient(u);
%calculate the velocity coefficient matrix du*px(or py)
%the .* operator causes only the corresponding two matrix positions to be
multiplied
%rather than real matrix multiplication
%clear all values of 11 below a minimum value
[m,n]=size(u);
maxu=max(max(u));
minu=maxul 1000;
for i = l:m,
for j = 1:n;
if u(i,j) < minu

 

194

u(i,j)=0;
end
end

end
ux=du*u.*px;
uy=du*u.*py;
mag=sqrt(ux.*ux+uy.*uy);
maxvel=max(max(abs(mag)));
minvel=maxvel/minv;

fori = l:m,
for j = 1:n;
if abs(mag(i,j)) < minvel
ux(i,j)=0;
UY(i,j)=O;
end
end
end

The program "retard4.m" calculates the global chemotactic response factor. An
example is Figure 25. The syntax is "[phi,den,num,dent]=retard4(u,s,kd,perct)" where u is
the cell matrix, s is the chemoattractant S matrix, kd is the dissociation constant for the
receptor-S complex, and perct is the lowest percent value of the maximum cell
concentration for which a value of the global chemotactic response factor will be
calculated. The global chemotactic response factor is given by the returned variable phi.

function[phi,den,num,dent]=retard4(u,s,kd,perct)

% Calculation of retardation factor.

% Retard4 calculates factor along y-direction for all x.

% perct is the minimum percentage of the maximum cell concentration above
% which the cell concentration will not be assumed to be 0.

% Right now, only works when gradient in direction of matrix columns.
% Calculation of s-gradient

[m,n]=size(u);

dy=5/m;

maxui=max(max(u));

minui=perct*maxui;

 

195

for i=1:n,
grads=abs(gradient(s(:,i)',dy)');
% Calculate (1+s/kd)) term.
grp1=(1+s(:,i)/kd);
% Calculate (1+s/kd)"(-2)*gradient(s)
gsgrp1=grp1."(-2).*grads;
% Calculate integrand of numerator of retardation factor
num=gsgrp1.*u(:,i);

for j: 1 :m,
if u(i,i) < minui
num(j)=0;
end
end

% Calculate integrand of denominator of retardation factor
den=grads.*u(:,i);

% Evaluate integrals with trapezoid method. Calculate phi.

% phi=zeros(size(pos));

%Make den a function of i, so average maximum flux can be reported.
dent(i)=trapz(den);
phi(i)=trapz(num)/trapz(den);

end

APPENDIX G

 

 

APPENDIX G

Dimensionless model

Over the course of the competition modeling studies, it became apparent that there
was an almost infinite number of parameter combinations to be tested. Many times, it is
possible to reduce the number of parameters in a model by making it dimensionless, and
then varying the dimensionless groups.

The competition model was made dimensionless by employing the following

scales (the ' indicates the dimensional quantity):

 

t = vnHt’ (57)
-' 58
ui = u, for 1=a,b ( )
uiO
. °’ . (59)
_]=—_" forj=H,S,Q
.10
D (60)
V = V’ —-H-
VaH

When these scales were substituted into the dimensional equations and the

equations were rearranged, the new dimensionless equations were:

196

 

 

 

197

a (61)
But“ :1, Vzua—X35$V.[[___G_IS_2]URVS —}ea83QV. _Q‘Q—z. 11an + H 11,,
(Gas +8) (eaQ + Q) Wait '1' H

 

 

 

 

Bub 0 9 H (62)
=?\. Vzu —?»5 V- —LuVS —?\.5 V— ——bQ—uV + n"
at b b b w [[(o,,+s)2] b b bQ (9,,Q+Q)2 b Q WbH+Hub

 

 

as S S (63)
—=o V28—

at S Bus Was-1's ua_ 1tl-IBbS—I—Wbs"1'SUb
519_ 2 __9_ Q (64)
at —GQV Q BaQ WaQ'l'Qua—TCHBbQ wa+Qub

8H H H (65)
__ ____ V2 __ _ __—

at H 1’ tllafl+Hua RH¢b tllbfl+Hub

where the dimensionless variables are defined as:

 

 

- . . . . . 66
l, = E‘— for i=a,b Measure of random motllity to nutr1entd1ffus1on ( )

H
Xoij . . . . . . _ ( 67)

5,1. = —- for 1=a,b; j=S,Q Measure of chemotaXIS to random 1not1t111ty
Hij
.. ( 68)
0,]. - ,le for i=a,b
Jo

198

\Ilij = "fl for i=a,b; j=S,Q Saturation constant over initial concentration

0

V .
71:]- = —b’ for j=S,Q,e,d Max. growth rate of b on j to max. growth rate of a on j
at

Vi'uio . . , . .
13,,- = J— for 1=a,b; j=S,Q,H,e Compares max uptake of1 on j to rate of 1 on H
1H 0

.l

D.
o. = D—’ for j=S,Q Ratio of diffusivity of j to diffusivity of H
H

 

(l)i = '0 for 1=a,b; St01chiometr1c concentration ratio
HOYiI-l

( 69)

( 70)

(71)

( 72)

( 73)

By comparing the dimensionless equations (Equations ( 61)-( 65)) to the

dimensional equations (Equations (38)-(40)), it is obvious that the number of parameters

has been reduced by only one. The introduction of the dimensionless groups has not

significantly simplified the model.

 

 

198

ij . . . . . . .
“’0' = ,— for i=a,b; j=S,Q Saturation constant over initial concentration

0

v .
it]. = —b’ for j=S,Q,e,d Max. growth rate of b on j to max. growth rate of a on j
at

i'uio . .
[3,). = J for i=a,b; j=S,Q,H,e Compares max uptake of1 on j to rate of1 on H

1H0

 

l

D.
G. = D-J- for j=S,Q Ratio of diffusivity of j to diffusivity of H
H

d), = ‘0 for 1=a,b; Steichiometric concentration ratio
HOYiH

 

( 69)

( 70)

(71)

( 72)

( 73)

By comparing the dimensionless equations (Equations ( 61)-( 65)) to the

dimensional equations (Equations (38)—(40)), it is obvious that the number of parameters

has been reduced by only one. The introduction of the dimensionless groups has not

significantly simplified the model.

 

 

 

 

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