WNW

l

i

5

$43 W

 

.THS

 

 

LIBRARY
Michigan State
University

 

 

 

This is to certify that the
dissertation entitled

KINESIN-MICROTUBULE INTERACTIONS: TRANSPORT
AND SPINDLE FORMATION

presented by

Zhiyuan Jia

has been accepted towards fulfillment
of the requirements for the

Doctoral degree in Mathematics

 

 

[341%

 

V Major Professor’s Signature

AL\T\L$\' 9-7 1 20061
(J 1

 

Date

MSU is an Affinnative Action/Equal Opportunity Employer

01-0-n-0-I-n-n-u-O-n---I-Q-I-0-1-.-.-0—o-l-l-0-c-o-.-I-o-l—a----—-— -

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5/08 KzlProleccaPres/ClRC/DateDue.indd

 

KINESIN-MICROTUBULE INTERACTIONS: TRANSPORT AND SPINDLE
FORMATION

By

Zhiyuan J ia

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Mathematics

2009

ABSTRACT

KINESIN-MICROTUBULE INTERACTIONS: TRANSPORT AND
SPINDLE FORMATION

By

Zhiyuan J ia

This thesis consists of two parts. The first part concerns the detailed modeling
of kinesin locomotion along microtubules. The second concerns modeling the self-
organization process of kinesin and microtubules.

Kinesin-l is composed of two identical heavy chains forming the two motor do-
mains, called heads by biologists. The neck linker connects the head and the coiled-coil
stalk. Kinesin-l converts the chemical energy from Adenosine triphosphate (ATP)
hydrolysis into locomotion along the microtubule by alternately exchanging the trail-
ing and the leading head. Kinesin-l takes 8 nm for each step by consuming one ATP
molecule. We carried out detailed simulations for the different chemical and mechan-
ical processes of the two heads of kinesin. Furthermore, simulations are performed
with different lengths of the neck linker and the mean speed of kinesin movement is
obtained. Our analysis and simulation shed light on understanding the processivity
of kinesin, the estimation of the tension in neck linkers and further the role of tension
in regulating the chemical states of tWo heads.

In the second part [36], Monte Carlo type simulations were implemented for the
self-organization of microtubules interacting with molecular motors. Microtubules
are treated as stiff polar rods of equal length exhibiting anisotropic diffusion in the
plane. The molecular motors are implicitly introduced by specifying certain prob—
abilistic collision rules resulting in realignment of the rods. This approximation of
the complicated microtubule-motor interaction by a simple instant collision allows us
to by-pass the computational bottlenecks associated with the details of the diffusion

and the dynamics of motors and the reorientation of microtubules. Consequently, we

are able to perform simulations of large ensembles of microtubules and motors on a
very large time scale. This simple model reproduces all important phenomenology
observed in in vitro experiments: formation of vortices for low motor density and

ray-like asters and bundles for higher motor density.

ACKNOWLEDGMENT

I am grateful for the help and guidance in those years from my mentor Professor
Peter W Bates. I benefit tremendously from working with him from which I learned
how to conduct scientific thinking, to extract information from experimental results,
to propose an idea, and to transform it into mathematical equations. I appreciate
every effort Professor Bates has made to help me meet his expectations.

I appreciate the help and encouragement from my committee members. I am
grateful to the many suggestions from Professor Chichia Chiu. I enrolled and enjoyed
the class of Professor Thomas Pence and I learned a lot from our discussions of
the microtubule pattern modeling in spring, 2006. The enthusiasm and ambition of
Professor Moxun Tang for the mathematical biology definitely influenced me doing
the research in this field. The methods and skills learned from the classes of Professor
Guowei Wei turned out to be important in my research.

I appreciate the help of Professor Weil for providing the support for using Latex
in the writing of this thesis.

I own my parents a debt of gratitude for their constant trust in me and encour-
agement to move forward.

Last but not least, I thank my wife Wan and my son Will, who give me the love,

the support, and the hope, which inspire me to overcome difficulties on the way.

iv

TABLE OF CONTENTS

List of Tables ................................. vi
List of Figures ................................ vii
1 Introduction to Kinesin and Microtubules ............ 1
1.1 Microtubules ............................... 1
1.2 General Results with Kinesin ...................... 2
1.3 Literature Review ............................. 6
1.3.1 Theoretical Modeling Work of Kinesin ............. 6
1.3.2 Interactions of Kinesins and Microtubules ........... 13

2 Regulation of Tensions of neck-linkers in Chemomechanical Pro-
cesses .............................. 19
2.1 Experimental Results of Kinesin with Extended Neck-linkers ..... 19
2.2 Bias of Kinesin Walking ......................... 22
2.3 Processivity of Kinesin Walking ..................... 25
2.4 Biochemical Reaction Cycle of Kinesin ................. 30
2.5 Tension Estimate of the neck—linkers .................. 32
2.6 Algorithm ................................. 37
2.7 Simulation Results ............................ 43
2.8 Discussion ................................. 58
3 Interactions between Microtubules and Molecular Motors . . . . 61
3.1 Essentials of the Model .......................... 61
3.2 Algorithm Description .......................... 66
3.3 Coarse-grained Variables ......................... 68
3.4 Pattern Characterization ......................... 68
3.5 Simulation Results ............................ 72
3.6 Conclusion ................................. 78
4 Summary and Future work ................... 80
A Pseudo Code of the Algorithms ................. 82
Bibliography .......................... 85

2.1

2.2

2.3

LIST OF TABLES

The reaction rate constants ....................... 31

The total lengths, persistence lengths and natural lengths of the neck-
linkers .................................... 33

The spring constants and the natural lengths of the neck-linkers . . . 36

vi

1.1

1.2

1.3

1.4

2.1

2.2

2.3

LIST OF FIGURES

Schematic representations of microtubule. The dark monomer denotes
fl tubulins and the light monomer is for a tubulins. ..........

Schematic representations of kinesin-1 (shown by permission from Cell
Press). The two motor domains (around 5 nm) are shown in the left
hand end and the two cargo binding domians are shown in the right
hand end. The middle coiled-coil part is the stalk, around 70nm. . i. .

Illustration of the kinetic diagram for a motor with N = 3 chemical
states. The squares represent the lattice sites on the track with d,
being the step size of the motor. Here we show two consecutive lattice
sites labeled by Id and (1+ 1)d. The chemical reaction cycle of a motor
consists of three states denoted byO ll and 2l. Ol represents the
empty state. ll In represent the AT}ll bound state and 2l represents

the ADP bound state. (shown by permission from Physica A [24].)

Experiment results of the pattern formation in [62](shown by permis-
sion from Science). They used fluoresence to highlight the accumula-
tion of the motors. The positions where there are more motors are
bright. ...................................

Illustration of the mutants and the results of run length, speed, AT-
Pase rate and coupling ratio. This figure is figure 1 in [95](shown by
permission from Cell Press) .......................

The experimental result: the histograms of stepsize of the wild type
and mutant kinesins ...........................

Simulation results from (2.1) in graph a and from (2.3) in graph b.
c = 1 / 200 is used for these graphs ....................

vii

10

14

26

2.4 A Chemomechanical cycle of kinesin. The letters represent the nu-
cleotide states of a kinesin catalytic core; E is for the empty state, T
is for the ATP bound state, D is for the ADP bound state, DP is for
the intermediate state after the ATP molecule is hydrolyzed. The dark
solid oval represents head2 and the light solid oval represents headl. .

2.5 Illustration of the binding sites for the wild type kinesin. Three vertical
stripes represent three protofilaments of the microtubule. Assume that
kinesin can only bind to the sites on these three neighboring protofil-
aments. The oval with X inside denotes the bound head, i.e., head2
in the algorithm and the dark head in Figure 2.4. The oval represents
headl in the algorithm, the light head in Figure 2.4. The five for-
ward binding sites for headl are represented by squares. The number
of binding sites for the mutants will increase depending on the reach-
able range of headl of the mutants. Notice that the binding sites are
arranged to reflect the helical structure of the microtubule .......

2.6 Speed of the wild type, 0P-26P and 14GS mutants computed by for-
mulas (2.24, 2.26, 2.28) ..........................

2.7 The speed of wild type kinesin vs ATP concentration (11M) ......

2.8 The average dwelling time of the wild type, 0P-26P and 14GS mutants

28

38

44

45

46

2.9 The average diffusion time of the wild type, 0P-26P and 14GS mutants 47

2.10 Speed of the wild type, 0P-26P and 14GS mutants ...........
2.11 Coupling ratio of the wild type, 0P-26P and 14GS mutants ......

2.12 Runlength of the wild type, 0P-26P and 14GS mutants. The mean run
length is shown in the insets ........................

48

49

50

2.13 Trajectory samples of wild type kinesin and 0P, 2P, 4P and 6P mutants. 51

2.14 Trajectory samples of wild type kinesin and 13P, 19P, 26F and 14GS
mutants ...................................

2.15 Stepsize histogram of the wild type and mutant kinesins. .......
2.16 The histogram of stepsize from the simulation results for the wild type,
6P, 13, 19P, 26P, and 14GS mutants. The histograms of experimental

results are shown in Figure 2.15 ......................

viii

52

53

54

2.17 Trajectory samples of wt, 6P, 13P, 19P, 26P, and 14GS from the ex-

2.18

3.1

3.2

3.3

3.4

3.5

perimental results. These trajectories have more or less the same slope
because they are obtained from different ATP concentrations. See the
simulation results in Figure 2.18 .................. _. . .

Simulation results for the trajectories of wt, 6P, 13P, 19P, 26P, and
14GS. The same ATP concentration, 1 mM, is used in the simulation.
See the experiment trajectories in Figure 2.17 .............

Schematics of an alignment event (inelastic collision) between two mi-
crotubules interacting with one multi-headed molecular motor. The
black dots represent the center of mass of the microtubules. (a) A
multi-headed molecular motor cluster attached at the intersection point
of microtubules moves from the negative (—) towards the positive (+)
end of the microtubules. (b) After the interaction, the orientational
angles ch2 and the corresponding positions of the midpoints R12

become aligned ...............................

Snapshots illustrating the patterns developing in a configuration of
6,000 rods for different motor densities, i.e., different values of P0.
Arrows represent microtubules, circles depict the cores of vortices or
asters. (a) vortices, t = 620 , 6 = 1.0, P0 = 0.08 (low motor density);
(b) asters, t = 602, fi = 0.95,PO = 0.10, (high motor density); (0)
bundles, t = 400, 6 = 1.0,P0 = 0.15. See also [100] for movies # 1,2
illustrating the self-organization process. ................

Coarse-grained images corresponding to parameters of Fig.3.2 for vor-
tices (a) and asters (b). Arrows represent the orientation field 7’. The
color (grey levels) shows the density p, red (bright) corresponds to the
maximum of p, and blue (dark) to its minimum. See also [100] for
movies # 3,4. ...............................

Averaged number of asters (squares), anti-asters (diamonds), and vor—
tices (circles) as a function of the interaction probability P0 for two
different values of parameter 6. The data for 6 = 0.35 is shown in
dashed lines, open symbols, and for 6 = 0.95 is shown in dotted lines,
closed symbols. ..............................

Phase diagram of various regimes as a function of the motor density
P (the horizontal axis) and the anisotropy parameter 6 (the vertical
axis). The disordered region is blue (black) here; the vortex region is
green (grey); the transition from vortex to aster happens at the yellow
region (white) and red (dark grey) denotes aster regions. The dashed
line denotes the boundary where the rods become bundled .......

ix

55

56

63

69.

75

76

77

Chapter 1

Introduction to Kinesin and

Microtubules

1. 1 Microtubules

The microtubule is one of three cytoskeletons in a cell. The other two are actin
and intermediate filaments. The microtubule is the most rigid among them. The
cytoskeletons can form the scaffolds to support and maintain the shape of a cell. In
cell movement, the structure and the distribution of the cytoskeletons will adapt to
facilitate moving. Actin filaments and microtubules are also the tracks for molecular
motors to move on carrying cargoes such as mRNA, neuro—transmitters, etc. Micro-
tubules are also an indispensable part of the spindle, the machinery of cell division.
Microtubules have long rigid cylindrical structures (length tens of microns and
diameter approximately 25 nm) comprising of heterogeneous tubulin dimers, each
dimer consisting of an a and a [3 tubulin, which self-assemble, 13 protofilaments
being required side-to-side to form the circular cross section (see Figure 1.1). The
length of a tubulin dimer is 8 nm so a microtubule filament can be seen as a linear

periodic track with periodicity 8 nm. Since the microtubule is polymerized by (16

1

tubulin dimers, one end of it exposes fl tubulin. The other end exposes a tubulin
and the ,6 tubulin end has high polymerization speed and the or end has low speed.
Biologists designate the end with fast polymerization speed as the plus end and the

other end the minus end.

1.2 General Results with Kinesin

There are many molecular motors in each cell conducting different tasks to maintain
the functions of the cell. For instance, DNA polymerase and RNA polymerase are the
motors moving along the DNA strand performing the replication and the transcription
of the DNA correspondingly. In this thesis, we will focus on a particular molecular
motor, kinesin [33, 79], which moves on microtubules. Kinesin can carry cargoes from
one place to another within the cell and can work with other motors in the cell to
facilitate the division process.

Kinesin converts chemical energy, hydrolyzed from ATP (Adenosine triphosphate)
molecules, into mechanical movements in a walking process. Most members of the
kinesin family walk toward the plus end of microtubules. Only one subfamily, kinesin—
14, N CD, a representative member, walks toward the minus end [64]. In the first part
of this dissertation we will conduct detailed modeling of the walking of conventional
kinesin, also called kinesin-1, since it has been investigated most extensively by biol-
ogists. From now on, for brevity, we always use kinesin for kinesin-1. In the second
part of the dissertation, we will model the interactions between kinesins and micro-
tubules and reproduce the self-organization process of the microtubules into different
patterns.

Kinesin is composed of two identical heavy chains, each of them includes a N—
terminal motor domain, in which there is an ATP binding site. The neck-linker is

the segment in each heavy chain connecting the head to the coiled—coil stalk (See

2

 

Figure 1.1: Schematic representations of microtubule. The dark monomer denotes fl
tubulins and the light monomer is for a tubulins.

 

Figure 1.2: Schematic representations of kinesin-1 (shown by permission from Cell
Press). The two motor domains (around 5 nm) are shown in the left hand end and the
two cargo binding domians are shown in the right hand end. The middle coiled-coil
part is the stalk, around 70nm.

Figure 1.2). Growing from the other end of the coiled-coil stalk are the two light
chains (arms), which can hold cargoes (e. g. mRNAs, protein complexes). Kinesin has
different microtubule binding affinities when in different nucleotide states [89], that is,
when its core contains either ATP, ADP, or is empty. Kinesin has the weakest binding
strength when its catalytic core contains an Adenosine diphosphate (ADP) molecule.
Kinesin binds to the microtubule strongest when in the ATP bound state. The
microtubule binding affinity has an intermediate strength when it is in the nucleotide
free state. Kinesin has been demonstrated to walk in a hand-over-hand manner[93,
8]. The two heads of the kinesin molecule alternately bind to and unbind from the
microtubule with mechanisms that provide a bias to the Brownian motion expected.
The center of mass of the kinesin moves 8 nm with each step, which is exactly the
length of one (16 tubulin dimer. Kinesin consumes one ATP molecule each step,
meaning that kinesin tightly couples a chemical reaction to a mechanical movement
[84, 17, 34, 83]. Kinesin walks processively on a microtubule, with experimental
results indicating that it can walk continuously for over 100 steps without falling off.

Kinesin primarily walks toward the plus end of microtubule while it will walk
backward more likely when a backward-pointing force of sufficient strength is applied
to it. The stall force of kinesin, around 7 picoNewton (pN), is the force where kinesin
has the same probability to walk either forward or backward. Thus, at the stall force,
the walking speed of kinesin is zero [13].

Currently, a consensus model regarding the walking of kinesin is proposed as

follows [94, 11].

1. Starting from a two head bound state where the leading head is in the nucleotide

free state and the trailing head has an ADP molecule bound in its catalytic core.

2. The trailing head detaches from the microtubule and begins a tethered diffusion
process. An ATP molecule comes and binds with the leading head. This ATP

4

binding releases energy which triggers part of the neck-linker to bind toward
the front of the leading head and become immobile (called zipping). The length
of the docked part of the neck-linker is about 2 nm at most. This neck-linker
docking [74], together with another mechanism, arising from the asymmetric
steric effect [87], such as the shape of the head and the shape. of the binding
site, provides a bias for the trailing head to step toward the next binding site

in the positive direction of the microtubule.

. After the tethered trailing head reaches the next binding site, it binds to the
microtubule tightly with the release of ADP. This tight binding induces a strain
on the new leading head to prevent the binding of an ATP molecule. Then the
ATP molecule in the trailing head hydrolyzes and a P2- is released. The hydrol-
ysis energy facilitates the unbinding of the trailing head and the intramolecular

strain caused by the binding of the leading head now is also released.

. Now the leading head is in the empty state and is ready for the binding of an
ATP molecule and the trailing head is in the weak binding state with ADP in

its catalytic core. This completes one Chemomechanical cycle of kinesin.

In step 2 of the above process, when the trailing head detaches from the micro-

tubule, it will diffuse subject to the physical restriction of the total length of the

neck-linkers and their flexibility. In this process, it can temporary bind to a rearward

binding site but this binding is weak because the trailing head is still in the ADP

bound state. There are two hypotheses for the trailing head not to release the ADP

when it binds backward. Its 6 sheet is in an upright position that inhibits the release

of the ADP molecule [45]. There may exist a specific configuration between the neck-

linker and the head which plays a role in preventing the release of ADP when the

neck-linker is pointing forward. Therefore the trailing head will eventually unbind

again, diffuse and bind to a forward binding site while ADP remains bound to it.

5

After the neck-linker of the bound head is zipped, the tethered head cannot reach
the rearward binding sites because of the shortened neck-linker.

With regard to the bias of kinesin movement, there are basically two models for
it. One argues that ATP-dependent neck-linker docking throws the tethered head
forward to the next binding site. The other more emphasizes the diffusive search of
the tethered head for the next binding site with more likelihood of binding forward
because the neck-linker is zipped towards the front of the bound head. Both of them
conjecture that the forward binding of the tethered head, accompanied by the release
of ADP, is strong so that it completes a step. Certainly these two models are not
mutually exclusive and they actually work together in our model.

Steric asymmetry is another source of bias [87]. The evidence of steric bias can
be seen in a series of experiments where the kinesin can walk toward either the plus
or the minus end depending on the applied external force when there is no ATP
at all, only ADP or AMPPNP(Adenylyl-imidodiphosphate, a nonhydrolyzable ATP
analog). The X-ray crystallography of a kinesin-microtubule complex suggests the
different binding conformations of leading and trailing heads when kinesin is in a
two-head-bound state [45, 80]. The leading head is in a tilted configuration and the
trailing head is in a upright configuration. The backward binding of the trailing head
favors an upright conformation due to the forward tension. It is believed that the

tilted configuration is required for the release of ADP.

1 .3 Literature Review

1.3.1 Theoretical Modeling Work of Kinesin

Biological experiments have stimulated many theoretical Works to elucidate the dif-
ferent aspects of the walking mechanism of kinesin, such as the bias and the proces-

sivity of the movement. Basically there are two different approaches to the theoretical

6

modeling. One is the continuum ratchet method, using a damped Langevin equation
[32, 41] or a set of coupled Fokker-Planck equations [32, 41] to describe the move-
ment of kinesin by assuming that kinesin is subject to different potentials when it
is in different chemical states. The potentials are chosen to be asymmetric over an
8 nm periodic interval to generate the biased movement of kinesin. The transitions
among the different potentials depend on the chemical states, the concentration of
ATP(ADP), and the external force and can be described by the transition rates, which
are tuned to coordinate the potentials. In this method, it is challenging to derive re—

alistic potential functions,(see for instance [38, 40, 42, 44, 59, 57, 65, 68, 73]).

In the following, we examine an example from [38], where the authors used the
Langevin model to discuss the speed dependence on ATP concentration, the stall
force, the trajectory, and the processivity of kinesin. The plus end of the microtubule
is taken to be the positive direction of the 3: axis and the coordinates of the tethered
head of the kinesin are (x, y), where y represents the one-dimensional displacement

of the head perpendicular to the microtubule. The Langevin system reads as

 

8H 6H -
- _ ratchet bistable a: /__dW£E
’73: _ —' 627- _ "" 6x + Fext + ZKBT’Y—dt ,

1.1)
6H 6H . dW (

. _ _ ratchet _ bistable y / __3_/.

7y - 6y 3y +Fext + ZKBT7 dt ’

 

where 7 is the drag coefficient, dag—l is white noise, H 7, at C h et is a periodic func-
tion in :c, i.e., Hratchetm + 2L,y,t) = Hratchet(:c,y,t), and L = 8nm is the

period of the microtubule. Specifically, H7. at C h at is given by

Hmtchetww, t) = 5(t)Cle($)Hy(3/)l - ay, (12)

where C = 0.8 eV and a = 0.044 eV are constants, S (t) is a switch function

7

1, on state
S(t) =
0.1, off state

which is supposed to produce a flashing ratchet. The authors select a number
p f l a 8 Mn. 9 between 0 and 1 in advance for the simulation, representing the prob—
ability of random arrivals of the ATP molecules. The motor is in the on state as a
simulation begins. A uniformly distributed random variable C is generated in each
s1mulat10n step to compare it With p fl a S hin g. If pf] as hing > C, then the state

of the motor is switched off. Otherwise, it will stay in the on state.

The function H :13 is given by

5

5
Hz(.’13)=a0+ Z amCOS(2T;;$)+ Z: bmsin(2;:$), (1.3)

 

 

which is the truncated Fourier expansion of the following asymmetric potential func-
tion 10

a: :1: :1: a:
1er ‘ [53]), at ‘ [EL-l < ”-91

we — is) — 3'— - is] >

U(:r) =

where [7%] is the integer part of the ratio.

The function Hy is given by

2 _ _
H’s/(y) =esvp(--£ij—fl-yfl)-) ~2e$p(y fiyO), (1-4)

 

where 6 = 5L = 40 nm and yo = L = 8 nm were used. The term Hy is supposed to

model the van der Waals interaction between the head of kinesin and the microtubule.

The function H bi 8t abl e is given by

I

”(W-@Y

where Ar is the distance between the two heads, CI = 5.4eV represents the

Hbz'stablemrl 2 Cl

 

 

coupling strength of the two heads, 6 = 0.75L, and 28 is the distance between the two

minima of the potential.

By solving the above 2D Langevin equation (1.1), the authors produced trajecto-
ries of the two heads of kinesin in a hand-over-hand walking process. They also tested
the relation between the speed and the external force and found the value of the stall
force to be 6.4 pN, which is in close to the experimental result, around 7 pN. The au-
thors also tested the processivity of the model and claimed that the motor described
by their model does have processivity . The potentials and parameter values used in
this model have not been justified, however. For instance, they pointed out that the
last term ay in (1.2) is critical to have the processivity of the motor but they could

not give a physical or chemical reason for the use of this term.

From this example we can see that, to have a realistic continuum ratchet model,
it is critical to construct potential functions which can reflect the structure of the
kinesin heads in different chemical states and the interactions with the protofila—
ments of microtubules. For this we need further information about the structures

and interactions.

The second approach uses discrete chemical kinetic networks to model the free
energy transduction in the walking process [70, 23, 24, 71, 25, 52, 54, 88]. An example

from [23, 25] can help illustrate the basic idea of the stochastic discrete kinetic method.

A schematic illustration of the walking of a motor on a linear periodic track is
shown in Figure 1.3. Corresponding to Figure 1.3, the sequential kinetic equation can

be described as follows.

ll
slls
l

 

 

 

 

 

 

Figure 1.3: Illustration of the kinetic diagram for a motor with N = 3 chemical states.
The squares represent the lattice sites on the track with (1, being the step size of the
motor. Here we show two consecutive lattice sites labeled by id and (l -l- 1)d. The
chemical reaction cycle of a motor consists of three states denoted by 05, ll and 2].
OZ represents the empty state. ll m represent the ATP bound state and l represents

the ADP bound state. (shown by permission from Physica A [24].)

U0 U1 UN—z UN—i
O = 1 = 2 : N —1 : .
lwl l’UJQ l wN_1( )l we Ol‘l'l’ (15)

where the lattice sites on the track are labeled by l (= 0,i1, :l:2, . . .) and the
chemical states are denoted by j = 0 for the free state, i.e., no ATP bound state, and
j = 1,2,. . . , N — 1 for the other various bound states, where N is the number of the
total chemical states. Thus j 1 represents the situation where the motor lands at the
1th lattice site and is in the jth chemical state. The distance between two lattice
sites, I and l + 1, equals the step size d as is shown in Figure 1.3. The reaction rates
uj and wj are independent of the lattice position. Let Pj(l, t) be the probability of

finding the motor at the site I with the state 3' at time t. The time evolution equation

for Pj(l, t) is

10

a
Egrey-(I, t) = uj_1Pj_1(l, t) + wj+1Pj+1(l,t) — [u]. + wj]Pj(1,t), (1,6)

for j = 0,1,... ,N - 1 with periodic conditions

P__1(l,t) = PN_1(l—1,t), PN(l,t) = P0(l+1,t), (17)

u_1= uN_1 and wN = 1110.
According to [21], the above equation can be solved explicitly to obtain the drift

velocity in terms of the reaction rates,

 

N—lw.
V=—]-25‘L 1— J- (18)
N j=0 J
while
N—l N—l k . .
1 wj-l-Z
R — r r —— 1+ 19
Ar 2: , . E: ) ( )
= ‘7 ‘7 “J( k—1z‘-_—.1“J+Z

 

VS +U d

=( N2 N —l(N+2)V) i (1.10)

R N

N
with

N—I N—I N—I
SN: 2 Sj Z (k+1)Tk+j+1,UN= Z ujTij, (1.11)

3:0 k=0 j=0

while the supplementary coefficients are

11

sj=% 1+ 2 11—55—11 . (1.12)

Among those chemical steps, if one or multiple steps are assumed to be load de-
pendent from the experimental observation, then one can derive the relation between

velocity and the external load.

The above example gives the basic idea of stochastic reaction network method
where a two-headed motor is simplified as one head without considering the coordi-
nation between the two heads. In [54], the authors considered the discrete reaction
network with two heads and produced the velocity-force relation. There are more ar-
ticles concerning the ratchet continunn method and the stochastic chemical reaction

network method cited in [47].

A great deal of mathematical analysis has been inspired by attempts to provide
rigorous results for the Brownian ratchet model of molecular motors, see [9, 12, 15,
16, 46, 66, 67] and the reference in these papers. Those mathematical analyses were
done for continuum ratchet models. For example, A Fokker-Planck equation was
considered in [15]. Assume the motor moves along a linear track and :1: denotes the
position of the motor. Let p = (p1,p2) be the probability densities of the motor at
different states, say i = 1, 2, representing two states here. Assume that the two states
are subject to different potentials. For instance, the state 1 and 2 might correspond
to the zipped and unzipped state of the neck-linker. The time evolution of p is given

by the following coupled Fokker-Planck equation.

3'01 3 391 I

“a" = a; (“a + W1) ‘ ”1P1+ ”2"? .

apz 8 3,02 1n (2, t > 0 (1.13)
I

a: = a; (as; + “Oz/’2) + ”191 " ”m

12

with the boundary conditions,

3P1 I
05 + 901p1= an 4
3P2 I on , t > 0 (1.1 )
“a? ”’2‘? =
and
(2207.0) = p? > 0,
in Q, z'=1,2 (1.15)

where Q = (0,1).

In [15], the authors proved that the stationary distribution of the probability
density described by equations (1.13, 1.14, 1.15) decays exponentially.

To the best of our knowledge, there is no detailed simulation of the walking pro-
cess of kinesin that faithfully follows the experimentally established biochemical and
mechanical processes. In this thesis, we develop algorithms to model this chemo-
mechanical process and apply the algorithm to elucidate some fundamental issues
surrounding the walking of kinesin, such as the bias and the processivity. Inspired
by [95], we modeled neck-linkers as entropic springs, discussed the tension estimate
of the neck-linkers, and furthermore clarified the role of tensions of the neck-linkers

of kinesin.

1.3.2 Interactions of Kinesins and Microtubules

Organization of complex networks of long biofilaments such as microtubules and actin
filaments in the course of cellular processes and division is one of the primary functions
of molecular motors [32]. A number of in vitro experiments were performed [86, 90, 62,
81, 35, 63] to study the interaction of molecular motors and microtubules energized

by the hydrolysis of ATP in isolation from other biophysical processes simultaneously

13

>

One motor

 

 

>-
Motor concentration (11M)

Figure 1.4: Experiment results of the pattern formation in [62] (shown by permission
from Science). They used fluoresence to highlight the accumulation of the motors.
The positions where there are more motors are bright.

occurring in vivo.

In particular, the experiments in [62] used microtubules and kinesin as a model
system to investigate the self-assemble process of the spindle formation. Indeed, in
the cells of some organisms, for instance, plant cells, the randomly distributed mi-
crotubules are self-organized into the spindles via the interactions with the molecular
motors kinesin and dynein (see [37] and the reference therein). The experiments [62]
clearly demonstrated that at large enough concentration of molecular motors and
microtubules, the latter organize into ray-like asters and rotating vortices depend-
ing on the type and concentration of molecular motors. These experiments spurred
numerous theoretical studied addressing various aspects of self-organization of active
filaments systems [51, 61, 55, 14, 19, 50, 2, 3, 92, 49, 96].

The experiments [62, 81, 35, 63] suggested the following qualitative picture of

14

motor-filament interaction. After a molecular motor has bound to a microtubule at
a random position, it marches along it in a definite direction until it unbinds without
appreciable displacement of microtubules (since the size of a molecular motor is small
in comparison with that of the microtubule). However, if a molecular motor binds
to two microtubules (some molecular motors (e.g., kinesin) form clusters with at
least two binding sites), it exerts significant torques and forces, and can change the
positions and orientations of the microtubules significantly, leading eventually to the

onset of large-scale ordered patterns.

In [51], a set of field equations were used to model the pattern formation observed
in the experiment [62, 81]. In a 2D square domain, let U be the local orientation
of microtubules and m be the concentration of the motors. The following equation
is introduced in [51] to describe the evolution of the orientation of microtubules and

the concentration of motors with respect to time.

%?=v2m_e( a)
_. (1.16)

which is subject to the reflecting boundary conditions Ulbounda'ry = —fi, where if
is the normal outward vector at the boundaries. Their simulation results shown the
formation of the aster and the vortex patterns. Typically there are both aster and
vortex patterns at low motor concentration and the vortex becomes dominant at the
high motor concentration contrary to experimental evidence. The author failed to
produce the transition from vortex to aster dominance when the motor concentration

increases.

Following basically the same idea, the authors in [78, 60] divided the motor popu-
lation into two fractions, the free diffusion motors and the microtubule-bound motors.

The concentration of the free motors fluctuates due to the binding of the free motors

15

and the unbinding of the microtubule-bound motors. So the first equation in (1.16)
was replaced by two equations to describe the evolution of the free motors and the

bound motors.

Bmf 2 _
_8t =DV mf—lyf-fibmf-i-fyb—tfmb
6m

b_ —+
a?

_—> -—>2 2—> -—> -—>—-—> 2—> —>
Tit—_U(I_IU| )+me U+eVmb-VU+<V U+€Vmb,

where 7b __) f is the transition rate for the motor to change from the rod bound state
to the free state, vice versa for 7 f __) b‘ C, e, c and 5 are parameters. Essentially,
the author obtained similar results to those in [51] and unfortunately also could
not produce the transition from vortex to aster patterns as the motor’s concentration
increased. In their simulation results, it seems what pattern will emerge relates to the
boundary conditions used. When all the other parameter values are the same, aster
patterns are more likely to appear with the reflecting boundary conditions and vortex
patterns are more likely to be seen in the parallel boundary conditions, meaning that
the orientation of the microtubules are perpendicular to the normal outward direction

of the boundaries.

Small-scale molecular dynamics simulations were performed to elucidate the na-
ture of self-organization [62, 81]. In these simulations the microtubules were modeled
by semi-flexible rods diffusing in viscous fluids. Molecular motors were correspond-
ingly modeled by short stiff linear springs with a large diffusion coefficient. Once
the motor diffuses to within a certain small distance from the intersection point of
two microtubules, it attaches to them with a certain probability P071. and marches
along with velocity v. The action of the motor is to exert forces and torques on

the microtubules, resulting in their mutual displacement and realignment. Then the

16

motor detaches with a probability p 0 f f To model the dwelling effects of the motors
on the end-points of microtubules, observed for some types of molecular motors, an
additional probability p an d to leave the end-point was assigned. The corresponding
typical dwelling time t en (1 is of the order 1 /pen (1' The simulations in [62, 81] in-
deed reproduced certain features of the observed phenomenology, such as the stability
of patterns and transitions between vortices and asters. However, in this approach
many fundamentally different time scales had to be simultaneously resolved compu-
tationally (e.g., fast diffusion of the motors and very slow pattern formation). As a
result, the method is very CPU-intensive, and only a small number of microtubules
were studied numerically, leaving many important questions, such as the nature of

the transition and structure of the phase diagram, unanswered.

In Refs. [2, 3] a continuum probabilistic model of alignment of microtubules
mediated by molecular motors was developed. The theory was formulated in terms
of a stochastic master equation governing the evolution of the probability density
of microtubules with a given orientation at a given location. The theory is based
on a number of simple assumptions on the interaction rules between microtubules
and molecular motors. In particular, only binary instant interactions of microtubules
called inelastic collisions are considered. These are mediated by molecular motors in
a two-dimensional microtubule-motor mixture of constant motor density. The motors
are implicitly introduced into the model by specifying the probability of interaction
of intersecting microtubules. Despite all the above simplifications of the biological
process of self-organization of the cytoskeleton, the model reproduced, on a qualitative
level, key experimental observations, such as the onset of an oriented (polar) phase
above a critical density of motors, formation of asters for large density of motors and
vortices for lower density, direct transition towards asters from the isotropic state for
large dwelling times of the motors at the end of microtubules, and a density instability

and the onset of bundle formation at very high motor density.

17

However, due to significant complexity of the derived stochastic master equation
governing the evolution of the probability density of microtubules, the analysis in
Refs. [2, 3] was carried out in a relatively narrow range of parameters, namely, in
the vicinity of the orientational instability, which allowed rigorous reduction of the
stochastic master equation to a set of much simpler amplitude or Ginzburg—Landau
type equations for the local coarse-grained density and orientation of microtubules.
This approach yields some insights into the self-organization process, but, it obviously
has its own limitations.

We perform Monte Carlo type simulation studies of self-organization of micro-
tubules interacting with molecular motors. Instead of modeling the self-organization
process in all details as was done in Refs. [62, 81], we use simplified interaction rules
suggested in the works [2, 3]. This simplification allows the elimination of fast time
scales associated with the diffusion and motion of the motors. Consequently, one may
focus on relevant time and length scales associated with large-scale pattern formation
and evolution. We studied very large ensembles of microtubules and addressed ques-
tions related to the structure of the corresponding phase diagram and the transitions
between various patterns. In agreement with the early experiments, we were able
to reproduce aster-like structures for a high motor density and vortices for a lower
density, as well as transitions to bundles. Our approach provides direct access to the
dynamics of the stochastic master equation and obtains insights far beyond the am-
plitude equations approach. Moreover, our method provides an efficient and fast tool
for simulation of complex biological process of cytoskeleton self-organization and can
be possibly extended to rather different systems, such as anisotropic granular media

and systems of self-propelled particles.

18

Chapter 2

Regulation of Tensions of
neck-linkers in Chemomechanical

Processes

2.1 Experimental Results of Kinesin with Extended

N eck-linkers

In order for kinesin to walk over 100 steps without falling off the microtubule, there
must be at least one head bound to the microtubule at all times in the walking process.
So there must be a mechanism for those two heads to prevent both of them being in
the ADP bound state when both are attached to the microtubules. In other words,
there should be mechanisms for them to keep their chemical states out of phase. The
tension of the neck-linkers is such a candidate in the regulation of the chemical states

of the two heads [31, 77].

To test how the tensions in the neck-linkers help coordinate the chemical states

of the two heads, Amet Yildiz et al [95] did an experiment in which they inserted

19

KKPPPP ...... PPPG

R

...KSTLLFGQRAKTIKNT\/SVNVE LTAEQWKKK...
W \M/

Motor Head Neck Linker
4 nm
WT
4.7 nm
2P

4P r3554nm
i6 m

n
6P
10 nm
19F
12 nm
26p A
~6 nm

14GS

Figure 2.1: Illustration of the mutants and the results of run length, speed, ATPase
rate and coupling ratio. This figure is figure 1 in [95](shown by permission from Cell

Press)

Neck Coiled-Coil

C

.N
or

run length (#m)

0.0
WT 0P 2P 4P 6P 13P19P26P14GS

speed(nmlsec)
'8‘ s “s“ s s

 

0
WT 0P 2P 4P 6P 13P19P26P14G

_.
U)

(ATPs/secltnobr)
o s s s s

WTOP 2P 4P 6P 13P19P26P14GS

:"
0

coupling ratio
.0 .0 .0 .0
N b- 0) m

0
WTOP 2P 4P 5P 13P1W26P14GS

different materials with different lengths into the native neck-linker. As is shown in
Figure 2.1, they inserted three amino acids, represented by K(lysine)KG(glycine) in
part A in Figure 2.1, at the junction of the neck-linker and the coiled-coil stalk so that
the joint between the two neck-linker parts is more flexible. This mutant is labeled
0P. From the mutant 0P, they inserted polyproline (P) helices with different lengths
at the position between KK and G to form the mutants 2P, 4P, 6P, 13P, 19P and 26P.
Finally 14GS is the mutant formed by inserting glycine-serine (GS) repeats instead

of polyproline.

Their experiment results shown that all of the mutants can still walk over 100
steps and have more or less the same run length as the wild type. However, the
speed of these mutants decreases as the mutants’ neck-linker becomes longer and the
inserted material becomes soft. The measured ATPase rates of the mutants are the
same as the wild type but much more futile cycles (where ATP is used but no step is
taken) were found in the mutants. The coupling ratio of ATP, defined as the quotient
of the number of steps over the number of hydrolyzed ATP molecules, decreases as
the similar tendency as the speed. The coupling ratio for the wild type is about 80%
but the 14GS has a ratio of only 10%.

The motility of kinesin is tested at different nucleotide conditions. If there is no
ATP at all, under external forces, kinesin can walk toward either the plus or the minus
end of the microtubules depending on the direction and the magnitude of the external
forces. A force of 3 pN (6 pN) is required for the motor to walk toward the plus (minus)
end. In this situation, there is no neck-linker docking at all and the direction of the
movement is decided by the direction of the external force. The authors also tested the
motor with only ADP or AMPPNP (Adenylyl-imidodiphosphate, a nonhydrolyzable
ATP analog) present and found that the amplitude of the external forces for the
motor to walk toward the plus end is 1 pN in ADP solution and 9 pN in AMPPN P
solution. A force of 2 pN in ADP solution and 12 pN in AMPPNP solution are

21

required for the motor to walk toward the minus end. Therefore, the amplitude of
the external forces for the kinesin to walk toward the plus end is different in the
different nucleotide solutions, 1 pN in ADP solution, 3 pN in no nucleotide condition,
and 9 pN in AMPPNP solution. This evidence verifies the dependence of binding
affinities on the nucleotide states. On the other hand, the amplitude of the external
forces for the kinesin to walk toward the minus end is also different in the different
nucleotide solutions, 2 pN in ADP solution, 6 pN in no nucleotide condition, and 12
pN in AMPPNP solution. These differences in the amplitude of the external forces
applied to the kinesin for the plus end walking and the minus end walking at the
same nucleotide condition clearly indicate that there exists asymmetric steric binding
aflinities for kinesin, in addition to the binding affinities on the nucleotide states
The step size histogram of the mutants indicates that kinesin still continues to
walk forward with the extended neck-linkers. There is only about 5% more backward
steps for mutants compared with the wild type. With the long neck—linker and soft
GS inserted, 14GS has much more lateral binding, meaning that the motor binds to
the sites on the protofilaments adjacent to the one where the bound head is on, (see

Figure 2.15).

2.2 Bias of Kinesin Walking

What is the origin of directional bias in the movement of kinesin along a microtubule?
Rice et a1. [74, 75, 76] proposed a mechanism called the neck-linker docking to explain
the unidirectional stepping of kinesin. Actually neck-linker docking alone cannot
explain the results in Yildiz paper [95].

Assume that kinesin walks on a protofilament of the microtubule. Its bound head
is at the origin and the other head is in the tethered state. After the neck-linker

is docked toward the plus end of the microtubule, the root of the free neck-linker is

22

 

pBW=o.o2g . ‘ 5N8,ep=31o

6P

 

13P

19P

 

26P

   

14GS

 

 

   

-16 -8 o 8 16 24 32 40 48
step size (nm)

.32 -24

Figure 2.2: The experimental result: the histograms of stepsize of the wild type and
mutant kinesins

23

located at :1: = 2 nm. Assume the length of neck-linkers is long enough so that the
tethered head can reach the forward binding sites a: = 8 and a: = 16 and the backward
binding site :3 = —8 and a: = —16. The distance from the root of neck-linker in the
bound head to those four position is (11 = 6, d2 = 10, d3 = 14 and d4 = 16. The
possibility for the tethered head to reach those sites decreases with respect to the

at2

distance and can be modeled as e_ i, i = 1, 2,3,4, where c is the parameter to
adjust the probabilities. After the normalization, we obtain the probability for the

tethered head to bind to one of those sites

 

2
P.=e 2,11: Ze—Cdi’z'=1,2,3,4, (2.1)

The probabilities for the tethered to bind to the position at :1: = —32, —24, — 16, —8,
0,8, 16, 24,32 are p_32 = P}, p_24 = 2134132, p_16 = P3, p_8 = 2134191420 =
2193104 + 2131132, 198 = 2192193, 1316 = P2, p24 = 2P1P4, p32 = P12. Graph a
in Figure 2.3 showed these probabilities and it does not have peaks at :1: = 16, 24
no matter how we adjust the value of parameter c in the formula. Therefore there
must be another mechanism to bias the kinesin walking toward the plus end of micro-
tubules. Entropy effects were investigated in [87] and showed that there is a 6K BT
free energr difference per step between walking toward the plus end and toward the
minus end, i.e., Eigee — E51266 = —6KBT, where fw and bw denote forward

and backward. According to Arrhenius transition rate theory, this difference in free

energy favors a biased forward walking with

6 1
P = —. 2.2
bw 66 +1 ( )

e
p :—
fw 66+1,

Indeed, Arrhenius transition rate theory [32] gives

24

free free ree ree
_ E f w -Eactivation _ Ebfm —nggizzgtion
_. K BT = K BT
kfw — Ae and kbw Ae

Then the probability for the forward and backward reactions are

k
fw+kbw

kbw
kfw+kbw-

 

P fw = k and wa =
Simple computation gives us (2.2).
After we take into account the bias for forward walking from the entropic effect

in the above formula, we arrive at the following modifications to (2.1):

2
waeCdl,3 waeCdgA
P123:——H_’P214=—_II——’H=Pf

e6le +eCd§ )+wa (eCdg +eCd3 ).
(2.3)

By using the probability from (2.3) to compute the binding probabilities for the
different binding sites, we produce a similar graph to that from the experimental

results, (see 2.2). Therefore this simple argument indicates that neck-linker docking

and entropic bias work together to be able to generate the bias seen in the experiments.

2.3 Processivity of Kinesin Walking

For kinesin to walk along a microtubule for a few micrometers without falling off,
the kinesin must coordinate the chemical states of their two heads to be out of phase
so that one of the heads is always bound to the microtubule. The mutants with
longer neck-linkers can walk over 100 steps as the wild type. The mutants having

long and(or) soft neck-linkers have small tensions in their neck-linkers. On the other

25

 

E

Probability
O
N

.O
.3
1
1

 

 

l I l l

-40 -30 -20 -10 O 10 20 30 40
Position

 

.°
or

 

o
uh
I

I

.°
00
I
1

Probability
P
N

P
d
l

 

—4o -30 -2o -10 70 1o 20 30 40
Position

 

 

Figure 2.3: Simulation results from (2.1) in graph a and from (2.3) in graph b.
c = 1 / 200 is used for these graphs

hand, if tension is the only factor to regulate the processivity, then small tensions in
the mutants should lead to the loss of processivity. Therefore, the processivity of the
mutants actually indicates there are other mechanisms in coordinating the chemical
states.

In Figure 2.4, H1 is represented by the light head and H2 is represented by the
dark head. We now examine how the chemical states of the two heads change during
one step in the walking process. Starting from the moment when kinesin is in the
two-head-bound state with the trailing head in the ADP bound state and the leading
head in the empty state, and ending at the moment when the trailing head has taken
one step forward and has become the leading head in the empty nucleotide state and
the leading head has become the trailing head in ADP bound state, we can decompose

this time period into several subdivisions according to the temporal order as follows:

' THI = TdMT + Tdiffusion + TdADP + THIATp
° TH2 = TATP + TATPhydro + Tap,-
° Tffl = TdMT + Tdiffusion + TdADP

T d MT is the time period between the moment when H1 acquires ADP and the
moment when the head unbinds from the microtubule. Td’l f fusion is the time of
diffusion of H1 until it binds again to the microtubule. Td A D P is the time taken for
H1 to release its bound ADP after H1 binds to a forward position. T H1 is the
ATP
time period between the moment when ADP is released until the moment when H2
releases a phosphate. T AT P is the time period for H2 to bind an ATP molecule and
the ATP molecule is trapped, i.e., the neck-linker is docked. TATP hydro is the
time elapsed from when the ATP is trapped in H2 until the ATP is hydrolyzed into
ADP - P. Finally, TdP- is the time needed for H2 to release the Pi after hydrolysis.
1.
If T1331 > TH2’ then both heads will be in the ADP bound state at the same time,

27

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.4: A Chemomechanical cycle of kinesin. The letters represent the nucleotide
states of a kinesin catalytic core; E is for the empty state, T is for the ATP bound
state, D is for the ADP bound state, DP is for the intermediate state after the ATP
molecule is hydrolyzed. The dark solid oval represents head2 and the light solid oval
represents headl.

28

which has the weakest binding affinity to the microtubule and so the kinesin is likely
to fall off. Therefore processivity requires Tffl S T H2'

Given the inequality Tfkfl S THZ’ we next determine how Till and T H2
change with the length and the tension of the neck-linkers. There are two hypotheses
concerning the tension of the neck-linkers: the front-gated—head model [30, 77] and
the rear-gated-head model [31]. The front-gated-head model postulates that in the
two-head-bound state with the leading head in its empty nucleotide state and the
trailing head in its ADP bound state, the tension in the leading head neck-linker
prevents ATP molecules binding to the leading head until the trailing head detaches
from the microtubule. The rear-gated-head model [31] postulates that the tension
in the trailing head neck-linker favors the dissociation of the trailing head from the

microtubule. These hypotheses imply the following conclusions, respectively.

0 TAT P \ as the tension\

0 Td MT /' as the tension\

On the other hand, if the length of the neck—linker is longer, then we know

0 Tdi f f’u. Si 0n /' as the neck-linker length /'

Therefore we have
0 T291 /' and T H2 \ as the tension\, and the neck—linker length /'

These changes with respect to the tension and the neck-linker length may break
the inequality Till S TH2 and further induce the loss of processivity. To restore
. * . *
the balance, we could e1ther decrease TH1 or 1ncrease THZ' To decrease THI’
Td A D P ought to decrease as the tension is small. This conclusion so far has not
been supported by experimental results. To increase THZ’ we can increase either
TATP hydro or Tsz‘ or both. Under the condition of saturated ATP, the rate-

limiting step in the ATP hydrolysis process turns out to be the P2- release. Therefore

29

it is reasonable to postulate that TdP' is regulated by tensions. This is the other part
2
in the updated rear-gated—head model [85]. The tension of the neck-linker enhances

the release of Pi after the ATP molecule is hydrolyzed.
o TdP- depends on tensions and TdP' /' as tension \.
Z 2

Although these two models are not mutually exclusive, the front-gated-head model
has obtained more support, especially because it is more consistent with new data
[30]. Here our analysis shows that the rear-head—gated model should work with the

front-gated-head to guarantee the processivity of kinesin.

2.4 Biochemical Reaction Cycle of Kinesin

The biochemical reaction pathway of kinesin can be described as follows. E represents
kinesin and M is the microtubule. M-E means that the kinesin is bound to the

microtubule.

M-E+ATP=M-E-ATP:M-E*-ATP:: ( )
2.4

M-E*-ADP-P:M-E*-ADP+P7; :M-E+ADP

Here an ATP binding process is divided into two steps. First an ATP molecule
arrives and binds to the catalytic core weakly and is easy to dissociate. If this weak
binding induces a conformational change of the catalytic core, denoted by E*, then
the ATP molecule is trapped into a tight binding state. After the ATP is trapped,
it will go through the hydrolysis process. Theoretically every biochemical reaction
is reversible. Because some reverse reaction rates of the hydrolysis process are very
small, we ignore them and come up with the following biochemical reaction process

for our model.

30

 

 

 

 

 

 

 

 

Reaction rate s—T
U
EMT 6". 1
k AT P 3pM s
k AT P 150
k”. 700
2n)
kATPhydro 100
0
k d P, 120
deDP 300

 

 

 

 

Table 2.1: The reaction rate constants

k+ k.” k
M-E+ATP "EPMEATP 3PM,E*.ATP ATflzydro
ATP (2.5)

kdp.
M-E*-ADP-P JM-E*-ADP+Pikd’iPPM-E+ADP

The neck-linker zipping takes place in the step where the ATP molecule is trapped,

changing from the weak binding state to the strong.

The reaction rates used in the simulation are found from the other experimental

results [18, 91] and the authors in [95] didn’t measure these reaction rates.

Considering the regulation of the reaction rates by tension [32, 69], we adjust the

reaction rates according to k (1 MT = kgMTeng/KBT, k zip = kgipe—ng/KBT,

2
k , = k0 ech/KBT, where the force F is computed as a scaler by (2.9) and
6.11% dP,

62, i = 1, 2, 3 are the characteristic distances along the chemical reaction coordinates.

In the simulation 6% = 0.7, 63 = 2.0, and 6% = 1.0.

31

2.5 Tension Estimate of the neck-linkers

The estimate of the tension in the neck-linkers is critical in our modeling. The neck-
linker is considered as an entrOpic spring, either a freely-jointed chain (FJC) or a
worm-like chain (WLC) [32, 69]. When it is modeled as a FJC, the force-extension
relation is determined only by the entropic effects. This is given by

KT 7fi%%3-f%?2)

C

(B = B .
FFJC() b “(If , (26)

where b is the length of a monomer of the freely-jointed chain, a: is the extension of
the polymer, here computed as the end-to—end distance of the polymer, [C is the total
length of the polymer, [C = Nb for a polymer with N monomers, K B is Boltzmann’s

constant, and T is the absolute temperature.

If the neck-linker is treated as a worm-like chain which considers both the elastic
and entropic effects of the polymer molecule in the force-extension formula. Then we

have [69]

1
FWLC(‘”) = (,0 4(1_ $2 ‘ z + a , (27)

 

where 3p is the persistence length of the polymer related to the material property
and the shape of the cross section of the polymer. The persistence length of a polymer
is a quantity used to measure how rigid the polymer is. Given a polymer chain, if we
take a segment from it with the arc length (arc and the tangent angles of the two
ends of the segment are denoted by 61,62, then we have the following formula for

the correlation of these two angles [32, 69]

-€arc
(003(01 — 02)) := e 2619

32

 

 

 

 

 

 

 

 

 

 

 

Material types [C(nm) 25mm) sn(nm)

Wild type 5.7 1.4 5.29
0P 6.84 1.4 5.86

2P 7.46 2.10 7.33

4P 8.08 2.28 7.95

6P 8.70 2.43 8.53

13P 10.87 2.82 10.33

19P 12.73 3.05 11.69

26F 14.90 3.25 13.13
14GS 12.16 1.08 7.08

 

 

 

 

 

 

Table 2.2: The total lengths, persistence lengths and natural lengths of the neck-
linkers.

The correlation relation decays exponentially with the arc length (arc [32, 69]. In
this thesis, the persistence lengths of the native neck-linker, the proline insertion

, €pr0lzne = 4.4nm, and

and the GS insertion in the experiment are [fit = 1.4nm
£95 = 0.8nm, [95]. The persistence length of the microtubule is about 1 mm.

When there is no external force applied to a worm-like chain, its mean square

_1+££)
(p

The mean square end-to—end length of an F JC with N monomers is (R2) = N b2,

end-to—end length is given by [32, 69]

(2.8)

(R2) = 22% (e—tc/l’p

[22, 32, 69].

Molecular dynamics simulations [36] were used to estimate the internal tension
stored in the neck-linker and found to be about 12-15 pN when both heads are bound.
Calculation using (2.7) shows that the tension is 9.7 pN when the motor is in its two-
head-bound state with :1: = 8nm if we use K BT = 4.2 pN - nm and choose 15 as
the total number of amino acids comprising each neck-linker. The experiments in
[95] suggest that kinesin will easily dissociate from the microtubule when the applied

external force is larger than around 10 pN. Therefore, we truncate the force of (2.7)

at 25 pN.

33

In Table 2.2, the total length of one neck-linker is given for the wild type and each

of the mutants. The persistence lengths and natural lengths are also given.

In the supplemental documents of [95], the authors used the length of each amino
acid to be 0.38 nm and the length of a coil of the polyproline helix to be 0.31 nm
in their computation. They did not include the lengths of the three amino acids,
KKG, in their length estimate. In the thesis, we include KKG in the total length
computation. Therefore the EC values in Table 2.2 are neither the same as the values

in in Figure 2.1 nor the same as those in the supplemental documents of [95].

In the supplemental material of [95], they estimated the tension stored in the neck-
linkers when the kinesin is in two-head-bound state. Somemistakes can be easily
found in their computation. They used 63 = 11.4nm, €p = 1.4nm, a: = 8 — En =
8 — 2.05 = 5.95 z 6 and KbT = 4.1 in (2.7), En the natural length of the polymer,
and found 3.9 pN as the tension for the kinesin in two head bound state with the
neck-linker undocked. It is not clear how to arrive at 2.05 nm for Zn. One possibility
is that it is from V 29 x 0.382, which is the formula for the mean square end-to-end
length of a freely-jointed chain and 29 = 15 + 14 is the sum of the total number
of amino acids for the 14GS mutants. They followed the same idea to estimate the

tension when the neck-linker is docked.

It is easy to see that FFLC,WLC(KC) = 00. This is consistent with the en-
tropic nature of these forces because there is only one configuration when the exten-
sion equals the total length, corresponding to the minimal entropy state. Actually,
for the worm-like chains, the entropic effect becomes dominant when the extension
approaches to the total length. The entropic force is not a real force acting in the
polymer but a quantity to describe a tendency for the polymer to restore to the maxi-
mal entr0pic state. Therefore it is not accurate to estimate the tension of neck-linkers
by using the formulae (2.6) and (2.7) when the extension is large. A reasonable com-

promise is to cut off the force at 25 pN and to introduce the square root of the mean

34

square end-to-end length (2.8) as the natural length tn by modifying formula (2.7)

 

 

as follows.
0, 0 S S < [n
K T _.
3p 4(1_ 33—8 )2 4 dc - En
Zc-Zn
25, F(a:) > 25

where fp is the effective persistence length of the neck-linkers of the mutants and is

computed by

_ tg’wt t tgmutant
6],, = 6],” if Hg?“ an L7;— (2.10)

If two neck-linkers of the mutants are viewed as a whole worm-like chain, then the

natural length is calculated as the square root of (R2):

 

3n = <R2> = J22?) (e—2ec/Zp — 1+ %), (2.11)

which is shown in Table 2.2.

In our simulation, we model the motion of the tethered head in three-dimensional
space. The formula (2.9) only gives us the magnitude of the force. We use XHI to
denote the position vector of the tethered head and XH2 to denote the position vector

of the bound head. The extension 12 in (2.9) is equal to Euclidean norm of the vector

XH2 — XHI’ i.e., a: = [[XH2 — XHlll. The orientation is a = 11%.

Therefore the force vector acting on the tethered head is

F(XH1) .—_ F(a:)n (2.12)

35

 

 

 

 

 

 

 

 

 

 

Material types R(pN/nm) ln(nm)
Wild type 0.91 5.29
0P 0.91 4.92
2P 0.65 4.99
4P 0.50 5.20
6? 0.41 5.47
13F 0.25 6.98
19P 0.19 8.38

26? 0.14 10.05
14GS 0.56 6.21

 

 

 

 

 

Table 2.3: The spring constants and the natural lengths of the neck-linkers

The neck-linker can also be approximated as a linear spring, in which the restoring
force is the product of the spring constant and the extension relative to the natural
length. The spring constant can be estimated from the linearization of (2.7), using K.

0F :1: 3K T
to denote the s i co sta t, = ‘51—) _ = . A s in co osed of
prng n n K. :8 [513—0 ij— pr g mp
two different materials, with the spring constants rel and n2, has an effective spring

constant

a = H. (2.13)
K1 + n2

The formula (2.10) can be derived from (2.13) as follows. For the extended neck-

linker of the mutant kinesin, denote its effective persistence length by 8p. According

to (2.13), we have

3K T BKBT
w w mutant mutant
3KBT _ 2 ,, 2e, 6,,

C

 

 

 

harp" “ 3K T + 3KBT (2'14)
2 rev 151 222nutantegnutant

The formula (2.10) is obtained by solving (2.14) for 8p.
The engineered neck-linker of kinesin mutants can be idealized as a freely-jointed
chain composed of the inserted part and the native part that are treated as worm-like

chains, but freely jointed to each other. This treatment gives the following formula

36

for the natural length of the neck-linkers of the mutants.

 

in = \/2 (R2>wt + 2 (R2)mu (2.15)
0, 0 _<_ S S In

F613) = (2.16)
75(3) - In), :13 2 In

Using the above formulas, we have the parameter values for the spring constants
and natural length in Table 2.3.
For the wild type kinesin, we model its two neck-linkers as a whole worm-like

chain and the natural length is equal to

 

rift = lift = \J2612, (e_2€C/€P — 1+ £769). (2.17)

2.6 Algorithm

The whole simulation process can be described as follows. Here, C is used to denote

different random numbers with uniform distribution between 0 and 1.
1. Headl dissociation

0 Start from a two-head-bound state on the microtubule, with the leading
head (the head close to the plus end of microtubule) in its nucleotide free

state and the trailing head in its ADP bound state, see figure 2.4 and 2.5.

o Headl dissociation. Test for the random dissociation of headl from the
microtubule and ATP binding in head2. The neck-linker zipping induced
by ATP binding in head2 may or may not imply the detachment of headl

from the microtubule depending on the length of neck-linkers.

37

J
x“. _,../ plus end

——-1

'--~

x ’ ~ mlnus end

 

 

 

 

Figure 2.5: Illustration of the binding sites for the wild type kinesin. Three vertical
stripes represent three protofilaments of the microtubule. Assume that kinesin can
only bind to the sites on these three neighboring protofilaments. The oval with X
inside denotes the bound head, i.e., head2 in the algorithm and the dark head in
Figure 2.4. The oval represents headl in the algorithm, the light head in Figure 2.4.
The five forward binding sites for headl are represented by squares. The number
of binding sites for the mutants will increase depending on the reachable range of
headl of the mutants. Notice that the binding sites are arranged to reflect the helical
structure of the microtubule.

38

2. Tethered diffusion.

o Headl experiences a 3D diffusion process during which it might bind to a
rearward binding site. If so, this binding is weak and headl will detach
again. If headl diffuses to a forward binding position and binds to it, then
it can release ADP rapidly, bind to the microtubule strongly and completes

a step.

0 The movement of the tethered head is modeled by a Langevin equation.
Let XHI be the position vector of the tethered head. Newton’s second

law gives the equation for the motion of XHI’

.. . dW
"’le = —’)’XH1 + F(XH1) + ’/2KBT7—dt—, (2.18)

where m is the mass of the head and '7 is the drag coeflicient. gal/[1

represents white noise and W(XH1, t) is Brownian motion. F(XH1) is the
sum of the entropic force and any external force acting on the head. The
external force in our simulation is zero and the entropic force is computed
0—17

by (2.12). The order of magnitude of the mass is 1 and the inertial

10—10

time scale is defined as m/ry and m/y z s, which is so small that
the inertial term can be ignored. The above equation becomes a Langevin

equation

. dW
7XH1 = F(XH1) + 2KBT’7-C-i-t—. (2.19)

Solving this stochastic differential equation by using the Euler scheme, we

have the following iteration formula for (2.19) from tn to tn+1 = tn + dt,

x’fisfl = xifil + $F(X’I}I1)dt + \/2Ddt(Wn+1 — w”) (2.20)

39

where we have used Einstein relation D = Egg-I.

c When the distance between the tethered head and some binding site is less
than a given threshold, r cut 0 f f = 2.5, a binding probability is considered

by using the following formula:

e—fld2
p . . =
binding e-fld2(

c036, wild type

)2 (2.21)

c030 , mutants

where d is the distance between the tethered head and the binding site
and c056 = 121 - 122, where 111 is the orientation of the plus end of the
microtubule and U2 is the orientation of the binding site to head2. 6 = 1.5
is used in the simulation to approximate the swing process of headl induced

by the neck-linker docking.

Other binding probability formulas are tested. The following one is to

model the binding through the electrostatic attraction:

where
1, d S 0.1
Pd = _ c2 (2.23)
1— cle “B”, 0.13 d g 2.5,
and P0 = e_C3l3in(a/2)l, where o: = 0 + C4bw and w is a random

619

variable with a normal distribution of mean zero and standard deviation

one. c1 and 02 are chosen to make (2.23) a continous function. We used
c1 =1.3 and c2 = —4.2rcut0ffln(1/c1).

o The chemical state of head2 stochastically update. If it has not bound

40

an ATP, then continue to test for random ATP binding and then neck-
linker zipping in head2. The random test of ATP binding is as follows. If
C < kZTPIATP’dt is true, then an ATP molecule binds to the catalytic
core of head2, where [ATP] represents the concentration of ATP. If ATP
has bound, then test for random ATP hydrolysis and Pi release until head2

arrives at the ADP bound state.

0 When head2 is in its ADP state after ATP hydrolysis, kinesin may fall
off or just release the bound ADP molecule. If headl happens to be in a
rearward bound state, then head2 will either release the ADP molecule and
not unbind from the microtubule or stay in its ADP bound state for the
next update. If headl is in the tethered diffusion state, head2 will either
release the ADP or possibly unbind from the microtubule depending on
the relative position of headl to head2. Let the positive direction of y
in (x,y, z) coordinates point to the plus end of microtubule, (see Figure

2.5). Xilflun) and Xlgl2 are the y—coordinate of headl and head2 at time

.7! < y , . .
tn. If XH1(tn) _ XH2 + Cgattngec’ then ADP release 1n the head2 1S
considered, i.e., that random event is tested. We use 0 = 0.2 in

gating
the simulation. Otherwise unbinding of head2 will compete with the ADP

release in head2. If unbinding of head2 takes place, then the kinesin falls
off. If the ADP is released in head2, then a futile ATP hydrolysis cycle is
recorded.

3. Forward binding.
0 Pb’in ding is tetsted agaist a random number in [0, 1] and if it is greater,

then headl binds to the binding site immediately.

0 After headl binds to a forward binding site, it releases the ADP quickly

and then is in a strong binding state. Head2 continues the process from

41

step 2 and eventually arrives in the ADP state. Next ATP binding does
not occur in head] until ATP hydrolysis is complete in head2 and head2
is in a weak binding state. Now the headl and the head2 have exchanged

their trailing and leading roles and are ready for a new step.

To compute the average speed and run length of kinesin and its mutants, we run
5000 Chemomechanical cycles for each of them. All the continuous paths are detected
from those 5000 steps and assume that there are Npat h continuous paths. The run
number, Nrn, of a path is defined as the total number of steps in this continuous
path and the run length, Lrun, of it is defined as the total distance traveled. Three

different ways are used to compute the average speed. The first formula is given by

The total distance traveled in Np at h paths

 

V1 = (2.24)

The total time spent 1n Np at h paths
For each path, we find the corresponding run number and its median value of all

the run numbers, Nrflfdmn.

_ The length of the ith path with Nrn Z NTWLedian

 

Vi h (2.25)
The corresponding time for the it path
V2 = <Vi> ,over all the paths with Nrn Z Nmedian (2.26)

To compare the result of run length between the simulation and the experiment,
we follow the method used in supplemental material in [95] where the mean run length

is defined as follows.

Lmean _ ZLrunZSOOLrun

_ 2.2
run The total number of the paths with Lrn Z 500 ( 7)

 

The third way to compute the speed is

42

(2.28)

 

-th

v _ <The length of the ith path with Lmn 2 500>
3 The corresponding time for the i ,

path

The average speed calculated from these three different definitions are very close,

see Figure 2.6.

2.7 Simulation Results

First of all, the ratio of the speed of the kinesin to the concentration of ATP is
calculated and the result with the Michaelis—Menton chemical kinetics, (see Figure
2.7).

Mutants with extended neck-linkers have smaller tension when taking an 8 nm
step and so they tend to take more time to detach from the microtubule, according
to the rear-gated head hypothesis. Also the long neck-linker mutants can reach more
backward binding sites so that they may have more backward temporary binding.
More backward binding needs more time for them to dissociate from the microtubule.
Our result supports this conclusion, Figure 2.8 Showing clearly the average dwelling
time to be an increasing function over the mutants. Here the dwelling time is defined
as all the time when headl is not in the diffusion state in one Chemomechanical cycle.

Our detailed simulation of the stepping process of kinesin and its mutants with
extended neck-linkers has reproduced the experimental results qualitatively. There
are mainly two differences between our simulation results and the experimental out-
comes. The magnitude of the speed from the simulations is large although it does
clearly show the small speed for the mutants with the longer neck-linkers. The second
difference is that the speed of 14GS is larger than the speed of 19P and (or) 26P in
some simulations.

Speed in Figure 2.10 shows the decreasing tendency as is seen in the experimental

result. In our simulation, the speed among the mutants 13P,19P,26P and 14GS does

43

CD
CD
:3

Speed. nmls
0|
8

 

1 2 3 4 5 6 7 8 9
Wild type, 0P-26P polyllne mutants, 1463

<3
<3
C)

Speed, nmls
0|
8

 

5 6 7 8 9

1 2 3 4
Wild type, 0P-26P polyllne mutants, 14Gs

 

    

1 2 3 4 5 6 7 8 9
Wild type. 0P-26P polyline mutants, 1463

Figure 2.6: Speed of the wild type, 0P-26P and 14GS mutants computed by formulas
(2.24, 2.26, 2.28)

44

 

550

500*

450r

&
O
0

Speed, nmls
0)
8'

 

300

250

 

 

20 0

Figure 2.7: The speed of wild type kinesin vs ATP concentration (pM)

200

400

45

600

ATP concentration 11 M

800

1000

 

Average dwelling time

 

 

 

1 2 3 4 5 6 7 8 9
Wild type, 0P-26P polyline mutants, 1468

Figure 2.8: The average dwelling time of the wild type, 0P-26P and 14GS mutants

46

 

0.04

0.035 - 1

0.03 - 1

Average diffusion time
O
S

 

 

 
     

1 2 3 4 5 6 7 8 9
Wild type, 0P-26P polyline mutants, 1463

Figure 2.9: The average diffusion time of the wild type, 0P—26P and 14GS mutants

47

 

Speed, nmls
U
8

N
O
O

.5
O
O

 

 

 

1 2 3 4 5 6 7 8 9
Wild type, 0P-26P polyllne mutants, 1463

Figure 2.10: Speed of the wild type, 0P-26P and 14GS mutants

48

 

Coupling ratio

 

 

    

1 2 3 4 5 6 7 B 9
Wild type, 0P-26P polyline mutants, 1463

Figure 2.11: Coupling ratio of the wild type, 0P-26P and 14GS mutants

49

 

 

 

 

 

 

 

.
.. .. 2P
5 5 10
o o
O U 5
o 1 1.5 2 2.5
Run Length 11M
15 0.82pm
‘5 E 13P
3 3 10
o o
o o 5
o 1 1.5 2 00.60.8 1 1.21.41.61.8 3.5 1 1.5
Run Length 11M Run Length 11M Run Length 11M
10 15
20 0.76pm 0.69pm
‘15 15 19P E 26P E 10 1468
3 3 5 3
0 1o 0 o
o o o 5
5
0 0
0.6 0.8 1 1.21.41.6 0.6 0.8 1 0.6 0.8 1 1.2 1.4
Run Length 11M Run Length uM Run Length pM

Figure 2.12: Runlength of the wild type, 0P-26P and 14GS mutants. The mean run
length is shown in the insets.

50

 

 

  

  
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

zoo . . . 200 . e e
E 150’ E 150*
c c 100 i
g 100’ g
5 5 5° 1
o , , o
50 -——-WT 0 ——m .
—"'°" """'""2P
on 0.2 0.4 0.6 0.3 ”500 0.2 0.4 0.6 0.8
Time (second) Time (second)
200 200
150’ 150'
E E
c 100' i 1: 100’
8 8
E E
.9 so» [ .2 so
a o
o 0 "-""'WT -l
—6P
.. . -50 -
500 0.5 1 0 0.5 1
Time (second) Time (second)

Figure 2.13: Trajectory samples of wild type kinesin and 0P, 2P, 4P and 6P mutants.

51

 

 

 

 

      
  

Distance nm
3
0
Distance nm
a

 

 

    
  
 

—wr, —wr.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
  

 

 

 

 

 

 

 

—-— 139 -—- 19P
'500 0.2 0.4 0.3 0.8 1 '500 0.5 1 1.5
Time (second) Time (second)
250 ~ . - 200
200'
150’ «
E 150 E
8 0
1: 100 8 100.
.2 5.. is
O O 50,
0 ——wr . ““WT
50 . - —26P 0 _ ‘ —— 1468
o 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1
Time (second) Time (second)

Figure 2.14: 'Ifajectory samples of wild type kinesin and 13P, 19P, 26F and 14GS
mutants.

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3”-
. 150
‘3 32°" '3
O O o 100
0 U 100 0 50
0 0 0
8 10 12 14 16 6 8 10 12 14 16 6 8 10 12 14 16 18
Stepsize nm Stepsize nm Stepsize nm
150 60 13?
§ 100 5 ‘0
o o
o 50 U 20
0 0
10 15 20 25 10 20 30 10 20 30
Stepsize nm Stepsize nm Stepsize nm
L25P go 1464
.- 40 on
'c' 1: 1: 60
3 3 3
e on + o“
20
0 10 20 30 40 0 10 20 30 40 0 10 20 30
Stepsize nm Stepsize nm Stepsize nm

Figure 2.15: Stepsize histogram of the wild type and mutant kinesins.

53

 

 

 

 

 

 

 

 

400
‘5' E
o 200 3
O U
0 0
8 1O 12 14 16 10 20 30 40
Stepsize nm Stepsize nm
Illfl
@ ,0
.5 100 E
8 50 8 20
0 0
10 15 20 25 30 10 20 30 4O
Stepsize nm Stepsize nm
60
80
‘g‘ 40 g 60
8 20 8 :3
0 o
10 15 20 25 30 10 15 20 25 30
Stepsize nm Stepsize nm

Figure 2.16: The histogram of stepsize from the simulation results for the wild type,
6P, 13, 19P, 26P, and 14GS mutants. The histograms of experimental results are
shown in Figure 2.15.

54

 

.. “‘TTZ’TI‘TTTZ‘TTTTT 13F

 

time (sec)

Figure 2.17: Trajectory samples of wt, 6P, 13P, 19P, 26P, and 14GS from the experi-
mental results. These trajectories have more or less the same slope because they are
obtained from different ATP concentrations. See the simulation results in Figure 2.18

55

 

350 *

Distance nm

    

 

—-— wr
--— 6P
— 19P
26P
1468

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Time (second)

 

 

 

 

 

 

Figure 2.18: Simulation results for the trajectories of wt, 6P, 13P, 19P, 26P, and 14GS.
The same ATP concentration, 1 mM, is used in the simulation. See the experiment
trajectories in Figure 2.17

56

not decrease dramatically and seems more or less to be the same. One of the possible
reasons is that the tension of 14GS calculated by (2.9) is actually larger than that
of 19P and 26F and this large tension comes from the entropy effect of 14GS since
it is softer, i.e., with a small persistence length. We also observed from Figure 2.2
that the 14GS mutant has more small step lateral walking. The extended neck-linker
length of 14GS is 12.13 nm, very close to 12.76 nm, the neck-linker length of 19P,
but the histogram, Figure 2.15, of 14GS is different from that of 19P where 14GS
does not have a peak at 24 nm. We tested an idea to reduce the entropy effect
so that the speed of 14GS may become small. We use the binding probability of
14GS Pb’in ding = e_ad2 which only considers the attraction distance without
considering the orientation term 0036 in (2.21). This consideration comes from the
observation of the histogram of 14GS in which there are more small lateral steps
because the soft 14GS segment in the neck-linker makes the mutant much more flexible

so that it can more likely bind to the lateral binding sites. With this modification,
the speed of 14GS becomes smaller than the speeds of 19P and 26P mutants.

The trajectory samples are shown in the Figures 2.13 2.14 and 2.18. Each sample
represents the trajectory of a ten-step run of one head of the kinesin. Because those
samples are taken randomly, the slopes of these trajectories do not represent the
average speeds of each type of kinesin, which is shown in the Figure 2.10. The results
in the Figures 2.13 and 2.14 indicate that the speed of wild type is greater than that

of the mutants.

The histograms of step sizes of all motors are shown in the Figure 2.15. The
simulation results are shown in the Figure 2.16. The simulation results are consistent

with the experimental results when any backward walking is ignored.

57

2.8 Discussion

In the front-gated head hypothesis, it is suggested that ATP does not bind to the
empty front head until the rear head dissociates from the microtubule [77]. If we
assume that the movement of the tethered head is purely diffusional, then this front-
gated head assumption may lead to a forward step of kinesin without consuming
an ATP molecule. Indeed, a kinesin head can be seen as a sphere of radius around
3 nm. According to the Stokes’ law [32], the diffusion constant of such a sphere
is around D = 6.7 x 107 nm2/s, which is very close to a reported experimental
measured value D = 2.24 x 107 nm2/s. For wild type kinesins, they mainly walk
on the axis of a protofilament so the walking can be modeled as one dimensional
diffusion process. For a 1D diffusion, if the particzle is not subject to external forces,
the first passage time [32] is equal to t fp = %2, which gives the average time
for the tethered head to diffuse through the distance dstep and bind to a front
binding site, completing one step. Next we examine what the probability is for the
motor to bind an ATP molecule during the period t fp' In consideration of the
diffusion constant, the time step size used in the simulation is alt = 10—8 second.
Correspondingly, the first passage time t f]? needs around N step = [fill] = 1143
time steps. On the other hand, on average, the probability of an ATP molecule to
bind is q = dt[ATP]kXT P = 10"8 x 3 x 103 = 3 x 10"5 when [ATP] is 1000,1114
and kj‘IT P takes the value in Table 2.1. Given N step 2 1143, the probability for the
kinesin to bind one ATP molecule is 1— (1 —q)NSt‘3P+1 = 1— (1 —3 x10-5)1144 =
0.034. This result indicates that the diffusion is so fast that the kinesin could have
finished one forward step even without an ATP binding, not to mention neck-linker
docking. Block in [11] pointed out that it takes less than 100113 for a kinesin to finish
a diffusive search process. A time of 100113 corresponds to 10000 iteration steps.

Given N step = 10000, the probability for the kinesin to bind one ATP molecule is

58

1 — (1 — q)10001 = 1 — (1 — 3 x 10_"5)10001 = 0.26, which is much larger than
0.034 but it is still not large enough. Therefore this simple calculation suggests it is
unlikely for the empty front bound head not to bind ATP molecules until the rear

head unbinds from the microtubule and starts a diffusive process.

To solve this puzzle, we may assume that the front bound head of the kinesin
begins ATP binding before the rear head detachment. The timing point may be
reasonablely assumed at the moment when the Pi is released and the rear head is
in the weak binding state. This assumption does not conflict with the result from
[56] where kinesin spends most time of a cycle in a two—head-bound state and it
quickly moves to the next front binding position in company with the ATP binding
when there is a high ATP concentration. Certainly another scenario is that the rear
head remians parked somewhere after it unbinds from the microtubule and rapidly
swings to a forward site with a force provided by the neck-linker docking that is
induced by ATP binding. This is the polymer gating mechanism in [1, 7, 27] where
the tethered head parks in front of the microtubule bound head and does not bind to
the tubulin until an ATP binds to the bound head. But the data in [56] suggest that
the tethered head parks behind the bound head instead of in the front of it. Surely,

more experiments are expected to elucidate the details of the polymer gating.

The movement of the tethered head is believed to involve a swing process induced
by the neck-linker docking. The experiments [56, 95] suggest that kinesin spends
most of the time in a two-head-bound state and it quickly swings to the next binding
site when the neck linker is zipped induced by the ATP binding. To model this
process, we suggest an idea where an overdamped beam equation may be employed
to describe this swing movement. We consider the neck-linkers as an elastic beam, in
which the potential is stored when the kinesin is in the two-head-bound state in which
we postulate that the neck-linkers are somehow twisted. And this is where the stored

potential energy may reside. When the tethered head unbinds from the microtubule,

59

the released potential, coming from the release of the strain, immediately changes the
orientation of the tethered head such that it cannot easily bind back to the initial
site again and throws the tethered head to the next front binding position. We may
set the root of the neck-linker in the bound head at the arclength 0 and the the
root of the neck-linker in the tethered head at arclength to. We need to determine
the appropriate boundary conditions to describe these two end points in the swing

process. This is a future direction of our work.

60

Chapter 3

Interactions between Microtubules

and Molecular Motors

3.1 Essentials of the Model

We model microtubules as stiff polar rods of equal length l exhibiting anisotropic
diffusion in the plane. Diffusion of the rod is characterized by three diffusion coeffi-
cients, diffusion parallel to the rod orientation D“, perpendicular to its orientation
DJ.’ and rotational diffusion Dr. In the following we assume D“ = 2D J. [22] for

stiff rods diffusing in a viscous fluid.

The key ingredient in the theory proposed in Refs.[2, 3] was the approximation of
the complicated process of interaction of one molecular motor with two microtubules
by a simple instant alignment process, see Fig. 3.1. We focus on the two dimensional
situation, and describe the orientation of microtubules by the planar angles (p12.
The microtubules before the collision posses initial angles to??? The action of the
molecular motor binding simultaneously to two microtubules results in their mutual

alignment, and the angles after interaction become

61

b b
«p‘f = <23 = ——801:‘p2- (3-1)
By analogy with the physics of inelastically colliding grains, we call this kind of
process fully inelastic collision (see e.g. [4]). Such an inelastic collision is a simple and
reasonable approximation of the complicated interaction process [3], and is, in fact,
an effect of simultaneous action of several motors or motors and static crosslinking
polymers. An analysis of the interaction of two stiff rods with one motor shows that
the overall change in the angle between the rods is rather small: the angle decreases
only by 25—30 % on average [3, 43]. However, simultaneous action of a static crosslink,
serving as a hinge, and a motor moving along both filaments results in a fast and com-
plete alignment of the filaments [97]. This justifies the assumption of fully inelastic
and instantaneous collisions for the rods interaction. Complete alignment also occurs
for the case of a simultaneous action of two motors moving in the opposite directions,
as in the experiments with kinesin-NCD (gluththione—S—transferase—nonclaret disjunc-
tional fusion protein) mixtures [81]. The same is true of two motors of the same type

moving in the same direction but with different speeds, where the variation in speed is

due to the variability of motor properties and the stochastic character of the motion.

We set the molecular motor concentration m to be uniform in space. This assump-
tion is not satisfied in reality in the presence of defects such as asters or vortices. Since
the motor convection along the filaments is faster than thermal diffusion, the motors
tend to congregate at the aster or vortex centers. However, as shown in [3], the motor
inhomogeneity has only a quantitative effect on the self-organization process near a
phase transition and does not affect the qualitative features, such as the morphology
or the phase diagram and the nature of the transitions. If it were necessary, however,
the effects of large-scale motor distribution inhomogeneities could be easily incorpo—

rated into our model by modification of the interaction rules according to the local

62

 

 

 

 

 

a b

Figure 3.1: Schematics of an alignment event (inelastic collision) between two micro-
tubules interacting with one multi-headed molecular motor. The black dots represent
the center of mass of the microtubules. (a) A multi—headed molecular motor clus-
ter attached at the intersection point of microtubules moves from the negative (——)
towards the positive (+) end of the microtubules. (b) After the interaction, the orien-
tational angles <p1,2 and the corresponding positions of the midpoints R12 become

aligned.

63

density and orientation of microtubules. At the same time, the assumption of motor
homogeneity significantly simplifies the calculations. In the future we plan to study

the effect of motor convection explicitly.

This assumption is justified by a large value of the motor diffusion Dm compared
to the microtubule diffusion, Dm z 20pm2/s z 500D”. Due to the advection of the
motors along the microtubules there is some accumulation of motors at the centers of
asters and vortices, (see e. g., [63]). However, for a typical motor speed of v z 1pm/s
and the microtubule length l x 1pm, the (effective) Péclet number Pe = % z 0.05,

implying that diffusion dominates advection.

The motor concentration m affects the probability of interaction Pint between two
microtubules in a given period of time At. In the following we set Pint ~ o2mAt,

2

where o is the effective interaction cross-section. The value of a z 30 — 50 nm, the

order of size of a kinesin-type molecular motor.

In addition, we assume that the interaction probability Pint depends on the po-
sition of the intersection point, see Figure 3.1. The intersection position is indicated
by the signed distance .912, —% S 312 S %, from the midpoint of each rod. Ac—
cording to Refs. [2, 3], due to the polar nature of the microtubules, the dependence of
the interaction probability on the intersection (and, correspondingly, the attachment
position) results in an anisotropic probability kernel in the collision integral in the
master equation. The anisotropy of the kernel, characterized by the parameter 6,

ranging between -1 and 1, depends on the motor dwelling time t which is small

end ’
for kinesin-type motors and large for NCD-type motors. In Ref. [3] the relation be-
tween the kernel anisotropy 6 and the motor dwell time at the end of the microtubule
tend 1n the 11m1t of tend >> 1 was estlmated as follows: 6 ~ (v—const/tend)/p0ff,
where v is the motor speed, 10 0 f f is the motor unbinding rate. Thus, one sees that

6 increases with the increase in t en d' In contrast, the motor attachment rate P077.

has little effect on the kernel anisotropy, in agreement with experiments [62, 63]. As

64

was shown in Refs. [2, 3], the anisotropy parameter controls the transition between
asters and vortices; in the continuum model no vortices were observed for large values

of the kernel anisotropy.

In order to accommodate the anisotropy effects, that is, dwelling of the motors
at the end of microtubules, in our model we introduce the following dependence of

interaction probability on the attachment positions:

+
Pint— _ P0 (1+ 51—1182). (3.2)

Here the parameter P0 = CmoQAt encodes the aforementioned dependence of the
interaction probability on the motor concentration m, the interaction cross-section o
and the elapsed time At (here C is a constant). Since a is a fixed physical parameter,
and At is fixed throughout our simulations (see below), the range of parameter values
0 < P0 3 0.5 reflects different values of the uniform motor concentration m. The
value (and the sign) of the other parameter, 6, depends on the type of motor. We
believe that this generic linear dependence on the distances 31,2 captures the qual—
itative effects of the kernel anisotropy. Our experiments with different dependencies

of the probability Pint on 31 2 yielded qualitatively similar results.

After the interaction we postulate that not only the angles, but also the midpoint

positions of the microtubules R1 2 coincide
7

9...,

a
R1=R2= 2

(3.3)
This approximation is reasonable in the case of large dwelling times t en (1 of the
motors, which guarantees that after the interaction the end points of the microtubules
will coincide. Then, together with the alignment interaction, this effect will justify the

assumption on the alignment of the midpoints as well. A large value of the dwelling

65

t1me t en d IS a reasonable approx1mat1on for NCD motors, however t en d 1S small for
kinesin-type motors. As we will show later, the midpoint alignment assumption may
produce under some conditions specific effects, such as layering of the microtubules,
or “smectic ordering” [20]. In our future work we plan to introduce more realistic

rules for the midpoint displacements.

3.2 Algorithm Description

We performed simulations on a two-dimensional square domain with periodic bound-
ary conditions. Initially, microtubules are randomly distributed over the domain. At
each time step, (e.g., from tn to tn+1)’ the update of the positions and orientations
of the microtubules is comprised of one substep processing anisotropic diffusion and

one substep processing inelastic collision. The total timestep size was set at At = 0.1.

The diffusion of rigid rods in a viscous fluid is characterized by three diffusion
coefficients, parallel D” , perpendicular DJ. and rotational Dr. We used the follow-
ing relations between the diffusion coefficients from Kirkwood’s theory for polymer
diffusion in three dimensions, D” = 2D_L, D7- : 262D”. We used c = 1.5 and l = 0.5
in our simulation. For a three—dimensional fluid the coefficient c ”~V 24 [22]. However,
the value of c rapidly decreases for quasi-two dimensional thin film and membranes,
see [53] We verified that the value of coefficient c does not change the qualitative
behavior of the system, it affects only the position of the transition points. The
“diffusion substep” is introduced as an anisotropic random walk of the microtubules’
center position R = (:r,y) and random rotation of its orientation go. The positions

and orientations are updated at each such substep as follows:

66

Rn+1 = Rn+C1AllUn+C2AiNn

<Pn+1 = 90n+43Ar (3-4)

where Ci 6 (—0.5, 0.5) are three uniform random numbers generated each time,
and A7; = M, where Di is either D _L ,D“ or Dr, and vectors Up, =
(cos gon,sin (pn), Nn = (—sin (pn,cos (on) are directed along (U) and perpendic-
ular (N) to the orientation of the microtubule. The factor 24 in the expression for At

ensures that the effective diffusion has the correct value of D7: based on the variance

At the “collision step”, after diffusion, we check whether any pairs of microtubules
intersect, and if so we locate the intersection points of the microtubules and assign an
interaction probability to those pairs according to (3.2). In all of those intersections,
some of them are simple binary intersections, but others may be multiple intersections,
that is, a microtubule intersecting with more than one other microtubule. Certainly,
at a low density of microtubules binary collisions are more typical. Regardless of
whether intersections are binary or multiple, we calculate all interaction probabilities
and sort them in descending order. Starting with the greatest Pint’ we compare
it with a randomly generated number (4 6 [0,1]. If Pint > (4, then we update
that pair of microtubules according to the collision rules Eqs.(3.1) and (3.3). If
either of these two microtubules had other intersections, they are ignored, that is,
these interaction probabilities are set to zero. We then proceed with the next largest
interaction probability, repeating until all have been acted upon. Note that the
diffusive substep size coincides with the total timestep size, so that collisions are

assumed to take pace instantaneously.

67

3.3 Coarse-grained Variables

In our simulations, the rods move freely within the domain and with fluctuations in
both position and orientation of the rods, it is difficult to identify relatively stable
patterns. For this reason, and as an aid for computing divergence and curl, we
impose a square grid on the domain with the mesh length d and introduce a coarse-
graining procedure to extract observable values, such as the local orientation 7' and
local density p. Using W to denote the two-dimensional position vector of a grid
point (Xi’ Yj)’ we calculate the number of rods N whose midpoint positions are in
the box [Xi — d, X2- + d] x [Yj — d, Y- + d]. The following coarse-grained functions

.7
are employed to compute 7' and p at this grid point(Xi, Yj):

2],; @(IW — Rk|)Uk

zivzlwwmknvk
p(W) = N (35)

 

T(W) =

Here | - | is Euclidean length and d) is a weighting function. We take

0(3) = e"? (3.6)

where l is chosen to be the length of each microtubule. In the simulation we also
have to include contributions from “image particles” originating from the periodic

boundary conditions.

3.4 Pattern Characterization

We computed the discrete divergence, p = V - T, and the curl, to = V x 7', of the

coarse-grained field of the pattern from the last 3000 iterations. Here p and to depend

68

‘ ’. l
((7- M37
. “yr.
'41
-x..'.‘.

. . 813'-
akin“
"i

1.1mm

 

Figure 3.2: Snapshots illustrating the patterns developing in a configuration of 6, 000
rods for different motor densities, i.e., different values of PO. Arrows represent mi-
crotubules, circles depict the cores of vortices or asters. (a) vortices, t = 620 ,
6 = 1.0,P = 0.08 (low motor density); (b) asters, t = 602, 6 = 0.95,P = 0.10,
(high motor density); (c) bundles, t = 400, 6 = 1.0,P0 = 0.15. See also [100] for
movies # 1,2 illustrating the self-organization process.
on the mesh size of the coarse-grained field. By using the central difference scheme,
the extrema of p and to can be —4 or 4 for an ideal aster or vortex under the 40 x 40
grid on the 20 x 20 domain.

The basic idea for pattern characterization is that an aster would have its local
divergence greater or less than a threshold value at the center. Similar observations

apply to a vortex and its curl. To realize the pattern characterization, we implemented

the following procedure:

0 Firstly, using the snapshot at t = 700, determine the local extrema of p and 0.)

with values sufficiently far from zero. Specifically:

1. Compute the minimal value of the divergence 11. Suppose that it occurs

at (M);

[\D

. Eliminate the surrounding square area consisting of (2q + 1) x (2q + 1)
mesh points. We chose q = 4 in our computation, that is, temporarily set
p(k,l) =0,i—4S k Si+4,j—4 S l Sj+4. Locate the next minimal

value of p from the remaining region;

69

3. Repeat step 2 on the remaining region until p > —2.5;

4. Use the above three steps to find the maxima of p with p > 2.5;

5. Go through steps 1 to 3 to locate the local minima of to with w < —2.5;
6. Apply a similar procedure to find local maxima of to with w > 2.5;

7. If two of the selected extrema of [p] and |w| occur in one of the selected
squares, then we discard the square that is not centered at a point where

the greater of [pl and Iw| occurs.

0 Secondly, take the local square area consisting of (2q +1) x (2q+1) mesh points
for each of the extracted locations, and compute four quantities, “min’ (Imam, ”min
and wmag; in this patch for each snapshot from t = 401 to t =-—- 700.

e Thirdly, compute the time averages of those four quantities for each patch
from these 300 snapshot values, denoted as fim’in = 36-0 232% [pmin(i)|,
Hmaa: = 311m 222% #maxfl). 0min = 135-0 232% lwminml’ and 57mm; =
3&0 Eggg wmax(i). To distinguish between a vortex and an aster, we intro-
duced an additional parameter 6 = 0.6, whose use is explained below.

0 Finally, to determine the type of pattern in each local square area, we decide

according to the following criteria.

— If flmin 2 3.0, Emax < 3.0, 5min < 3.0 and 71mm: S éflmin, it is
an aster.

— If 6min 2 3-0,&7ma:1: 2 3-0 (5min _>_ 3-0), 5min S 66mm (amass S
{fimZ-n) and flmax 5 @min’ it is an intermediate form between an

aster and a vortex and we assign it an aster-vortex pattern.

— If :4.me 2 3.0, Umag; < 3.0, 5min < 3.0 and 72min S éflmax, the

directions of the rods point outwards and it is an anti-aster pattern.

70

— Ifflmax 2 3.0, Gmag; 2 3.0 (norm-72> _ 3. O), wmin < Email: (wmax <
@mam) and fimin 5 @max, it is an antiaster-vortex pattern.

— If 3min < 3.0 and 'flmax < 3.0 and 5min 2 3.0 (Umax S $777,271) or
Umax 2 3.0 (5min S {D—mam), then it is a vortex pattern.

— in any other case, it is isotropic.

The parameter space (P0,6) is in the range 0.01 3 P0 g 0.15 and 0.0 S 6 _<_ 1.0.
We made a grid with stepsizes APO = 0.01 and A6 = 0.05 so that we had 15 x 21 =
315 mesh points. For each pair of values, we used three different initial conditions
for the simulations, using the characterization of the final states described above. We
obtained the numbers of asters Na, aster—vortices Nay, antiasters Naa, antiaster-
vortices Naa'u and vortices N1) for each (P0,6) and we found that Naa = 0 and
Naav = 0. At each parameter grid point we computed two values according to the

following formula:

1i+IJ+1
N1: 2( Na+Naa) (3.7)
1i— lj— —1
1 z+1j+1
11411013) N— Z ZUV’U) (38)
li—lj—l

For the boundary points, the summations in (3.7) and (3.8) are taken only over the
neighboring points around (i, j) within the parameter domain. N1 in (3.7) and (3.8)

is the number of points in the summation. From Ma(i, j) and Mru(i, j), we calcu-

lated 'm(i,j)= M a“ ”11"“? (7: j) and 7v(i,) j) =M aGAfi’Sfiflva, 3) Finally we

generated a matrix, 1p, whose entries give the pattern information at that parameter
point.
e If :1)y(Ma(i,j) + MrU(i,j)) g 1.5, then it belongs to disordered region and
Ip(i,j) = -—1.0.

71

o If 7a(i, j) 2 0.6 and 'yrU(i,j) S 0.4, then it belongs to aster region and Ip(i, j) =
1.0.

o If 'm(i,j) S 0.4 and 7v(i,j) 2 0.6, then it belongs to vortex region and

Ip(i, j) = 0.0.
0 Otherwise, it belongs to a transition region and Ip(i, j) = 0.5.

We used Ip matrix to produce a pseudo—color phase diagram. The pixels with
Ip(i, j) = 1 are assigned red, the pixels with Ip(i, j) = 0.5 yellow, the pixels with
Ip(i,j) = 0.0 green and the pixels with Ip(i,j) = —1.0 blue.

To identify the bundled region, we calculated the density of the rods at each
grid point, which is defined as the number of rods whose positions are in the square
box with the grid point as the center. Next we computed the global minimal and
maximal density in the domain at each time slice. Those minimal and maximal
densities were averaged over 300 slices and then over three samples, i.e., fimin =
23(300213-0 31pmmc» Pmaas— — 323(310 23.11131 Pmaac( )1 If pm,” <
0.2 and Pmax > 60, then this point is marked bundled. In the bundled region the
rods formed several, with these stripes sometimes forming concentric circles. Moreover

asters appear to dominate vortex structures.

3.5 Simulation Results

We applied our model to 6,000 microtubules in a 20 X 20 domain varying parameters
P0 and fl in a wide range with 7,000 time steps in the simulation for each choice of
(P0, fl). It took approximately 69 minutes to complete each run on a SGI Altix 3700
3x2 with 1.6GHz ItaniumZ processors, which is an improvement over the explicit
dynamics approach of [81].

We impose a 40 x 40 grid on the domain to calculate the coarse-grained field.

72

A snapshot was taken every 10 iterations and so 700 snapshots were obtained for
each simulation process. For most of the parameter values chosen, it took about 300
snapshots (3000 time steps) to relax towards relatively stable large scale patterns,
and more than 500 snapshots (5000 steps) to become stationary. Some simulations
clearly shown a pattern of asters and/ or vortices while others resulted in ambiguous
patterns. Moreover, the clear-cut distinction between asters and vortices appears to
be difficult because of fluctuations. To examine the parameter space (P0, 0) where
there are transition regions between asters and vortices, we have devised a pattern
characterization scheme. The simulation results obtained from the first four thousand
iterations were ignored as they represent transient states. For the last three thousand
frames of data, we performed the pattern characterization algorithm given in the last

section.

Select simulation results are shown in Figs 3.2, 3.3, 3.4 where D H = 1/ 120 in all
simulations. In agreement with the experiments [62, 81] and the theoretical models
[2, 3], we obtained an isotrOpic phase for low motor densities (not shown), and then
vortices, transient aster-vortices (structures which resemble vortices near the core and
aster far from the core), asters, and bundles with gradual increase of the motor density.
Representative snapshots of the rod configurations for three different values of the
motor density P0 are shown in Fig. 3.2 and the two corresponding coarse-grained
snapshots of them superimposed with the rod density field are shown in Fig. 3.3. As is
evident from our simulation results, a transition from an isotropic (disordered) phase
to an oriented phase happens with the increase in the motor density characterized by
the parameter P0. While due to the small size of the system (only 6,000 particles)
we have very strong fluctuations in the number of vortices, asters and anti-asters
(structures similar to asters but with the opposite orientation of microtubules; see Fig.
3.4), a general trend can be identified: with the increase in the interaction probability

Po the average number of vortices decreases while the number of aster increases. For

73

small values of the anisotropy parameter ,8 asters and anti-asters appears to occur
with equal probability. However, with the increase in fl the number of anti-asters
rapidly decreases while the number of aster increases. For very high motor densities
we observed an additional instability resulting in the formation of dense bundles of
filaments with the same orientation (see Fig. 3.20). The bundles are also associated
with a certain layering (smectic ordering) of the filaments. This ordering is due to the
microscopic interaction law which results in the alignment of the rod midpoints as in
Eq. (3.3). While this might be the case for the NCD motors with a large dwelling
time, for the kinesin motors the bundles may have a different structure which is not
necessarily captured by these simulations. These results are in good agreement with

earlier theoretical predictions [2, 3].

The phase diagram delineating various regimes of self-organization is shown in
Fig. 3.5. It bears a strong resemblance to the experimental observations [62, 81]
and the theoretical model of Refs. [2, 3]. While the boundaries are quite blurred
due to strong fluctuations (see Fig. 3.4), there is a transition from vortices to asters
with the increase of the interaction rate P0. Due to strong fluctuations, pattern
characterization is rather difficult, and even sometimes ambiguous. In particular,
we often observed anti-aster, i.e. structures with the orientation of rods opposite
to that determined by the motion of the motors. Thus, when we calculated the
phase diagram, we had to take into account the number of anti-asters and anti-aster-
vortices. We also noticed that the rod density in aster regions is greater than that
in anti-aster regions. Specifically we found that there were about 80 more rods on
average in an aster region than in an anti-aster region. Moreover, the domain of
stability of vortices decreases with the increase of the anisotropy parameter [3 related
to the dwell time of the motors, as observed experimentally and in agreement with
the continuum model of Refs. [2, 3]. However, we need to emphasize that all the

boundaries shown in Fig.3.5 are rather blurred; instead of sharp phase transitions we

74

 

Figure 3.3: Coarse-grained images corresponding to parameters of Fig.3.2 for vortices
(a) and asters (b). Arrows represent the orientation field 7'. The color (grey levels)
shows the density ,9, red (bright) corresponds to the maximum of p, and blue (dark)
to its minimum. See also [100] for movies # 3,4.

observed only smooth crossovers between different regimes due to strong fluctuations
and relatively small number of particles in the system It is known that in related
two—dimensional X Y models there is no well-defined second order phase transition
from isotropic to ordered phase; the mechanism is related to unbinding of Kosterlitz-
Thouless vortices by fluctuations, see[29]. However, sharp phase transition occurs in

three dimensions.

The coarse-graining allows for easier identification of aster and vortex structures
(see Fig. 3.3). In the movies made using coarse-grained fields we are able to follow
the formation, interaction and evolution of asters and vortices. A typical scenario of
the dynamical evolution of the system is that small vortices and asters can coalesce
to form a larger vortex or aster (see the movies in [100] for parameters P0 = 0.12,
)6 = 1.0). In accordance with the experiments, vortices have suppression of the

microtubule density in the center (holes) and asters lead to an increase of the density

75

 

g‘ \ I T l I I

‘1’: - Q\ o
- I \\ X G O VOTIICES «
,’ 7Q Q G +1 asters
5 J \ \ <> <> anti-asters _,

 

 

 

 

int

Figure 3.4: Averaged number of asters (squares), anti-asters (diamonds), and vortices
(circles) as a function of the interaction probability PO for two different values of
parameter ,6. The data for 6 = 0.35 is shown in dashed lines, open symbols, and for
6 = 0.95 is shown in dotted lines, closed symbols.

76

 

 

Figure 3.5: Phase diagram of various regimes as a function of the motor density
Po (the horizontal axis) and the anisotropy parameter 6 (the vertical axis). The
disordered region is blue (black) here; the vortex region is green (grey); the transition
from vortex to aster happens at the yellow region (white) and red (dark grey) denotes
aster regions. The dashed line denotes the boundary where the rods become bundled.

77

of microtubules. We have also observed the transformation of vortices into asters in
the course of the simulations, likely due to fluctuation and fine size effects.

We followed the trajectory of individual rods in the vicinity of the vortex core in
the steady-state. We have found that the particles generally do not rotate around
the vortex core. This stems from the fact that in our binary collision algorithm the
center of mass of two interacting rods is not displaced in the course of collision, Eq.
(3.3). This restriction suppresses directed motion of the rods, and, consequently,
global rotation. Thus, the rotation of microtubules seen in experiment [62] is likely
related to the interaction with the substrate or the boundary of the container [3, 49],
or, possibly is related to multi-particle interactions and anisotropic interaction with
the fluid [50] neglected in our model.

In our simulations we also observed that the centers of the asters typically exhibit
a drift, reminiscent to the acceleration instability of aster cores predicted in Ref. [3].
This phenomenon especially appears at the stage of formation of asters. However,
the precise nature of the drift is still an open question since it could be also due to

fluctuation effects.

3.6 Conclusion

A Monte Carlo type stochastic approach has been developed conduct the study of
self-organization of microtubules mediated by molecular motors. The approach allows
us to bypass the fast time scales associated with the diffusion and the motion of
individual molecular motors and concentrates on the relevant features of the long-
time and large-scale behaviors associated with the self—organization phenomena.
While a direct comparison with the earlier algorithms introduced in Ref. [63] is
not always possible due to the different nature of the approximations, some rough

estimates are useful. The total simulation time reported in Ref [63] was 1500 sec.

78

The characteristic time scale of the simulations of the order of 1 sec can be inferred
from the density of microtubules (about 0.05 rim—2, or about 500 microtubules in
a box 100 x 100 microns) and the motor diffusion ( D = 20pm2/s), which roughly
corresponds to 103 dimensionless units of time. Our simulations, performed with
much higher number of microtubules (6000) and in bigger boxes were performed for
about 1000 dimensionless time units, that is, about the same order of magnitude as
in Refs. [62, 63].

Our method can be easily adapted to new experimental settings, such as a mo-
tor and microtubule system with a fraction of the motors permanently bound to
the substrate [49]. Our results are complimentary to the analytical studies of self-
organization in the framework of amplitude equations derived from the stochastic
master equations, and provide valuable tests for a variety of phenomenological con-
tinuum theories of cytoskeleton formation [51, 55, 50, 92]. Moreover, our simulations
shed a new light on the microscopic details of self-organization not available in the
continuum formulation. We anticipate that somewhat similar approaches can be ap-
plied to a broad range of systems, such as networks of actin filaments interacting with
myosin motors [35], patterns emerging in granular systems with anisotropic particles

[10, 48, 6, 60, 5], and systems of self-propelled objects [99, 28].

79

Chapter 4

Summary and Future work

In the first part of the thesis, we have developed a set of algorithms which tightly
follow the mechanochemical transition process of kinesin motors. The mechanical
moving process is described by a 3D Langevin equation, solved numerically by the
Euler scheme of stochastic differential equations. The chemical reaction process is
simulated by a Monte Carlo method. These two processes are coupled in the simula-
tion by following the consensus walking model of kinesin so far obtained by biologists
(See [11] and reference therein). We carried out a detailed simulation of the walking of
the wild type and its mutants with extended neck-linkers and obtained results in line
with the experimental results [95]. In this process, we discussed different approaches
for the estimate of the tension in the neck-linkers by using models from polymer
science. We explored the binding mechanism by working out and testing different
binding probability formulas for the tethered head. In our analysis of the processivity
of kinesin, we also clarified the role of the front-gated-head and the rear-gated-head
hypotheses in the regulation of the processivity. Our conclusion is that both of them

should work together.

Based on the algorithm developed, we can further take into account the backward

walking of the kinesin which is not included in our model because the probability of

80

backward stepping is small. Furthermore, it is interesting to derive an overdamped
beam equation to describe the swing process of the tethered head triggered by the
neck-linker docking. For those mutants with longer neck-linkers, the potential stored
in the neck-linkers is small and thus it takes more time for the tethered head to be
swung to a forward binding site.

In the second part of the thesis, motivated by the experiment [62] and the theoret-
ical work [2, 3], we performed Monte Carlo simulations of a large system (including
6000 thousand microtubules) in a large parameter space while the numerical simula-
tion in [2, 3] was carried out in a small neighborhood of parameter space. We devised
an algorithm to select binary collisions in the cases of multiple intersections. We
also deve10ped a procedure to characterize the patterns. Our results have reproduced
the consecutive transitions from the disordered state to the vortex state, to the aster
state, and then to the bundled state when the motor concentration increases. This
phase transition is in agreement with the experimental results [62], while the other

models [51, 78] failed to reproduce these experimental results.

81

Appendix A

Pseudo Code of the Algorithms

First, we introduce some indexes used in the pseudo codes.

0 H1010 is the index of headl backward binding. 1=the bound state, 0=the

unbound state. H1 bw = 1 when the kinesin begins a new step.

0 H1 fw is the index of headl forward binding. 1=the bound state, 0=the un-
bound state. The default value of H1 fw is 0. It will be 1 when headl binds to

a forward site.

0 H2 Si is a 1 x 5 index matrix. The first column is for bound state, 1=the bound
state 0=the unbound state; the second is for ATP bound state, 0=the ATP
empty state, 1: the ATP bound st; the third column is for neck-linker zipping,
0=the unzipped state, 1=the unzipped state; the fourth column is for ATP
hydrolysis, 0=ATP not hydrolyzed, 1: ATP hydrolyzed; the fifth column is
for P2- release, 0=Pi not released, 1=Pi released and head2 is in microtubule
bound with the ADP in the catalytic core. H2 st = [1, 0, 0, 0, 0] when the kinesin

begins a new step.

82

 

Algorithm 1 This is the part one of the whole algorithm.
Hlb’w =1,H1f,w = 0, and H28t(1, 1) =1
while Hlf’w = 0 and H2St(1, 1) = 1 do
while Hlbw = 1 do
Compute the tension in the neck-linkers by (2.9)
F6}
K BT

 

 

__ 0
de T “ deTe
_ F5503
K Br

 

, _ 0
kzzp — kzipe
if H2st(1,3) = 0 then

if C S deTdt then

Hlbw = 0
end if
if H28t(1,2)=1 then

if C s It; AT Pdt then
H2St(1’ 2) = 0
else if kcfATPdt < C S (led-ATP + kzip)dt then

The neck-linker is docked. For the wild type. this docking induce the
dissociation of headl from the microtubule, i.e., H1 = 0, because of
the restriction of the neck-linkers. For the mutants, this docking may
not trigger the detachment of headl from microtubule.
end if
else
if c s 193147. Pdt then
H2 St(1, 2) = 1
end if
end if
end if
if H2St(1’3) = 1 then
if C S deTdt then
H1 .bw = 0
end 1f
Consider the ATP hydrolysis in head2
end if
end while
end while

 

83

 

Algorithm 2 This is the part two of the whole algorithm, following the part 2
while Hlf’w = 0 and H2St(1, 1) = 1 do
if H28t(1,3) = 0 then
if H2St(1,2) = 1 then
if C s k; AT Pdt then
H28t(1’ 2) = 0
. — — 0
< .
else 1f deTPdt < C _ (deTP + [€2.1th then
The neck-linker is docked.

 

end if
else
if g g kg] AT Pdt then
H2 st(1: 2) = 1
end if
end if

Compute the tension in the neck-linkers by (2.9) and (2.12).
Update the position of headl according to (2.20).
end if
if H28t(1’3) = 1 then
Compute the tension in the neck-linkers by (2.9) and (2.12).
Update the position of headl according to (2.20).
Consider the ATP hydrolysis in head2
end if
if the distance between headl and a binding site is less than 2.5 nm then
Consider the possibility of the binding to that site by using the formula (2.23)
if headl binds to a backward binding site then
H1 = 1
end if
if headl binds to a forward binding site then
H] fw = 1
end if
end if
end while
Consider the ADP release in headl. Headl does not bind an ATP molecule until
the Pi is released in head2. When headl is in the empty state and head2 is in the
ADP bound state, a complete Chemomechanical cycle is finished.

 

84

BIBLIOGRAPHY

[1] Alonso, M.C., Drummond, D.R., Kain, S., Hoeng, J., Amos, L. and Cross, RA,
(2007). An ATP gate controls tubulin binding by the tethered head of kinesin-1.
Science. 316: 120—123.

[2] Aranson, LS. and Tsimring, LS, (2005). Pattern formation of microtubules and
motors: Inelastic interaction of polar rods. Physical Review E. 74: 050901.

 

[3] Aranson, LS. and Tsimring, LS, (2006). Theory of self-assembly of microtubules
and motors. Physical Review E. 74: 031915.

 

[4] Aranson, LS. and Tsimring, LS, (2006). Patterns and collective behavior in
granular media: Theoretical concepts. Review of Modern Physics. 78: 641.

 

[5] Aranson, I.S., Volfson, D., and Tsimring, LS, (2007). Swirling motion in a
system of vibrated elongated particles. Physical Review E. 75: 051301.

 

[6] Aranson, LS. and Tsimring, LS, (2003). Model of coarsening and vortex forma-
tion in vibrated granular rods. Physical Review E. 67: 021305.

 

[7] Asenjo, AB. and Sosa, H., (2009). A mobile kinesin-head intermediate during
the ATP-waiting state. Proc. Natl. Acad. Sci.. 106: 5657-5662.

[8] Asbury, C.L., Fehr, AN. and Block, S.M., (2003). Kinesin moves by an asym-
metric hand-over-hand mechanism. Science. 302: 2130-2134.

[9] Blanchet, A., Dolbeault, J ., and Kowalczyk, M., (2009). Stochastic Stokes’ drift,
homogenized functional inequalities, and large time behavior of Brownian ratch-
ets. SIAM J. Math. Anal: 41: 46—76.

[10] Blair,D. L., Neicu, T., and Kudrolli, A., (2003). Vortices in vibrated granular
rods. Physical Review E. 67: 031303.

 

[11] Block, S. M., (2007). Kinesin motor mechanics: binding,stepping,tracking, gat-
ing, and limping. Biophys J .. 92: 2986—2995.

85

[12] Budhiraja, A. and Fricks, J ., (2006). Molecular motors, brownian ratchets, and
reflected diffusions. Discrete and Continuous Dynamical Systems. 6: 711—734.

 

[13] Carter, N. J. and Cross, R. A., (2005). Mechanics of the kinesin step. Nature.
435: 308-312.

[14] Chakravarty, A., Howard, L., and Compton, DA, (2004). A mechanistic model
for the organization of microtubule asters by motor and non-motor proteins in a
mammalian mitotic extract. Molecular biology of the cell. 15: 2116—2132.

 

[15] Chipot, M., Hastings, S., and Kinderlehrer, D., (2004). Transport in a molecular
motor system. M2AN Math. Model. Numer. Anal. 38: 1011—1034.

[16] Chipot, M., Hilhorst, D., Kinderlehrer, D., and Olech, M., (2009). Contrac-

tion in L for a system arising in chemical reactions and molecular motors.
Differ. Equ. Appl.. 1: 139—151.

 

[17] Coy, D. L., Wagenbach, M., and Howard, J ., (1999). Kinesin takes one 8-nm step
for each ATP that it hydrolyzes J. Biol. Chem. 274: 3667—3671.

[18] Cross, R. A., (2004). The kinetic mechanism of kinesin. Trends Biochem. Sci..
29: 301—309.

[19] Cytrynbaum, E.N., Rodionov, V., and Mogilner, A., (2004). Computational
model of dynein-dependent self-organization of microtubule asters. J. Cell Sci..
117: 1381—1397.

[20] de Gennes, P.-G. and Prost, J ., (1995). The Physics of Liqid Crystals. Claredon
Press, Oxford.

 

[21] Derrida, B., (1983). Velocity and diffusion constant of a periodic one-dimensional
hopping model. J. Stat. Phys.. 31: 433—450.

 

[22] Doi, M. and Edwards, SF. (1988). The Theory of Polymer Dynamics. Clarendon
Press, Oxford.

 

[23] Fisher, ME. and Kolomeisky, A.B., (1999). The force exerted by a molecular
motor. Proc. Natl. Acad. Sci.. 96: 6597-602.

[24] Fisher, ME. and Kolomeisky, A.B., (1999). Molecular motors and the forces they
exert. Physica A. 274: 241—266.

[25] Fisher, ME. and Kolomeisky, A.B., (2001). Simple mechanochemistry describes
the dynamics of kinesin molecules. Proc. Natl Acad. Sci.. 98: 7748—7753.

[26] Gennerich, A., Carter, A.P., Reck-Peterson, S.L., and Vale, RD, (2007). Force-
induced bidirectional stepping of cytoplasmic Dynein. ELI. 131: 952—965.

86

[27] Gennerich, A. and Vale, RD. (2009). Walking the walk: How kinesin and dynein
coordinate their steps. Current Opinion in Cell Biology. 21: 59—67.

 

[28] Grégoire, G. and Chaté, H., (2004). Onset of collective and cohesive motion.
Physical Review Letter. 92: 025702.

 

[29] Gupta, R., DeLapp, J ., Batrouni, G.G., Fox, G.C., Baillie, CF, and Apostolakis,
J., (1988). Physical Review Letter. 61: 1996.

 

[30] Guydosh, NR. and Block, S.M., (2006). Backsteps induced by nucleotide analogs
suggest the front head of kinesin is gated by strain. Proc. Natl. Acad. Sci.. 103:
8054-8059.

[31] Hancock, W.O. and Howard, J ., ( 1999). Kinesins processivity results from me-
chanical and chemical coordination between the ATP hydrolysis cycles of the
two motor domains. PNAS. 96: 13147-13152.

[32] Howard, J ., (2000). Mechanics of Motor Proteins and the Cytoskeleton.
Springer, New York, 2000.

 

[33] Howard J., (1996). The movement of kinesin along microtubules.
Annu. Rev. Physiol.. 58: 703—729.

 

[34] Hua, W., Young, E.C., Fleming, M.L., and Gelles, J. (1997). Coupling of kinesin
steps to ATP hydrolysis. Nature. 388: 390-393

[35] Humphrey, D., Duggan, C., Saha, D., Smith, D. and K515, J., (2002). Active
fluidization of polymer networks through molecular motors. Nature. 416: 413—
16.

[36] Hyeon, C. and Onuchic, J.N., (2007). Internal strain regulates the nucleotide
binding site of the kinesin leading head Pro. Natl. Acad. Sci.. 104: 2175—2180.

[37] Hyman, AA. and Karsenti, E., (1996). Morphogenetic properties of microtubules
and mitotic spindle assembly _C_e_ll. 84: 401—410.

[38] Jamali, Y., Lohrasebi, H. and Rafli-Tabar, H., (2007) Computational modeling
of the stochastic dynamics of kinesin biomolecular motors. Physica A. 381: 239—
254.

[39] Jia, Z., Karpeev, D., Aranson, I., and Bates, P., (2008). Simulation studies of
self-organization of microtubules and molecular motors. Physical Review E. 77:
051905.

 

[40] Julicher, F., Ajdari, A., and Frost, J., (1997). Modeling molecular motors.
Rev. Mod. Phys.. 69: 1269-81.

 

[41] van Kampen N.G., (1981). Stochastic Processes in Physics and Chemistry.
North-Holland Publishing Company.

 

87

[42] Kanada, R, and Sasaki, K., (2003). Theoretical model for motility and proces—
sivity of two—headed molecular motors. Phys. Rev. E. 67: 061917.

 

[43] Karpeev, D., Aranson, I., Tsimring, L., and Kaper, H., (2007). Interactions of
semiflexible filaments and molecular motors. Physical Review E. 76: 051905.

 

[44] Keller, D. and Bustamante, C., (2000). The mechanochemistry of molecular mo-
tors. Biophysical Journal. 78: 541—556.

 

[45] Kikkawa, M., (2008). The role of microtubules in processive kinesin movement.
Trends in Cell Biology. 18: 128—135.

 

[46] Kinderlehrer, D. and Kowalczyk, M.,(2002). Diffusion-mediated transport and
the flashing ratchet. Arch. Ration. Mech. Anal. 161. no. 2: 149-179.

[47] Kolomeisky, A. and Fisher, M., (2007). Molecular motors: A theorist’s perspec-
tive. Annu. Rev. Phys. Chem.. 58: 675—95.

 

[48] Kudrolli, A., Lumay, G., Volfson, D., and Tsimring, L. S., (2008). Swarming
and swirling in self-propelled polar granular rods. Physical Review Letter. 100:
058001.

 

[49] Kraikivski, P., Lipowsky, R., and Kierfeld, J., (2006). Enhanced ordering of
interacting filaments by molecular motors. Physical Review Letter. 96: 258103.

 

[50] Kruse, K., Joanny, J.F., Julicher, F., Prost, J., and Sekimoto, K., (2004).
Asters, vortices, and rotating spirals in active gels of polar filaments.
Physical Review Letter. 92: 078101.

 

[51] Lee, H.Y. and Kardar, M., (2001). Macroscopic equations for pattern formation
in mixtures of microtubules and molecular motors. Phys. Rev. E. 64: 056113.

 

[52] Leibler S and Huse, DA, (1993). Porters versus rowers: a unified stochastic
model of motor proteins. J. Cell Biol.. 121:1356-68.

[53] Levine, A. J., Liverpool, T. B., and MacKintosh, F. C., (2004). Phys. Rev. E.
69: 021503.

 

[54] Liepelt, S. and Lipowsky, R., (2007). Kinesin’s network of Chemomechanical
motor cycles. Phys. Rev. Lett.. 98: 258102.

 

[55] Liverpool, TB. and Marchetti, M.C., (2003). Instabilities of Isotropic Solutions
of Active Polar Filaments. Phys. Rev. Lett.. 90: 138102.

 

[56] Mori, T., Vale, RD, and Tomishige, M., (2007). How kinesin waits between
steps. Nature. 450: 750-754.

[57] Munarriz, J ., Mazo, J .J ., and Falo, F., (2008). Model for hand-over-hand motion
of molecular motors. Phys. Rev. E. 77: 031915.

 

88

[58] Mather, W.H. and Fox, RF, (2006). Kinesin’s Biased Stepping Mechanism:
Amplification of neck-linker Zippering. Biophys J .. 91: 2416—2426.

[59] Mogilner, A, Fisher, Al and Baskin, R.J., (2001). Structural changes in the
neck—linker of kinesin explain the load dependence of the motor’s mechanical
cycle. J. Theor. Biol.. 211: 143-57.

[60] Narayan, V., Menon, N., and Ramaswami, S., (2006). Nonequilibrium steady
states in a vibrated-rod monolayer: tetratic, nematic, and smectic correlations.

J. Stat. Mech.. P01005.

[61] Nédélec, F.J., (2002). Computer simulations reveal motor properties generating
stable antiparallel microtubule interactions. J. Cell Biology. 158: 1005-1015.

 

[62] Nédélec, F.J., Surrey, T., Maggs, A.C., and Leibler, S., (1997). Self-organization
of microtubules and motors, Nature. 389: 305—308.

 

[63] Nédélec, F., Surrey, T., and Maggs, A.C., (2001). Dynamic concentration of
motors in microtubule arrays. Physical Review Letter. 86: 3192.

 

[64] Nicholas, F.E., Yoshioka, C., Milligan AR, and Vale, DR, (2006). A lever-
arm rotation drives motility of the minus-end-directed kinesin ch. Nature. 439:
875—878.

[65] Parmeggiani, A., Julicher, F., Ajdari, A. and Prost, J ., (1999). Energy transduc-
tion of isothermal ratchet: Generic aspects and specific examples close to and
far from equilibrium. Phys. Rev. E. 60: 2127-2140.

 

[66] Perthame, B. and Souganidis, RE, (2009). Asymmetric potentials and motor
effect: a large deviation approach. Arch. Ration. Mech. Anal. 193: 153—169.

[67] Perthame, B., (2007). Transport Equations in Biology. Frontiers in Mathematics.
Birkhauser, Boston.

[68] Peskin, GS. and Oster, G. (1995). Coordinated hydrolysis explains the mechan-
ical behavior of kinesin. Biophys. J .. 68: S202-11.

[69] Phillips, R., Kondev, J., and Theriot, J. (2009). Physical Biology of the Cell.
Taylor & Francisc Group.

 

[70] Qian, H., (1997). A simple theory of motor protein kinetics and energetics.
Biophysical chemistry. 67: 263—267.

 

[71] Qian, H., (2000). A simple theory of motor protein kinetics and energetics. II.
Biophysical chemistry. 83: 35—43

 

[72] Reck-Peterson, S.L., Yildiz, A., Carter, A.P., Gennerich, A., Zhang, N., and
Vale, RD, (2006). Single molecule analysis of dynein processivity and stepping
behavior. _Qe_ll 126: 335—348.

89

[73] Reimann, P., (2002). Brownian motors: noisy transport far from equilibrium.
Phys. Rep.. 361: 57-265

[74] Rice, 8., Lin, A.W., Safer, D.C., Hart, L., Naber, N., Carragher, B.O., Cain,
S.M., Pechatnikova, E., Wilson-Kubalek, E.M., Whittaker, M., Pate, E., Cooke,
R., Taylor, E.W., Milligan, AR, and Vale, RD, (1999). A structural change in
the kinesin motor protein that drives motility. Nature. 402: 778-784.

 

[75] Rice, 8., Cui, Y., Sindelar, C., Naber, N ., Matuska, M., Vale R., and Cooke,
R., (2003). Thermodynamic properties of the kinesin neck-region docking to the
catalytic core. Biophys. J .. 84: 1844-1854.

[76] Rosenfeld, S.S., Jefferson, GM. and King, RH, (2001). ATP reorients the neck-
linker of kinesin in two sequential steps. J. Biol. Chem.. 276: 40167-40174.

[77] Rosenfeld, S.S., Fordyce, P.M., Jefferson, G.M., King, RH, and Block, S.M.,
(2003). Stepping and stretching: how kinesin uses internal strain to walk proces-
sively. J. Biol. Chem.. 278: 18550-18556.

[78] Sankararamn, 8., Menon, GI, and Sunial Kumar, RB, (2004). Self-organized
pattern formation in motor-microtubule mixtures. Physical Review E. 70:
031905.

 

 

[79] Schliwa, M. and Woehike, G., (2003). Molecular motors. Nature. 422: 759—765.

[80] Sindelar, CV. and Downing, K.H., (2007). The beginning of kinesin’s force-
generating cycle visualized at 9-A resolution. J. Cell Biology. 177: 377—385.

 

[81] Surrey, T., Nédélec, F., Leibler, S., and Karsenti, E. (2001). Physical properties
determining self-organization of motors and microtubules. Science. 292: 1167—
1171.

 

[82] Svoboda, K. and Block, S.M., (1994). Force and velocity measured for single
kinesin molecules. Qcfl. 77: 773—784.

[83] Schnitzer, M.J., Visscher, K. and Block, S.M., (2000). Force production by single
kinesin motors. Nature cell biology. 2: 718-723.

 

[84] Schnitzer, M.J. and Block, S.M., (1997). Kinesin hydrolyses one ATP per 8-nm
step, Nature. 388: 386—390.

[85] Schief, W.R., Clark, R.H., Crevenna, AH, and Howard, J ., (2004). Inhibition of
kinesin motility by ADP and phosphate supports a hand-over-hand mechanism.
PNAS. 101: 1183—1188.

[86] Takiguchi, K., (1991). Heavy-meromyosin induces sliding movements between
antiparallel actin filaments. J. Biochem.. (Tokyo). 109: 250.

90

[87] Taniguchi, Y., Nishiyama, M., Ishii, Y., and Yanagida., T., (2005). Entropy
rectifies the Brownian steps of kinesin. Nat Chem Biol.. 1: 342-347.

[88] Tsygankov, D., Linden, M., and Fisher, M.E., (2007). Back—stepping, hidden
substeps, and conditional dwell times in molecular motors. Physical Review E.
75: 021909.

 

[89] Uemura, S., kawaguchi, K., Yajima, J., Edamatsu, M., Toyoshima, Y.Y., and
Ishiwata, S., (2002). Kinesin—microtubule binding depends on both nucleotide
state and loading direction. Proc. Natl. Acad. Sci.. 99: 5977—5981.

[90] Urrutia, R., McNiven, M.A., Albanesi, J.P., Murphy, DB, and Kachar, B.,
(1991). Purified kinesin promotes vesicle motility and induces active sliding be-
tween microtubules in vitro. Proc. Natl. Acad. Sci.. 88: 6701.

[91] Valentine, MT. and Gilbert, SP, (2007). To step or not to step? How
biochemistry and mechanics influence processivity in Kinesin and Eg5.
Curr. Opin. in Cell Bio.. 19: 75-81.

 

[92] Voituriez, R., Joanny, J .F., and Prost, J ., (2006). Generic phase diagram of active
polar films. Physical Review Letter. 96: 028102.

 

[93] Yildiz, A., Tomishige, M., Vale, RD, and Selvin, PR, (2004). Kinesin walks
hand-over-hand. Science. 303: 676—679.

 

[94] Yildiz, A. and Selvin, RR, (2005). Kinesin: walking, crawling or sliding along.
Trends in cell biology. 15: 112—120.

 

[95] Yildiz, A., Tomishige, M., Gennerich, A., and Vale, RD, (2008). Intramolecular
strain coordinates kinesin stepping behavior along microtubules. Qe_ll. 134: 1030—
1041.

[96] Ziebert, F. and Zimmermann, W., (2005). Nonlinear competition between asters
and stripes in filament-motor systems. Eur. Phys. J. E. 18: 41.

 

[97] Ziebert, F., Aranson, 1.8., and Tsimring, LS, (2007). Effects of cross-links on
motor-mediated filament organization. New Journal of Physics. 9: 421.

 

[98] Hahlen, K., Ebbing, B., Reinders, J., Mergler, J., Sickmann, A., and Woehlke,
G., (2006). Feedback of the Kinesin-1 neck-linker Position on the Catalytic Site,
J. Biol. Chem. Vol: 281, no: 27: 18868-18877.

[99] Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, 1., and Shochet, O.,
(1995). Novel type of phase-transition in a system of self-driven particles.
Physical Review Letter. 75: 1226.

 

91

[100] See EPAPS Document No. E—PLEEE8—77—016805 for computer animations
of simulations results. A direct link to this document may be found in the
online article’s HTML reference section. The document may also be reached
via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html) or from
ftp.aip.org in the directory /epaps/. See the EPAPS homepage for more infor-
mation.

92

1293 03063 0267

H
H
H
I"
H
T”
H
I“
N“
I
|
T“
H
H
S"
u
'1'

3