Collective force generated by multiple biofilaments can exceed the sum of forces due to individual ones

We first discuss a simple toy model which is analytically tractable and hence demonstrates the phenomenon exactly. We consider N filaments, each composed of subunits of length d, collectively pushing a rigid wall, with an external force f acting against the growth direction of the filaments (see figure 1). Consistent with Kramers theory, each filament has a growth rate $u={{u}_{0}}{{{\mbox{e}}}^{-\tilde{f}\delta }}$ when they touch the wall, and ${{u}_{0}}$ otherwise; here $\tilde{f}=fd/{{k}_{{\mbox{B}}}}T$ and $\delta \in \left[ 0,1 \right]$ is the force distribution factor [9, 23]. Each filament can be in two distinct depolymerization states 1 (blue) or 2 (red) (figure 1(a)) giving rise to four distinct states for a two-filament system (figure 1(b)). Filaments in state 1 and 2 depolymerize with intrinsic rates ${{w}_{10}}$ and ${{w}_{20}}$ respectively. When only one filament belonging to a multifilament system is in contact with the wall, these rates become force-dependent, and is given by ${{w}_{1}}={{w}_{10}}{{{\mbox{e}}}^{\tilde{f}\left( 1-\delta \right)}}$ or ${{w}_{2}}={{w}_{20}}{{{\mbox{e}}}^{\tilde{f}\left( 1-\delta \right)}}$ (figures 1(c), (d)). When more than one filament touch the wall simultaneously (figure 1(e)), depolymerization rates are force independent (similar to [9]) as a single depolymerization event does not cause wall movement for perfectly rigid wall and filaments. Furthermore, any filament as a whole can dynamically switch from state 1 to 2 with rate ${{k}_{12}}$, and switch back with rate ${{k}_{21}}$ (see figure 1(a)). Below we focus on $\delta =1$, consistent with earlier theoretical literature [9, 14, 17] and close to experiments on microtubules [23].

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For a single filament, the average velocity is given by ${{V}^{\left( 1 \right)}}=\left( u-{{w}_{1}} \right){{P}_{1}}+\left( u-{{w}_{2}} \right){{P}_{2}}$, where ${{P}_{1}}$ and ${{P}_{2}}$ denote the probability of residency in states 1 and 2. Following figure 1(a), or using Master equations for the microscopic dynamics (see appendix A) it can be shown that the detailed balance condition ${{P}_{1}}{{k}_{12}}={{P}_{2}}{{k}_{21}}$ holds in the steady state. Along with the normalization condition ${{P}_{1}}+{{P}_{2}}=1$, this yields:

$$\begin{eqnarray}&&{{V}^{(1)}}=[(u-{{w}_{1}}){{k}_{21}}+(u-{{w}_{2}}){{k}_{12}}]/({{k}_{12}}+{{k}_{21}}).\end{eqnarray} \tag{ 1 }$$

Setting ${{V}^{\left( 1 \right)}}=0$, the stall force of the single filament is $f_{s}^{\left( 1 \right)}=\frac{{{k}_{B}}T}{d}{\rm ln} \left[ \left( {{k}_{12}}+{{k}_{21}} \right){{u}_{0}}/\left( {{k}_{12}}{{w}_{20}}+{{k}_{21}}{{w}_{10}} \right) \right]$, which is independent of the value of δ.

For two filaments, let ${{P}_{11}}$, ${{P}_{12}}$, ${{P}_{21}}$, and ${{P}_{22}}$ denote the joint probabilities for filaments to be in states $\left\{ 1,1 \right\}$, $\left\{ 1,2 \right\}$, $\left\{ 2,1 \right\}$, and $\left\{ 2,2 \right\}$, respectively (figure 1(b)). Using the microscopic Master equations (see (B.1) and figure B1) the following steady state balance conditions may be derived: ${{k}_{21}}\left( {{P}_{12}}+{{P}_{21}} \right)=2{{k}_{12}}{{P}_{11}}$, ${{k}_{12}}\left( {{P}_{12}}+{{P}_{21}} \right)=2{{k}_{21}}{{P}_{22}}$, and ${{k}_{12}}{{P}_{11}}+{{k}_{21}}{{P}_{22}}=\left( {{k}_{12}}+{{k}_{21}} \right){{P}_{12}}$. The latter conditions, in addition to the normalization condition ${{\sum }_{i,j}}{{P}_{ij}}=1$, solve for ${{P}_{11}}=k_{21}^{2}/{{\left( {{k}_{12}}+{{k}_{21}} \right)}^{2}}$, ${{P}_{22}}=k_{12}^{2}/{{\left( {{k}_{12}}+{{k}_{21}} \right)}^{2}}$ and ${{P}_{12}}={{P}_{21}}={{k}_{12}}{{k}_{21}}/{{\left( {{k}_{12}}+{{k}_{21}} \right)}^{2}}$. The average velocity of the wall for the two-filament system is ${{V}^{\left( 2 \right)}}={{P}_{11}}{{v}_{11}}+{{P}_{12}}{{v}_{12}}+{{P}_{21}}{{v}_{21}}+{{P}_{22}}{{v}_{22}}$. Here ${{v}_{11}}$ and ${{v}_{22}}$ are the velocities in homogeneous states $\left\{ 1,1 \right\}$ and $\left\{ 2,2 \right\}$ respectively. These are known from previous works [9, 17]:

$$\begin{eqnarray}\begin{array}{rcl} {{v}_{11}} & = & 2\left( u{{u}_{0}}-{{w}_{1}}{{w}_{10}} \right)/\left( u+{{u}_{0}}+{{w}_{1}}+{{w}_{10}} \right), \\ {{v}_{22}} & = & 2\left( u{{u}_{0}}-{{w}_{2}}{{w}_{20}} \right)/\left( u+{{u}_{0}}+{{w}_{2}}+{{w}_{20}} \right), \\ \end{array}\end{eqnarray} \tag{ 2 }$$

where i = 1, 2. The velocities in the heterogeneous states ($\left\{ 1,2 \right\}$ or $\left\{ 2,1 \right\}$) have not been calculated previously; solving the inter-filament gap dynamics (see (B.2)) we get

$$\begin{eqnarray}&&{{v}_{12}}={{v}_{21}}=\frac{2u-\left( u+{{w}_{1}} \right){{\gamma }_{1}}-\left( u+{{w}_{2}} \right){{\gamma }_{2}}+\left( {{w}_{1}}+{{w}_{2}} \right){{\gamma }_{1}}{{\gamma }_{2}}}{\left( 1-{{\gamma }_{1}}{{\gamma }_{2}} \right)}.\end{eqnarray} \tag{ 3 }$$

Here ${{\gamma }_{1}}=\left( u+{{w}_{20}} \right)/\left( {{u}_{0}}+{{w}_{1}} \right)$ and ${{\gamma }_{2}}=\left( u+{{w}_{10}} \right)/\left( {{u}_{0}}+{{w}_{2}} \right)$ are both $<1$ for the existence of the steady state. Combining all these, we find that the velocity of the two-filament system, switching between two states, is

$$\begin{eqnarray}&&{{V}^{(2)}}=[k_{21}^{2}{{v}_{11}}+k_{12}^{2}{{v}_{22}}+2{{k}_{12}}{{k}_{21}}{{v}_{12}}]/{{\left( {{k}_{12}}+{{k}_{21}} \right)}^{2}}\end{eqnarray} \tag{ 4 }$$

with ${{v}_{11}}$, ${{v}_{22}}$, ${{v}_{12}}$ and ${{v}_{21}}$ given by equations (2) and (3). Note that various limits (${{w}_{10}}={{w}_{20}}$, ${{k}_{12}}=0$, or ${{k}_{21}}\to \infty$) retrieve the expected result ${{V}^{\left( 2 \right)}}={{v}_{11}}$.

The equation (4) is valid for any δ. To obtain stall force we set ${{V}^{\left( 2 \right)}}=0$ and this leads to a cubic equation (for $\delta =1$) in ${{{\mbox{e}}}^{\tilde{f}}}$ whose only real root gives $f_{s}^{\left( 2 \right)}$ analytically (see (B.3)). The analytical result for ${{V}^{\left( 1 \right)}}$ and ${{V}^{\left( 2 \right)}}$ (equations (1) and (4)) are plotted in figure 2 (main figure) as continuous curves, and the data points obtained from kinetic Monte-Carlo simulations, with the same parameters, are superposed on them. Most importantly we see that the scaled force $f_{s}^{\left( 2 \right)}/f_{s}^{\left( 1 \right)}$ for which ${{V}^{\left( 2 \right)}}=0$ is clearly $>2$. For $N>2$ filaments, we do not have any analytical formula for the model, but we obtain stall forces from the kinetic Monte-Carlo simulation. We plot the excess force ${{\tilde{\Delta }}^{\left( N \right)}}=\tilde{f}_{s}^{\left( N \right)}-N\tilde{f}_{s}^{\left( 1 \right)}$ for different N in the inset of figure 2, for $\delta =1$ and $\delta =0$. As can be seen, ${{\tilde{\Delta }}^{\left( N \right)}}>0$ increases with N, and seems to saturate at large N—this is true for all $\delta \in \left[ 0,1 \right]$. Thus we have shown that dynamic switching between heterogeneous depolymerization states lead to $f_{s}^{\left( N \right)}\ne Nf_{s}^{\left( 1 \right)}$.

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We now proceed to show that the introduction of switching between distinct chemical states (${{w}_{1}}\ne {{w}_{2}}$, ${{k}_{12}}/{{k}_{21}}\ne 0$ or ${{k}_{12}}/{{k}_{21}}\ne \infty$) produces non-equilibrium dynamics embodied in the violation of the well known detailed balance condition. To demonstrate this for the single-filament toy model, in figure 3(a), we consider a loop of dynamically connected configurations (charaterized by its length and state): $\left\{ l,1 \right\}\rightleftharpoons \left\{ l+1,1 \right\}\rightleftharpoons \left\{ l+1,2 \right\}\rightleftharpoons \left\{ l,2 \right\}\rightleftharpoons \left\{ l,1 \right\}$. The product of rates clockwise and anticlockwise are $u{{k}_{12}}{{w}_{2}}{{k}_{21}}$ and ${{k}_{12}}u{{k}_{21}}{{w}_{1}}$, respectively. For the condition of detailed balance to hold (in equilibrium) the two products need to be equal (Kolmogorovʼs criterion), which leads to ${{w}_{1}}={{w}_{2}}$. In figure 3(b), we plot ${{\tilde{\Delta }}^{\left( 2 \right)}}=\tilde{f}_{s}^{\left( 2 \right)}-2\tilde{f}_{s}^{\left( 1 \right)}$ against ${{w}_{2}}$ (with fixed ${{w}_{1}}=1\;{{{\mbox{s}}}^{-1}}$)—we see that ${{\tilde{\Delta }}^{\left( 2 \right)}}>0$ for all ${{w}_{2}}$ except ${{w}_{2}}={{w}_{1}}$ (the equilibrium case). This indicates that the collective phenomenon of excess force generation is tied to the departure from equilibrium. This should be compared with [18], where it was shown that for a biofilament model involving no switching (which is unrealistic), the relationship $f_{s}^{\left( N \right)}=Nf_{s}^{\left( 1 \right)}$ follows from the condition of detailed balance. The effect of non-equilibrium switching is further reflected in the variation of ${{\tilde{\Delta }}^{\left( 2 \right)}}$ as a function of switching rates. If ${{k}_{12}}$ is varied (keeping ${{k}_{21}}$ fixed), we see in figure 3(c) that ${{\tilde{\Delta }}^{\left( 2 \right)}}>0$ always, except in the limits ${{k}_{12}}/{{k}_{21}}\to 0$ or $\infty$. These limits correspond to the absence of switching and hence equilibrium.

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Is our toy model comparable to real cytoskeletal filaments with ATP/GTP hydrolysis? In real filaments the tip monomer can be in two states—ATP/GTP bound or ADP/GDP bound—similar to states 1 or 2 of our toy model. However, in real filaments the chemical states of the subunits may vary along the length, and the switching probabilities are indirectly coupled to force-dependent polymerisation and depolymerization events [6, 11–15, 24]. Thus study of more complex models with explicit ATP/GTP hydrolysis are warranted to get convinced that the interesting collective phenomenon is expected in real biofilament experiments in vitro.

In the literature, there are three different models of ATP/GTP hydrolysis, namely the sequential hydrolysis model [11, 14] and the random hydrolysis model [6, 15, 25], and a mixed cooperative hydrolysis model [13, 26, 27]. In this section, we investigate the collective dynamics within the random model as it is a widely used model and is thought to be closer to reality [7]. We also present different variants of the random model to show that our results are robust. In appendix C, interested readers may find similar results (with no qualitative difference) for the sequential hydrolysis model.

In figure 4 we show the schematic diagram of the random hydrolysis model. In this model, polymerization of filaments occurs with a rate $u={{u}_{0}}{{{\mbox{e}}}^{-\tilde{f}}}$ (next to the wall) or ${{u}_{0}}$ (away from the wall). Note that ${{u}_{0}}={{k}_{0}}c$ where ${{k}_{0}}$ is the intrinsic polymerization rate constant and c is the free ATP/GTP subunit concentration. The depolymerization occurs with a rate ${{w}_{T}}$ if the tip-monomer is ATP/GTP bound, and ${{w}_{D}}$ if it is ADP/GDP bound. There is no f dependence of ${{w}_{T}}$ and ${{w}_{D}}$ (i.e. $\delta =1$). In the random model, hydrolysis happens on any subunit randomly in space [15] (see figure 4) with a rate r per unit ATP/GTP bound monomer. Here, as argued in [14, 28], we consider effective lengths of a tubulin and G-actin subunits as 0.6 nm and 2.7 nm respectively to account for the actual multi-protofilament nature of the real biofilaments (see appendix D for details). We did kinetic Monte-carlo simulations of this model using realistic parameters suited for microtubule and actin (see table 1).

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Table 1.  Rates (actin [1, 3] and microtubules (MT) [1, 4, 29])

	${{k}_{0}}$ ($\mu {{M}^{-1}}{{{\mbox{s}}}^{-1}}$)	${{w}_{T}}\left( {{{\mbox{s}}}^{-1}} \right)$	${{w}_{D}}\left( {{{\mbox{s}}}^{-1}} \right)$	r (${{{\mbox{s}}}^{-1}}$)
Actin	11.6	1.4	7.2	0.003
MT	3.2	24	290	0.2

We first numerically calculated the single microtubule stall force $f_{s}^{\left( 1 \right)}$, and then we checked that for a two-microtubule system; the wall moves with a positive velocity at $f=2f_{s}^{\left( 1 \right)}$ (see figure 5(a) (top)). We find the actual stall force $f_{s}^{\left( 2 \right)}$ at which the wall has zero average velocity (figure 5(a) (bottom)) is greater than $2f_{s}^{\left( 1 \right)}$. In figure 5(b)(■) the velocity against scaled force for two microtubules show clearly that $f_{s}^{\left( 2 \right)}>2f_{s}^{\left( 1 \right)}$, and the resulting ${{\Delta }^{\left( 2 \right)}}=0.09\times f_{s}^{\left( 1 \right)}=1.51\;$ pN (where $f_{s}^{\left( 1 \right)}=16.75$ pN for $c=100\;\mu {\mbox{M}}$). Another interesting point is that at any velocity, even away from stall, the collective force with hydrolysis is way above the collective force without hydrolysis—a comparison of the force-velocity curves without hydrolysis (r = 0; figure 5(b), (▴) and with random hydrolysis (figure 5(b), (■) demonstrate this. Similar force-velocity curve for two-actins is shown in figure 5(c) and we calculated ${{\Delta }^{\left( 2 \right)}}=0.038\times f_{s}^{\left( 1 \right)}=0.119$ pN (where $f_{s}^{\left( 1 \right)}=3.134$ pN for $c=1\;\mu {\mbox{M}}$). In figures 6(a) and 6(b), we show that the excess stall force ${{\Delta }^{\left( N \right)}}$ increases with N, both for microtubule and actin. For microtubule, the excess force is as big as 6.5 pN for N = 8, while for actin it goes up to 0.6 pN for N = 8.

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We now investigate the dependence of the excess force on various parameters. For N = 2, within the random model, we show the deviations (${{\Delta }^{\left( 2 \right)}}$) and percentage relative deviation (${{\Delta }^{\left( 2 \right)}}/2f_{s}^{\left( 1 \right)}$) as a function of free monomer concentration (c) for actin (figure 7(a)) and microtubule (figure 7(b)). The absolute deviation (${{\Delta }^{\left( 2 \right)}}$) increases with c and goes upto 0.13 pN for actin and 1.61 pN for microtubules. The percentage relative deviation is ≈ 5% (for actin) and 12% (for microtubules), at low c. Given that there is huge uncertainty in the estimate of ${{w}_{T}}$ for microtubule [4], and that in vivo proteins can regulate depolymerization rates, we systematically varied ${{w}_{T}}$. In figures 7(c)–(d) we show that ${{\Delta }^{\left( 2 \right)}}$ increases rapidly with decreasing ${{w}_{T}}$ and can go up to 1.5 pN (${{\Delta }^{\left( 2 \right)}}/2f_{s}^{\left( 1 \right)}\approx 9\%$) for actin and 9 pN (${{\Delta }^{\left( 2 \right)}}/2f_{s}^{\left( 1 \right)}\approx 19\%$) for microtubules. We did a similar study of ${{\Delta }^{\left( 2 \right)}}$ as a function of ${{w}_{D}}$ (see figures 7(e)–(f)), and find that ${{\Delta }^{\left( 2 \right)}}$ increases with increasing ${{w}_{D}}$. The important thing to note is that ${{\Delta }^{\left( 2 \right)}}$ increases as we increase ${{w}_{D}}$ (at constant ${{w}_{T}}$) and decrease ${{w}_{T}}$ (at constant ${{w}_{D}}$), i.e. the magnitude of ${{\Delta }^{\left( 2 \right)}}$ is tied to the magnitude of difference of ${{w}_{D}}$ from ${{w}_{T}}$ (just as in our toy model in section 2.1). This also suggests that changes in depolymerization rates, typically regulated by microtubule associated (actin binding) proteins in vivo, may cause large variation in collective forces exerted by biofilaments.

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The case ${{w}_{D}}={{w}_{T}}$ effectively corresponds to the absence of switching, since dynamically there is no distinction between ATP/GTP-bound and ADP/GDP-bound subunits. When we move away from this point (i.e. when ${{w}_{D}}\ne {{w}_{T}}$), the effects of switching manifest. In these cases, the hydrolysis process violates detailed balance as it is unidirectional: ATP/GTP $\to$ ADP/GDP conversion is never balanced by a reverse conversion ADP/GDP $\to$ ATP/GTP. Thus, similar to our toy model, we expect the non-equilibrium nature of switching (hydrolysis) to be related to the phenomenon of excess force generation. In figure 8 we plot ${{\Delta }^{\left( 2 \right)}}$ as a function of ${{w}_{D}}$ (for smaller values of ${{w}_{D}}$ compared to figure 7(f)) for two-microtubule system, and find that indeed at the point ${{w}_{D}}={{w}_{T}}$, ${{\Delta }^{\left( 2 \right)}}=0$. Moreover, for biologically impossible situations of ${{w}_{D}}<{{w}_{T}}$ (see table 1), we find ${{\Delta }^{\left( 2 \right)}}<0$.

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We now proceed to discuss the above phenomenon in further realistic variants of the random hydrolysis model. In reality actin hydrolysis involves two steps: ATP $\to$ ADP-${{{\mbox{P}}}_{i}}\to$ ADP [5–7]. Our two-state models above approximate this with the dominant rate limiting step of ${{{\mbox{P}}}_{i}}$ release [11, 14]. To test the robustness of our results,we study a more detailed 'three-state' model,which is defined by the following processes (as depicted in figure 9(a)) and rates (taken from [6, 7]): (i) addition of ATP-bound subunits (${{u}_{0}}={{k}_{0}}c=11.6\;{{{\mbox{s}}}^{-1}}$ with concentration $c=1\;\mu {\mbox{M}}$), (ii) random ${\mbox{ATP}}$ $\to$ADP-${{{\mbox{P}}}_{i}}$ conversion (${{r}_{DP}}=0.3\;{{{\mbox{s}}}^{-1}}$), (iii) random ADP-${{{\mbox{P}}}_{i}}$ $\to$ADP conversion ($r=0.003\;{{{\mbox{s}}}^{-1}}$), (iv) dissociation of ATP-bound subunits (${{w}_{T}}=1.4\;{{{\mbox{s}}}^{-1}}$), (v) dissociation of ADP-${{{\mbox{P}}}_{i}}$-bound subunits (${{w}_{DP}}=0.16\;{{{\mbox{s}}}^{-1}}$), and (vi) dissociation of ADP-bound subunits (${{w}_{D}}=7.2\;{{{\mbox{s}}}^{-1}}$). For this model we first calculate the single-filament stall force $f_{s}^{\left( 1 \right)}=4.49$ pN. Simulating this model for two actin filaments we find that the wall moves with a positive velocity at $f=2f_{s}^{\left( 1 \right)}$ (see figure 9(b) (top))—this proves $f_{s}^{\left( 2 \right)}>2f_{s}^{\left( 1 \right)}$. We then calculate the two-filament stall force $f_{s}^{\left( 2 \right)}=9.10$ pN (also see figure 9(b) (bottom) for few traces of the wall position at stall). Consequently, we obtain the excess force ${{\Delta }^{\left( 2 \right)}}=f_{s}^{\left( 2 \right)}-2f_{s}^{\left( 1 \right)}=0.12$ pN—this value is same as that of the two-state random hydrolysis model (see table D1, actin).

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Table D1.  Comparison of stall force and excess force values between simplified random hydrolysis model (figure 4) and One-layer (multi-protofilament) random hydrolysis model (figure D2), at fixed concentrations $c=100\;\mu {\mbox{M}}$ for microtubules and $c=1\,\mu {\mbox{M}}$ for actins. Other parameters are specified in table 1.

	Simplified random hydrolysis			One-layer (multi-protofilament)
	$f_{s}^{\left( 1 \right)}$ (pN)	$f_{s}^{\left( 2 \right)}$ (pN)	${{\Delta }^{\left( 2 \right)}}$ (pN)	$f_{s}^{\left( 1 \right)}$ (pN)	$f_{s}^{\left( 2 \right)}$ (pN)	${{\Delta }^{\left( 2 \right)}}$ (pN)
Actin	3.13	6.38	0.12	3.14	6.38	0.10
Microtubule	16.75	35.01	1.51	17.06	35.06	0.94

In [7] a variant of the above three-state model is discussed where ADP-${{{\mbox{P}}}_{i}}$ $\to$ADP conversion happens at two different rates—with a rate ${{r}_{{\mbox{tip}}}}$ at the tip, and with a rate r in the bulk. In this model ${{r}_{{\mbox{DP}}}}=\infty$ i.e. as soon as an ATP subunit binds to a filament it converts to ADP-${{{\mbox{P}}}_{i}}$ state—this implies that effectively this model is a two-state model. We have simulated this model for actin filaments using the rates given in [7]: ${{u}_{0}}={{k}_{0}}c=11.6\;{{{\mbox{s}}}^{-1}}$ with concentration $c=1\mu {\mbox{M}}$, ${{r}_{{\mbox{tip}}}}=1.8\;{{{\mbox{s}}}^{-1}}$, $r=0.007\;{{{\mbox{s}}}^{-1}}$, ${{w}_{DP}}=0.16\;{{{\mbox{s}}}^{-1}}$, and ${{w}_{D}}=5.8\;{{{\mbox{s}}}^{-1}}$. The stall forces for single filament and two filaments are $f_{s}^{\left( 1 \right)}=3.03$ pN and $f_{s}^{\left( 2 \right)}=6.17$ pN respectively, implying $f_{s}^{\left( 2 \right)}>2f_{s}^{\left( 1 \right)}$. The resulting excess force ${{\Delta }^{\left( 2 \right)}}=0.11$ pN remains almost unchanged compared to the simple two-state random hydrolysis model (see table D1, actin).

Although the multiple protofilament composition of actin and microtubules do not appear explicitly in any of the above models, we used effective subunit lengths to indirectly account for that. This drew from the fact that the sequential hydrolysis model can be exactly mapped to a multi-protofilament model (called the 'one-layer' model) with strong inter-protofilament interactions [11, 14]—studies of two such composite filaments are discussed in D.1. We further studied a new 'one-layer' model with random hydrolysis in D.2, and confirmed that simulation of the multi-protofilament model yields similar results (even quantitatively) as the simpler random hydrolysis model discussed in this section.

In the two-filament model, we have four different states (see figure B1 (top)). In the steady state, the system obeys probability flux balance conditions, which may be intuitively derived following figure B1 (bottom). Below we provide a more systematic derivation of these equations starting from the microscopic Master equations. Following the mathematical procedure of [17], we define: ${{P}_{ij}}\left( l,l-k;t \right)$, the joint probability that, at time t, the top filament touching the wall (like in figure B2(a)) is of length l and in state i, and the bottom filament of length $l-k$ ($l>k$) is in state j. Here l and k are natural numbers and $i,j$ = 1 or 2. We write (using the rates shown in figure B2) the following four Master equations satisfied by these probabilities corresponding to four different joint states (see figure B1 (top)):

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$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{11}}(l,l-k;t)}{{\mbox{dt}}} & = & u{{P}_{11}}(l-1,l-k)+{{u}_{0}}{{P}_{11}}(l,l-k-1)+{{w}_{10}}{{P}_{11}}(l,l-k+1) \\ {} & {} & \quad +{{w}_{1}}{{P}_{11}}(l+1,l-k)+{{k}_{21}}({{P}_{12}}(l,l-k)+{{P}_{21}}(l,l-k)) \\ {} & {} & \quad -(u+{{u}_{0}}+{{w}_{1}}+{{w}_{10}}+2{{k}_{12}}){{P}_{11}}(l,l-k), \\ \end{array}\end{eqnarray} \tag{ B.1 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{22}}(l,l-k;t)}{{\mbox{dt}}} & = & u{{P}_{22}}(l-1,l-k)+{{u}_{0}}{{P}_{22}}(l,l-k-1)+{{w}_{20}}{{P}_{22}}(l,l-k+1) \\ {} & {} & \quad +{{w}_{2}}{{P}_{22}}(l+1,l-k)+{{k}_{12}}({{P}_{12}}(l,l-k)+{{P}_{21}}(l,l-k)) \\ {} & {} & \quad -(u+{{u}_{0}}+{{w}_{2}}+{{w}_{20}}+2{{k}_{21}}){{P}_{22}}(l,l-k), \\ \end{array}\end{eqnarray} \tag{ B.2 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{12}}(l,l-k;t)}{{\mbox{dt}}} & = & u{{P}_{12}}(l-1,l-k)+{{u}_{0}}{{P}_{12}}(l,l-k-1)+{{w}_{20}}{{P}_{12}}(l,l-k+1) \\ {} & {} & \quad +{{w}_{1}}{{P}_{12}}(l+1,l-k)+{{k}_{12}}{{P}_{11}}(l,l-k)+{{k}_{21}}{{P}_{22}}(l,l-k) \\ {} & {} & \quad -(u+{{u}_{0}}+{{w}_{1}}+{{w}_{20}}+{{k}_{12}}+{{k}_{21}}){{P}_{12}}(l,l-k), \\ \end{array}\end{eqnarray} \tag{ B.3 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{21}}(l,l-k;t)}{{\mbox{dt}}} & = & u{{P}_{21}}(l-1,l-k)+{{u}_{0}}{{P}_{21}}(l,l-k-1)+{{w}_{10}}{{P}_{21}}(l,l-k+1) \\ {} & {} & \quad +{{w}_{2}}{{P}_{21}}(l+1,l-k)+{{k}_{12}}{{P}_{11}}(l,l-k)+{{k}_{21}}{{P}_{22}}(l,l-k) \\ {} & {} & \quad -(u+{{u}_{0}}+{{w}_{2}}+{{w}_{10}}+{{k}_{12}}+{{k}_{21}}){{P}_{21}}(l,l-k). \\ \end{array}\end{eqnarray} \tag{ B.4 }$$

Next, we define ${{P}_{ij}}\left( l-k,l+1;t \right)$, the probability that, at time t, the top filament of length $l-k$ ($l\geqslant k$) is in state i, and the bottom filament of length $l+1$ touching the wall (see figure B2(b)) is in state j. Here l and k are natural numbers and $i,j$ = 1 or 2. These probabilities satisfy the following four Master equations:

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{11}}(l-k,l+1;t)}{{\mbox{dt}}} & = & u{{P}_{11}}(l-k,l)+{{w}_{10}}{{P}_{11}}(l-k+1,l+1)+{{u}_{0}}{{P}_{11}}(l-k-1,l+1) \\ {} & {} & \quad +{{w}_{1}}{{P}_{11}}(l-k,l+2)+{{k}_{21}}({{P}_{12}}(l-k,l+1)+{{P}_{21}}(l-k,l+1)) \\ {} & {} & \quad -(u+{{u}_{0}}+{{w}_{1}}+{{w}_{10}}+2{{k}_{12}}){{P}_{11}}(l-k,l+1), \\ \end{array}\end{eqnarray} \tag{ B.5 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{22}}(l-k,l+1;t)}{{\mbox{dt}}} & = & u{{P}_{22}}(l-k,l)+{{w}_{20}}{{P}_{22}}(l-k+1,l+1)+{{u}_{0}}{{P}_{22}}(l-k-1,l+1) \\ {} & {} & \quad +{{w}_{2}}{{P}_{22}}(l-k,l+2)+{{k}_{12}}({{P}_{12}}(l-k,l+1)+{{P}_{21}}(l-k,l+1)) \\ {} & {} & \quad -(u+{{u}_{0}}+{{w}_{2}}+{{w}_{20}}+2{{k}_{21}}){{P}_{22}}(l-k,l+1), \\ \end{array}\end{eqnarray} \tag{ B.6 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{12}}(l-k,l+1;t)}{{\mbox{dt}}} & = & u{{P}_{12}}(l-k,l)+{{w}_{10}}{{P}_{12}}(l-k+1,l+1)+{{u}_{0}}{{P}_{12}}(l-k-1,l+1) \\ {} & {} & \quad +{{w}_{2}}{{P}_{12}}(l-k,l+2)+{{k}_{12}}{{P}_{11}}(l-k,l+1)+{{k}_{21}}{{P}_{22}}(l-k,l+1) \\ {} & {} & \quad -(u+{{u}_{0}}+{{w}_{2}}+{{w}_{10}}+{{k}_{12}}+{{k}_{21}}){{P}_{12}}(l-k,l+1), \\ \end{array}\end{eqnarray} \tag{ B.7 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{21}}(l-k,l+1;t)}{{\mbox{dt}}} & = & u{{P}_{21}}(l-k,l)+{{w}_{20}}{{P}_{21}}(l-k+1,l+1)+{{u}_{0}}{{P}_{21}}(l-k-1,l+1) \\ {} & {} & \quad +{{w}_{1}}{{P}_{21}}(l-k,l+2)+{{k}_{12}}{{P}_{11}}(l-k,l+1)+{{k}_{21}}{{P}_{22}}(l-k,l+1) \\ {} & {} & \quad -(u+{{u}_{0}}+{{w}_{1}}+{{w}_{20}}+{{k}_{12}}+{{k}_{21}}){{P}_{21}}(l-k,l+1). \\ \end{array}\end{eqnarray} \tag{ B.8 }$$

Next, let ${{P}_{ij}}\left( l,l;t \right)$ be the probability that, at time t, both filaments have same length l, and they are in a joint state $\left\{ i,j \right\}$. One such situation is shown in figure B2(c). These probabilities satisfy the following Master equations:

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{11}}(l,l;t)}{{\mbox{dt}}} & = & {{u}_{0}}{{P}_{11}}(l-1,l)+{{w}_{1}}{{P}_{11}}(l,l+1)+{{u}_{0}}{{P}_{11}}(l,l-1)+{{w}_{1}}{{P}_{11}}(l+1,l) \\ {} & {} & \quad +{{k}_{21}}({{P}_{12}}(l,l)+{{P}_{21}}(l,l))-2(u+{{w}_{10}}+{{k}_{12}}){{P}_{11}}(l,l), \\ \end{array}\end{eqnarray} \tag{ B.9 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{22}}(l,l;t)}{{\mbox{dt}}} & = & {{u}_{0}}{{P}_{22}}(l-1,l)+{{w}_{2}}{{P}_{22}}(l,l+1)+{{u}_{0}}{{P}_{22}}(l,l-1)+{{w}_{2}}{{P}_{22}}(l+1,l) \\ {} & {} & \quad +{{k}_{12}}({{P}_{12}}(l,l)+{{P}_{21}}(l,l))-2(u+{{w}_{20}}+{{k}_{21}}){{P}_{22}}(l,l), \\ \end{array}\end{eqnarray} \tag{ B.10 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{12}}(l,l;t)}{{\mbox{dt}}} & = & {{u}_{0}}{{P}_{12}}(l-1,l)+{{w}_{2}}{{P}_{12}}(l,l+1)+{{u}_{0}}{{P}_{12}}(l,l-1)+{{w}_{1}}{{P}_{12}}(l+1,l) \\ {} & {} & \quad +{{k}_{12}}{{P}_{11}}(l,l)+{{k}_{21}}{{P}_{22}}(l,l)-(2u+{{w}_{10}}+{{w}_{20}}+{{k}_{12}}+{{k}_{21}}){{P}_{12}}(l,l), \\ \end{array}\end{eqnarray} \tag{ B.11 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{P}_{21}}(l,l;t)}{{\mbox{dt}}} & = & {{u}_{0}}{{P}_{21}}(l-1,l)+{{w}_{1}}{{P}_{21}}(l,l+1)+{{u}_{0}}{{P}_{21}}(l,l-1)+{{w}_{2}}{{P}_{21}}(l+1,l) \\ {} & {} & +{{k}_{12}}{{P}_{11}}(l,l)+{{k}_{21}}{{P}_{22}}(l,l)-(2u+{{w}_{10}}+{{w}_{20}}+{{k}_{12}}+{{k}_{21}}){{P}_{21}}(l,l). \\ \end{array}\end{eqnarray} \tag{ B.12 }$$

We also define the probability of residency in joint state $\left\{ i,j \right\}$ as: ${{P}_{ij}}\equiv {{\sum }_{l,k}}\left[ {{P}_{ij}}\left( l,l-k;t \right)+{{P}_{ij}}\left( l-k,l+1;t \right) \right]$. The normalization of probabilities leads to ${{\sum }_{\left\{ i,j \right\}=1,2}}\sum _{l,k=0}^{\infty }\left[ {{P}_{ij}}\left( l,l-k;t \right)+{{P}_{ij}}\left( l-k,l+1;t \right) \right]={{P}_{11}}+{{P}_{22}}+{{P}_{12}}+{{P}_{21}}=1$. Now one can take the sum over all l and k in the Master equations (B.1)–(B.12), and set the time-derivatives to zero to get the steady-state ($t\to \infty$) balance equations satisfied by the probabilities ${{P}_{ij}}$:

$$\begin{eqnarray}&&{{k}_{21}}({{P}_{12}}+{{P}_{21}})=2{{k}_{12}}{{P}_{11}},\end{eqnarray} \tag{ B.13 }$$

$$\begin{eqnarray}&&{{k}_{12}}({{P}_{12}}+{{P}_{21}})=2{{k}_{21}}{{P}_{22}},\end{eqnarray} \tag{ B.14 }$$

$$\begin{eqnarray}&&{{k}_{12}}{{P}_{11}}+{{k}_{21}}{{P}_{22}}=({{k}_{12}}+{{k}_{21}}){{P}_{12}},\end{eqnarray} \tag{ B.15 }$$

$$\begin{eqnarray}&&{{k}_{12}}{{P}_{11}}+{{k}_{21}}{{P}_{22}}=({{k}_{12}}+{{k}_{21}}){{P}_{21}}.\end{eqnarray} \tag{ B.16 }$$

Note that one of the equations (B.13–B.16) is redundant as it may be derived from the other three. From equations (B.15) and (B.16) we find the symmetric relationship ${{P}_{12}}={{P}_{21}}$. Solving the above, along with the normalization condition, we get

$$\begin{eqnarray}\begin{array}{rcl} {{P}_{11}} & = & k_{21}^{2}/{{({{k}_{12}}+{{k}_{21}})}^{2}}, \\ {{P}_{22}} & = & k_{12}^{2}/{{({{k}_{12}}+{{k}_{21}})}^{2}}, \\ {{P}_{12}} & = & {{P}_{21}}={{k}_{12}}{{k}_{21}}/{{({{k}_{12}}+{{k}_{21}})}^{2}}. \\ \end{array}\end{eqnarray} \tag{ B.17 }$$

The probabilities obtained above may be used to calculate the two-filament velocity ${{V}^{\left( 2 \right)}}={{P}_{11}}{{v}_{11}}+{{P}_{22}}{{v}_{22}}+{{P}_{12}}{{v}_{12}}+{{P}_{21}}{{v}_{21}}$. The velocities ${{v}_{11}}$ and ${{v}_{22}}$ for homogeneous cases (i.e. when both filaments are in the same state) can be directly read off from the result in [9] (see equation (2) in the main text). But we need to calculate afresh the velocities of heterogeneous cases, namely ${{v}_{12}}$ and ${{v}_{21}}$, which are same by symmetry.

The velocity ${{v}_{12}}$ of the two-filament system with the top filament in state 1 and the bottom in state 2 is an average obtained by sampling all possible microscopic filament configurations in the $\left\{ 1,2 \right\}$ state. In this state, let ${{p}^{\left( i \right)}}\left( k,t \right)$ be the probability that there is a gap of k monomers between the two filaments and the filament which is in state i ($i\ =$ 1 or 2), is touching the wall. Evidently we have: ${{p}^{\left( 1 \right)}}\left( k,t \right)={{\sum }_{l}}{{P}_{12}}\left( l,l-k;t \right)$ for $k\geqslant 1$, and ${{p}^{\left( 2 \right)}}\left( k,t \right)={{\sum }_{l}}{{P}_{12}}\left( l-k,l+1;t \right)$. We also define: $p\left( 0,t \right)={{\sum }_{l}}{{P}_{12}}\left( l,l;t \right)$, the probability of having zero gap between the filaments. Now since the filaments are not switching between the states (the top is always in 1 and the bottom in 2), equations (B.3), (B.7), (B.11) for ${{P}_{12}}\left( l,l-k;t \right)$, ${{P}_{12}}\left( l-k,l+1;t \right)$, and ${{P}_{12}}\left( l,l;t \right)$ can be used after setting ${{k}_{12}}={{k}_{21}}=0$. This leads to the Master equations satisfied by ${{p}^{\left( i \right)}}\left( k,t \right)$ for $k\geqslant 2$:

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{p}^{(1)}}(k,t)}{{\mbox{dt}}} & = & ({{u}_{0}}+{{w}_{1}}){{p}^{(1)}}(k+1)+(u+{{w}_{20}}){{p}^{(1)}}(k-1) \\ {} & {} & \quad -\;(u+{{u}_{0}}+{{w}_{20}}+{{w}_{1}}){{p}^{(1)}}(k), \\ \end{array}\end{eqnarray} \tag{ B.18 }$$

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\mbox{d}}{{p}^{(2)}}(k,t)}{{\mbox{dt}}} & = & ({{u}_{0}}+{{w}_{2}}){{p}^{(2)}}(k+1)+(u+{{w}_{10}}){{p}^{(2)}}(k-1) \\ {} & {} & \quad -\;(u+{{u}_{0}}+{{w}_{10}}+{{w}_{2}}){{p}^{(2)}}(k), \\ \end{array}\end{eqnarray} \tag{ B.19 }$$

and for k = 1 and 0,

$$\begin{eqnarray}&&\frac{{\mbox{d}}{{p}^{(1)}}(1,t)}{{\mbox{dt}}}=({{u}_{0}}+{{w}_{1}}){{p}^{(1)}}(2)+(u+{{w}_{20}})p(0)-(u+{{u}_{0}}+{{w}_{20}}+{{w}_{1}}){{p}^{(1)}}(1),\end{eqnarray} \tag{ B.20 }$$

$$\begin{eqnarray}&&\frac{{\mbox{d}}{{p}^{(2)}}(1,t)}{{\mbox{dt}}}=({{u}_{0}}+{{w}_{2}}){{p}^{(2)}}(2)+(u+{{w}_{10}})p(0)-(u+{{u}_{0}}+{{w}_{10}}+{{w}_{2}}){{p}^{(2)}}(1),\end{eqnarray} \tag{ B.21 }$$

$$\begin{eqnarray}&&\frac{{\mbox{d}}p(0,t)}{{\mbox{dt}}}=({{u}_{0}}+{{w}_{1}}){{p}^{(1)}}(1)+({{u}_{0}}+{{w}_{2}}){{p}^{(2)}}(1)-(2u+{{w}_{10}}+{{w}_{20}})p(0).\end{eqnarray} \tag{ B.22 }$$

In the steady state, the above equations (B.18)–(B.22), along with the normalization condition: ${{\sum }_{n}}{{p}^{\left( 1 \right)}}\left( k \right)+{{\sum }_{n}}{{p}^{\left( 2 \right)}}\left( k \right)+p\left( 0 \right)=1$, can be solved exactly and one gets the following distributions for the gaps:

$$\begin{eqnarray}&&{{p}^{\left( 1 \right)}}\left( k \right)=p\left( 0 \right)\gamma _{1}^{k},\quad {{p}^{\left( 2 \right)}}\left( k \right)=p\left( 0 \right)\gamma _{2}^{k},\quad {\mbox{for}}\ {\mkern 1mu} {\mbox{k}}\geqslant 1.\end{eqnarray} \tag{ B.23 }$$

Here, $p\left( 0 \right)=\left( 1-{{\gamma }_{1}} \right)\left( 1-{{\gamma }_{2}} \right)/\left( 1-{{\gamma }_{1}}{{\gamma }_{2}} \right)$, ${{\gamma }_{1}}=\left( u+{{w}_{20}} \right)/\left( {{u}_{0}}+{{w}_{1}} \right)$, and ${{\gamma }_{2}}=\left( u+{{w}_{10}} \right)/\left( {{u}_{0}}+{{w}_{2}} \right)$. We must have ${{\gamma }_{1}}<1$ and ${{\gamma }_{2}}<1$ for the existence of the steady state. Now, the average velocity in the $\left\{ 1,2 \right\}$ state is given by

$$\begin{eqnarray}\begin{array}{rcl} {{v}_{12}} & = & (u-{{w}_{1}})\mathop{\sum }\limits_{k=1}^{\infty }{{p}^{(1)}}(k)+(u-{{w}_{2}})\mathop{\sum }\limits_{k=1}^{\infty }{{p}^{(2)}}(k)+2up(0) \\ {} & = & p(0)\left[ \frac{{{\gamma }_{1}}(u-{{w}_{1}})}{1-{{\gamma }_{1}}}+\frac{{{\gamma }_{2}}(u-{{w}_{2}})}{1-{{\gamma }_{2}}}+2u \right], \\ \end{array}\end{eqnarray} \tag{ B.24 }$$

and after simplification this leads to

$$\begin{eqnarray}&&{{v}_{12}}=\frac{2u-\left( u+{{w}_{1}} \right){{\gamma }_{1}}-\left( u+{{w}_{2}} \right){{\gamma }_{2}}+\left( {{w}_{1}}+{{w}_{2}} \right){{\gamma }_{1}}{{\gamma }_{2}}}{\left( 1-{{\gamma }_{1}}{{\gamma }_{2}} \right)},\end{eqnarray} \tag{ B.25 }$$

which is equation (3) in our main text.

Using the explicit expressions for all ${{P}_{ij}}$ (equation (B.17)) and ${{v}_{11}},{{v}_{22}}$ (equation (2), main text), ${{v}_{12}}$ (equation (B.25)) we calculate the two-filament velocity: ${{V}^{\left( 2 \right)}}={{P}_{11}}{{v}_{11}}+{{P}_{22}}{{v}_{22}}+2{{P}_{12}}{{v}_{12}}\;=\left[ k_{21}^{2}{{v}_{11}}+k_{12}^{2}{{v}_{22}}+2{{k}_{12}}{{k}_{21}}{{v}_{12}} \right]/{{\left( {{k}_{12}}+{{k}_{21}} \right)}^{2}}$. At stall force $\tilde{f}=\tilde{f}_{s}^{\left( 2 \right)}$, ${{V}^{\left( 2 \right)}}$ is zero. So setting ${{V}^{\left( 2 \right)}}=0$, and choosing $\delta =1$ (i.e. $u={{u}_{0}}{{{\mbox{e}}}^{-\tilde{f}}}$, ${{w}_{1}}={{w}_{10}},$ and ${{w}_{2}}={{w}_{20}}$) we obtain

$$\begin{eqnarray}&&{{\left. \frac{k_{21}^{2}\left( u_{0}^{2}-w_{10}^{2}{{{\mbox{e}}}^{\tilde{f}}} \right)}{{{u}_{0}}+\left( {{u}_{0}}+2{{w}_{10}} \right){{{\mbox{e}}}^{\tilde{f}}}}+\frac{k_{12}^{2}\left( u_{0}^{2}-w_{20}^{2}{{{\mbox{e}}}^{\tilde{f}}} \right)}{{{u}_{0}}+\left( {{u}_{0}}+2{{w}_{20}} \right){{{\mbox{e}}}^{\tilde{f}}}}+\frac{2{{k}_{12}}{{k}_{21}}\left( u_{0}^{2}-{{w}_{10}}{{w}_{20}}{{{\mbox{e}}}^{\tilde{f}}} \right)}{{{u}_{0}}+\left( {{u}_{0}}+{{w}_{10}}+{{w}_{20}} \right){{{\mbox{e}}}^{\tilde{f}}}} \right|}_{\tilde{f}=\tilde{f}_{s}^{\left( 2 \right)}}}=0,\end{eqnarray} \tag{ B.26 }$$

which is clearly a cubic equation in ${{{\mbox{e}}}^{\tilde{f}}}$ whose only real root gives the two-filament stall force $\tilde{f}_{s}^{\left( 2 \right)}$.

Microtubules and actin filaments are structures consisting of multiple proto-filaments that strongly interact with each other. Actin filaments are two-stranded helical polymers while microtubules are hollow cylinders made of 13 protofilaments [1, 2, 4]. In this section we discuss the equivalence between the single-filament picture that we have been using, and the multiprotofilament nature of cytoskeletal filaments. In [14], it has been shown that, within the sequential hydrolysis, a multiprofilament model called 'one-layer' model can be exactly mapped to the single-filament picture we used. Below we discuss the one-layer model and show that our measured stall forces are exactly the same as in the sequential hydrolysis model (see appendix C above).

One layer model makes use of two known experimental facts: (1) there is a strong lateral interaction between protofilaments (inter-protofilament interaction, which is as strong as $\approx -8{{k}_{B}}$ T for microtubules). (2) Each protofilament is shifted by a certain amount from its neighbor. We take $\epsilon =b/m$ (see figure D1) where b is the length of one tubulin/G-actin monomer, and m is the number of protofilaments within one actin/microtubule (m = 2 for actin, and m = 13 for microtubule). Fact (1) would imply that any monomer binding on to a cytoskeletal filament (say, microtubule) would highly prefer a location that would form maximal lateral (inter-protifilament) bonds. This would lead to a situation where a growing cytoskeletal filament will be in a conformation where distance between any two protofilament tip will never be larger than b (see [11, 17, 37] where this model is discussed in detail).

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The above-mentioned restrictions would lead to the following rules for growth dynamics: (i) addition of a monomer can happen only at the most trailing tip at a rate $u={{u}_{0}}{\rm exp} \left( -f\epsilon /{{k}_{B}}T \right)$—for example, a monomer only can bind at protofilament 3 in figure D1), (ii) dissociation of a monomer only takes place at most leading protofilament at a rate ${{w}_{T}}$ (when tip is ATP/GTP-bound) or ${{w}_{D}}$ (when tip is ADP/GDP-bound)—for example, a monomer only can dissociate from protofilament 1 in figure D1, and (iii) a hydrolysis event only happens at the most trailing T–D interface at a rate R—for example, a hydrolysis only takes place at protofilament 2 in figure D1. It has been shown analytically [14] that this one-layer model exactly maps for one actin filament (m = 2) to a simple one-filament sequential hydrolysis model by taking $d=\epsilon =b/m$, where d is the length of a subunit in sequential hydrolysis model. Note that this mapping is expected since in the one-layer model the right wall (see figure D1) only moves by an amount of after each association/dissociation event. We further numerically find that this mapping exactly works for one microtubule and two microtubules and actin filaments. All the stall forces $f_{s}^{\left( 1 \right)}$, $f_{s}^{\left( 2 \right)}$, and thus the excess force ${{\Delta }^{\left( 2 \right)}}$ are exactly same in real units (with the above mapping) as in sequential hydrolysis model (see table C1).

In the spirit of 'one-layer' sequential hydrolysis model [11, 17, 37] we propose a one-layer version within the random hydrolysis. We consider a biofilament made of m protofilaments, and each protofilament is shifted by an amount $\epsilon =b/m$ from its neighbour (see figure D2). Here b is the length of a Tubulin/G-actin monomer. To simulate the system we apply the following rules for polymerization and depolymerization and hydrolysis dynamics: (i) addition of a monomer can happen only at the most trailing protofilament tip with a rate $u={{u}_{0}}{\rm exp} \left( -f\epsilon /{{k}_{B}}T \right)$—for example, a monomer can bind only at protofilament 3 in figure D2, (ii) dissociation of a monomer takes place only at the most leading protofilament with a rate ${{w}_{T}}$ (when tip is ATP/GTP-bound) or ${{w}_{D}}$ (when tip is ADP/GDP-bound)—for example, a monomer can dissociate only from protofilament 1 in figure D2, and (iii) a random hydrolysis event happens only at that protofilament which has maximum number of ATP/GTP-bound subunits with a rate r — for example, a hydrolysis event takes place only at protofilament 3 in figure D2 (at any random location). Numerically we find that the results for stall forces and excess forces in this model are very close to that of the random hydrolysis model (see table D1)—to make the correspondence, we set the subunit-length $d=\epsilon =b/m$ in the random hydrolysis model. Thus, d = 8 nm $/13=0.6\;$ nm for microtubule and d = 5.4 nm $/2=2.7{\mkern 1mu}$nm for actin within random hydrolysis model.

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