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

Collective dynamics and force generation by cytoskeletal filaments are crucial in many cellular processes. Investigating growth dynamics of a bundle of N independent cytoskeletal filaments pushing against a wall, we show that chemical switching (ATP/GTP hydrolysis) leads to a collective phenomenon that is currently unknown. Obtaining force-velocity relations for different models that capture chemical switching, we show, analytically and numerically, that the collective stall force of N filaments is greater than N times the stall force of a single filament. Employing an exactly solvable toy model, we analytically prove the above result for N=2. We, further, numerically show the existence of this collective phenomenon, for N>=2, in realistic models (with random and sequential hydrolysis) that simulate actin and microtubule bundle growth. We make quantitative predictions for the excess forces, and argue that this collective effect is related to the non-equilibrium nature of chemical switching.


Introduction
Biofilaments such as actin and microtubules are simple nano-machines that consume chemical energy, grow and generate significant amount of force. Living cells make use of this force in a number of ways-e.g., to generate protrusions and locomotion, to segregate chromosomes during cell devision, and to perform specific tasks such as acrosomal process where an actin bundle from sperm penetrates into an egg [1,2]. Understanding the growth of cytoskeletal filaments provides insights into a wide range of questions related to collective dynamics of biomolecules and chemo-mechanical energy transduction.
Actin and microtubules grow by addition of subunits that are typically bound to ATP/GTP. Once polymerized, ATP/GTP in these subunits get hydrolyzed into ADP/GDP creating a heterogeneous filament with interesting polymerisation-depolymerisation dynamics [3,4]. In actin, the subunits are known to also exist in an intermediate state bound to ADP-P i [5][6][7]. Even though the growth kinetics of actin and microtubules are similar, under cellular conditions they show diverse dynamical phenomena such as treadmilling and dynamic instability. They also have very different structures: actin filaments are two-stranded helical polymers while microtubules are hollow cylinders made of 13 protofilaments [1,2,4].
These filaments growing against a wall can generate force using a Brownian ratchet mechanism [1]. The maximum force these filaments can generate, known as 'stall force', is of great interest to experimentalists and theorists alike [1,8,9]. In theoretical literature, growth of a single bio-filament and the resulting force generation has been extensively studied [1,6,10,11]. Explicit relations of velocity versus force (or monomer concentration) have been derived assuming either simple polymerization and depolymerization rates [1] or more realistic models that take into account ATP/GTP hydrolysis [6,[11][12][13][14][15][16]. In fact without considering hydrolysis, the observed velocities and length fluctuations of single filaments cannot be explained [6,[11][12][13].
Even though single filament studies teach us useful aspects of kinetics of the system, what is relevant, biologically, is the collective behaviour of multiple filaments. However, the theory of collective effects due to N (⩾2) filaments pushing against a wall is poorly understood. Simple models of filaments with polymerization and depolymerization rates have been studied: for two filaments, exact dependence of velocity on force is known [9,17], while for ⩾ N 2, numerical results and theoretical arguments [9,18] show that the net force f ( ) s N to stall the system is N times the force f ( ) s 1 to stall a single filament. A similar result, namely ∝ f N ( ) s N , was obtained for multiple protofilaments with lateral interactions in the absence of hydrolysis [19]. Under harmonic force, experimental studies claimed collective stall forces to be lesser than N times single filament stall force for actin [8], and proportional to N for microtubules [20]. Based on the understanding of single filaments, it has been speculated that hydrolysis might lower the stall force of N actin filaments [8]. In a recent theoretical study within a two-state model, under harmonic force [21], it was shown that the average polymerization forces generated by N microtubules scale as N. However, for constant force ensemble, there exists no similar theory for multiple filaments that incorporates the crucial feature of ATP/GTP hydrolysis, or appropriate structural transitions [22], systematically. How, precisely, the ATP/GTP hydrolysis influences the growth and force generation of N filaments is an important open question.
Motivated by the above, in this paper, we investigate collective dynamics of multiple filaments, using a number of models that capture chemical switching. These models extend the work of Tsekouras et al [9] by adding the 'active' phenomenon of ATP/GTP hydrolysis. The first one is a simple model in which each filament switches between two depolymerization states. Within the model, we analytically show that ; this result extends to > N 2. We then proceed to study numerically two detailed models involving sequential and random mechanisms for hydrolysis [6,11,14,15]. Using parameters appropriate for the cytoskeletal filaments, we show that indeed > f Nf

Models and results
2.1. An exactly solvable toy model that demonstrates the relationship f (2) s > 2f (1) s analytically 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  and δ ∈ [ ] 0, 1 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 . 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 δ = 1, consistent with earlier theoretical literature [9,14,17] and close to experiments on microtubules [23]. For a single filament, the average velocity is given by 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 = Pk Pk 1 12 2 21 holds in the steady state. Along with the normalization condition + = P P 1 1 2 , this yields: 2 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 with v 11 , v 22 , v 12 and v 21 given by equations (2) and (3). Note that various limits ( = w w 10 20 , 11 . The equation (4) is valid for any δ. To obtain stall force we set = V 0 ( ) 2 and this leads to a cubic equation (for δ = 1) inẽ f whose only real root gives f ( ) s 2 analytically (see (B.3)). The analytical result for V ( ) 1 and V ( ) 2 (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 f 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 Δ =˜−f Nf We now proceed to show that the introduction of switching between distinct chemical states ( ≠ w w 1 2 , ≠ k k 0 12 21 or ≠ ∞ k k 12 21 ) produces non-equilibrium dynamics embodied in the violation of the well known detailed balance condition. To demonstrate this for the singlefilament toy model, in figure 3(a), we consider a loop of dynamically connected configurations (charaterized by its length and state): 1 1, 1 1, 2 , 2 , 1 . The product of rates clockwise and anticlockwise are uk w k 12 2 21 and k uk w the condition of detailed balance to hold (in equilibrium) the two products need to be equal (Kolmogorovʼs criterion), which leads to = w w 1 2 . In figure 3 always, except in the limits → k k 0 12 21 or ∞. These limits correspond to the absence of switching and hence equilibrium. 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 boundsimilar 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][12][13][14][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 = −ũ u e f 0 (next to the wall) or u 0 (away from the wall). Note that = u k c 0 0 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 tipmonomer 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. δ = 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).
We first numerically calculated the single microtubule stall force f ( )    pN for μ = c 100 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 Δ In figures 6(a) and 6(b), we show that the excess stall force Δ ( ) N 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.
We now investigate the dependence of the excess force on various parameters. For N = 2, within the random model, we show the deviations (Δ ( ) 2 ) and percentage relative deviation ) as a function of free monomer concentration (c) for actin (figure 7(a)) and microtubule (figure 7(b)). The absolute deviation (Δ ( ) 2 ) 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 Δ ( ) 2 increases rapidly with decreasing w T and can go up to 1.
) for actin and 9 pN ( ) for microtubules. We did a similar study of Δ ( ) 2 as a function of w D (see figures 7(e)-(f)), and find that Δ ( ) 2 increases with increasing w D . The important thing to note is that Δ ( ) 2 increases as we increase w D (at constant w T ) and decrease w T (at constant w D ), i.e. the magnitude of Δ ( ) 2 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.
The case = w w D 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 cases, the hydrolysis process violates detailed balance as it is unidirectional: ATP/GTP → ADP/ GDP conversion is never balanced by a reverse conversion ADP/GDP → 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 Δ ( ) 2 as a function of w D (for smaller values of w D compared to figure 7(f)) for two-microtubule system, and find that Moreover, for biologically impossible situations of 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 → ADP-→ P i ADP [5][6][7]. Our two-state models above approximate this with the dominant rate limiting step of 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 ). For this model we first calculate the single-filament stall force pN. Simulating this model for two actin filaments we find that the wall moves with pN (also see figure 9(b) (bottom) for few traces of the wall position at stall). Consequently, we obtain the excess force pN-this value is same as that of the two-state random hydrolysis model (see table D1, actin).
In [7] a variant of the above three-state model is discussed where ADP-P i →ADP conversion happens at two different rates-with a rate r tip at the tip, and with a rate r in the bulk. In this model = ∞ r DP i.e. as soon as an ATP subunit binds to a filament it converts to ADP-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 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.

Discussion and conclusion
The study of force generation by cytoskeletal filaments has been an active area of research for the last few years [8,9,14,18,20,30]. Earlier theories like our current work have studied the phenomenon of force generation by biofilaments and their dynamics in a theoretical picture of filaments growing against a constant applied force [9,14,18]. These theories either neglected ATP/GTP hydrolysis, or looked at single (N = 1) filament case, and hence outlined a notion that stall force of N biofilaments is simply N times the stall force of a single one i.e. Then we proceeded to study the phenomenon in realistic random hydrolysis model and many of its variants (based on features of hydrolysis suggested from recent experimental literature). Even the sequential hydrolysis model is shown to exhibit similar phenomenon (see appendix C). Thus our extensive theoretical case studies suggest that the phenomenon of excess force generation is quite general and convincing. The question remains that how our results can be observed in suitably designed experiments in vitro. There have been a few in vitro experiments which studied the phenomenon of collective force generation by biofilaments [8,20]. In [8] authors directly measured the force of parallel actin filaments by using an optical trap technique, and found that the force generated by eight-actin filaments is much lesser than expected. We would like to comment that this experimental result can not be compared to theoretical predictions like above due to the fact that the experiment is done under harmonic force, and not in a constant force ensemble as in theory. Secondly the experimental filaments had buckling problems, which are unaccounted for in theory. A later experiment [20] on multiple microtubules (which were not allowed to buckle using a linear array of small traps) found that the most probable values of forces generated by a bundle of microtubules appear as integral multiples of certain unit. This led to an interpretation that multiple microtubules generate force which grow linearly with filament number. We would like to comment that the theoretical stall forces mean maximum forces, which are not the most probable forces (as studied experimentally). Secondly like [8], experiment of [20] also had harmonic forces, unlike constant forces in theories. To validate our claim of excess force generation, new in vitro experiments should work within a constant force ensemble, ensure that filaments do not buckle, and the averaging of wall-velocity is done over sufficiently long times, such that the effect of switching between heterogeneous states is truly sampled.
A simple way to prevent the buckling of the filaments is to keep their lengths short as the filaments under constant force will not bend below a critical length [31]. The estimates of the critical length for buckling, at stall force, for microtubule is μ − m 4 17 (for μ = − c 10 100 M), and for actin is ). Since stall force does not depend on the length one can prevent buckling by choosing lengths well below the critical lengths of the filament, as done in [20].
Apart from the direct measurement of the excess force Δ ( ) N , there is yet another way to check the validity of the relationship > f Nf For example, as seen in figure 5(b), at = f f 2 ( ) s 1 two microtubules with random hydrolysis have a velocity ∼1 subunit/s-equivalent to a growth of 500 nm in less than 15 min. In figure 5(c), one can see that, for two-actin within random hydrolysis, this velocity is ∼0.03 subunits − s 1 . This would imply a growth of ∼300 nm in 1 h. These velocities can be even higher for other ATP/GTP concentrations. Such velocities (and the resulting length change) are considerable to be observed experimentally in a biologically relevant time scalethus the claim of the force equality violation may be validated.
Even if the magnitude of the excess force is small, it can be crucial whenever there is a competition between two forces. For example, it is known that an active tensegrity picture [32] can explain cell shape, movement and many aspects of mechanical response. The core of the tensegrity picture is a force balance between growing microtubules and actin-dominated tensile elementsD microtubule has to balance the compressive force exerted on them. So, in such a scenario, where two forces have to precisely balance, even a small change is enough to break the symmetry.
Our toy model has potential to go beyond this particular filament-growth problem: the model suggests that a simple two-state model with non-equilibrium switching can generate an interesting cooperative phenomenon and produce excess force/chemical potential than expected. In biology there are a number of non-equilibrium systems that switch between two (or multiple) states. For example, molecular motors, active channels across cell membrane, etc. Following our finding, it will be of great interest to test whether such a cooperative phenomenon will arise in other biological and physical situations.
In this paper we did not address the issue of dynamic instability in the presence of force. This is an interesting problem in itself, and recent theoretical and experimental studies have addressed various aspects of this problem [20,21]. Our models are also capable of exhibiting this phenomenon, and interested reader may look at our recent work [33] where we have studied the collective catastrophes and cap dynamics of multiple filaments under constant force, in detail, for the random hydrolysis model, and compared our results to recent experiments [20]. The literature of two-state models [21,34] have shown that catastrophes arise in multiple filaments having force-dependent growth-to-shrinkage switching rates [21]. In microscopic models like ours, effective force-dependent catastrophe rates emerge naturally; for example, in random hydrolysis model (see [15]), it is known that catastrophe rates are comparable to the results of Janson et al [35] and Drechsel et al [36]. The toy model too can show dynamic instability with catastrophe and rescues (see appendix E, and figure E1(a)). Even with constant (force-independent) switching rates k 12 and k 21 , the toy model exhibits the phenomenon of force-dependent catastrophe; in figure E1(b) we have shown that the rate ( +− k ) of switching from growing state to shrinking state, computed from simulation, increases with force (also see appendix E). To test how the system behaves under explicit force-dependent switching, we made =( ) k k f exp ( ) 12 12  [21], where they show that the average polymerization force of N microtubules grows linearly with N when rescues are permitted, for filaments in harmonic trap, would be interesting to explore in detail in future.
In summary, we have studied collective dynamics of multiple biofilaments pushing against a wall and undergoing ATP/GTP hydrolysis. Quite contrary to the prevalent idea in the current literature [8,9,18], we find that hydrolysis enhances the collective stall force compared to the sum of individual forces-i.e. the equality = f Nf Using the above, after summing over all l, we get The normalization condition is ∑ , 1 . So the velocity of the wall is: In the steady state ( → ∞ t ), 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 l l k t , ; ij , 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)): ) ( ) P l k l t uP l k l w P l k l u P l k l wP l k l k P l k l P l k l u u w w k P l k l ) ( ) P l k l t uP l k l w P l k l u P l k l w P l k l k P l k l P l k l u u w w k P l k l )( ) P l k l t uP l k l w P l k l u P l k l w P l k l k P l k l k P l k l u u w w k k P l k l  Next, let ( ) P l l t , ; ij be the probability that, at time t, both filaments have same length l, and they are in a joint state { } i j , . One such situation is shown in figure B2(c). These probabilities satisfy the following Master equations: P l l t u P l l w P l l u P l l w P l l k P l l P l l u w k P l l d , ; P l l t u P l l w P l l u P l l w P l l k P l l P l l u w k P l l d , ; P l l t u P l l w P l l u P l l w P l l k P l l k P l l u w w k k P l l d , ; We also define the probability of residency in joint state { } i j , as: . The normalization of probabilities leads to 1,2 , 0 11 22 12 21 . 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 ) balance equations satisfied by the probabilities P ij :  The probabilities obtained above may be used to calculate the two-filament velocity   10 20 In the steady state, the above equations (B.18)-(B.22), along with the normalization condition: , can be solved exactly and one gets the following distributions for the gaps: depolymerization rate is w T if there exists a finite ATP/GTP cap (like the top filament in figure C1(a)); otherwise it is w D if the cap does not exist (like the bottom filament in figure C1(a)). The hydrolysis rate (the rate of T becoming D) is R and it can happen only at the interface of the ADP/GDP(bulk)-ATP/GTP(cap) regions. The switch ADP/GDP → ATP/GTP at the tip can happen only by addition of free T monomers-there is no direct conversion of ADP/ GDP → ATP/GTP within a filament. For sequential hydrolysis the stall force of single filament is exactly known [14], which is while for two filaments we need to calculate it numerically as no exact formula is available. Given the exactly known single filament stall force f ( ) 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 ≈− k 8 B T for microtubules).
(2) Each protofilament is shifted by a certain amount ϵfrom its neighbor. We take ϵ = 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 (interprotifilament) 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). 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 -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 onelayer model exactly maps for one actin filament (m = 2) to a simple one-filament sequential hydrolysis model by taking 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 ϵ 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 ( )  (see  table C1).

D.2. One-layer multi-protofilament model with random hydrolysis
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 ϵ = 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 -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/GTPbound subunits with a rate rfor 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 ϵ = = d b m in the random hydrolysis model. Thus, d = 8 nm = 13 0.6 nm for microtubule and d = 5.4 nm = 2 2.7nm for actin within random hydrolysis model. Appendix E. Force-dependence of the growth-to-shrinkage switching rate within the toy model As discussed in the literature [34], microtubule dynamics can be classified into two dynamical phases-(i) bounded growth phase (average filament velocity v = 0) and (ii) unbounded growth phase ( > v 0). For forces greater than the stall force f ( )  Figure D2. Schematic diagram of a single filament made of m protofilaments in onelayer model with random hydrolysis. Blue and red colors refer to ATP/GTP-bound and ADP/GDP-bound subunits respectively. Rules for the growth/shrinkage dynamics are discussed in the text below.

Simplified random hydrolysis
One-layer (multi-protofilament) wall position (x versus t) is shown in figure E1(a)-we clearly see the filament collapsing to zero length frequently. Following figure E1(a), we define a 'peak' to be the highest value of x between two successive zero values. We then define the growth time + ( ) T as the time it takes to reach a 'peak' starting from the preceding zero (see the regions shaded grey in figure E1(a)). We construct a switching rate from growth to shrinkage as = +− + k T 1 , where + T is measured by averaging over a long time window. In figure E1(b) we show that the rate +− k increases with the force.  , and δ = 1.