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Quantitative Investigations of Neuronal Architecture

and the Structural Role of the Cytoskeleton

presented by

Timothy John Dennerll

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MSU Is An Affirmative Action/Equal Opportunity Institution

QUANTITATIVE INVESTIGATIONS OF NEURONAL ARCHITECTURE

AND THE STRUCTURAL ROLE OF THE CYTOSKELETON

BY

Timothy John Dennerll

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physiology

1989

geolt77

ABSTRACT

QUANTITATIVE INVESTIGATIONS OF NEURONAL ARCHITECTURE
AND THE STRUCTURAL ROLE OF THE CYTOSKELETON

BY

Timothy John Dennerll

The elastic properties of PC-12 neurites were assessed
by applying force with a calibrated glass needle for 1-2
seconds and measuring the resulting changes in neurite
length and needle deflection. We observed a linear
relationship between neurite tension and length change and
were thus able to determine neurite spring constants and
initial rest tensions. We observed positive neurite rest
tensions ranging over three orders of magnitude but
clustering around 30—40 microdynes. Treatment with an anti-
actixi drug reduced neurite rest tensions while treatment
with an anti-microtubule drug increased tensions. These
observations suggest a complementary force interaction
underlying axonal shape: the cortical actin network under
tension and the microtubule core under compression.

We also used force-calibrated glass needles to measure
PC-12 neurite lengthening in response to experimentally
applied tensions over time periods of 30-60 minutes. We
found that initial applied tensions of 100 ndcrodynes or
less caused a two phase, viscoelastic neurite length

increase. This behavior is modelled with a trio of

classical mechanical elements: two springs and a dashpot.
Computer simulations based on the expected behavior of this
model closely match experimental observations. At applied
tensions of greater than 100 udynes, there is an additional
third phase of elongation with an average rate linearly
dependent on applied force. we tentatively interpret this
third phase to be towed axonal growth as observed by Bray.
In the presence of 1 uM taxol, only the third elongation
phase is inhibited, suggesting its dependence upon
microtubule assembly.

The viscosity of F-Actin and microtubule suspensions
was measured as a function of shear rate with a Weisenberg
rheogoniometer. At shear rates of less than 1.0 see“1 the
viscosity of suspensions of these two structural proteins is
inversely related to shear rate. These results are
consistent with previous in-vivo measurements of the
viscosity of cytoplasm. This power law implies that shear
stress is independent of shear rate; that is, shear stress
is a constant at all shear rates less than 1.0 sec’l. This
flow property may explain several aspects of intracellular
motility in living cells. Possible explanations for this
flow property are based on a recent model for semidilute
suspensions of rigid rods or a solid friction model for

liquid crystals.

 

 

 

 

DEDICATION

To my father Donald Dennerll, who taught me to be
critical in thinking, so that I might not be fooled into
settling for anything but the truth.

To my mother Jane Weiser, who believed I could achieve
any goal I set my heart on and thus gave me the courage to
be successful.

To my wife Jeanie, a princess who spied a frog in the
pond and kissed him so that he might become a man.

To my daughter Diane, who instructs me daily in the
important things in life and is definitely the best stinker
I know.

To my son Raymond, whose smile melts my heart every day.

iv

ACKNOWLEDGMENTS

Mom, Jerry, Dad and Evie, because of the terminology,
you may find the meaning of the rest of this dissertation to
be somewhat less than clear. Yet I'm sure that you
appreciate the amount of work it represents. My success is
no accident; it is the result of my being supported and
coached by many outstanding people. Along the way I
discovered something quite remarkable, that each one of us
wants to make a contribution to another, but we usually
aren’t willing to be contributed to. Learning how to let
myself be supported has been one of the most valuable (and
necessary) discoveries I made during my graduate program.

I also want you all to know that through this experience (as
well as my becoming a parent myself), I have become deeply
aware of your love for me, and the knowledge that this was
the source of what I feel to be my greatest strength: my
love for myself and God which in turn sources my ability to
contribute to the lives of those around me.

I am unable to acknowledge either my wife Jeanie or my
advisor Steve Heidemann enough for their commitment to my
success, one that many times exceeded my own and without
which I would have given up long ago. Steve, you have been
a true mentor to me in many ways. Your enthusiasm for your
work has inspired me to try to be my very best. I
especially thank you for greeting me like the prodigal son
after my disappearances; it allowed me to begin to trust
others in helping me find my way. Jeanie, your patience and
willingness to support me through school during almost our
entire marriage so far has meant more to me than you’ll ever
know, I only hope that I will have the opportunity to return
this gift to you some day.

I'm very lucky to have Robert Buxbaum as both an
advisor and friend. Bux, you have clearly made it possible
for me to successfully pursue my interest in physics in a
biological setting. I also thank my other committee members
Ron Meyer, Mel Schindler and Birgit zipser for their
thoughtful contributions to my education and this
dissertation. other faculty I would like to acknowledge are
Sherwood Haynes, Seth Hootman, Jack Krier, Clarence Suelter,
Lynne Weaver, and John Wilson.

There are many other people whom I have come to call
friends and would like to thank for their support. In
particular I wish to acknowledge Greg Adams, Peter Baas,
Mary Lynn Bajt, Jean Foley, Harish Joshi, Phil Lamoureux,
Jerry Lepar, Charles Liang, Shelly Maiorana, Vivian Steel,
Leigh White, and Jing Zhen.

There is one more person who I need to acknowledge for
her love and support which has gone far beyond the call of
duty. Helen, you have made it possible for me to have it
all. I truly appreciate your partnership in caring for our
children as well as the many delicious meals, washed clothes
and house cleanings. I also thank you for giving me such a
beautiful and charming daughter for my wife.

V

TABLE OF CONTENTS

List of Figures

General Introduction

Chapter 1.

Chapter 2.

Chapter 3.

Appendix A.

Appendix B.

Quantitative Measurements of
Tension and Compression in the
Cytoskeleton of PC-12 Neurites.
Introduction

Materials and Methods

Results

Discussion

Tension, Viscoelasticity, and Growth
in PC-12 Neurites.

Introduction
Materials and Methods
Results

Discussion

F-Actin and Microtubule Suspensions
as Indeterminate Fluids.

Introduction

Materials and Methods

Results and Discussion

Method for analytical determination
of neurite mechanical constants from
observed distension behavior.

Logic underlying the computer

simulation based on our viscoelastic
model for neurite distension.

List of References

13

26

31

33

35

48

55

56

58

71

72

78

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

10.

11.

12.

13.

14.

LIST OF FIGURES

Method of measurement of neurite
tension by lateral displacement.

Three typical plots of neurite axial
tension vs. change in neurite length.

Frequency distribution for neurite
rest tensions.

Frequency distribution for neurite
spring constants.

Neurite tension as a function of
length change before and after
addition of 0.1% DMSO.

Neurite tension as a function of
length change before and after
addition of 2 ug/ml cytochalasin D.

Summary of the effect of
cytochalasin D on neurite rest tension.

Neurite tension as a function of
length change before and after
addition of 1 ug/ml nocodazole.

Summary of the effect of nocodazole
on neurite rest tension.

Method of measurement of neurite
tension during long duration
distension.

Neurite length and tension as a
function of time for a 2-phase
neurite distension.

Viscoelastic model of neurite
mechanical response to long duration
distension.

Computer simulation of observed
neurite distensions using our
viscoelastic model.

Computer simulation of hypothetical
neurite distensions.

vii

11

14

15

17

19

21

22

23

24

34

36

38

39

40

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

Neurite length and tension as a
function of time for a 3-phase
neurite distension.

Low and high initially applied
tensions on the same neurite.

Threshold tension for phase 3
neurite elongation.

Taxol inhibition of phase 3
neurite elongation.

3-position controller model for the
tension dependence of neurite length.

Viscosity as a function of shear
rate for F-actin and microtubule
suspensions.

Effect of shear history on viscosity
of F-actin.

Viscosity of F-actin compared with
viscosity measurements of cytoplasm
by various in-vivo methods.

Shear stress as a function of shear
rate for cytoskeletal proteins and
the viscosity standard.

Polarization micrographs of sus-
pensions of actin and microtubules.

viii

42

43

45

46

52

59

61

62

64

70

INTRODUCTION

Fundamental to the design of all structures is a
strategy to resist deformation and collapse from forces that
originate both externally from the surrounding environment
and internally from active and passive processes. Both the
mechanical properties of :materials used. as well as the
manner in which individual elements are structurally related
bear on this problem. Unlike most man-made structures which
are composed of solid, static elements, the living cell is
structurally dynamic at both the organizational and
molecular levels (2). In addition and probably in part as a
result, the cell seems to behave mechanically with both
'solid-like’ and ’fluid-like’ properties (44,46,73,80).

The shape, polarity, and motility of all eucaryotic
cells is thought to be the result of the organization and
control of the cell’s cytoskeleton (2). This "cell
skeleton" is generally considered to be composed of three
filamentous protein polymers: F-actin, microtubules and
intermediate filaments. Although the cytoskeleton is
presumed to provide the structural support for the cell,
very little work has been done to describe mechanical
behavior or the architectural role for individual
cytoskeletal elements. Towards a better understanding of
these problems, this dissertation attempts to explore the
following basic questions. What are the mechanical

properties of the cell at the level of individual elements

 

2

and in complete integrated structures? How is the
cytoskeleton organized in supporting both internal and
external forces such that it is able to determine and
maintain cell shape?

There is convincing evidence that external force plays
a role in determining cell shape, polarity, and motility. A
wide variety of cell types have been clearly shown to exert
force on their environment (6,39,107). Some of the forces
exerted by cells are involved in the crawling movement and
stationary shape of cells (1,39). There is widespread
agreement that these forces must be generated and supported
by the cellular cytoskeleton (107). External force applied
to many cells has a major effect on their morphology and
motile behavior. Perhaps best known is the work of Bray
(11) who has shown that axonal growth can be elicited by
experimentally applied mechanical tension. Vandenburg (110)
has shown that chick skeletal myofibers will orient
themselves to tension applied while the cells are growing.
Also experimentally applied tensile stress causes an
increase in the migratory activity of isolated corneal
epithelia of embryonic chick (104).

Neurons make good models for investigating the role of
the cytoskeleton in determining cell shape, polarity, and
motility. These cells develop their highly asymmetric shape
through a tip growth process: an advancing growth cone
appears to initiate the formation of both axons and

dendrites. The importance of the cytoskeleton in

determining neuronal motility and elongation is widely
accepted (13,60,63,84,108). The available evidence (55,60)
suggests that axonal elongation may be divided into two
phases; 1) actin based forward motility of the growth cone
and 2) actin and microtubule based elongation of the axon.
There is widespread agreement that the axon elongates as a
result of a "pulling" growth cone (10,11,60,63,84,108). In
particular, Bray has provided convincing evidence that
tension on the neurites (processes of cultured neurons) is
sufficient to cause their elongation. He showed that the
vector sum of all neurite segments of a single cultured
neuron nearly always formed a closed loop, indirectly
indicating that the cell body was suspended by a network of
neurites under tension (9). In addition, using a "towing
motor" attached to the growth cone, he was able to "pull
out" a morphologically and ultrastructurally normal neurite
(11). This suggests that the integrative coupling between
growth cone movement and axonal elongation is solely one of
mechanical force. Campenot (20) observed cell body
movements when neurites either elongated or shortened. He
similarly concluded that nerve fiber elongation is normally
controlled by tension generated by growth cone movements.
In further support, Letourneu (61) showed that neurites will
preferentially navigate into regions of greater adhesivity.
In light of Bray’s findings that neurites could be guided by
tension alone (9), Letourneau (62) reinterpreted his 1975

findings as a tensile competition between weaker and

stronger adhesion sites. That is, the greater tension
supported by the stronger adhesion breaks the weaker
adhesions, and thus directs the growth cone to the more
adhesive surface. In summary, both phases of axonal
growth are thought to be dependent on the organization and
function of the cytoskeleton. These experiments suggest
that external force plays a role in the regulation of the
axonal cytoskeleton during axonal elongation.

The structural organization of the cytoskeletal
elements within the axon should reflect the forces they
support. Previous work from our lab as well as other’s have
shown that anti-cytoskeletal drugs reveal a complementary
"division of labor" within the neurite cytoskeleton. Anti-
microtubule drugs cause the neurite to collapse but have
little effect on the ruffling activity of the growth cone
(14,27,40,51,53,98). Conversely, anti-actin drugs inhibit
growth cone motility but neurites remain extended
(14,40,53,98). In addition, drugs that disrupt actin
networks stabilize the neurite to retraction, while drugs
that disrupt microtubules cause retraction (53,65,98).
These complementary drug effects may reflect a complementary
force interaction in the neurite cytoskeleton: an actin
network in tension supported in part by microtubule
compression and in part by compression of the underlying
substrate.

This dissertation is composed of three chapters, each

one aimed at quantitatively measuring the relationship

5

between mechanical force and deformation of the cell or
cytoskeleton. I was interested in investigating both the
structural properties of individual elements and the
structural relationships between them. The first two
chapters deal with quantitative force measurements made at
the cellular level on PC-12 neurites using force calibrated
glass needles. This work has significance regarding both
cytoskeletal architecture and neurobiology. For short
duration deformations a classical Hookian ’solid-like’
behavior is observed, allowing the quantitative measurement
of both neurite tensions and spring constants. With the aid
of microtubule and actin inhibitors, the complementary
structural roles of these two cytoskeletal elements is
confirmed: actin in tension and microtubules in compression.
In the second chapter, long duration distensions are used to
uncover some additional ’fluid-like’ behaviors of the cell.
This allows neurite structure to be modeled from classical
mechanical elements: springs and dashpots. In addition, a
tension induced neurite elongation, similar to the growth
response observed by Bray (11), was observed and found to
have a minimum force threshold, with elongation rate
linearly related to applied force.

The elastic and viscoelastic measurements discussed in
the first two chapters concern the mechanical behavior of
the combined structural elements in the living cell. Toward
a better understanding of this behavior, we also

investigated the mechanical behavior of individual elements.

6

Cytoplasmic structure has long been thought to be a key
element in understanding cell structure and motility (96,2).
Recent investigators have focused. on. suspensions of“ the
purified cytoskeletal filaments, because these suspensions
(particularly of actin filaments) are found to have many of
the properties of cytoplasm. Both cytoplasm and filament
suspensions are shear thinning (71,79,88,89,119,120,). An
initial stress "overshoot", indicative of a gel to sol
transition, is also generally observed in both cytoplasm
(26,90,109) and actin suspensions (71,79,88). Chapter 3
investigates the rheological properties of F-actin and
microtubule suspensions to obtain a clearer understanding of
cytoplasmic structure. This is done by measuring the forces
associated with varying rates of shear applied to these
suspensions. Surprisingly, all rates of shear produced the
same amount of force for a particular sample. This unusual
result is consistent with the combined force measurements
made by others in-vivo on cytoplasm and provides some
insight into the mechanics of cytoplasmic structure and cell

motility.

CHAPTER 1. Quantitative Measurements of
Tension and Compression in the

Cytoskeleton of PC-12 Neurites.
INIBQDHQIIQN

Since the pioneering studies of Yamada et al (117),
many investigators have found a clear cut "division of
labor" in the role of the cytoskeleton in growth cone
motility and axonal elongation. Anti-microtubule drugs
cause neurites to collapse but have no effect on growth cone
motility functions, eg. ruffling, microspike activity, etc.
(14,27,40,51,53,98). Conversely, anti-actin drugs inhibit
growth cone motility functions but neurites remain extended
(14,40,53,98). Further, drugs that disrupt the actin
network stabilize the neurite to retraction, while drugs
that disrupt microtubules (MTs) cause retraction. Also,
drugs that augment actin assembly cause retraction while
drugs that augment MT assembly stabilize neurites to
retraction and sometimes cause extension (24,53,65,98).
Similarly, the networks move at different rates in slow
axonal transport (8). Apparently, while microtubule-based
and actin-based elements of neuronal growth interact, they
are structurally and functionally distinct (60).

Neurites of cultured neurons have been shown by
indirect means to be under tension (9,53). Bray (11)
elicited apparently normal neurite elongation by "towing"

7

8

neurites with an appropriately paced motor in a process very
similar to the "towed growth" long known to occur in_§fitu
(112). Tension has also been shown to markedly alter the
motility at the edge of fish skin cells (57), align the
division plane of plants (67) and dramatically orient
growing chick myofibers (110). These and other examples
(107) of mechanical force affecting cell growth, shape and
polarity, processes widely regarded to depend on
cytoskeletal function (2,28), suggest that tension may play
a role in regulating the cytoskeleton. Previous results
from our lab (53) caused us to propose that actin and
microtubules in PC-12 neurites are in a complementary force
interaction: the actin network being under tension that is
partly supported by microtubule compression and partly
supported by the underlying substrate. This postulated
complementary force interaCtion, similar to the interaction
of structural elements in "tensegrity" architecture (34),
provides, in principle, a means for integrating microtubule
assembly with the advance of the growth cone.

In an effort to better understand the structural roles
of actin and microtubules in axonal shape and to test our
complementary force model, we directly measured mechanical
force in PC-12 neurites and the effect of cytoskeletal drugs

on that force.

MATERIALS AND METHODS

Cell Culture

PC-12 cells were grown as previously described (53),
primed in 50 ng/ml 7s nerve growth factor (NGF) for 4-6 d,
then replated on untreated 60 mm tissue culture dishes in 78
NGF media, allowing neurites to regenerate for 2-3 days
prior to tension measurements. The microscope was mounted
on a vibration isolation table and cultures were maintained
at 37°C on the microscope stage with an air curtain
incubator (Sage Instruments, Jamaica Plains, MA).
Nocodazole (Aldrich Chemical Co., Inc., Milwaukee WI) and
cytochalasin D (Sigma Chemical Co., St. Louis, MO) stock
solutions were prepared at 1000x concentration in DMSO
(Sigma Chemical Co., St. Louis, MO), with final experimental
concentrations of 1 ‘ug/ml :nocodazole and 2ug/ml
cytochalasin D with 0.1% DMSO for each. Neurite tensions
were measured (see below) both prior to and 10-15 mins
following addition of drug. Only those neurites that
remained attached throughout an experimental series of
deflections could be analyzed for the effect of the drug.
Drug treated neurites showed length changes of less than 5%,
generally less than 1%. Control experiments were done in a

similar manner in 0.1% DMSO.

10

Force Needle Calibration

Glass needles were fabricated and calibrated for force
as described by Nicklas (78), with the following
modifications. TWO relatively stiff "reference" needles
were made from 8 and 16 mm lengths of uniform diameter 25 um
chromel wire (Omega Engineering, Inc., Stamford, CN). Each
was calibrated for force by hanging weights from their tips.
A third 24 mm wire needle could then be calibrated
indirectly by using' the following standard relation from
beam theory: Force = (constant)(tip displacement)/(needle
length)3. Values obtained for the constant from the two
shorter (stiffer) needles differed by less than 1% and gave
a calculated ‘value of 90.9 udynes/um deflection for the
third needle. More flexible glass needles were then
calibrated by bending them against this wire needle, and
subsequently glass needles against each other, to get
"working" needles with stiffness calibration constants (Cal)

of 1-10 udynes/um.

Neurite Tension Measurement

To make a tension measurement, a force calibrated
needle is placed at the center of the neurite and rapidly
moved perpendicular to the neurite axis, using a Narishige
hydraulic micromanipulator, causing both neurite distension
and needle deflection (see figure 1). After holding for 2-3
secs, the needle is lifted; both needle and neurite assume

rest positions almost instantaneously. This procedure is

\\

 

 

 

 

needlexx‘
L°:23 LS=28 '
a 3
growth
cone /\ -- xk
INITIRL PULL RELERSE

Figure 1. Method for measurement of neurite tension by
lateral displacement.

A force calibrated glass needle is placed at the
neurite center (left), micromanipulated orthogonal to the
neurite axis and held for 1-2 seconds (center), and then
lifted allowing the neurite to straighten and the needle tip
to assume its equilibrium position (right). Following the
experiment, measurements are made from recorded video images
of neurite rest length (Lo), neurite lateral displacement
(b) and force calibrated needle deflection (d). These
measurements are used to calculate neurite axial tension and
change in neurite length (see Materials and Methods).

12

repeated 10 to 20 times at various displacements in random
order. Video recording allows direct measurement from the
video screen of neurite rest length (Lo), lateral neurite
displacement (b), and needle deflection (d) for each "pull".
Screen measurements were converted into actual distances
through the use of a stage micrometer. These values were

used to calculate axial neurite stretch tension (T8) in

udynes and corresponding neurite stretch lengths (Ls) for
each pull by the following analysis: Since c2 = a2+b2 ,

L8 = 2c = 2 ia2+b2 . A vector force balance is then used to

determine axial forces in the stretched neurite:

T8 = To+k1(Ls-Lo) = force parallel to neurite axis

d(Cal) force perpendicular to neurite

d(Cal) = Zsine{To+k1(Ls-Lo)}
sine = b/c = 213/1.s

d(Cal) = 4b{To+k1(LS-Lo)}/Ls To = neurite rest tension

d(Cal)Ls/4b = To+k1(Ls-Lo) k1 = neurite spring constant

Distended neurite axial tension d(Cal)Ls/4b is plotted as a

function of change in neurite length (Ls-L Linear plots

0"

indicate Hookian elastic behavior ‘where the y-intercept

equals the neurite rest tension (T tension at zero

0'

distension) and the slope equals the spring constant (k1).

13

RESULTS

Neurites are Under Tension

PC-12 neurites were multiply distended in a direction
orthogonal to the neurite axis using a force calibrated
glass needle to measure neurite mechanical properties (see
figure 1 and materials and methods). Tensions were measured
on 82 neurites and figure 2 shows the results for three
typical tension experiments. Regression coefficients (R) of
0.80 or better were observed in over 90% of the neurite
length-tension plots. Initial neurite rest lengths varied
between 34 and 176 um, and maximum strains varied
between 4.1% and 16.6% of the initial rest length. The y-
intercept of the length—tension plot represents the tension
on the neurite with no displacement (neurite rest tension,
To) and the slope is the neurite spring constant (k1). The
observed rest tensions for 82 neurites varied over 3 orders
of magnitude (see figure 3) but most neurites exhibited rest
tensions in the neighborhood of 35 udynes, and only 1 rest
tension value was less than zero (-2 udynes). We found no
correlation between neurite length and rest tension.
Interestingly, we found a visual cue that correlates to
neurite rest tension. A small roughly spherical bleb in the
neurite just proximal to the growth cone is typically
associated with low observed rest tensions (1-25 udynes).
This bleb becomes more ovoid in shape in neurites with

intermediate rest tensions (25—200 udynes), disappearing

14

 

 

 

 

 

3;" Lo: 96 um

,0; R: .906
Q)
C
>\
'0
i3 0
C 1‘ L0: 176
O .
a R: .957
C
0.)
I— ..
3 .. Lo= 52
E = .845
0)
Z

0 ° . . . . .

o 5 10

Neurhe Length Change (unfi

Figure 2. Three typical plots of neurite axial tension vs.
change in neurite length.

Neurite axial tension calculated as described in
material and methods is plotted as a function of neurite
length change. Each data point represents a single "pull"
as shown in figure 1. Straight lines were fixed to the data
by a least squares fit regression. In all such plots, the
slope of the line represents neurite spring constant (k1)
and the y-intercept indicates the tension on the neurite
with no displacement (neurite rest tension, To)°

15

 

 

 

 

 

 

 

 

25 0
20; FREQUENCY DISTRIBUTION
E for
15; Neurite Rest Tensions (To)
Frequency
10% “=82 .‘L
5-} 7%? __
o<~ /%% W%‘
0 <3 <3 c: o 0 <3 c: O O
o o
RestTension v (I) T “I3 a? E 8 I?) g “’03 .9 8
(d ) O 8 3 8 ('3 ('3 C', I I I I
u ynes 00 o o 8 8 8
— OJ r) v— to
Figure 3. Frequency distribution for neurite rest tensions.

Summary of results for 82 neurite rest tensions derived
from plot intercepts as in figure 2.

16

into the neurite body at greater rest tensions. The highest
observed tensions (1-3 millidynes) are comparable to those
observed in the first cleavage furrow of the echinoderm egg
(85).

Neurite spring constants varied over a narrower range
than rest tensions. Figure 4 shows the frequency
distribution for spring constants for 75 of the 82 measured
neurites. The mean value was 24.4 t 2.2 (std. error)
udynes/um with 85% of values falling between 5 and 40
udynes/um. The 7 neurites not included here were those very
high tension neurites for which we were unable to obtain
accurate spring constant measurements. There was no
significant correlation between spring constant and neurite

length or tension.

Control Treatment

Experiments involved multiple distensions before and
after treatments with anti-cytoskeletal drugs dissolved in
DMSO at a final concentration of 0.1%. As control
experiments for the effects of DMSO and multiple
distensions, we compared length-tension plots before and
after treatment with DMSO. We initially deflected neurites
various distances in random order to give an untreated
reference line. A typical series of 15-20 "pulls" on a
neurite would last 3 mins. DMSO was added to the culture
dish to a final concentration of 0.1% and allowed to sit for

10-15 mins. The same neurite was then subjected to an

17

 

   
 
   

 
 

 

 

 

25I Q FREQUENCY DISTRIBUTION __
20-:- \ N 't s r'forConstonts (k1) 33
1O;- §§§ N=75
Frequency5 §~§§ .
o: §§§ ‘ m m
fuodzsigimg ; g g I: g g g g E

Figure 4. Frequency distribution for neurite spring
constants.

Summary of results for 75 neurite spring constants
derived from plot slopes as in figure 2.

18

additional 15-20 pulls of random distensions for comparison
with the above. Figure 5 is typical of these control
experiments. Both within and between series of distensions,
equal displacements produced nearly identical tension
measurements. However, regression lines varied somewhat
between pre- and post—DMSO plots producing small differences
in rest tensions (y-intercept) and spring constants (slope).
In 8 experiments, we found that rest tensions following DMSO
treatment varied within 50% of their pre-treatment values;
the average result was 99.8% of their pre-treatment rest
tensions. Similarly, the spring constants averaged 110% of
their reference values following DMSO treatment.
Measurements from three neurites subjected to two bouts of
multiple distensions fifteen minutes apart but without DMSO
were indistinguishable from those with added DMSO. We
conclude that neither the DMSO carrier nor the distensions
themselves significantly affected the neurite mechanical

properties measured.

Cytochalasin Reduces Neurite Tension

We tested the effect of the anti-actin drug
cytochalasin D (16,45) on the neurite mechanical properties.
Following a reference series of distensions, neurites were
treated with 2 ug/ml cytochalasin D (Cyt-D) for
approximately 10 mins. After treatment, additional series
of random deflections were carried out on the same neurite.

Figure 6 shows the results of such a Cyt-D experiment, with

19

 

     

 

 

 

250 ‘_ I 4 I I I I ..

200 -- Reference —

E : (/” 3

c .. , DMSO _;_

“g 150 ".— (12 mins):
C

.9 “I
(D

C ..

0) ..

I— ..

I?

o " I I I 4 I 4.
O I 2 3 4 5 6 7

NRT Length Change (um)

Figure 5. Neurite tension as a function of length change
before (0) and after (4) addition of 0.1% DMSO (Control
Experiment).

Neurite axial tension was plotted as a function of
neurite length change (see figure 2) for 2 series of
displacements on the same neurite.

20

one pre-treatment plot and two plots showing the effect of
Cyt-D at 10 and 16 mins. As shown in this figure, the
neurite rest tension and spring constant declined with
increasing ‘time of drug treatment" However, it was not
possible to wait for a steady state value because the
neurite behavior became primarily viscous and does not
recover from distension. As summarized in figure 7, all
Cyt-D treated neurites showed significant declines in rest
tension; the maximum rest tension was 31.6% of the reference
value. In two cases (including fig. 6) rest tension values
following Cyt-D treatment fell below zero indicating a net
neurite compression. The average result for all 7 neurites
was a 114% decline in the initial rest tension, indicating
net neurite compression. The neurite spring constants also
declined, from a mean 15.2 udynes/um to 2.4 udynes/um after

cytochalasin treatment, 15.8% of the reference value.

Nocodazole Increases Neurite Tension

We also tested the effect of the anti-microtubule drug
nocodazole (29) on the neurite mechanical properties.
Following a reference series of deflections, neurites were
treated with l ug/ml nocodazole (NOC) for approximately 10
mins followed by an additional series of random deflections
on the same neurite. Figure 8 shows the results of a
typical NOC experiment, with one pre-treatment plot and one
plot showing the effect of NOC at 11 mins. As summarized in

figure 9, all NOC treated neurites showed an increase in

 

21

 

  
   
 

 

 

 

 

250, . . . . . _
200": Reference -_
7x? I ’
2 I50: :
>. .
U .
3 . .
V 100: Cyt—D (10 mins) ’ , T
.5 I ’Q , ’ ’ ’ Z
(I) .1 0’9. ’ ’ O _
E 50: O , 90 , o :
0: A , , 0‘ U o CO .._Q_.._.._.._.._.._.._.._...o._..$>----<>~I
i__,.,_._.._.._.- Cyt—D (16 mins)
_50 , , , . , . . . . I . . . . r . . . .
O 5 IO 15 20

NRT Length Change (um)

Figure 6. Neurite tension as a function of length change
before (0), 10 minutess after (0) and 16 minutes after
(0) addition of 2 ug/ml cytochalasin D.

Neurite axial tension is plotted as a function of
neurite length change (see figure 2) for 3 series of
displacements on the same neurite. In this trial, growth
cone detachment ended the last series of displacements after
4 pulls. This is neurite #6 in figure 7.

22

 

Zug/UHCyb43.10—15rnms I
1Ll..- _ _ _ _._ _ _ _ _._._._ _ _ _ _ _ _._.__.:

Nonnafizedgg
Rest

Tension I %

 

 

 

 

Neurhe #

Figure 7. Summary of the effect of cytochalasin D on
neurite rest tension.

The effect of cytochalasin D on rest tension is plotted
for each of 7 neurites. The initial rest tension for each
neurite is normalized to 1 and the graph shows the relative
change for that neurite’s rest tension after 10-15 mins drug
exposure.

23

 

  
  

600‘ I . . . I I . . . l . . '
3 NOC(11mins)/ / ’;
. I] x C] '
500: D/// B o T
: E ,z .
400; D //’ '_
2 D . '

Reference

Tension (udynes)
(.4
o
o
A A l L -

 

 

 

0 5 10

NRT Length Change (um)

Figure 8. Neurite tension as a function of length change
before (0) and after ((3) addition of 1 ug/ml nocodazole.
Neurite axial tension is plotted as a function of

neurite length change (see figure 2) for 2 series of
displacements on the same neurite. This is neurite #8 in
figure 9.

24

 

5;: 1 ug/nfl NOC ,10—15 nuns
4e:-
NonnaHzedIZ

%
3:3. Z i
%

2%}

I-::---

 

 

 

 

 

 

h)—
(.14
.p
()1
0,-
\I
(I)

Neurhe #

Figure 9. Summary of the effect of nocodazole on neurite
rest tension.

The effect of nocodazole on neurite rest tension is
plotted for each of 8 neurites. The initial rest tension
for each neurite was normalized to 1 and the graph shows the
relative change for that neurite’s rest tension after 10-15
mins drug exposure.

25

rest tension ranging from 153% to 560% of the reference rest
tension with an average value of 282%. These values
appeared to be short lived plateaus: all neurites eventually
retracted in response to the increased tension. However the
neurite spring constants did not show a consistent change:
four went up, two went down and two remained the same. The
shift in the mean was negligible, going from a reference
value of 20.5 udynes/um to 20.0 udynes/um in the drug

treated neurites.

26

DISCUSSION

We found that growing neurites of PC-12 behaved as
simple elastic solids for short duration distensions (2-3
secs). That is, we consistently found a linear relationship
between applied force (axial tension) and the induced length
change, and a high degree of reproducibility in measured
force in multiple trials of the same distension (figure 2).
In the presence of anti-cytoskeletal drugs, the linear
regression coefficients decreased somewhat. This may
reflect time dependent changes in spring constants during
drug treatment of the neurites. As expected from
observations of other cells (47,73,80,81), neurites behave
viscoelastically under longer duration distensions.

The majority (67%) of our measurements reported in
figures 3 and 4 and all of our drug experiments were on
isolated, non-branching neurites with active, filopodia
bearing growth cones attached to the dish. We observed 23
such neurites for 30 min and found an average rate of
elongation of 5.3 um/hr, supporting the premise that this
population is in an active. growth phase. The remaining
tension ‘measurements were on neurites that terminated on
other cells (23%), those that did not have visible
filopodial activity (9%) and branched neurite segments (1%).
Measurements from these neurites were indistinguishable from
the majority of tension measurements. The most highly

selected neurites were those which survived nocodazole

27

treatment. Most PC-12 neurites retract after treatment with
nocodazole (40,53) but this tendency can be decreased (4,51)
by allowing neurite outgrowth for periods longer than the 24
hours used previously (40,53). Approximately 30% of the
cells regrown for 2-3 days and subjected to reference
distensions remained extended long enough after nocodazole
treatment to allow an additional series of distensions.

Direct. mechanical measurements reported here confirm
the hypothesis that neurites are under tension (9,53). The
rest tension values were greater than zero in all but 1 of
82 growing neurites (figures 2 & 3). Clearly, the
maintenance of this tension requires structural support.
Our ability to measure a net tension in the highly
asymmetric neurite at its "rest" length indicates that there
is some tensile element in the neurite itself, and that this
tension is at least partially supported externally, i.e. by
compression of the dish. We are unable to interpret the
wide range of observed neurite rest tensions (figure 3) at
this time. However, experiments in progress should enable
us to determine whether rest tension is related to some
other variable such as neurite circumference, cortical actin
network cross sectional area (Albrecht-Buhler, pers. comm.),
total cross sectional area (9) and/or rate of elongation
(6).

Neurite rest tensions may be manifestations of "surface
forces" or "cortical tensions" observed by others in a

variety of cell types (3,22,44,73,80,81,95). Cortical

28

tension may be the passive result of an osmotic force, an
occasionally activated "motor" (15) or could be actively
generated continuously within the cortical actin network
(3,15). Alternatively, the neurite tension may be the
passive result of tension generated by the growth cone
(12,16,20,60,65,108). Although we do not know the source of
the tension, we assume for a number of reasons that the
measured tension is held passively. Previous results
indicate that combined inhibitors of glycolysis and
oxidative phosphorylation do not affect neurite retraction
in PC-12 (31). The majority of evidence in neurons supports
a pulling growth cone and we are unaware of studies
suggesting continuously generated cortical force in
neurites. Also, the assumption of continuously generated
tension requires a number of ad hoc assumptions, not needed
for passive tension models, to explain the tension changes
following drug treatments. Passive tension allows us to
interpret the effect of drugs by a simple force balance
between elastic cytoskeletal elements; actin filaments and
microtubules being measurably elastic in vitro (75,118).
Direct mechanical measurements demonstrate (figures 6 &
7) that cytochalasin D significantly reduces neurite rest
tension. This indicates that anti-actin drug-sensitive
elements are in tension. We presume the drug-sensitive
structure correlates with the highly crossed—linked, actin-
rich subplasmalemmal region of neurites (47,94). Addition

of cytochalasin D disrupts actin networks in vivo (23,92)

29

dissipating tension in the network. If microtubules (MTs)
are providing internal compressive support (53), their
depolymerization should transfer additional tension support
to the dish causing an increase in rest tension. Indeed,
our measurements before and after depolymerization of MTs by
nocodazole (29) demonstrate a significant increase in
neurite rest tension in the presence of nocodazole (figures
8 & 9), supporting our claim that MTs are in compression and
lending support to our previous claim (53) that nocodazole
causes neurite retraction by increasing tension. The
existence of an internal compressive support in the neurite
is also suggested by the two cases where neurite rest
tension fell below zero (and became a positive compression)
in the presence of cytochalasin D. Under conditions of
stationary neurite attachment to a stationary substrate, the
change in neurite force from net tension to net compression
indicates that some internal element is under compression.
Based on changes in cell shape caused by cytoskeletal drugs,
chick embryo fibroblasts grown in collagen matrix appear to
have a similar complementary force interaction within the
actin and microtubule elements of their cytoskeleton (106).
The effects of the anticytoskeletal drugs were
quantitatively inconsistent. The apparent tension ix: the
actin network (the tension measured in the presence of
nocodazole) was greater than the sum of the apparent
compression in the MTs (measured in the presence of

cytochalasin) and the compression of the dish (the rest

 

 

 

30

tension). The explanation we favor is that the surface
interactions between actin and MTs (47,94) provide a
stabilization, ie. a surface compression, whose magnitude is
the difference observed above. Such an effect was expected
and a formal model for the tension and compression
relationships within the cytoskeleton includes a term for
this force (17). An alternative explanation that we cannot
rule out at present is that depolymerizing MTs causes a net
increase in tension generated by the actin network.

The observed complementary force interaction between a
surrounding tensile network and an internal compressive
support, similar to the "tensegrity" architecture of
Buckminster Fuller (34), has also been proposed by Ingber
and others to explain cell shape (49,50). Tension in such
architectures has the counter-intuitive effect of
stabilizing elongated forms such as masts (34). Axons
resemble tensegrity structures (34) in their very large
ratio of length to cross-sectional area, their elasticity
and strength, and the finding that the putatively tensile
actin network is continuous beneath the plasma membrane
while the compressive microtubules are arrayed as discrete,

internal fascicles (47,94).

CHAPTER 2. Tension, Viscoelasticity and Growth

in PC-12 Neurites.

INTRODUCTION

The cytoskeleton is widely thought to provide the
mechanical basis for cell shape (2) in general and for
neuronal cell shape in particular (60). Neurons are good
systems for studying animal cell architecture because their
exaggerated anisotropy facilitates quantitative mechanical
measurement (30). In addition, the growth processes
underlying axonal elongation are thought to reflect general
dynamic processes determining cell shape (108).

External force has been shown to affect cell shape,
polarity, and motility. Specifically, the role of tension
in axonal elongation has long been recognized (112). More
recently, Bray has demonstrated that tension alone is
sufficient to cause axonal elongation (11). Under special
conditions, Campenot observed cell body migrations that
suggested both tension induced neurite growth and tension
induced retraction that appeared to be very much the reverse
of growth (20). For these reasons, we are interested in
further investigating the role of tension in the control of
neurite length.

Previous studies by our lab, using force calibrated
glass needles to make short duration (1-2 sec) distensions
on PC—12 neurites, indicated an Hookian elastic response to

31

32

tension (30). We also reported preliminary observations
that longer duration distensions elicited an additional
viscoelastic mechanical response. Clearly, to further our
understanding of the role of force in determining neurite
length, we need to characterize and distinguish these
passive mechanical responses from active growth and
retraction. Toward this goal, the present study will
quantitatively investigate neurite mechanical response to

long duration distensions.

33

MATERIALS AND METHODS

PC-12 cells were cultured as previously described
(30,53). Neurite rest tensions and spring constants were
measured by short duration distensions as described (30).
Using a modification of this method, the neurite was
subsequently exposed to a long duration distension (see
figure 10). Specifically, an initial rapid micro-
manipulation of the force needle orthogonal to the neurite
center was followed by periods of 25 to 85 minutes during
which the micromanipulator was undisturbed to observe
neurite response. A trial was ended by lifting the force-
needle and thereby releasing the neurite. From the video
record of this distension, neurite axial tensions and
distended lengths were determined as a function of time
using the trigonometric method previously described (30).
Measurements of orthogonal neurite distension (b) and
corresponding needle deflection from equilibrium (d) were
taken at intervals of 1-2 minutes. In some experiments, a
stock solution of taxol in DMSO (gift of Dr. Mathew
Suffness, Developmental Therapeutics Program of the National
Cancer Institute) was added to the culture media to achieve
a final concentration of 1 uM taxol and 0.1% DMSO. long
duration distensions were begun 5-10 minutes after addition

of the drug.

34

     

 

Force
needle \\ a
Lo=Za
a
rowth ,fl’”
gone /K
INITIRL STRETCH ELONGRTION RELERSE

Figure 10. Method of measurement of PC-12 neurite tension
during viscoelastic distension.

A force calibrated glass needle is initially placed at
the neurite center and then deflected in an orthogonal
direction causing an initial elastic stretch. Over time,
the neurite elongates further. An experiment is ended by
lifting the needle and releasing the neurite. After the
experiment, measurements are made from recorded video images
of neurite rest length (Lo), and at intervals of 1-2 minutes
of neurite lateral displacements (b1,b2...bn) and force
calibrated needle deflections (d ,d2...d ). These values
are used to determine neurite ax1al tenSIOns and lengths as
a function of time.

35

RESULTS

Viscoelastic Response to Neurite Distension

In 20 trials on 13 PC-12 cells, neurites were
continuously distended in a direction orthogonal to the
neurite axis for periods of 25-85 minutes (see figure 10 and
Materials and Methods). For eight trials initial applied
forces were 100 udynes or less; under these conditions we
observed. a 2-phase ‘viscoelastic lengthening response (see
figure 11). Initially, there was a rapid increase in
neurite length and tension (phase 1), followed by a slow
damped increase in length and decrease in tension to plateau
values (phase 2). Due to the dependence of force-needle
deflection on neurite length following the initial
micromanipulation of the needle, plots of neurite tension
over time are mirror the plots of neurite length (see figure
11).

Recovery from such distensions following release was
rapid; the neurite typically became visibly straight and
normal in appearance within 1 minute. In all 8 trials, a
series of short duration distensions were made 1 to 5
minutes before and 1 to 10 minutes after a long duration
distension to determine if it had an effect on the neurite’s
elastic properties. Following long duration distensions,
rest tensions averaged 108% and spring constants averaged
147% of initial values.

These data indicate that PC-12 neurites respond to low

36

 

120- ’-
110- ~
100- . ' -

.' phase 2
90 7° -

hose 1
80- }p -

70 .T....... .....,- .
0 10 20 30 4O 50

Neurite Length (um)

 

 

 

Time (mins)

150‘ l A I l . l

 

125~ ’-
100-
75~ - ’-

50- -

Neurite Tension (udynes)

25- -

 

 

O T ' l l I
0 10 20 30 40 50

 

Figure 11. Tension induced 2-phase neurite elongation.

Resulting neurite length (above) and tension (below) is
plotted as a function of time for an initial applied force
of 99.7 udynes. Typical for applied tensions of 100 udynes
or less, an initial elastic stretch (phase 1) is followed by
a damped viscous elongation to a plateau (phase 2).

 

 

 

 

37

force, long duration distension as visco-elastic solids. By
this we mean that the deformation respose of neurites to
applied force has both fluid-like and solid-like properties
and at long time scales ther is a final equilibrium between
force and deformation as in a solid. We have modelled this
behavior with a combination of 3 classical mechanical
elements (see figure 12): a relatively stiff spring (spring
constant k1) in series with a Voight element consisting of a
less stiff spring (k2) in parallel with a dashpot (viscosity
constant H). Values for these three mechanical constants
were determined analytically (see Appendix A). As shown in
figure 13, computer simulations of the expected behavior of
our proposed mechanical model (see Appendix B) using the
analytically determined spring and dashpot constants show a
close match between expected and observed behavior in
neurites of both PC-12 and chick sensory dorsal root
ganglia. A deviation from theoretical tension values
follows the first 10 minutes of distension in neurite 4; we
interpret this deviation as a third phase of elongation
(growth) as described below. It is impossible to fit all
the data for this neurite by manipulating the three
mechanical constants above. A more intuitive feel for the
contributuion of individual model elements is gained by
simulating the behavior of a family of hypothetical neurites
that vary only in one mechanical element; this is done in

figure 14.

 

 

 

 

 

 

 

line = I] secs 1 sec SI] mins

Figure 12. Viscoelastic model of neurite mechanical
response to long duration distansion.

We have modelled neurite viscoelastic behavior using
classical mechanical elements as shown above. A relatively
stiff "free" spring (spring constant k1) is in series with a
Voight element consisting of a second spring (k2) in
parallel with a dashpot (viscosity constant H). Pictured
above is the behavior of such a system following an
transient stretch which is then held stable. Initially, the
system is at equilibrium at low tension (left). Immediatly
following distension (center), tension rises and is held
completely by spring 1. Over time, the dashpot flows
allowing spring 2 to elongate and hold more of the applied
tension while spring 1 shortens as the total tension of the
system declines. Finally, a new equilibrium is reached
(right) at a longer length and at somewhat higher than
initial tension. Changes in tension held by the system can
be easily seen as relative changes in the length of
spring 1.

39

Neurite 1 Neurite 2

 

   
 

   
  

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

KOO ‘fifi
3 400- 73‘ 400-
C C
6 6‘
3 k1=1100dnes m
v 300. k1 = 49 udynes/um 3’ 300. k2 _ 2 y /U
E k2 = 3.9 udynes/um S - .3 udynes/um
VI '5 = .
E 200‘ H ___ 6500 udyne-sec/um E 200‘ H 12.500 udyne sec/um
3 u:
5 . t
2 100- 23 100‘
I" ‘ V v 1* n
O 10 20 30 4O 50 60 0 10 2'0 30 40 50 60
Time (mins) Time (mins)
Neurite 3 Neurite 4
1 in I I 7“" I I

”u? 125 < "‘
g I} 200.
>. E.
:3: 100. k1 = 12.9 udynes/um g H = 35 udynes/urn
c k2 = 0.66 udynes/um V 150. k2 = 1.7 udynes/um

_ C
g 75. H — 600 udynesec/um % H = 500 udynelsec/urn

C
Iv 0 100 -
T, so 7.
S 5
é’ 25‘ 2 5° ' ' ' ' ° - .
n V . V 1 n
o 10 2o 30 40 50 o 10 20 30 4'0 50 60
Time (mins) Time (mins)

Figure 13. Computer simulation of observed neurite
distensions using our viscoelastic model.

In each graph, individual points represent observed
neurite tensions during a distension experiment. Reported
values for spring constants k1 and k2 and dashpot viscosity
constant H are emperically determined from length and
tension changes over time as described in Appendix 1. The
solid lines are theoretical tension values over time from a
computer simulation of distension based on our viscoelastic
model (see Appendix 2) using the reported mechanical
constants. Plots for neurites 1 and 2 represent results
from chick DRG sensory neurons (distension data courtesy of
Phil Lamoureux). A two phase PC—12 distension is plotted
for neurite 3 and a three phase PC-12 distension is plotted
for neurite 4.

 

 

 

 

40

 

Effect of Variations in K1 EHect of Variations in K2

400-

 

300 -

200 ‘

Neurite Tension (udynes)

100‘

Neuriie Tension (udynes)

   

 

 

 

 

 

 

 

10 20 3'0 40 so 60 10 20 3‘0 4‘0 so 60
Time (mins) Time (mins)

 

Effect of Variations in H

H=100.000 (100K)

 

 

Neurite Tension (udynes)

 

 

 

o 10 20 so 40 5.0 60
Time (mins)

Figure 14. Computer simulation of hypothetical neurite
distensions.

Hypothetical neurite tensions are plotted as a function
of time where each plot shows the effect of varying the
three mechanical constants (spring constants k and k and
dashpot viscosity constant H) independently. in all hree
plots, the single dashed curve represents a "standard"
neurite with k = 9 udynes/um, k2 = 4 udynes/um and H = 6000
udyne'sec/um while the family of solid curves represent
variations in the single labeled constant.

41

Tension Induced Phase 3 "Growth"

In 9 of the total 20 trials on PC-12 neurons, neurites
were subjected to initial applied forces greater than 100
udynes and responded with an additional third fluid-like
lengthening phase. By this we mean phase 1 and 2 proceeded
as in the low force distension discussed above and was
followed by a third period of elongation and tension
decrease that either did not plateau during the experiment
or plateaued only after neurite axial tension fell below a
critical value of 50-100 udynes (see figure 15). In two
cases, a neurite was exposed to both a low and high initial
applied force; these are shown in figure 16. A clear
qualitative difference between the low force (<100 udynes),
2-phase response and the high force (>100 udynes), 3-phase
response is seen in both neurites.

Following release, neurites which had exhibited this 3
phase elongation typically lay flaccid and curvy on the dish
for several minutes; the result of experimentally applied
forces was an obvious elongation but no obvious thinning.
Within 10-15 minutes, these neurites shortened and evetually
tightened between the growth cone and cell body dish
attachments. In 8 trials, when a series of short duration
distensions were successfully made both before and after
recovery from a long duration distension, final rest
tensions averaged 42% and final spring constants averaged
84% of initial values.

For convenient analysis, we tentatively defined phase 3

42

 

 

 

 

 

 

 

 

160‘ J a * l 1 -
150,3 . . o . o 0 o o o o o o ’_
E 140-: , . ' . . ° 3
V : ...o phose3 I
.5 130‘: .0” L
O" ‘ .0 r
s - . =
_l 120:3 phase2 '.
.2 i! i
g 1104? :_
z 1 }phase1
100;» ;
90‘ . T . . , . . . . , . . . . , . . . . , . . . . '
O 10 20 30 4O 50
Time (mins)
300‘ . . . . 1 A . . . 1 l l l . l . . . . 1 i .
1,? 250? ’-
0 o .
C It >
>‘ 4: ’
3 200-1; L
s 3". '
.2 150: .0” T
a.) 2 ~00. '
l..- ‘ . O o i
3 100: 0 ‘ ° 0 . .
S : ° ° 0 o o o o 0 o . ’
o . .
z 50‘. . To = 28.4 udynes f
.r. ____________________ .
Oi I 1 ' I 1 I r
0 1O 20 3O 4O 50

Time (mins)

Figure 15. Tension induced 3-phase neurite elongation.
Resulting change in neurite length (above) and tension
(below) is plotted as a function of time for an initial
applied force of 233 udynes. Typical for applied tensions of
greater than 100 udynes, phase 1 and 2 elongation is
followed by a viscous-like third phase of elongation.

43

 

 

 

 

 

 

Neurite 1

120 P; A I . I A . . . 1 . . . . g ‘ . A A 1 . A . A ‘ A . A A P

L

a:

1 10 O O O o O O o O O L.

E 0 O 0 g 0 O 0 O O ’
3 100 o o o o ° ° ;
L- o initial applied force =- 195 udynes I
2’ 90 .° 3
0 O .
_J a ’
3 80 L
g 000.000..000000000...... :
Z 70 initial applied force - 52.1 udynes i

60 I T Y ' ' I I . I ' I ' 1 v f I v v v r 1fifi v v 1 v v v .
0 10 20 30 4O 50 60
Time (mins)
Neurite 2

150 PL . . A 1 . . l A l . l . . 1 . . . . 1 . . y . J . . J A ’

140 o o o o O o GE.

A o o o o o o o o o :
E 130 o ° ° ° 7
e . . :
120 0 ° initial applied force = 216 udynes -

c h
+1 O :
2‘ 110 o r
3 o E
3 100 0 r
.c t
3 90 0°° initial applied force =- 66 udynes ;
Z . a . a o . :
80 O. . . . . O . . . . . O . . . . . . O . O . . E

70 I ' V ' ‘ I j j ‘ ‘i I T f f V I ' ti V | Y Y V v 1 v v r 1
0 1O 20 3O 4O ' 50 50

Time (mins)

Figure 16. Low and high initially applied tensions on the
same neurite.

Plots of length change vs. time are shown above for two
neurites that underwent both a low (<100 udynes) and high
(>100 udynes) initial applied force.

44

elongation as beginning 10 minutes after the initial
application of experimental tension. Figure 17 plots the
resulting average rate of elongation during phase 3 against
the initial amount of applied force for all 20 long duration
distension trials. This plot shows a threshold value of
approximately 100 udynes below which there is no positive
phase 3 elongation rate, and a rough linear dependence of
this elongation rate on applied force above 100 udynes.
However in 3 of the total 20 trials on PC-12 neurons, the
neurites responded to initial applied forces above 100
udynes by exhibiting a two phase lengthening response and
rapid (< 1 minute) recovery typical of the response to

initial applied forces below 100 udynes (see figure 17).

Taxol Inhibition of Phase 3 Elongation

In 5 trials on 5 PC-12 neurons, neurites were exposed
to initial applied tensions above 100 udynes after the
addition of 1 uM taxol, a potent stimulator of microtubule
assembly (91). In these neurites, phase 3 elongation is
completely eliminated with little or no qualitative effect
on either phase 1 or 2 (see figure 18). Experiments
done in the presence of taxol exhibited fast recoveries
following release from a high initial applied force; the
neurite became visibly straight and appeared normal within 2
minutes of release of tension. In 3 out of 5 of these taxol
trials, we were able to perform a series of short duration

distensions both before and after the long duration

45

 

 

 

 

 

 

3; r
e a-
3 :
‘9 :
o _
a: .
C :
.9 T
*6 E
03 .—
C I
u—‘j A A:

__1Q:ll,. . ., . - .. , . . .. , .A¢\An ,. . . ., . .. . ’

O 100 200 300 600 700 800

hfifialAppHed Force (udynes)

Figure 17. Threshold tension for phase 3 neurite
elongation.

The average rate of elongation following the first ten
minutes of distension (i.e. phase 3) is plotted against
initial applied force. A least squares fit regression was
done for all trials with initial applied forces between 100
and 300 udynes (filled circles) that gives a correlation of
0.871. All trials with initial applied force of 100 udynes
or less (open circles) have near zero elongation rates. For
unknown reasons, three very high initial applied tension
trials (open triangles) exhibited non-typical behavior.

46

 

 

 

 

 

  
 

 

 

 

 

130 ‘ ' ' ‘ ‘ ' 500
1i i A
A 120 - ' 50° 8
E D : 5.
£1 - u
110 a tension ' 400 3,
5 '0 e . e . e e e e e I
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0 ° °°°°o° ° °°Oooooo°°oo°O° €
2 90‘ I th ' 3
eng :100 z
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C 20 - -
.9
g. 10 - -
C
.9
Lu 0 e e '
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-10 v . I . I
O 100 200 300 400

lnitiol Applied Force (udynes)

Figure 18. Taxol inhibition of phase 3 elongation.

Upper plot: resulting change in neurite length and
tension in the presence of 1 uM Taxol is plotted as a
function of time for a single trial with an initial applied
force of 287 udynes. Lower plot: The average rate of
elongation in the presence of 1 uM Taxol following the first
ten minutes of distension (i.e. phase 3) is plotted against
initial applied force for all 5 trials.

47

distension. In these cases, final rest tensions averaged
176% and final spring constants averaged 160% of initial
values. These behaviors are distict from five separate
trials on five PC-12 neurons exposed to a series of short
duration neurite distensions in the presence of 1 uM Taxol.
These experiments showed little or no effect on neurite
tension or spring constants; rest tensions averaged 124% and

spring constants averaged 95% of pre-treatment values.

48

DISCUSSION

We have found that PC-12 neurites behave as
viscoelastic solids in response to long duration (25-85
minute) distensions at a low continuous applied force
(figure 11). Specifically, these neurites respond with an
initial elastic stretch (phase 1) followed by a damped
viscous-like elongation that plateaus (phase 2). Recovery
from these biphasic elongations is rapid, returning to
original linear morphology within a minute following
release. Rest tensions and spring constants (measureable in
1-10 minutes) also recover to near pre-distension values.
Studies on sea urchin eggs (44,46,73) and lymphocytes (80)
also indicate viscoelastic solid behavior when mechanical
measurements are made on the intact cell.

We interpret the neurite’s biphasic distension as a
passive mechanical response to force and have modeled
neurite viscoelastic behavior with a combination of three
classical mechanical elements: two springs and a dashpot
(figure 12). The free spring (k1) is able to respond
instantaneously to an applied load by lengthening in
proportion to the load. The same elastic element appears
responsible for both the spring constant measured in short
duration distensions (30) and for the initial rapid
elongation (phase 1) in long duration distensions. It
appears that there is a Voight element in series with this

free spring, ie. a second spring in parallel with a dashpot.

49

This Voight element also responds to force like a spring
with a load proportional elongation, but does so in a
delayed (damped) fashion due to the viscous drag of the
dashpot. This pair of elements models the damped approach
to plateau length and tension (phase 2) seen in long
duration distensions. The validity of this mechanical model
is strongly supported by the close match between
experimentally observed responses to long duration
distensions and computer simulations using analytically
determined values for spring and dashpot constants for a
particular neurite (figure 13 and Appendices A and B).
Earlier work with cytochalasin D (30) points to a strong
association of the polymerized actin network with the free
spring (k1). Unfortunately, most drug studies are too crude
in effect to be of much value. Future studies in our lab
are aimed at developing more specific perturbations of
cytoskeletal elements to better define functional roles of
individual structures in the cell for each of these
measurable mechanical constants.

PC-12 neurites exposed. to initial applied forces of
greater than 100 udynes showed an additional third phase of
elongation at a constant rate (figure 15) which. we will
interpret as "towed" neurite growth similar to that seen by
Bray in response to experimentally applied tension (11).
This interpretation is supported by the observation that
neurite lengths increased up to 218% of initial length

during distension, with no observable changes in diameter

 

 

 

50

and no obvious structural damage. These neurites recover
their original morphology after release, but rather slower
than neurites exhibiting a two phase elongation response.
This putative towed growth appears to occur above a tension
set point of approximately 100 udynes. Above this tension
load, towed growth occurred in 8 out of 12 neurites with a
roughly linear dependence of growth rate on applied force
(figure 17). Below this level, all 8 neurites showed near
zero elongation rates. The existence of a set point is
further supported by the different elongation behavior of
two neurites that experienced initial applied forces below
and above 100 udynes (figure 16). We investigated the role
of microtubule assembly in phase 3 elongation by observing
the effects of 1 uM taxol, a known stimulator of microtubule
assembly (91). In long duration distensions with an initial
applied force above 100 udynes, taxol treatment showed no
effect on phase 1 or 2 elongation, but the phase 3
elongation rate was strongly inhibited. This result
suggests that tension induced (phase 3) growth is dependent
upon microtubule assembly. Taxol may be disrupting the
normal microtubule assembly necessary for neurite growth in
a manner similar to the observed taxol inhibition of normal
neurite growth, as reported by Letourneau and Ressler (64).
This could be the result of tubulin dimer depletion, or of a
structural interference in the growth process caused by the
addition of the "taxol microtubules".

Others in our lab have shown that the neurites of chick

51

sensory dorsal root ganglion neurons also respond
viscoelastically' to externally' applied force (figure 13).
In addition, they are able to actively generate tension in
response to decreases in applied force (41). When these
neurites are slackened, they respond with a biphasic
shortening and tension recovery that looks like the inverse
of the 2-phase length and tension response to distension
discussed above for PC-12. Significantly, in greater than
60% of these experimental trials, tension recovery exceeded
(and sometimes even doubled) initial pre—slackened values.
This behavior clearly suggests active tension generation by
the neurite shaft. A historically interesting parallel here
is that some 60 years ago muscle was still being described
as a "viscous-elastic system" capable of exerting force by
shortening (66).

The story emerging from our data is that neurite (or
axon) length is sensitive to a continuum of axial tension.
At tensions above a particular value, the neurite grows in
length; at intermediate tensions it responds as a
viscoelastic solid; and when neurite tension falls below a
particular value the neurite shaft is stimulated to generate
tension and retract. These results suggest. a 3-position
controller for neurite length respose to changes in axial
tension (figure 19). The postulated 3-position controller
is distinct from one where the neurite may be responding to
changes in axial tension with a purely proportional

response. A proportional response occurs when there are no

 

 

52

     
 
    

 
 

'FORWQRD'
Neurite Passive
, visco- 3 INEUTRQL'
TenSlon elastic

 

  

 

 

\ .
active
ten51on. IREUERSEI
generatlon

\/

Figure 19. 3-position controller model for tension
dependence of neurite length.

The figure at left summarizes the three observed
tension-dependent neurite behaviors and the cartoons on the
right depict a hypothetical 3-position controller analogous
to a ’Frankenstein-type’ switch that would be used to
control an electric motor. This switch represents a
tension-sensitive controller that is able to initiate either
neurite assembly (towed growth) above an upper tension
threshold or neurite contraction (active tension generation)
and/or disassembly below a lower tension threshold. At
intermediate tensions, the switch is not activated and the
neurite behaves passively as a viscoelastic solid.

 

 

 

53

set points, just a continuum of response from shortening to
lengthening as tension increases. It is possible that
growth is proportionately controlled and that PC-12 neurites
that have zero growth rates in response to applied tension
(figure 17) were in a refractory mode and insensitive to
force stimuli. This seems unlikely because it would imply
that neurites at their "natural" rest tension (zero applied
force) are all either in the process of shortening or are
refractory. To the contrary (and as logic dictates), we
observed undisturbed PC-12 neurites in culture to have a
positive elongation rate averaging 5.3 um/hr (30).

The tension dependence of neurite length control is
significant for several reasons. Tension has been long
inferred to be responsible for the passive axonal elongation
during growth to adult body size that must follow embryonic
synapse formation (112) . Bray elegantly demonstrated that
tension is sufficient for axonal growth (11). In addition,
recent work by our lab indicates that the growth cone is
indeed a pulling motor (59) . Together these results
suggest that tension may be the cause of axonal elongation
generally. Mitchison and Kirschner (74) have pointed out
that tension is an attractive "second messenger" for growth
control based on a priori arguments. This behavior is also
consistent with our thermodynamic model for the regulation
of microtubule assembly by force (59). In this model, the
advancing growth cone transfers tension support

(compression) from the internal microtubule core onto the

 

 

 

54

underlying substrate. This results in both a net increase in
neurite tension and a decrease in microtubule compression,
decreasing the critical concentration of tubulin dimer for
assembly. Assuming equilibrium conditions prior to growth
cone advance, net microtubule assembly and neurite
elongation occur until equilibrium is restored. Similarly,
a loss of growth cone adhesion would increase microtubule
compression and cause disassembly and neurite shortening.
We have observed such a compression induced disassembly of
microtubules in PC-12 cells by releasing neurite adhesion
with surface active agents and measuring changes in the
solublezpolymerized tubulin ratio with quantitative

immunoblotting (30).

 

 

 

 

CHAPTER 3. F-Actin and Microtubule Suspensions as

Indeterminate Fluids.

INTRODUCTION

Actin and Microtubules are among the most widely
conserved structural proteins within eucaryotic cells. It
has been long suspected that the viscoelastic properties of
these polymers (particularly F-actin) play an important role
in the maintenance and development of cell shape and in
cellular motility (82,102,105). Mauryama et al. (71) found
that the measured viscosity of F—actin was precisely
inversely proportional to the shear rate. These workers
reported on the uniqueness of this inverse proportionality
and speculated that it may be the result of the formation of
an actin network. Subsequent rheological studies confirmed
that F-actin suspensions are shear thinning but did not find
the same shear—viscosity dependence or did not discuss this
relationship (87,88,120). We confirm ‘the observation of
Maruyama et al. for actin and observe a similar inverse
proportionality for suspensions of microtubules. Also, we
show that as a result of this dependence, the fluid flow
profiles for such suspensions are indeterminate. That is, a
given force does not fix the velocity of movement in these

fluids.

55

56

MATERIALS AND METHODS

Actin was purified from rabbit skeletal muscle by the
method of Spudich and Watt, or by their method and further
purification by chromatography on Sephacryl S-200
(Pharmacia, Piscataway, NJ) (68,101). Purified actin was
maintained in dialysis at 4°C against buffer A [2.0mM tris-
HCl, pH8.0, 0.2 mM ATP (adenosine triphosphate), 0.5 mM
mercaptoethanol, and 0.2 mM CaClz] for no more than 6 days.
Microtubule protein was purified from beef brain in the
presence of glycerol according to the method of Shelanski et
al. (97) with the modification that 0.1 M PIPES (piperazine
ethanesulfonic acid), pH 6.6, was used in all reassembly
buffers. A model R.18 modified (72) Weisenberg cone and
plate rheogoniometer was used for all rheometry experiments.
Actin was polymerized by adding KCl to a final concentration
of 50mM and MgSO4 to a final concentration of 2 mM to G
actin at various concentrations in buffer A. The mixture
was applied quickly to the bottom cone (2°, 5 cm in
diameter) of the rheogoniometer, the upper plate was
immediately lowered to the correct height above the cone,
and the actin was polymerized for 1 hour at 23°C before
measurements were made. Microtubule protein suspended in
0.1 M PIPES, pH 6.9, 1 mM Mgc12, 1 mM EGTA, 0.1 mM EDTA, and
0.514 GTP (guanosine triphosphate) was polymerized at 37°C
for 20 minutes and applied to the cone of the

rheogoniometer. A humidified atmosphere was maintained

 

57

around the cone and plate assembly by adding a small pool of
water to the floor of the instrument’s temperature control
chamber. For each sample of protein loaded into the
rheometer, 10 to 14 shear rates (drive settings) were chosen
to obtain torque readings. After each torque reading the
drive was turned off and the torque allowed to return to
baseline before we reset the drive for the next reading.
Measurements on a single protein sample took approximately 2
hours. The rheogoniometer was calibrated with a Newtonian
viscosity standard (R12000, Cannon Instrument Company).
Values for viscosity and shear rate were calculated from
rheogoniometer drive settings and instrument readings by
standard formulas. Polarization micrographs were taken with

a Leitz Ortholux I Pol microscope, at 1200x.

58

E U T D ISCUSS 0

Figure 20 is a Logarithmic plot of viscosity versus
shear rate for two runs of actin at different
concentrations, two runs of microtubules with and without
taxol [a stimulator of microtubule assembly (91)], and one
run of the viscosity standard. The values reported in Fig.
20 and throughout this report are the stable values obtained
after shearing the fluids for several seconds. As found
previously for actin (71) in both actin and microtubule
suspensions, the torque rises sharpLy as shear begins but
then declines to a stable value over a time period that
depends on the shear rate. In addition to the shear-
thinning behavior characteristic of all polymer suspensions,
these plots show that the power law dependency for actin and
microtubule suspensions is nearly -1 for all concentrations.
That is, actin and microtubule suspensions behave as power

law fluids with

v = A(ds/dt)n (Eq. 1)

where V = viscosity, A is a constant, ds/dt is shear rate,
and n is the power law exponent. In 12 runs of actin at
concentrations from 2 to 6 mg/ml the exponent varied from -
0.85 +/-0.02 (+/- standard error of the regression
coefficient) to -1.15 +/- 0.01 with a a mean of -1.00. Both

extreme values occurred for actin at 2 mg/ml, a

59

 

10 000 1 l l l ljlll l l l I 1.111 1 I J l 1111' 1 l J 1111]
1 I
l ‘ I.
C -
n

1,000 -:

  
 

 

 

 

 

”a? '
.9 D {3 -—a _
8- Viscosity Standard _
V
>~ 100'? 3
:fl : :
U) .
O
O
.9 .
> MT/SuM Taxol
10‘ MT-12 mg/ml _
‘ FA -6 mg/ml 3
FA —2 mg/ml *
0.001 0.01 0.1 1 10

Shear Rate (sec—1 )

Figure 20. Viscosity as a function of shear rate for F-
actin and microtuble suspensions.

Viscosity measurements at various shear rates on
suspensions of cytoskeletel proteins and a viscosity
standard at 23°C are shown on a log-lag plot. (0) F-actin,
6 mg/ml, slope = -1.00 +/- 0.02; ((3) chromatographed F-
actin, 2 mg/ml, slope = -0.98 +/- 0.02; ([5) polymerized
microtubule protein, 12 mg/ml, slope = -1.00+/- 0.01;

(A) microtubule protein 12 mg/ml, polymerized in the
presence of 5 uM taxol, slope = -o.9s +/- 0.3; ([3) R12000
viscosity standard, slope = 0.00 +/- 0.01.

60

concentration that produces a torque on the rheogoniometer
at its lower limit of resolution. In seven microtubule runs
the exponent varied from -0.90 +/- 0.01 to -1.03 +/- 0.03
with a mean of -0.94. The -1 slope (on log-log plots) was
found to be nearly independent of the shearing history of
the sample, as shown in Fig. 21. The slope of the line for
increasing shear (-1.00 +/- 0.02) is nearly identical to
that for decreasing shear (-1.01 +/- 0.02). In a run of
chromatographed actin (3 mg/ml) in which the shear rate was
varied at random the power law dependence was -0.99 +/-
0.09. The power law dependence of the microtubule
suspensions viscosity was similarly insensitive to shear
history. Our measurements varied only slightly. The small
standard errors of regression coefficients indicate little
variation of points around the calculated line. The slope
from run to run also varied little at nominally the same
conditions. In five runs of actin at 6 mg/ml from three
different actin preparations the mean slope was 1.01 +/-
0.04 (SEM). The possibility that our results are due to
slipping at rheometer surfaces is eliminated by the finding
that actin and microtubule suspensions pour easily and are
well mixed after the shearing process in the rheometer. The
observed viscosities for suspensions of these two
filamentous constituents of the cytoplasm are consistent
with previous in vivo measurements of cytoplasmic viscosity.
Figure 22 shows a comparison of data obtained from various

cytoplasmic studies with those obtained from an F-actin

61

 

 

 

 

 

10.000 : 1 1 1 111111 1 1 1 111111 1 1 1 11111.-

: e . I

. F—Actln 3 mg/ml .

-I r

1,000 3 :—

/~\ I :

0) ~ .

.9 -4 i-

o . .

o. - .
v

>N 10C>1 ;

.4: a :

m . .

C) ~ _

0 < L
:>

101 :'

3 E

1 I I I rIIIII I I I IIIII‘F f I I ITTTI

0.001 0.01 0.1 i

Shear Rate (sec--1 )

Figure 21. Effect of shear history on viscosity of F-actin.

Two shear protocols for chromatographed F-actin, 3
mg/ml, from low to high rates of shear (Q), slope a: -1.00
+/- 0.02 and from high to low rates of shear (CD), slope =
-1.01 +/- 0.02.

 

62

 

l l Ill]... j
100,000 4 ‘ ”“nl ’ 1 ‘1‘“" I 1111“.
d
I
‘
‘

10.000 [\3

III

1001:

Viscosity (poise)

O
1

I 11111!

_.I
1...]

 

 

 

0,1...m,m-...,........,
0.001 0.01 0.1 1 10

 

Shear Rate (sec.1 )

Figure 22. Viscosity of F-actin compared with viscosity
measurements of cytoplasm by various in-vivo methods.

(0) F-actin, 3 mg/ml; (O) pulmonary macrophages,
Valberg and Albertini (109); (0) particles in lungs, Nemoto
(77); (A) human neutrophils, Sung et al. (103): (A) sea
urchin egg, Hiramoto (45): (I) airway mucus, King and
Macklem (56); (:1) physarium, Sata et al. (90); (V) chick
fibroblasts, Crick and Hughs (26); (v) amoeba, Yagi (116).
The linear regression for the in-vivo data (---) gives a
slope of -1.02 +/- 0.18.

63

suspension at 3 mg/ml. This similarity of viscosity
function in vitro and in vivo is particularly interesting in
view of the lack of true cross-links within purified F-actin
(119) and the highly cross-linked nature of the cytoplasm
(43,93,115). Power law observations in this range are
significant because they contradict Graessley’s classic
random-coil-entanglement explanation that suggests the
minimum n value is -9/11 (37,38). Typically, data show n
between -0.4 and -0.85 for polymer solutions (7).
Significantly, since shear stress is viscosity multiplied by

shear rate

'r = V(ds/dt) = A(ds/dt)n(ds/dt) = A(ds/dt) (n+1) (Eq. 2)

where 1? is the shear stress (force per unit area). Our
observation that n = -1 means that shear stress is
independent of shear rate. In this case, fluid motion is
completely indeterminate. In Fig. 23 we plot shear stress
data for actin, microtubules, and the Newtonian fluid
standard. One’s intuition, for example, on the basis of
flooded rivers, is that increased shear rate increases shear
stress. As shown in Fig. 23, the Newtonian standard follows
the intuitive expectation. However, for both actin and
microtubules the shear stress remains constant at all shear
rates tested. This constant is the value A in equations 1
and 2.

In these experiments we controlled the shear rate and

64

 

     
 

 

 

30 +4 1 1 1 4 1 14
i
A '1
N
E
Q -
g 20 - Viscosity Standard _-
>‘ u
.0
v .,
(n . .
m ‘ h
0) < ,.
L. ., p
a 10‘ O F—Actin 3 mg/ml f
5 i A MT 6 mg/ml
a) .
.C
(f)
O . . . sis.“ . . r . l I... I-

 

 

0.001 0.010 0.100 1.000
Shear Rate (sec-1 )

Figure 23. Shear stress as a function of shear rate for
cytoskeletal proteins and the viscosity standard.

(0) F-actin, 3 mg/ml; (A) polymerized tubulin,
6 mg/ml; (E3) R12000 viscosity standard.

 

65

measured the force. In most biological (and other)
situations, however, the interaction is reversed: a pump, a
gravitational potential, or' a cell supplies a particular
shear stress, and the flow pattern accommodates it.
However, as shown in Fig. 23, a force alone cannot fix the
flow patterns for actin and microtubule (that is,
indeterminate) suspensions. For any geometry, only one
force pattern gives reasonable flow rates; push with less
force and there is no motion, push with more force and the
fluid velocity is limited by inertia alone.

Actin and microtubules are widely regarded as important
in determining the physical properties of cytoplasm
(82,102,105). Energy-dependent movement of particles through
the cytoplasm occurs in all eucaryotic cells, and cytoplasm
itself is capable of rapid, energy-dependent. movement in
some cells (54,86). Our data suggest that the velocity of
movement is not limited by force, and the low Reynolds
number characteristic of living systems suggests that
inertia is not responsible. By deduction then, the observed
velocities are fixed by the chemical kinetics of the
biological motors. For example, structures within cells
move suddenly over moderate ( ~30 um) distances at constant
velocity, and they halt equally suddenly (86). One
explanation is that no motion is occurs until a threshold
force is reached, then a constant velocity is maintained by
the stable kinetics of the mechanochemical motor.

The unusual field dynamics of nearly indeterminate

66

fluids 'may also explain the flow' properties that
characterize cytoplasmic streaming in various cells (54).
In these cells, shearing flows are contained by cell
membrane-cortex structures or, in the case of algae, by
cytoplasmic walls. The flowing cytoplasm and the containing
cytoplasm are fundamentally similar in their content of
water and proteins. Flow indeterminacy may explain how
these two regions of similar chemistry have such different
velocities while bearing the same stresses. Possibly, the
cell pushes the cytoplasm at constant force in some part of
the cell and drags on the cytoplasm in the rest of the cell
(necessarily with the same force per unit area). If the
area pushing is smaller than the area dragging, there will
be no cytoplasmic flow. If it is larger, flow is rapid.
Thus for finite flows and equal shears on the cytoplasm and
membrane, small differences in chemistry can produce
enormous differences in flow.

This viscosity relation may also play a role in slow
axonal transport, which involves a flow’ of‘ cytoskeletal
elements down the neural axon at a variety of different
rates characteristic of the different cytoskeletal elements
(48,113). If the fluid flow is nearly indeterminate and the
axonal pressure drop will greatly affect their transport
rate down the axon.

Another possibility from shear-thinning behavior that
does not require flow indeterminacy concerns organisms that

locomote upon a slime trail. In most instances this slime

67

is a polymer solution that would be shear thinning; most
crawling results from sustained oscillations of contraction
and expansion. Two models are possible: (i) an asymmetric
oscillation in space so that the moving area is always
smaller’ than the stationary area. or (ii) an asymmetric
oscillation in time both with the condition that the drag on
the front of the organism over the substrate is determined
by Newtonian viscosity (in water, for example) and the drag
on the rear of the creature is determined by shear thinning
slime. Then, as in Fig. 20, if the cell extends faster than
it contracts, a net force will move the organism forward.
Snail slime possesses "solid-like" properties at low shear
rates (31) in concert with a shear-thinning fluid model.

The rheology of actin suspensions has generally been
thought to be the result of network destruction by shearing
(69,71,119). Although microtubule suspensions do not form
gels or the looping entanglements required by the network
theories (37,38,87,88), they have the rheological properties
of actin suspensions. Further, the limiting power law
exponent for entangled polymers is calculated to be -9/11
(37,38). Our data (Fig. 21) also indicate that shearing
history has a relatively small effect on the viscosity of
actin and microtubule suspensions. These observations
suggest that network destruction is not primarily
responsible for the rheology of these filamentous proteins.
We have formulated two alternate models for these viscosity

observations.

68

One model follows a suggestion of Zanner and Stossel
(120) for actin suspensions and assumes that semidilute
suspensions of rigid rods are at concentrations below the
liquid crystal formation (that is, an isotropic distribution
of the rods in space before shearing). In this model, as
treated Doi and Edwards (32) and by Jain and Cohen (52), a
large obstructional effect is caused by the constraint that
rods cannot pass through each other. Assuming that the
frequency of rod-rod obstructions is due to shear
independent Brownian motion, Doi and Edwards (32) found
regions where shear stress is independent of shear rate. To
the extent that this model is appropriate, the phenomenon
that is typically called "gelation" (true solidification) in
the biological literature is seen as the "glass" (liquid
with solid properties) transition induced by this
obstructional constraint. This glass transition explains
why the viscosity of cytoplasm and actin suspensions is so
similar, that is , why crosslinkers in cytoplasm have so
little effect on viscosity, as shown in Fig. 22. It also
explains why actin "gels" behave as liquids. This "log jam
model" can predict the observed -1 power law dependence of
the viscosity and also that the power can be smaller than -1
over small ranges of shear.

We propose a second model to explain the observations
with relatively flexible actin filaments and to extend the
applicable concentrations above those for liquid crystal

formation. This model differs from the network model and

69

the rigid rod model in that the fluid is locally
anisotropic. Fluid motion is not continuous but occurs
among' discrete domains in, a iclassical analogy' to solid
surface friction or superfluidity. Anderson (5) has
proposed this model to describe large-scale density waves in
smectic B liquid crystals. If one domain slides over
another, two roughnesses encountering one another must
either be completely overwhelmed by the forces driving them
or' must stop the ‘motion entirely; A. roughness cannot
provide a continuous source of linear dissipation because
any roughness that causes a purely elastic deformation does
not dissipate energy. Instead, it returns as much
mechanical energy to the lattice as was present in the
original deformation. Viscous stress is thus viewed as the
(shear rate dependent) force necessary to substantially kink
or break those protein strands that extend from one solid
domain to another. The effect that biologists call gelation
is, presumably liquid crystal domain formation. This solid
friction analogy is supported by the finding of ourselves
and others (71) that the initial shear stress (analogous to
static friction of solids) is always greater than the steady
state. stress (analogous 'to sliding friction of solids).
Additional evidence for this model is shown in Fig. 24 ill
which actin and microtubule protein suspensions reveal
distinct domains under polarized light in contradiction

tothe assumptions of the rigid rod model described above.

70

F-ACTIN

 

MrcnorueLes

Figure 24. Polarization micrographs of suspensions of actin
and microtubules.

The upper left micrograph is of F-Actin at 25 mg/ml.
(Similar domains were seen at concentrations of 8 mg/ml but
the birefringence was to dim to photograph successfully).
The lower right micrograph is of microtubules at a
concentration of 6 mg/ml.

APPENDICES

 

Appendix A. Method for analytical determination of neurite
mechanical constants from observed distension

behavior.

.4
ll

Tension held by neurite, To = non-distended neurite rest tension

[—
ll

Neurite length, L0 = non distended neurite rest length

Short duratien elastie extensions;

** k1 = slope of Tension I; Length plot **
Lon dura i n V E x n i n ili ri m°

K(nrt) = (Teq-To)/(Leq-Lo)

1/K(nrt) = 1/k1 + 1/k2

** k2 = 1/( l/Knrt - 1/k1 ) **
Duin V E i n i n°

portion of elongation due to spring 1 is Axl

by Hook's law T-To = k1 x Axl , therefore Axl = (T-To)/k1

(spring 1 is "free")

portion of elongation due to Voight element is AxZ

AxZ = (L-Lo) - Axl

T(spring 2) = k2 x sz

AxZ' = time rate of change of x2 (Voight element length)

= [Ax2(time2) - Ax2(time1)] / (timeZ - timel)

H = dashpot constant

T-To = k1 x Ax1= (k2 x AxZ) + (H x AxZ')

** H = [(T-To) - (k2 x Ax2)] / sz' **

H is calculated for every time interval (2 mins) and averaged
To, k1, k2, Knrt and H are constants for a paticular neurite

Axl, sz, AxZ' and T are all varying during an applied distension

71

Appendix B. Logic underlying the computer simulation based
on our viscoelastic model for neurite

distension.

The computer simulation is written in the APL
programming language (STSC, Inc., Rockville, MD). String
variables were created for the following parameters: T
(neurite tension), L (neurite length), F (needle force), B
(height of distension triangle, equivalent to distance "b"
in figure 10), x1 (length increase of spring 1 relative to
its length at zero distension), x2 (relative length of Voigt
element) and t (simulation time). Individual string values
represent that parameter’s value as calculated over
successive intervals in "simulated time". Constants were
defined for the following parameters: Lr (neurite "rest"
length: length at zero distension), Tr (neurite "rest" axial
tension), Fb (beginning applied force held by the needle),
CAL (needle calibration constant), Tb (beginning neurite
axial tension), K1 (spring 1 elastic modulus), K2 (spring 2
elastic modulus), E (dashpot viscosity constant), dt (time
interval between calculated values in "simulated time"), and
N (number of time intervals calculated). Constants were
given new values prior to each simulation run based on
values from a particular long duration distension experiment

The computer simulation uses a numerical approximation
method that requires small values for dt and large values

for N to achieve satisfactory results. I typically used a

72

|f

 

73

value of 3 (secs) for dt which required a value of 600 for N
to give a net simulation time of 30 minutes. Resulting
simulation values were sequentially stored in the string
variables. Every tenth value (time interval of 30 seconds)
of the neurite tension variable (T) was plotted on the y-
axis against every tenth value of the time variable (t) and
connected with a smooth curve. This plot was combined with
the actual tension-over-time data for the particular neurite
distension to give each of the four graphs in figure 13.

The following paragraph states our viscoelastic model
in mathematical terminology. A free spring (K1) and a
Voight element {a spring (K2) in parallel with a dashpot
(3)} are in series and therefore each continuously hold the
entire neurite tension. A change in neurite axial tension
(T) equals the product of the change in length of spring 1
(KI) and its elastic modulus (K1). Also, a change in
neurite axial tension equals the sum of a) the product of
the change in length of spring 2 (x2) and its elastic
modulus (K2) and b) the product of the dashpot viscosity
constant (B) and the time rate of change of dashpot length
(x2) .

This same relationship is expressed symbolically below.
The string variable for tension is a series: T[1], T[2],
T[3], ... T[N]. T[l] is neurite tension at 0 seconds
(immediately following distension), T[2] is neurite tension
at 3 seconds, T[3] is neurite tension at 6 seconds, etc.

The same is true for the other string variables. The change

 

74

in neurite tension between arbitrary (i)th and (j)th values
of T is represented as T[j]-T[i]. I can therefore represent
the word description in the preceeding paragraph by the

following equations:

T[j]-T[i] = (x1)(x1[j]-x1[i]) Equation 1

T[jJ’TIi] = (K2)(32[j]'32[i]) + 3(32[jJ'XZTiJ/(tljl't[i]))
Equation 2

--------------------- IN REVIEW ---------------—--------

Constants

Lr = neurite rest length (um)
Tr = neurite rest tension (udynes)

K1 = elastic modulus of "free" spring 1 (udynes/um)
K2 = elastic modulus of Voight spring 2 (udynes/um)
E = dashpot viscosity constant (udyne)(sec)/(um)

CAL = needle calibration constant (udynes/um)

dt = simulated time increment (secs)

2
ll

number of computer iterations

Initial Conditions
Fb = initial applied lateral needle force (udynes)

Tb

initial resulting neurite axial tension (udynes)

75

Strtng Variables

B[i] = resulting lateral displacement distances
(height of distension triangle in um)
T[i] = resulting neurite tensions over time (udynes)

L[i] = resulting neurite lengths over time (um)

Kl[i] resulting spring 1 lengths over time

(relative to its "rest" length in um))

x2[i] = resulting Voight element lengths over time
(relative to its "rest" length in um)

t[i] = simulated time values in multiples of at (secs)

-------------------- PROGRAM LOGIC ---------------—------

The following assumptions are made for all
instagtaneeus changes in neurite tension (including the
initial distension):

a) only spring 1 length (x1) changes.

b) spring 2 and the dashpot length (82) do not respond
instantaneously. Their response requires some
finite amount of time (dt) to pass.

Program Steps
1) set all string variable values to zero, the iteration
counter (i) to 1, and all constants to their
respective values.

2) set F[1] = Pb and T[1] = Tb.

 

76

3) let spring 1 length (x1) increase in response to initial
increase in neurite tension.
x1[i] = (Tb-Tr)/K1 (from equation 1 above)
4) let neurite length increase by this amount
L[i] = Lr + x1[1]
START OF LOOP:
5) let iteration counter increase by 1
i = 1+1
6) let some small finite amount of time (at) pass
t[i] = t[i-l] + dt
(hold neurite tension and therefore spring 1 length
constant for now)
7) let Voight element length (x2) increase

. (T[i'll'T[i]) (at) + (H) (lei‘11)
x2[1] =

 

(K2)(dt) + 3
(this is derived by solving equation 2 above for x2)
8) let neurite length (L) increase by this amount
L[i] = Lr + x1[i-1] + x2[i]
9) as the neurite elongates, let triangle height (B)
increase...
B[i] = (0.5)(L[i]2 - Lr2)°°5
(this is derived by Pathagorean thereom)
10) ... and needle force (F) diminish
F[i] = F[i-1] - (CAL)(B[i]-B[i-1])
11) as a result, let neurite tension (T) fall
T[1] = (F[i])(L[i])/4(B[i])

(this equation was derived in Chapter 1)

 

 

12) let spring 1 length (:1) decrease to in response to

77

lower neurite tension

13) let neurite length (L) decrease by this amount (a small

21(1) = (T[i]-Tr)/K1

value compared to the increase in step 8).

14) using new values for neurite length (L) and tension (T),
return to step 5) and do this loop until total elapsed
time is sufficient to reach plateau values for L and

T.

[O]
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]

L[i] = L: + x1[i] + x2[i]

This effectively means to adjust the value of N.

L P 0 ram C de

TEN4

1+1

T[116Tb

X1[1]+(T[l]-Tr)+Kl

XZEIJeO

L[l]+Lr+X1[lJ

Ftller
BtlJ+O.SX((L[1]*2)-Lr*2)*0.5

STARTziei+1

X2Ei]6(((T[i-ll-Tr)th)+HXX2[i-l])+H+K2th
L[i]+Lr+Xl[i-1]+X2[i]
BEiJGO.SX((L[i]*2)-Lr*2)*0.5
FtileFti-lJ-CALx(B[iJ-B[i-l])
T[i]+(F[iJXL[i])+(4XB[i])
X1[i]+(T[i]-Tr)+Kl

L[ileLr+X1[i]+x2[iJ

+(i<N)pSTART

(from equation 1 above)

"V

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