dynamics of filament growth

7/30/2019 Dynamics of filament Growth

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Dual Degree Project Presentation

Dynamics of filament growthUnder the supervision of

Prof.Mandar Inamdar

Neeraj KookadaDepartment of Civil Engineering

IIT Bombay

May 2, 2013


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Flow of Presentation

Introduction Objectives

Stochastic Simulations of Chemical Reactions

Gillespie Algorithm

Degradation Reaction

Degradation and Production Reaction

Diffusion using Compartment-based approach Force generation by filament polymerization

Introduction

2-State Model

Theoretical Stall force for a single filament

Stall force for a two filament model

Pseudo code Simulation Results and Discussions

Conclusion and Future work

References


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Introduction Optimization problems in dynamic, stochastic environments are an increasingly

important part of biological systems.

Simulation methods can provide solutions for complex partial differential

equations and problems which involve uncertainty about models to draw

conclusions about the related variables.

The growth dynamics of fibres like microtubules and actin filaments play an

important role in cellular biology. These fibres generate forces that play a role in cellular motility processes such as

the motion of chromosomes during mitosis.

Recent experimental advances have allowed to determine the dynamics of growth

of cytoskeleton proteins with a single-molecule precision at different conditions.

These experiments suggest that the complex biochemical transitions and

intermediate states in the cytoskeleton proteins influence their dynamic properties

and functions and thus the control the force that a cell can produce.


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Objectives To develop stochastic models for processes like degradation, production and

diffusion which mimic the interactions between different components in molecularbiology

To apply the Gillespie algorithm as it turns out to be a more efficient model

reducing the computational time when compared with Monte Carlo simulations to

predict the time evaluation of processes.

To develop a two-state model for a single actin filament which assumes thepolymerization and depolymerisation, as well as switching of states, where the

filament is restrained to grow against a wall (movable) and the resulting force

generated is studied

The maximum force generated by the filaments stall force and its relations with

the velocity of the wall and length fluctuations of the filament is plotted using

Monte Carlo Simulations. The one filament model is extended to a two filament model and numerically it is

derived that the stall force for N filaments is N times the stall force for a single

filament.


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Gillespie Algorithm

Introduction

Gillespie Algorithm generates a statistically correct trajectory (possible solution) of

a stochastic equation

Gillespie algorithm allows a discrete and stochastic simulation of a system with fewreactants because every reaction is explicitly simulated.

A Gillespie realization represents a random walk that exactly represents the

distribution of the Master equation.

The physical basis of the algorithm is the collision of molecules within a reaction

vessel (well mixed). All reactions within the Gillespie framework must involve at most two molecules.

Reactions involving three molecules are assumed to be extremely rare and are

modeled as a sequence of binary reactions.


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Flowchart for Gillespie Algorithm

Initialize Start Initialize System time to zero

Event list

Create event list of all possible transitions

Calculate the time to-event outcomes

Time step

If events available, perform next event, otherwise stop

Advance time

Terminate

Re-compute the event list

Time reached

Stop


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Algorithm

1. Choose a small time step t. We compute the number of molecules A(t) at times t = i t, i =

1, 2, 3, . . . , as follows. Starting with t = 0 and A(0) = n0, we perform two steps at time t,

2. Generate a random number r uniformly distributed in the interval (0, 1).

3. If r < A(t)kt, then put A(tt) = A(t)1; otherwise, A(tt) = A(t).

4. Continue with step (1) for time t +t.

5. Generate a random number r uniformly distributed in the interval (0,1)6. is computed from eq (1.3)

7. Compute the number of molecules at time t by A (t) = A (t)-1.

8. Continue with step (4).

since r is a random number uniformly distributed in the interval (0, 1), the probability that

r < A(t)kt is equal to A(t)kt.

We use k = 0.1sec-1 and t = 0.005 sec. A(0) = 20, Time = 30s.

For the given input values, A(t)kt (0, 0.01).


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Result

Results of ten realizations of SSA (a2)(c2)(solid lines; different colours show different

realizations) and stochastic mean plotted by the dashed line

Since the function A(t)

has only integer values

{0, 1, 2, . . . , 20}, someof the computed

curves A(t) partially

overlap.

It would be very

unlikely that there

would be two

realizations giving

exactly the same

results.

Details of realization

A(t) depend on thesequence of random

numbers)

Average values give a

reproducible

characteristic of the

system


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Degradation & Production Reactions

Consider the chemical reactions

The first reaction describes the degradation of chemical A with the rate constant k1. It was

already studied previously as reaction (1.1). We couple it with the second reaction which

represents the production of chemical A with the rate constant k2. This reaction is similarto a random walk carried by a particle in the following case.

The particle in the given figure has three mechanisms r1,r2 and r3 possible at different

possibilities of p1,p2 and p3 respectively, where r1,r2 and r3 are set of mutually exclusive

events.

(2.4)


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Algorithm

(starting with A(0) = n0 at time t = 0):

1. Generate two random numbers r1, r2 uniformly distributed in (0, 1).

2. Compute 0 = A(t)k1 + k2.

3. Compute the time when the next chemical reaction takes place as t where from

(1.3)

4. = 1/0 ln(1/r1)

5. Compute the number of molecules at time t by

A(t ) =A(t) 1 if r2 < k2/ 0;

= A(t) 1 if r2 k2/ 0.


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Graphs

frequency over 10000 simulationsDistance v/s Time


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Diffusion using Compartment-based

approachDiffusion is the random migration of molecules (or small particles) arising from motion due tothermal energy. As shown by Einstein, the kinetic energy of a molecule (e.g. protein) isproportional to the absolute temperature. In particular, the protein molecule has a non-zeroinstantaneous speed at, for example, room temperature or at the temperature of the humanbody. A typical protein molecule is immersed in the aqueous medium of a living cell.Consequently, it cannot travel too far before it bumps into other molecules (e.g. water molecules)in the solution. As a result, the trajectory of the molecule is not straight but it executes a randomwalk.

Consider diffusion of 1000 molecules

.

We simulate directly the time evolution of 40 compartments. Considering a computational

domain [0,L] and dividing it into K = 40 compartments of length h = L/K = 25 m. Denote thenumber of molecules of chemical species A in the i-th compartment [(i 1)h, ih) by Ai, i = 1, . . .

,K. The rate of diffusion is assumed as d. Applying the Gillespie SSA provides a correct model of

diffusion provided that the rate constant d is chosen as d = D/h2 where D is the diffusion constant

and h is the compartment length.


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AlgorithmGenerate two random numbers r1, r2 uniformly distributed in (0, 1).

1. Compute propensity functions of reactions by i= A

i(t)d for i = 1, 2, . . . ,K. Compute

+

2. Compute the time at which the next chemical reaction takes place as

3. If < = , then for j = (1,2,,k-1),

<

=

=

4. The number of molecules at time t is given by:

Aj(t ) = Aj (t)-1

Aj+1(t ) = Aj+1 (t) + 1

Ai(t ) = A

i(t) for I j, I j 1


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5. If

= , then for j = (2,3,,k-1),

+

= <

+

=

=

=

6. The number of molecules at time t is given by:

Aj(t ) = Aj (t)-1

Aj-1(t ) = Aj-1 (t) + 1

Ai(t ) = Ai(t) for I j, I j 1


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We simulate 1000 molecules starting from position 0.4 mm in the interval [0,L]for L = 1 mm. We use K = 40.

Since 0.4 mm is exactly a boundary between the 16th and 17th compartment,the initial condition is given by A16(0) = 500, A17(0) = 500 and Ai(0) = 0 fori 16, i 17.

As D = 104 mm2 sec1, we have d = D/h2 = 0.16 sec1. The numbers Ai(t),

i = 1, . . . ,K, at time t = 4 min, are plotted. For example, only one chemical

reaction occurs per time step. Consequently, only two propensity functions change and need to be updated

here N = 1000 is the total number of molecules in the simulation (this numberis conserved because there is no creation or degradation of the molecules inthe system).

Hence, we need to re-compute only when there is a change in or ,i.e. whenever the boundary compartments were involved in the previousreaction

0 1 k


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Results

Compartment-based approach


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Force generation by filament

polymerizationIntroduction

Actin filaments and microtubules are key components of the cytoskeleton of

eukaryotic cells.

In order to understand the dynamical behaviour of single filaments such as actin or

microtubules and the force they can generate, discrete stochastic models havebeen developed that incorporate at the molecular level the coupling of hydrolysis

and polymerization.

Single polymerizing filaments can generate forces in the pico-newton range, as has

been demonstrated experimentally for microtubules (MTs). Such force generation

mainly relies on the gain in chemical bonding energy upon monomer attachment. An opposing force slows down filament growth, which finally stops at the stall

force representing the maximal polymerization force a filament can generate.

Therefore, the stall force is the essential quantity to characterize polymerization

forces.


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2-state Model

Assumptions

Consider two rigid surfaces: one fixed where filaments are nucleated (nucleating

wall) and one movable (barrier) whose position is defined to be the position of the

filament furthest away from the nucleating wall .

We do not model the internal structure of the filaments, and in particular we do

not account for hydrolysis.

After nucleation, the filaments grow or shrink by exchanging monomers with the

surrounding pool of monomers, which acts as a reservoir.

The filaments are coupled only through mechanical contact with the barrier.

The filament consists of two species of monomers (type blue and type orange, as

depicted in fig.(4.1). The filament is modelled as a two-state model depending onwhich type of monomer is in contact with the barrier. Only the filaments in contact

feel the force exerted by the barrier on them, and as a result, this changes their

polymerization rates as compared with free filaments.


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2-state Model

Assumptions

We assume that a type blue and type orange monomers can be added to any free

filament with a rate u0and removed with a rate w and w respectively. Similarly,

the monomers can be added to the filament in contact with the barrier with a rate

u(f) and removed with a rate w(f) and w(f) respectively. Assume that the barrierexerts a constant forcefon the filaments in contact


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2-State Model

Dynamics of a single filament

For a filament that has exchanged work with the barrier through addition or loss of

monomers, we use the following relation:

00

Where: u0 and w0 represent the polymerization and depolymerisation rates for monomers

attaching to free filaments (not bounded by the barrier),

u, w represent the polymerization and depolymerisation rates for monomers

attaching to bounded ends of the filaments.

Consider the switching of states (blue/orange = 1/2) with rates k12 and k21

respectively.


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For a single filament, the polymerization and depolymerisation rates are as

follows:

0 0 1

where, the parameter [0, 1] depends on the location of the energy barrierwhich the wall has to overcome in moving one monomeric step.

We have defined two states 1 and 2 for the filament. The state actually depends

on the last molecule in the filament. We define the rate of conversion from state 1

to 2 as k12 and vice versa. Let the probability of a filament of length l being in

state 1 or 2 at time t be P1(l,t) and P2 (l,t) respectively. The probabilities obey the

following :


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1 ,

1 1, + 11 + 1 , + 212 , + 1 + 12 1 ,

2 ,

2 1, + 22( + 1 , ) + 121(, ) + 2 + 21 2(,)

(1())/212 121

2 121 212()

Summing over all l, we get

For steady state, probabilities are independent of time.

1 212 121 0212 121


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The average position is given by

, + ,

=

The velocity is given by

(1) 1 ,

+2 ,

0

1 1 , + ( 2)2 , 1 1 + ( 2)2

In steady state

(1) 1 21 + 2 1212 + 21

Putting V(1) = 0 and u = ue-Fstall, we find the stall force for one filament system


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Stall force expression for a single

filament The stall force is defined as the value of the force applied on the barrier for which

the velocity given by equation (4.7) vanishes.

At stall force, the velocity can be assumed to be zero, which gives:

( ++ )


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Stall force for a two filament model

Dynamics of a two-filament system

Here, two filaments have been modeled as a two state model with the rates of

polymerization and depolymerization same as had been considered in the previouscase.


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Pseudo Code : Stall force v/s length of filament

function [ final_length ] = biofil( F )

%UNTITLED Summary of this function goes here

% Detailed explanation goes here

iter_max=10000;

k1=0.3; k2=0.15;

u0=1;

w1=0.2; w2=0.1; a1=50e3;

b1=50.1e3;

t_max=1; t=2;

iter=1;

stateA = 1; stateB=1; A=zeros(iter_max,2);

while(t>t_max)


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Pseudo Code

if(a1>b1)

s1=0;s2=1;

elseif(b1>a1)

s1=1;s2=0;

else

s1=u0*exp(-F)/(u0+u0*exp(-F));

s2=u0*exp(-F)/(u0+u0*exp(-F));

end

%molecules entering

r=rand*2;

if(s2*u0*exp(-F)+s1*u0>r)

a1=a1+1;

end

r=rand*2;

if(s1*u0*exp(-F)+s2*u0>r)

b1=b1+1;

end


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Pseudo Code

%state change

r=rand*2;

if(stateA==1)

if(k1>r)

stateA=2;

end

else

if(k2>r)

stateA=1;

end

end

r=rand*2;

if(stateB==1)

if(k1>r)

stateB=2;

end


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Pseudo Code

else

if(k2>r)

stateB=1;

end

end

%molecules leaving

r=rand*2;

if(stateA==1)

if(w1>r && a1>0)

a1=a1-1;

end

else

if(w2>r && a1>0)

a1=a1-1;

end

end


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Pseudo Code

r=rand*2;

if(stateB==1)

if(w1>r && b1>0)

b1=b1-1;

end

else

if(w2>r && b1>0)

b1=b1-1;

end

end

%loop exit conditions

if(iter>iter_max-1)

t=0;

end

A(iter,1) = a1; A(iter,2) = b1;


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Pseudo Code Stall Force v/s Velocity

function [ output_args ] = velocity( )

%UNTITLED Summary of this function goes here

% Detailed explanation goes here

v=zeros(23,1);

i=1;

f=linspace(0,4.4,23);

for fs=0:0.2:4.4;

v(i)=(biofil(fs)-50e3)/1000;

i=i+1;

end

plot(f,v);

end


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Simulation Results and Discussions

From the graph, it is evident that the position of the barrier isnt changing much with the

iterations. The numerical value of the force used in the code is 4.23 units which is nearly

equal to twice the stall force for a single element. Thus in absence of hydrolysis, the stall

force of a system of N filaments is equal to N times the stall force for each filament


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For forces below the stall force, the length of the filament goes on increasing with

time.

On the other hand, for forces above the stall force, the length of the filament goes on

decreasing with time.


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The mean velocity of the barrier becomes zero at stall force which from the monte carlo

simulations comes out to be around 4.23


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Conclusion and Future Work

We presented SSAs for systems of chemical reactions and molecular diffusion. Thealgorithms for simulating systems of chemical reactions were based on the Gillespie

algorithm.

We presented a model of diffusion in this paper based on the chain of chemical

reactions computing the time evolution of the numbers of molecules in compartments.

Based on the models, we obtained the exact result for the stall force of polymerizing

ensembles of rigid filaments with lateral interactions. The stall force is a linear functionof the number N of filaments in the ensemble and increases linearly with the lateral

interaction.

These results have been confirmed by simulations using the Gillespie algorithm.

Our results are relevant for the interpretation of experimental data on the force-

velocity relation in microtubule polymerization and in the polymerization of bundles of

interacting actin filaments or microtubules.

Future work would include extending the parameters like stall force, velocity for N-

filament structures. The study and effects of hydrolysis on the polymerization and

depolymerisation should be understood and a new exhaustive model should be

prepared


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Thank You

dynamics of filament growth

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