MEAN-FIELD MODEL FOR THE DYNAMICS OF ACTIN FILAMENTS IN THE

CONTRACTILE RING

In the following, we will give the mathematical details of the physical model used in the

main text.

Model definition

With our theory, we try to capture essential features of the ring dynamics, such as,

filament polarity, rules of interaction between filaments through molecular motors. Still, in

the following, we use the words actin filaments and myosin for their relevance in cytokinetic

rings.

Consider a ring of actin filaments such that the filaments align with the ring perimeter.

We denote the co-ordinate along the ring perimeter by x and describe the distribution

of (polar) actin filaments along x by the densities c+ for filaments with their plus-end

pointing clockwise and c for filaments of the opposite orientation. Two filaments of

opposite orientation can join their plus-ends forming a bipolar filament (Fig. 4a, top).

In cells, these joints might be established by protein complexes containing, for example,

formins and myosins (Figs. 2c, 3b). The distribution of bipolar filaments is denoted by

cbp giving the density of their centres. Actin filaments are assumed to have a length

`, bipolar filaments a length 2`. Consequently, the total actin density at a point x

is `0d (c+(x+ ) + cbp(x+ ) + c

(x ) + cbp(x )) . Bipolar filaments form at ratecc

+c, bipolar filaments can split into two filaments of opposite orientations at rate d

(Fig. 4a, top).

Let us now turn to the processes of filament dynamics we account for. Filaments are

assumed to grow at their plus-ends and to shrink at their minus-ends at the same rate .

This treadmilling dynamics leads to movement of actin filaments at velocity vtr = a into the

direction of their plus-ends, where a is the size of an actin monomer. Bipolar filaments are

not displaced by treadmilling. Myosin mini-filaments can induce relative sliding between

filaments. The corresponding velocities are between filaments of the same orientation

(Fig. 4a, middle) and for filaments of opposite orientations (Fig. 4a, bottom). We use these

parameters to quantify the strength of the motor-mediated filament-filament interactions.

1

We assume that myosins are located at the filaments plus-ends, such that c+ + c + cbp is

the distribution of myosin. Finally, fluctuations are accounted for by diffusion terms with

an effective diffusion constant D. The corresponding dynamic equations read:

tc+(x) =D2xc

+(x) x `0

d(c+(x+ ) c+(x )) c+(x)

x `0

d cbp(x+ )c+(x) + x

`0

d(c(x ) + cbp(x )

)c+(x)

xvtrc+(x) cc+(x)c(x) + dcbp(x) (1)

tc(x) =D2xc

(x) x `0

d(c(x+ ) c(x )) c(x)

+ x

`0

d cbp(x )c(x) x `0

d(c+(x+ ) + cbp(x+ )

)c(x)

+ xvtrc(x) cc+(x)c(x) + dcbp(x) (2)

tcbp(x) =D2xcbp(x) x

`0

d (cbp(x+ ) cbp(x )) cbp(x)

x `0

d(c(x+ ) c+(x )) cbp(x)

x `0

d(c+(x+ ) c(x )) cbp(x) + cc+(x)c(x) dcbp(x) (3)

For numerical solution of the dynamic equations, we used a first-order upwind scheme

with adaptive time stepping.

Calculation of the stress in the bundle

The stress in the bundle is defined as the sum of the stresses in the individual filaments.

Stresses in a filament are generated by motors that pull on the filaments and by friction

with the surrounding medium. Explicitly, force balance on a single filament gives

s =1

v + fmot. (4)

In this expression, s is the co-ordinate along the filament, the stress in the filament,

a mobility, v the filaments velocity, and fmot the force density exerted by motors on the

filament. Only the effects of motors cross-linking two filaments are accounted for. The stress

along a filament is thus piece-wise linear in s with slope v/, where v = or v = depending on the orientation and the relative position of the partner filament, the motor is

2

filament (length l)motor position smotstress profile

v l/

s

FIG. 1. Illustration of the stress distribution along a stiff slender rod of length l (the filament, red)

that is drawn by a point force at s0 into the direction of the arrow. The filament velocity is v, its

mobility . The black line indicates the stress profile that results from the applied force and the

filament friction with the environment.

connected to. If there is no motor at a filament end, then the stress vanishes at this point,

and the stress jumps by an amount |v|`/ at the positions smot, where motors are boundto the filament, see below. The total stress profile along the bundle is then obtained by

summing the stress profiles along all filaments in the bundle. Since the expressions are quite

involved, we refrain from giving them here explicitly.

3

Results and Observations

Model considering actin polar filaments This odel has already ee disussed i Atiely Cotratig Budles of Polar Filaets,PRL (Vol. 85 No. 8).However the model that we have used to describe dynamics includes treadmmilling

urret: x(vtrc+- .o I this odel as it's straightforard to see fro the graph that c >0 for all parameters (when vtr=0), which means unstability is due to interaction between parallel filaments

ith c Also it a e easily see that c increases with increasing D, diffusion coeff.(As

agitude oly i graph is atually */D here is legth of filaet oth filaments were assued of sae legth is dereasig , c ireases.As soo as e derease size of syste L, c ireasesshorteig of udles gies staility.Also c decreases with increasing c22 w.r.t. i.e. increasing filaments of same orientation.(graph Also Whe treadillig is aouted,c decreases which means treadmilling is also responsible for destabilization of system.Now it might

e said for that regio he c decreases with increase in D is due to dominance of treadmilling in those region.(see graph 2).

Graph 1 (Here coordinates on X-axis is and coordinates on Y-axis is .Curves with green color indicate more fraction of c22 compared to c11 and and red color indicate lesser fraction

c22 compared to c11.)

Results and Observations

Graph 2(Here in this case c11=.3 and c22=.7,Here coordinates on X-axis is and coordinates on Y-axis is

)

Model considering actin polar filaments and Bipolar filaments This model is an extension of previous model with consideration of Bipolar filaments which has

been mentioned in "Self-organization and mechanical properties of active filament bundles", Physical Review E 67 051913. However the model that we have used to describe dynamics includes

treadillig urret: x(vtrc+-) ). Also it a e easily see that c decreases with increasing wd ,rate of splitting of bipolar filaments to result in polar ones.(As agitude oly i graph is atually /d* here is legth of filaet oth filaets ere assued of sae legth is dereasig , c decreases).As soon as we decrease size of syste L, c first decreases then increases(shortening of bundles for stailtiy o has arrier .Also c ay irease ith irease i ot alays! I gree ure,for egatie part, c has very random patterns) and sometimes function is multivalued, So no ertaiity for all

Results and Observations

beta

alp

ha

c

-13.33 -6.67 0 6.67 13.33 20

1.67

3.33

5

6.67

8.33

10

L=5 with c0+=.3,c0-=.7L=5 with c+0=.3,c0-=1.5L=10 with c+0=.3,c0-=1.5

Constant Prameters: ( D=2,wd=1.5

,wc=0.5,vtr=1.5 ,=1) Here Also Whe treadillig is aouted,c initially increases rapidly with increase in vtr then increases very slowly (remains almost constant).

Results and Observations

Constant Prameters: (C0+ =.3 , C0- =.7, D=2, =.5 ,d=.7 ,wc=.3 ,=1)

Also c decreases with increase in wc ,So combination of polar filaments(antiparallel) resulting in bipolar filament causes unstability to system( wc is rate of combination of polar filaments).If this

result is in consistence with other observations, then it can be understood physically that "As

antiparallel filaments may cause stability to system,so making of bipolar filaments by

combination will bring unstability to system."(however this argument seems to be

mathematically wrong when wcwd which is not happening as you can see from graph curve with lesser has lesser stability for all times.By looking at graph ,one thing that is needed to be observed,for system to have stability

with lesser L (contraction) is not so straightforward.as it can be seen that for wc lesser than certain

value,system has more stability than system of lesser size(really ???).

Results and Observations

alp

ha

c

wc

2.5 5 7.5 10 12.5 15

1

2

3

4

5

6

7

8

9

10L=10with beta=1.0L=5 with beta=1.0L=5 with beta=.5(green)

Constant Prameters: (C0+ =.3 , C0- =.7, D=1,wd=2.5 ,vtr=.5 ,=1)

First Order Upwind Scheme for solving Actin Dynamics Equation

Till now we have been trying to understand it's behaviour using linear stability analysis or loosely speaking

geometric intuition.Now we use first order upwind scheme to get numerical solutions of these equations.

Results obtained for stability of system were in consistence with previous results.Also this method extends

it's analysis to initial states of any kind unlike steady state analysis in previous methods.Graphs can be seen

on my previous report describing analysis behind this method.

.

ModelResults