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Load sharing in the growth of bundled biopolymers

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Published 20 November 2014 © 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
, , Citation Ruizhe Wang and A E Carlsson 2014 New J. Phys. 16 113047 DOI 10.1088/1367-2630/16/11/113047

1367-2630/16/11/113047

Abstract

To elucidate the nature of load sharing in the growth of multiple biopolymers, we perform stochastic simulations of the growth of biopolymer bundles against obstacles under a broad range of conditions and varying assumptions. The obstacle motion due to thermal fluctuations is treated explicitly. We assume the 'perfect Brownian ratchet' model, in which the polymerization rate equals the free-filament rate as soon as the filament-obstacle distance exceeds the monomer size. Accurate closed-form formulas are obtained for the case of a rapidly moving obstacle. We find the following: (1) load sharing is usually sub-perfect in the sense that polymerization is slower than for a single filament carrying the same average force; (2) the sub-perfect behavior becomes significant at a total force proportional to the logarithm or the square root of the number of filaments, depending on the alignment of the filaments; (3) for the special case of slow barrier diffusion and low opposing force, an enhanced obstacle velocity for an increasing number of filaments is possible; (4) the obstacle velocity is very sensitive to the alignment of the filaments in the bundle, with a staggered alignment being an order of magnitude faster than an unstaggered one at forces of only 0.5 pN per filament for 20 filaments; (5) for large numbers of filaments, the power is maximized at a force well below 1 pN per filament; (6) for intermediate values of the obstacle diffusion coefficient, the shape of the force velocity relation is very similar to that for rapid obstacle diffusion.

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1. Introduction

Actin filament growth against the plasma membrane generates forces to drive cell migration, which is required for numerous phenomena including embryonic development, wound healing, immune response and cancer metastasis. Actin polymerization forces are also important for generating protrusions such as filopodia and lamellipodia. This has motivated experiments to measure the dependence of velocity on force. The results vary greatly according to the experimental conditions and the geometry of the filaments. Most of the measurements of force–velocity relations have been made for branched actin networks. Measurements of Listeria and actin-propelled beads [1, 2] found that the velocity drops rapidly with small forces and slowly with larger forces. Measurements of actin-propelled cantilevers [3] suggested that the velocity stays at a plateau at light loads, but decays quickly at heavier loads. In lamellipodia [4, 5], the velocity was found to have a rapid initial drop at low force followed by a slower decay with increasing force. Measurements for a collection of unbranched filaments confined between beads [6] found that the relative velocity of the beads drops rapidly with force compressing the beads together, but the polymerization rate of actin filaments is essentially unchanged. We are not aware of measurements of the steady-state force–velocity relation of actin filament bundles. However, the stall force of actin bundles nucleated by acrosomes has been measured [7]. Recently, dynamic forces generated by filopodia [8] and filopodia-like protrusions [9] have been measured. Amin et al [8] reported net forces (the difference between the polymerization force and the elastic restoring force from the membrane) up to about $4\;{\rm pN}$. Farrel et al [9] measured force changes believed to be due to polymerization, up to about $15\;{\rm pN}$, which were attributed to approximately six filaments each carrying $2.5\;{\rm pN}$ of force. The force–velocity relation has also been measured for microtubules [10], which are similar to actin bundles in that they are made up of 13 growing 'protofilaments'.

Theoretical analysis and simulations are useful here because they can help interpret or guide experiments, and because they can provide order-of-magnitude estimates where data are not available. Peskin, Odell and Osterʼs classic paper [11] proposed a model that we call the perfect Brownian ratchet (PBR). If an obstacle experiences an external force F, then its thermal fluctuations can open up gaps comparable to the monomer size δ. Gaps less than δ allow no polymerization, but polymerization will be unhindered for any gaps greater or equal to δ, driving the obstacle forward. By PBR we mean that the the polymerization rate reaches its free-filament value as soon as the gap exceeds δ. The authors treated obstacle motion implicitly, deriving the simple exponential formula ${\rm exp} (-F\delta /{{k}_{B}}T)$ for the slowing of polymerization by an obstacle in equilibrium with the tip of the filament.

Subsequent work, using more complete models, has confirmed the general features of the PBR results for single filaments. The calculations of [12] obtained the Fv relation of a single filament in 2D, treating monomer diffusion explicitly, and thus avoiding the PBR assumption. Using assumed monomer-monomer and monomer-obstacle potentials, it was found that the Fv relation is generally exponential, though with a larger decay rate than obtained by [11]. Burroughs and Marenduzzo [13] treated a model that included stochastic obstacle diffusion in 1D and a semi-flexible filament, using the PBR assumption. They found that the simulated force–velocity relation is almost identical to the exponential form ${\rm exp} (-F\delta /{{k}_{B}}T)$ predicted by Peskin et al [11].

However, there is less consensus about the growth of multiple filaments under opposing force. The distribution of obstacle heights relative to a given filament tip is strongly affected by the positions of the other filament tips, leading to complex interactions between polymerization events on different filaments. An intuitive assumption is that of 'perfect load sharing' (PLS), which means that a change in the number of filaments will not alter the polymerization rate if the opposing force per filament stays the same. Schaus and Borisy [14] performed a detailed study of load sharing in 2D lamellipodia using a combination of continuum diffusion and stochastic polymerization, by assuming that each filament has an individual growth rate depending on its tipʼs distance to the obstacle. They defined two limits: (i) the zero load sharing limit (ZLS), where the growth velocity is $\propto \;n{{{\rm e}}^{-F\delta /{{k}_{B}}T}}$; and (ii) the PLS limit. In their model, PLS gives a velocity proportional to ${{{\rm e}}^{-F\delta /n{{k}_{B}}T}}$, where n is the number of free filaments, and F is the total obstacle force. They concluded that load sharing for realistic models is between the two limits. Recent work expliclty including diffusive obstacle motion under zero load, pushed by a randomly staggered collection of filaments, found that extreme statistics had unexpected importance [15]. In addition, the problem of two coupled filaments growing under the influence of lateral interactions has been given an exact solution [16]. However, a force–velocity relation was not plotted.

The problem of load sharing of actin filaments is closely related to the sharing of the load between protofilaments in microtubules. This has been addressed in [17], where the growth rate of a protofilament was also taken to be an exponential function of its distance from the obstacle, whose motion was treated implicitly. It was found that the force–velocity relation remains exponential, but with a larger decay parameter than would have been expected. This work was subsequently generalized by [18], who also found that the force–velocity relation decays much more rapidly than would have been predicted from PLS [19]. Performed a Monte Carlo treatment of the growth of microtubules, including interactions at the tip, again assuming a polymerization rate that depends on average distance from an implicit obstacle. Subsequent simulations [20] treated the effects of attractive interactions between subunits on different filaments. They found that the stall force corresponds to equal load sharing, but did not explicitly address how load sharing affects the dynamics of polymerization. Recently, cooperative dynamics of mictrotubule ensembles have been studied using the PLS assumption [21].

Several papers not explicitly calculating load sharing have made varying assumptions about load sharing. In simulating filopodial dynamics, Papoian and collaborators [22, 23] assumed that filaments share the membrane force evenly, and that the growth rate for n filaments under a total membrane force F is $\propto \;{{{\rm e}}^{-F\delta /n{{k}_{B}}T}}$. This formula implies PLS, because the velocity depends only on the ratio $F/n$. Tsekouras et al [24] assumed PBR and ZLS, and focused on the particular case of low actin concentrations where the zero-force polymerization rate is about twice the off-rate. Using a low bulk actin concentration $\lt 0.25$ μM, the authors generated the Fv curve of a bundle of 100 actin filaments. The curve is concave at small forces and convex at forces $\gt 4$ ${\rm pN}$. This work was subsequently extended to include the effects of ATP hydrolysis [25]. Smith and Liu [26] recently investigated the growth of branched actin networks of a large number of filaments (up to $\gt 10\;000$), against nanonewton-scale forces. Their assumptions, when applied to our case of parallel filaments, amount to PBR and PLS together.

These models have played important roles in understanding actin Fv relations and paved paths for future studies. However, the type of load sharing has often been assumed, rather than explicitly calculated; the validity of the assumptions made, such as PLS or ZLS, are not clear. Furthermore, the obstacle motion has been assumed infinitely rapid and treated implicitly, so it has not been possible to treat the effect of slower obstacle motion. To our knowledge, there is no systematic treatment in the literature of the different regimes of load sharing as determined by a broad range of opposing forces, filament number, and obstacle diffusion coefficient. For example, it is not clear at what value of the obstacle diffusion coefficient one can safely assume it to be infinite. It is also not known at what force the crossover between the weak and strong force limits occurs. Finally, it is not known at what value of the obtacle diffusion coefficient one obtains significant corrections to the widely employed approximation of infinitely rapid obstacle diffusion.

For this reason, we perform a broad range of calculations of the extent of load sharing in a simple model system. We treat the motion of a hard obstacle explicitly, along with polymerization of multiple filaments, using stochastic simulations that make no a priori assumptions about the nature of load sharing. The results allow us to evaluate the regimes of validity of commonly made assumptions about load sharing. They clarify the relation between the growth velocity of multiple filaments, the external force on the load, and the obstacle diffusion coefficient. For rapid obstacle diffusion, we obtain explicit, portable formulas that cover the whole range of applied forces and possible filament numbers. These will be useful in future modeling efforts, and for analyzing experimental data. We also show that when the obstacle diffusion coefficient is of intermediate magnitude, qualitative changes to the load sharing at weak forces are seen.

2. Model

Our model system consists of an explicitly moving obstacle pushed by n force-generating filaments that are assumed to be held together by crosslinkers or nonspecific bundlers. We do not treat potential effects that these bundlers could have on the polymerization rate. In the PBR model that we implement, polymerization at filament tips occurs only when the tip to obstacle distance $h\geqslant \delta $ (figure 1). 'Perfect' means that monomer-addition events are calculated stochastically from the free-filament rate ${{k}_{0}}={{k}_{{\rm on}}}{{C}_{0}}$ as long as $h\geqslant \delta $, where kon is the on-rate constant and C0 is the free-actin concentration. Monomer diffusion is treated implicitly and no specific assumption beyond PBR, regarding filament and obstacle geometry, is made. Depolymerization is ignored. A filament is assumed to start at an initial length of four subunits, and filament nucleation and disappearance are not treated. Filaments are modeled as rigid straight lines of actin monomers, and the obstacle is modeled as a hard disk. Filaments cannot penetrate the obstacle. The diffusion of the obstacle is treated via stochastic jumps on a 1D lattice with a mesh size ${\rm d}x$. For computational efficiency, we adopt an algorithm in which the obstacle moves either up or down during each time step. Given the diffusion coefficient of the obstacle D, we choose a time step size ${\rm d}t={\rm d}{{x}^{2}}/2D$, in which ${\rm d}x$ is assigned a value small (maximum value used: 0.3 ${\rm nm}$) compared to the monomer size. Note that the simulated polymerization will not depend on the choice of the mesh size, as long as ${{k}_{0}}{\rm d}t\ll 1$; however, when D is very small, ${\rm d}t$ may become large enough that ${{k}_{0}}{\rm d}t\ll 1$ breaks down. In such cases, we reduce the value of ${\rm d}x$ as needed.

Figure 1.

Figure 1. Schematic of a basic Brownian ratchet model for a single filament. If $h\lt \delta $, no polymerization is allowed; if $h\geqslant \delta $, polymerization is allowed.

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The obstacle in the model moves by biased Brownian motion [27]. At each time step, the probability for the obstacle to hop upward (${{r}_{{\rm u}}}$) or downward (${{r}_{{\rm d}}}$) (unless blocked by a filament) by ${\rm d}x$ is determined by the force F. The requirement that the model reproduce the Boltzmann distribution of h in steady state implies that ${{r}_{{\rm d}}}{{{\rm e}}^{-F{\rm d}x/{{k}_{B}}T}}={{r}_{{\rm u}}}$ (i.e. upward transitions balance downward transitions). Also, ${{r}_{{\rm d}}}+{{r}_{{\rm u}}}=1$, so

Equation (1)

For $F{\rm d}x/{{k}_{B}}T$ sufficiently small, one readily shows that for unconstrained motion beginning at the origin, this algorithm correctly satisfies the relations $\langle x\rangle =FDt/{{k}_{B}}T$ and $\langle {{x}^{2}}\rangle ={{\langle x\rangle }^{2}}+2Dt.$ We have always chosen our step size to make sure this holds. We have also checked that the model correctly obtains the average first passage time tFP: the average time required for the obstacle to reach the height $z=\delta $ above the highest filament, starting from z = 0. Our simulation results reproduce the analytic result ${{t}_{{\rm FP}}}=[{{k}_{B}}T/{{F}^{2}}{{\delta }^{2}}][{{k}_{B}}T({{{\rm e}}^{F\delta /{{k}_{B}}T}}-1)-F\delta ]$ obtained using the 'first-passage time' equation [28].

Tables 1 and 2 list variables and key quantities in simulations. Each simulation was run until a number Npol of polymerization events had occurred for each individual filament (${{N}_{{\rm pol}}}\gt {{10}^{4}}$ for a single filament and ${{N}_{{\rm pol}}}\gt 1600$ for a bundle of filaments). Then the monomer addition rate per filament was evaluated as $k^{\prime} ={{N}_{{\rm pol}}}/t$, where t is the simulated time. Given Npol, the statistical errors are expected to be less than 1% for a single filament, and 2.5% for a bundle of filaments. We only consider one value of the monomer concentration C0. Since we ignore depolymerization, any change in the monomer concentration can be compensated by a corresponding change in the diffusion coefficient, without changing the force–velocity relation.

Table 1.  Simulation inputs.

Symbol Definition Value
δ Monomer size 2.7 ${\rm nm}$
${{k}_{B}}T$ Thermal energy 4.1 ${\rm pN}\cdot {\rm nm}$
D Obstacle diffusion coefficient Varies
${{C}_{0}}$ Bulk monomer concentration 10 $\mu {\rm m}$
${{k}_{{\rm on}}}$ Monomer on-rate constant 11.6 $\mu {{{\rm m}}^{-1}}\;{{{\rm s}}^{-1}}$[29]
$\tilde{D}=D/({{k}_{{\rm on}}}{{C}_{0}}{{\delta }^{2}})$ Dimensionless diffusion coefficient, to quantify obstacle diffusion relative to polymerization Varies

Table 2.  Other quantities used in the simulations.

Symbol Definition
n Number of filaments in bundle
F Total opposing force applied on the obstacle
f Average force per filament, equal to $F/n$
${{k}_{0}}$ Polymerization rate of a free filament at the bulk concentration; equal to ${{k}_{{\rm on}}}{{C}_{0}}$
$k^{\prime} $ Actual polymerization rate per filament, generally less than k0
$v\left( n,f \right)$ $k^{\prime} /{{k}_{0}}$, scaled velocity of n-filament bundle under a load F = nf
p ${{{\rm e}}^{-F\delta /{{k}_{B}}T}}$
pr ${{{\rm e}}^{-F\delta /n{{k}_{B}}T}}={{{\rm e}}^{-f\delta /{{k}_{B}}T}}$
P Power per filament
Pmax Maximum of P
fmax Value of f at which Pmax is achieved
h Distance between the filament tip and the obstacle

3. Results for load sharing

In presenting our results, we focus on the polymerization rate per filament, quantified by the dimensionless velocity $v=k^{\prime} /{{k}_{0}}$, where $k^{\prime} $ is the polymerization rate (in subunits per second) found in the simulations, ${{k}_{0}}={{k}_{{\rm on}}}{{C}_{0}}$ is the polymerization rate of a free filament at the bulk concentration, kon is the monomer on-rate constant, and C0 is the bulk monomer concentration. The speed of obstacle diffusion relative to polymerization is quantified by the dimensionless diffusion coefficient $\tilde{D}=D/({{k}_{{\rm on}}}{{C}_{0}}{{\delta }^{2}})$. We focus on $\tilde{D}$ values $\gt 1$, since a biologically relevant obstacle is unlikely to have a diffusion coefficient small enough to render $\tilde{D}$ much less than 1. For example, according to the Einstein relation and Stokesʼs law [30], a large bead of radius $R=5\;\mu {\rm m}$ would have a diffusion coefficient $D=({{k}_{B}}T/6\pi \eta R)=0.0044\;\mu {{{\rm m}}^{2}}\;{{{\rm s}}^{-1}}$, where we have taken the viscosity $\eta ={{10}^{-2}}\;{\rm Pa}\cdot {\rm s}$ to have ten times the value for water. A very rapidly polymerizing filament in an optical trap would have a polymerization rate k0 about $50\;\mu {\rm m}\times 11.6\;\mu {{{\rm m}}^{-1}}\;{{{\rm s}}^{-1}}\sim 580\;{{{\rm s}}^{-1}}$. Then $\tilde{D}=D/{{k}_{0}}{{\delta }^{2}}$ would be about 1.0. In most cases, D and k0 will be greater and lower than the values used above, respectively. Thus we focus on $\tilde{D}$ values in the intermediate and large range. For rapid diffusion, we use $\tilde{D}=D/{{k}_{0}}{{\delta }^{2}}\sim 70$ corresponding to a bead radius in the range of $0.3\;\mu {\rm m}$ to $0.4\;\mu {\rm m}$ and an actin concentration of $10\;\mu {\rm m}$. For slower obstacle diffusion we use $\tilde{D}=3.0$.

The distribution of load between the filaments at a given time will depend on how much the ends of the filaments are shifted relative to each other in the direction of growth [14, 17, 24]. There are many possibilities for the relative shifts. We consider two idealized types of initial height alignment for a bundle of n filaments: (i) unstaggered alignment (figure 2(a)) and (ii) staggered alignment (figure 2(b)). In the unstaggered alignment, all the filaments in a bundle have the same initial length. Then any two filaments can have the same length after several polymerization events. Staggered alignment means that the initial height of the shortest and longest filament are L and $L+(1-1/n)\delta $; and all filaments are uniformly distributed within $[L,L+(1-1/n)\delta ]$. For example, for five filaments, the shortest filamentʼs length is L, and the others have lengths $L+\delta /5,L+2\delta /5,L+3\delta /5,L+4\delta /5$. In this scenario, treated by [17], no filaments have the same length. This alignment is advantageous for force generation, since there always exist shorter filaments more prone to polymerize due to their tips being farther from the obstacle.

Figure 2.

Figure 2. Two idealized types of initial height alignment.

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We define three types of load sharing in table 3: super load sharing, PLS, and sub-PLS. Here f is the force per filament applied to the obstacle. This mathematical definition of PLS corresponds to the physical definition given in the introduction. Sub-PLS means that a bundle of filaments has a lower mean polymerization rate than a single filament if f is the same. We focus on the applied force f rather than the total force on the obstacle, which would include the drag force. We make this choice because f is more directly accessible in experiments, by for example changing the spring constant in an optical tweezer. One could also base the definition of load sharing on the number of filaments in contact with the obstacle. However, we see no way of measuring this quantity using existing light-microscopy methods, and thus no way to validate or disprove the theory. Therefore we base our definition of load sharing on the obstacle velocity and the number of filaments, both of which are experimentally accessible. Because of the thermal motion of the obstacle, a meaningful definition of the number of filaments in contact with the obstacle would involve an arbitrary threshold distance below which contact is considered to occur. If one chooses a threshold distance equal to the monomer size, the average number of filaments in contact ranges from less than 1 at very small forces (where obstacle fluctuations are large) to almost n at very large forces. The crossover occurs roughly at the force where where v = 0.5.

Table 3.  Types of load sharing.

Super load sharing Perfect load sharing Sub-perfect load sharing
$v\left( n,f \right)\gt v\left( 1,f \right),n\gt 1$ $v\left( n,f \right)=v\left( 1,f \right),n\gt 1$ $v\left( n,f \right)\lt v\left( 1,f \right),n\gt 1$

We focus our study on load sharing as it influences the growth velocity at forces substantially below the stall force. The neglect of depolymerization in our calculations means that there is no rigorously defined stall force, defined as the force where the velocity switches from positive to negative. However, we often observe a 'quasi-stall', where the velocity becomes very close to zero, which was also observed in [24]. This behavior is fairly independent of the off-rate. Calculation of the stall force is complicated by hydrolysis of the subunits at the growing filament tip to ADP-actin or GDP-tubulin, as discussed below under Critique of Assumptions. Near the stall force, polymerization is so slow that hydrolysis can occur before monomer addition. This greatly increases the off-rate. In order to avoid dealing with the complexities of hydrolysis, we focus our attention on forces below the stall force. We have also performed additional calculations including depolymerization in a two-state model without hydrolysis. We find that the stall force is proportional to n, consistent with [18].

3.1. Large $\tilde{D}$ ($\tilde{D}=70$) results

3.1.1. Unstaggered filaments

We first present approximate analytic results for two limits, assuming $\tilde{D}=\infty $: (i) weak external force and (ii) large external force. For (i) we treat the effect of the force to linear order. Then the time-averaged force carried by each filament is $f=F/n$, since for large $\tilde{D}$ the drag force can be ignored. (Note that the result f = F/n depends only on Newtonʼs third law and does not imply PLS, since in all of our models each filament will carry equal time-averaged force.) The growth velocity of a filament could in principle depend on the time-statistics of the force in addition to its time average. For example, in quadratic order, the filament velocities could depend on a functional of the time autocorrelation function. However, to linear order, time translation invariance implies that the only property of the force that can enter the time-averaged polymerization rate is the time-averaged force. Therefore the velocity of a given filament in the bundle is the same as that of an isolated filament carrying a force f, which is the definition of PLS. Thus PLS holds in the weak-force limit. Note that this issue was addressed in earlier work treating microtubules [17], using a different method. For the large-F limit (ii), at most times all of the filament tips will be at the same height. Once an obstacle fluctuation of height δ occurs, and a filament polymerizes a subunit, this filament will support the obstacle and the other filaments will rapidly polymerize to fill the gap. Thus $v\simeq n{{{\rm e}}^{-F\delta /{{k}_{B}}T}}$ is equivalent to n times the polymerization rate for a single filament carrying the entire load. This is the ZLS limit defined by [14].

To describe the entire range of forces and filament number, we develop the following formula based on interpolation between these two limits

Equation (2)

where $p={{{\rm e}}^{-F\delta /{{k}_{B}}T}}$ and ${{p}_{r}}={{{\rm e}}^{-F\delta /n{{k}_{B}}T}}={{{\rm e}}^{-f\delta /{{k}_{B}}T}}$. For small F, an expansion to linear order in $1-{{p}_{r}}$ yields $v\simeq {{p}_{r}}$, which is the PLS limit. For large F, p is small, and an expansion in p obtains $v\simeq np$, the ZLS limit. Schaus and Borisy [14] suggested that $np$ is the lower bound of load sharing. But the $(n-1)p$ term in equation (2) suggests v can be even smaller than $np$, showing that the lower bound is below $np$. From equation (2), we can approximately extract the crossover force Fc where the reduction relative to PLS reaches 50%, i.e. $v/{{p}_{r}}=0.5$. Calculating $v/{{p}_{r}}$ from equation (2), and replacing $n-1$ by n everywhere on the assumption that n is large, we obtain ${{F}_{{\rm c}}}\simeq ({{k}_{B}}T/\delta ){\rm ln} (n)$. The corresponding force per filament, ${{F}_{{\rm c}}}/n$, becomes small for large n.

Figure 3(a) shows the simulation results for the dependence of v on f and n along with the interpolation formula (2), and the PLS curve. We find that the interpolation formula gives an almost perfect match to the simulations. As predicted, the filaments have near PLS for small f. However, the load sharing performance drops for larger f, where v decays more rapidly than PLS. For larger bundles, the discrepancy relative to PLS becomes larger and sets in sooner. This occurs because at a given force per filament, a larger number of filaments corresponds to a larger total force. It is clear that for $f=1\;{\rm pN}$, even a bundle of five filaments already performs well below the PLS limit; the discrepancy becomes even larger for 9 and 20 filaments. Our results for unstaggered filaments differ from those obtained in [24], which employed simplifying assumptions including that of ZLS. Over almost the entire range of forces, our calculated v values are significantly above those of [24]. For example, for n = 100 and a total force $F=6\;{\rm pN}$, we obtain (including depolymerization) v = 0.62, while [24] obtains $v\simeq 0.2$ (also including depolymerization). We believe the discrepancy may be due to the assumption of ZLS.

Figure 3.

Figure 3. Force–velocity relations in the large $\tilde{D}$ limit. For frame (a), the formula is from equation (2); for frame (b), it is from equation (3). The 'perfect load sharing' curves are the force–velocity relation obtained by simulations for a single filament.

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3.1.2. Staggered filaments

For this case (figure 3(b)), v drops less rapidly with f and n, though essentially the entire range of forces gives sub-PLS. The reason for the better performance in this case is the 'subsidy effect' described in [17]: under a given force, at least one of the filaments can polymerize when the obstacle is only a fraction of δ away from the longest filament; in contrast, the unstaggered case will require the obstacle to be greater than δ above the longest filament(s). Consider two staggered filaments having lengths L and $L+\delta /2$. Then the shorter filament L only requires the obstacle to be $\delta /2$ above the longer one, to polymerize. Thus, staggered filaments will have more rapid polymerization. This also explains why the mean polymerization rate of a bundle of staggered filaments drops more slowly with f than the unstaggered filaments.

To interpret these results quantitatively, we again treat the limiting cases of strong and weak forces. In the small f limit, the argument from the unstaggered case applies equally well, and PLS will hold. In the larger-f limit, the load sharing can be approached in two ways: (i) when the force is very large, the obstacle is mostly in touch with the longest filament and only makes small excursions. Therefore, only the shortest filament can add monomers. Since the distance between the shortest and longest filament is $(n-1)\delta /n$ (see figure 2(a)), the shortest filament will only require a $\delta /n$ gap for polymerization. The corresponding probability is ${{{\rm e}}^{-F\delta /n{{k}_{B}}T}}={{p}_{r}}$. Since v is proportional to the total polymerization rate divided by n, we obtain the asymptotic behavior $v\simeq {{p}_{r}}/n$. (ii) If the force is of intermediate magnitude, and there are many filaments, the distribution of filament-obstacle distances can be regarded as continuous and uniform, and $v\simeq (1/n){{({{k}_{B}}T/f\delta )}^{2}}$ [17]. Our simulation data fit this result over a broader range of forces than the one for very large forces.

To describe the whole range of F and n, we have developed an interpolation formula analogous to equation (2). This formula is based on the weak- and strong-force limits discussed above, and we find that it also fits the intermediate range of forces well. The weak-force limit can be written as $v\simeq 1/{{{\rm e}}^{f\delta /{{k}_{B}}T}}\simeq 1/(1+f\delta /{{k}_{B}}T)=1/[1+|{\rm ln} ({{p}_{r}})|]$, while the strong-force limit is $v\simeq (1/n){{({{k}_{B}}T/f\delta )}^{2}}=(1/n){{[{\rm ln} ({{p}_{r}})]}^{2}}$. To match these limits, we developed the formula

Equation (3)

which matches the limits since ${\rm ln} ({{p}_{r}})$ is small at weak force and large at strong force. The sum-of-squares addition in the denominator was found to fit the simulation data better than a direct sum. As seen in figure 3(b), this formula fits the simulation data within a few percent over most of the range of f and n, although the fit is not as good as in the unstaggered case. In this case, the critical force for the velocity to be half its PLS value, obtained by a procedure similar to that for the unstaggered case, is ${{F}_{{\rm c}}}\simeq ({{k}_{B}}T/\delta )\sqrt{n}$. Again ${{F}_{{\rm c}}}/n$ becomes small for large n.

3.1.3. Force-power relation

Molecular motors driven by a power stroke, like myosin, have a force-power relation with the maximum power generated at a given force [31]. This holds for our simulated bundles as well. The maximum-power force may impact the design of force-generating organelles such as filopodia. Here we calculate the force and power at the peak-power point, and investigate how these quantitites depend on the number of filaments. Multiplying equation (2) by f, and multiplying by a factor ${{k}_{0}}\delta $ since v is dimensionless, we obtain the average power P for n unstaggered filaments

Equation (4)

figure 4(a) shows the simulation results and analytic theory for the power-force relations for multiple unstaggered filaments. P initially climbs up to its maximum at the force Pmax at fmax, and then drops off quickly. It is worth noticing that $\forall \ n\gt m$, ${{P}_{{\rm max} }}\left( n \right)\lt {{P}_{{\rm max} }}\left( m \right)$. In other words, given the same average force per filament, more filaments have a lower mean power than fewer filaments. Also, as the number of filaments increases, the corresponding fmax becomes smaller; for n = 20 it is only $\sim 0.2\;{\rm pN}$.

Figure 4.

Figure 4. Force-power relation in the large $\tilde{D}$ limit; ${{k}_{0}}\sim 116\;{{{\rm s}}^{-1}}$. For frame (a), the formula is from equation (2); for frame (b), it is from equation (3).

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The power-force relation for staggered filaments is shown in figure 4(b). The analytic fit of v in equation (3) gives the following formula for the power, plotted in figure 4(b)

Equation (5)

The general behavior of P(f) is similar to that in the unstaggered case, except for the long tails of P at large f values. These occur because v decays less rapidly than in the unstaggered case. For example at $f=2\ {\rm pN}$, v is about 10% for five filaments; whereas in the unstaggered case v is less than 1% for five filaments. The long tails suggest that the staggered alignment is a more efficient mode of force generation for multiple filaments under heavy loads. For n = 20, the peak in the relation is at about $0.4\;{\rm pN}$.

3.2. Intermediate $\tilde{D}$ ($\tilde{D}=3.0$)

3.2.1. Unstaggered filaments

Figure 5(a) shows the vf relations of unstaggered filaments from the simulations. Compared with the large $\tilde{D}$ (figure 3(a)) case, the v values are lower. In this case super load sharing occurs at small f : the bundle grows faster than a single filament with the same applied obstacle force per filament. This can be understood via the obstacle drag force. The time-averaged drag force from the Einstein relation is approximately ${{k}_{B}}Tv/D$, so the drag force per filament is ${{k}_{B}}Tv/nD$. This drops with increasing n, explaining why the growth rate increases. At larger f, the effect of the applied obstacle force F($=\;nf$) dominates, and super load sharing is eliminated because the effects leading to sub-PLS increase with opposing force.

Figure 5.

Figure 5. Force–velocity relation in the intermediate $\tilde{D}$ limit. The 'perfect load sharing' curves are the force–velocity relation obtained by simulations for a single filament.

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Another way of understanding this effect is via the nonequilibrium nature of the obstacle height distribution at intermediate $\tilde{D}$. Considering first the unstaggered case, the obstacle has to reach at least a height δ to allow polymerization. Therefore a lower bound to the time required to add a subunit per filament is given by the first-passage time tFP for the obstacle to reach a height δ. In the weak-force limit, ${{t}_{{\rm FP}}}={{\delta }^{2}}/2D$ [11], so that $\tilde{D}=1/2{{k}_{0}}{{t}_{{\rm FP}}}$ measures the ratio of the polymerization time to tFP. For a larger number of filaments, the additional time beyond tFP required for polymerization (and locking in the obstacle displacement) will be smaller because more polymerization sites are available. This contributes to super load sharing. In the case of staggered filaments, an additional contribution to super load sharing occurs because the obstacle height required for polymerization is smaller, reducing tFP. The slowing effects of tFP will dominate the sub-PLS effects of the external force, if the the external force is weak. Thus for any finite value of $\tilde{D}$, there is always a super load sharing regime at sufficiently weak external force. But it is invisible on the scale of figures 3(a) and (b), since $\tilde{D}$ is so large there that tFP is negligible in comparison to the polymerization time.

3.2.2. Staggered filaments

Figure 5(b) shows that v still decays with growing f and n, but at a slower rate. At $f=0.8\ {\rm pN}$, 9 unstaggered filaments (see figure 5(a)) have v close to zero, but 9 staggered filaments have $v\gt 0.2$. Thus, as before, the staggered alignment is less affected by increasing force and is a more efficient way of load sharing. Super load sharing is seen again at small f.

Thus in both the unstaggered and staggered cases, the load sharing for intermediate $\tilde{D}$ values is different from that for large $\tilde{D}$ values. However, the shape of the individual force–velocity relations is quite similar.

4. Biological implications

Our main results are given in figures 3(a), (b), 5(a) and (b). For large numbers of filaments, the force–velocity relation has an initial slope corresponding to PLS, and then drops more rapidly. The shape of the curves is affected very little by slowing of obstacle diffusion to intermediate values. We feel that the staggered case defines an upper limit to the growth velocity, and the unstaggered case defines a lower limit. Below we discuss some concrete applications of these general results.

4.1. Actin filaments in a filopodium pushing a bead will have substantially sub-PLS

Filopodia are thin protrusions extending from the leading edges of cells. Monomers enter a filopodium from the base and have to diffuse the whole length of the filopodium to the filament tips to be polymerized. Since monomers start out from a pool far away from the polymerization region at the filament tips, the local monomer concentration near the tips can be much lower than the bulk monomer concentration. A typical value of the polymerization rate in filopodia is 20 subunits per filament per second (estimated from the protrusion rate of $3.3\;\mu {\rm m}\;{{{\rm min} }^{-1}}$) [32]. Because our model assumes a rigid obstacle, it is probably not directly applicable to the tips of filopodia, which are wrapped by the flexible plasma membrane. However, it is possible to perform experiments where filopodia or filopodia-like extensions exert forces on plastic beads [8, 9], and in such experiments the obstacle may be viewed as rigid. In the experiments of [8], the bead size was $1\;\mu {\rm m}$, leading to an obstacle diffusion coefficient, from the Einstein relation argument used above, of $D\sim 0.022\;\mu {{{\rm m}}^{2}}\;{{{\rm s}}^{-1}}$. Using the $20\;{{{\rm s}}^{-1}}$ polymerization rate, we obtain a dimensionless diffusion coefficient $\tilde{D}=D/{{k}_{0}}{{\delta }^{2}}\simeq 0.022\;\mu {{{\rm m}}^{2}}\;{{{\rm s}}^{-1}}/(20\ {{{\rm s}}^{-1}}\times {{0.0027}^{2}}\;\mu {{{\rm m}}^{2}})\sim 150$. Ref. [9] used $4\;\mu {\rm m}$ beads, which would lead to $\tilde{D}\sim 40$. So $\tilde{D}$ should be large in both of these experiments. Hence, if the actin filaments are rigidly linked to each other they should exhibit sub-PLS if the force is sufficiently large. The magnitude of the total force reported in [8] was $4\;{\rm pN}$. This force is the difference between the actin polymerization force and the force required to pull out the membrane tether. Measurements of the latter force suggest values in the range 10–50 pN [33, 34], implying polymerization forces in the range 15–55 pN. Ref. [9] measured force changes believed to be due to polymerization of up to $15\;{\rm pN}$. If n = 20, a value of $F=15\;{\rm pN}$ would give $f\simeq 0.75\;{\rm pN}$, where we predict large effects. At this force, v for unstaggered filaments would be essentially zero, while for staggered filaments $v\simeq 0.2$ (see figures 3(a) and (b)). We are not able to compare these predictions for the steady-state force–velocity relation to the data of [8] or [9], since the forces in these experiments were rapidly changing.

4.2. Increasing the number of filaments changes the shape of the Fv relation

Treating larger numbers of filaments reveals changes in the shape of the Fv curve. Figure 6(a) shows the Fv relation for 1, 5, and 200 unstaggered filaments.We plot velocity as a function of total force F rather than force per filament f because this shows the features more clearly. For a single filament, v is exponential; for five unstaggered filaments, the curve shifts upwards and becomes more concave; for 200 unstaggered filaments v has a plateau before it drops at large F. This can be understood from equation (2). Since PLS holds at low forces, the initial slope ${\rm d}v/{\rm d}F$ at F = 0 is $-\delta /n{{k}_{B}}T$. On the other hand, one easily shows that v drops to $1/2$ at ${{F}_{1/2}}\simeq {{k}_{B}}T{\rm log} (n)/\delta $. Extrapolation of the initial linear behavior to ${{F}_{1/2}}$ would result in a v-value much higher than $1/2$, showing that on the scale of the figure the initial behavior is very flat. Figure 6(b) shows the simulated Fv relation for 1, 5, and 100 staggered filaments, where a similar plateau is seen. In actin-based protrusions, the number of filaments can vary greatly. For example, stereocilia generally have more filaments than filopodia. Therefore, the shape of Fv in stereocilia could be very different from that in filopodia, provided that in each case the protrusion is pushing on a hard obstacle.

Figure 6.

Figure 6. Fv relation for large n values, in the large $\tilde{D}$ limit. Symbols denote simulation points. Lines are intended to guide the eye.

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Figure 7.

Figure 7.  $F$$v$ relation for 13 protofilaments.

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4.3. F–v relation for microtubules

As in the previous work described in the introduction, we treat microtubules with 13 protofilaments polymerizing separately. A subunit (tubulin dimer) is 8 ${\rm nm}$ [35] long. We thus simulate the growth of 13 protofilaments with $\delta =8\;{\rm nm}$. The protofilaments are treated as the perfectly staggered, while $\tilde{D}=70$ and $\tilde{D}=3.0$ are used. Figure 7 compares our simulation results to experimental data. The squares (experimental data) and dashed lines (exponential fit) from [10] are normalized to the F = 0 value. The model provides an excellent fit to the data, for both $\tilde{D}=70$ and $\tilde{D}=3.0$. Comparable fits have been obtained by previous studies using a basically similar model with various simplifying assumptions [1720]. Our results validate the assumptions made in these calculations, over a broader range of $\tilde{D}$ values than might have been expected.

5. Critique of assumptions

In this article, several approximations have been made to simplify the calculations. We discuss their validity here, as well as the extent to which they might be addressed in future work.

5.1. Implicit treatment of monomer diffusion

This means that the depletion effects from monomer consumption, and the replenishment of new monomers by diffusion, will not be seen. In the simplest view, this would reduce the on-rate for all of the filaments, and the magnitude of the effect would decrease with increasing force. Thus the growth velocity at low force would be reduced more than at high force. However, more complex effects could also occur. For example, filaments in the center of the bundle may have their polymerization rate reduced more than those at the edges, and thus carry less force.

5.2. PBR approximation

Even when the barrier is more than a distance δ above a filament, it can provide some impediment to diffusion of the monomer to the filament tip. Previous work [12] has suggested that this effect can be important in 2D models. But it is not known how important it is in 3D.

5.3. Neglect of actin filaments' flexibility

We assume that actin filaments are rigid. This may be true for moderately long filaments since they are cross-linked. Mogilner and Rubinstein [36] estimated the F-actin persistence length to be $\sim 10\;\mu {\rm m}$. For a bundle of n filaments, the persistence length will be ∝ n2 if the filaments are strongly cross-linked, and $\propto \;n$ if loosely cross-linked [36]. So it is much longer than filament lengths in filopodia. Thus, this assumption about the rigidity may be valid over a substantial range of experiments.

5.4. Neglect of ATP hydrolysis to ADP in actin filaments and GTP hydrolysis to GDP in microtubules

The nucleotides ATP and GTP undergo hydrolysis in filaments, which changes the rate constants to favor depolymerization. It is generally believed that the tip will become ADP- or GDP-like when polymerization is very slow, and such a switch has been reported as a function of free-monomer concentration for actin [37]. Therefore our results will not be accurate very close to the stall force.

5.5. Neglect of depolymerization

Depending on the nucleotide state of actin filaments, depolymerization occurs at rates kd ranging from about 1.4 ${{{\rm s}}^{-1}}$ [29] to 5.4 ${{{\rm s}}^{-1}}$ [37]; for microtubules values between 0.16 ${{{\rm s}}^{-1}}$ and 0.8 ${{{\rm s}}^{-1}}$ have been suggested [18]. We find, in additional simulations, that inclusion of depolymerization reduces the growth velocity at small forces by an amount essentially equal to kd. At larger forces, the reduction is much smaller, an effect also seen in [18]. From our simulation results, we have a simple physical interpretation of this finding. A depolymerization event that occurs at any but the leading filaments will lead to a gap larger than δ, which will enhance polymerization if the obstacle is close to the leading filaments. Thus depolymerization is countered by increased polymerization at large forces if more than one filament is present. Our simulations using both of the above values of kd showed that for forces where load sharing is well below the PLS limit, the new polymerization caused by depolymerization almost completely cancels the depolymerization.

6. Conclusions

In this article, we have investigated growth and power generation by multiple filaments growing against hard barrier. We make no ad hoc assumptions regarding the form of load sharing; all velocities are generated directly by simulations with explicit obstacle motion. Our major findings are the following:

  • (i)  
    For a given force per filament, sub-PLS (more filaments leads to slower growth and lower power per filament) generally holds; however, super load sharing (more filaments leads to faster growth and higher power) can occur if the diffusion coefficient of the obstacle and the external force are small enough.
  • (ii)  
    The difference between a staggered alignment of filaments (where the initial heights of filaments differ by a fraction of the monomer size δ) and an unstaggered alignment can be quite large. For example, for 20 filaments the the staggered alignment grows about twice as fast as the unstaggered one already at $f=0.3\;{\rm pN}$.
  • (iii)  
    The closed-form expressions in equations (2) and (3) provide an accurate, portable description of the simulation results. They give lower and upper limits of the velocity, corresponding to unstaggered and staggered filament alignment.

Acknowledgements

This work was supported by the National Institutes of Health under Grant Number R01 GM107667.

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10.1088/1367-2630/16/11/113047