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Phys Rev E Stat Nonlin Soft Matter Phys. Author manuscript; available in PMC 2010 April 29.

Published in final edited form as:

Phys Rev E Stat Nonlin Soft Matter Phys. 2010 March; 81(3 Pt 1): 031914.

Published online 2010 March 22. PMCID: PMC2861361

NIHMSID: NIHMS195141

Department of Physics, Washington University, St. Louis, Missouri 63130, USA

The properties of actin network growth against a flat obstacle are studied using several different sets of molecular-level assumptions regarding filament growth and nucleation. These assumptions are incorporated into a multi-filament methodology which treats both the distribution of filament orientations and bending of filaments. Three single-filament force-generation mechanisms in the literature are compared within this framework. Each mechanism is treated using two different filament nucleation modes, namely spontaneous nucleation and branching off preexisting filaments. We find that the shape of the force-velocity relation depends mainly on the ratio of the thermodynamic and mechanical stall forces of the filaments. If the thermodynamic stall force greatly exceeds the mechanical stall force, the velocity drops abruptly to zero when the mechanical stall force is reached; otherwise it goes more gradually to zero. In addition, branching nucleation gives a steeper increase of filament number with opposing force than spontaneous nucleation does. Finally, the zero-force velocity of the obstacle as a function of the detachment and capping rates differs significantly between the different single-filament growth mechanisms. Experiments are proposed to use these differences to discriminate between the network growth models.

In actin-based cellular protrusion and locomotion, mechanical force is generated by the elongation of actin filaments inside the cell. The processes controlling force generation have been studied extensively over the past few decades, and include polymerization, creation of new filaments by branching or spontaneous nucleation, capping, severing, attachment, and detachment [1]. Although these processes are well established in general, the details of the underlying molecular-level mechanisms and the relevant rates are not well understood. For this reason, a number of force generation models have been developed which differ in their molecular-level assumptions regarding both the single-filament growth mechanism and the mode of nucleation for new filaments.

The single-filament growth mechanisms are of three main types (see Fig. 1).

- The tethered ratchet mechanism (TR) [2], which incorporates the earlier Brownian ratchet mechanism [3], assumes that some of the actin filaments are attached to the obstacle. Filaments are created with their barbed ends attached to the obstacle surface. While attached, they do not grow but rather pull the obstacle reactively. After detachment, they polymerize and generate pushing forces similar to those in the Brownian ratchet mechanism. Finally, they become capped and are left behind by the obstacle’s motion.
- The passive-processive mechanism (PP) [4] assumes that filaments can grow while their barbed ends are attached to the obstacle. For each filament, the attachment point switches thermally between the two terminal subunits, without ATP hydrolysis. Actin monomers can attach to the barbed ends if enough space is available between the filament tip and the obstacle. Free filaments provide additional pushing force.
- The end-tracking mechanism (ET) [5, 6] treats filaments attached to the obstacle surface through a barbedend tracking protein. The rate with which the tracking protein moves along the filament depends on the filament’s ATP hydrolysis rate, and thus growth is coupled to ATP hydrolysis. Therefore a high protrusion force of up to a few tens of pN is thermodynamically possible. Free filaments can provide an additional, smaller pushing force.

Three single-filament growth mechanisms. For each mechanism, a free growing filament, a capped free filament and an attached filament are shown.

These three mechanisms make different predictions for the force-velocity relation of single filaments. In the TR mechanism, the growing filaments follow the exponentially decaying Brownian-ratchet force-velocity relation [3], which is concave-up. In the PP mechanism, the force-velocity relation for attached filaments has a sigmoidal shape. The velocity increases to an asymptotic value if a pulling force is present and decreases exponentially for large pushing forces. In the ET mechanism, the velocity of attached filaments remains constant at low force, and drops rapidly when the force reaches a threshold, forming a concave-down shape. Due to lack of experimental data on the force-velocity relation of single filaments, the above predictions have not yet been evaluated definitively. A major goal of this work is to evaluate to what extent the differences between the single-filament growth mechanisms persist in more complete models.

The possible modes of filament nucleation are of two main types (see Fig. 2a):

a) Filament nucleation modes. Dashed filaments indicate newly created ones. b) Schematic of a filament pushing against an obstacle, illustrating of filament deformation *x*. Obstacle velocity: *v*_{b}. Filament velocity: *v*. Components of *f*_{x} force relative to **...**

Force generation and other properties of actin networks within some combinations of these single-filament growth mechanisms and nucleation modes have been studied theoretically in previous works [2, 9-18], incorporating some, but not all, of the effects treated here. However, there has been no comprehensive treatment systematically evaluating the differences between the growth mechanisms and nucleation modes within the context of realistic network-growth models.

The combination of three growth mechanisms and two nucleation modes gives altogether six network models. One can of course define many more models by treating additional “fine-tuning” assumptions, but the range of models covered here encompasses most of the existing literature and we address the importance of additional assumptions below. One would like to distinguish between these six models by performing suitable experiments. Measurements of the force-velocity relation [19-23] and the force-filament number relation of actin networks as measured by fluorescence [19, 22, 23] are in principle promising. Other types of experiments which could be relevant are the dependence of the growth velocity on the concentrations of key actin-binding proteins. It is the purpose of this paper to evaluate the connection between such experimental measurements and the molecular-scale assumptions of the six network models. Thus we calculate growth properties of the six models taking into account contributions from both free and attached filaments, the distribution of filament orientations, transport of filament tips away from the obstacle, and filament bending. For each model, we calculate the force-velocity relation, the force-filament number relation, and the dependence of the obstacle’s zero-force velocity on the filament’s capping and detachment rates. We find that shape of the force-velocity relation depends on the relative magnitudes of the thermodynamic stall force (the opposing force at which polymerization and depolymerization balance for a rigid filament) and mechanical stall force (the force required to bend a filament to be parallel to the obstacle surface). Larger thermodynamic stall forces tend to give force-velocity relations with a sudden drop near the maximum force, while lower thermodynamic forces give smoother behavior. The mechanical stall force is determined by a “cross-linking distance” *l _{c}*. For

We note that a continuum elastic model [24] based on larger length scales has been used to calculate the force-velocity relation for actin-based propulsion. It treats the actin network as an elastic gel, and emphasizes the internal stress instead of the properties of individual filaments. It successfully explained the experimental force-velocity curve for biomimetic beads in a pure-protein solution [19], and explained an actin network instability which cannot be explained by the single-filament growth mechanisms [25]. However, the effect of the single-filament force-velocity relation and the filament nucleation mode on force generation by actin networks have not been explored within this framework.

We treat an actin network polymerizing against a flat rigid obstacle moving in the *X*-direction. The use of a flat obstacle simplifies the connection between the molecular-level assumptions and network growth properties because it reduces the effects of network deformation. The bulk of our calculations are performed using a three-dimensional geometry most relevant to cantilever experiments of the type described by Parekh *et al*. [20]. We feel that the actin gel surface in these experiments is reasonably flat since the filaments are nucleated mainly from the cantilever, rather than the atomic-force-microscopy tip on the cantilever. For concreteness, we take the obstacle to have an area of 180 *μ*m^{2} as in Ref. [20]. We also perform a few simulations using a two-dimensional geometry, which is most relevant to the lamellipodial protrusion experiments such as those of Prass *et al*. [21]. For mathematical simplicity, we assume that the actin is stationary and the obstacle moves, but by a simple change of reference frame these calculations could handle the case of a stationary obstacle and moving actin network. Because we treat a flat obstacle, we do not include shearing deformations of the actin gel. This means that the results are not directly relevant to the propulsion of *Listeria* or protein-coated beads, where such deformations are important [26].

Our mathematical approach is based on the coarse-grained densities of filaments as a function of their orientation and the local deformation of the actin gel, rather than on stochastic simulation. To motivate our treatment of the local deformation, we note that actin filaments growing at different angles to the substrate will have different *X*-components of the growth velocity. If some of the growing filaments are attached to the obstacle, accommodation of the velocity differences will require local deformation of the actin gel. We treat this effect by allowing the base of each filament to be displaced relative to the surrounding gel. The forces arising from this displacement are described by an effective spring constant *k*_{el}. The value of *k*_{el} is determined by the elastic modulus of the actin gel and the average spacing *l _{c}* between crosslinks in the network. As described in Appendix A, we take

We assume that new filaments are nucleated in an attached undeformed state, so that *x* = 0. Therefore the rate of change of *P*_{a} due to filament nucleation can be written as

$${\phantom{\frac{{P}_{\mathrm{a}}t\mathrm{nuc}}{=}}}_{}$$

(1)

where *k*_{nuc} = *k*_{nuc}(*θ*) is the nucleation rate of filaments with orientation *θ* as defined below, and *δ*(*x*) is the Dirac delta function. There is also a possibility that new filaments are first created unattached, then attach to the object at a later time. However, we find that the reduction in the number of attached filaments resulting from unattached nucleation can be reproduced by simply changing the attachment rate.

The filament nucleation rate, for either spontaneous or branching nucleation, will depend on the local concentrations of formins or activated Arp2/3 complex. It will also depend strongly on the local free-actin concentration at the obstacle, because the critical nucleus is expected to comprise several monomers [27]. The local concentrations will in general be reduced, because depletion of proteins near the obstacle due to nucleation and filament growth will not be completely compensated by diffusion [28]. The magnitude of the depletion effect is described by reduction factors *u _{s}* (spontaneous nucleation) and

$${k}_{\mathrm{nuc}}\left(\theta \right)=\{\begin{array}{cc}{\scriptstyle \frac{1}{2}}{k}_{\mathrm{s}}{u}_{\mathrm{s}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}\theta \hfill & \text{spontaneous},\hfill \\ {k}_{\mathrm{b}}{u}_{\mathrm{b}}{\int}_{-\pi /2}^{\pi /2}K(\theta ,{\theta}^{\prime}){n}_{\mathrm{c}}\left({\theta}^{\prime}\right)\mathrm{d}{\theta}^{\prime}\hfill & \text{branching},\hfill \end{array}\phantom{\}}$$

(2)

where *k*_{s} and *k*_{b} are the spontaneous and branching nucleation rates without depletion effects, the factor of $\frac{1}{2}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}\theta $ in the upper equation comes from the range of filament orientations (−*π*/2 to *π*/2) in spherical coordinates, and *K*(*θ,θ*’) is a three-dimensional branching kernel given by

$$K(\theta ,{\theta}^{\prime})=\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}\theta /\left(\pi \phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{\mathrm{br}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}{\theta}^{\prime}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}\varphi \right),$$

(3)

where *θ*_{br} = 70° is the branching angle and *ϕ* is an azimuthal angle determined (see Ref. [9]) by *θ* and *θ*’:

$$\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\theta =\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{\mathrm{br}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}{\theta}^{\prime}+\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{\mathrm{br}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\varphi \phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}{\theta}^{\prime}.$$

(4)

To account for the observed spread [8, 29] in *θ*_{br}, we use a weighted integral of the form given by Eq. 4, calculated using a Gaussian distribution of *θ*_{br} of width Δ*θ* = 10°. In our two-dimensional calculations, we use a kernel of the form

$$K(\theta ,{\theta}^{\prime})=\left[\mathrm{exp}\phantom{\rule{thinmathspace}{0ex}}\left(-{(\theta -{\theta}^{\prime}-{\theta}_{\mathrm{br}})}^{2}/2\Delta {\theta}^{2}\right)+\mathrm{exp}\phantom{\rule{thinmathspace}{0ex}}\left(-{(\theta -{\theta}^{\prime}+{\theta}_{\mathrm{br}})}^{2}/2\Delta {\theta}^{2}\right)\right]/\sqrt{8\pi}\Delta \theta .$$

(5)

In Eq. 2, *n _{c}*(

$${n}_{\mathrm{c}}\left(\theta \right)={\int}_{0}^{\infty}\left[{P}_{\mathrm{a}}\right(x,\theta )+{P}_{\mathrm{f}}(x,\theta \left)\right]{\mathrm{e}}^{-{f}_{}}$$

(6)

where *f*_{} = *f*_{}(*x,θ*) is the force along the actin filament (given below), *f*_{0} = *k*_{B}*T*/*δ* ≈ 1.5 pN, and *δ* ≈ 2.7 nm is the filament length increment per subunit. The exponential factor describes the reduction of the association rate of branching proteins to the filaments’ tips. This force reduction factor is present because the nucleation of a new filament requires insertion of Arp2/3 complex. We do not know the exact magnitude of the force reduction, but for convenience we have assumed the exponential form appropriate for insertional assembly of actin filaments [3].

We assume that the attached filaments are protected from capping by steric constraints. The rate of change of free filaments from capping is

$${\phantom{\frac{{P}_{\mathrm{f}}t\text{cap}}{=}}}_{}$$

(7)

where *k*_{cap} = *k*_{cap}(*x,θ*) is the free filaments’ capping rate. Like the branching rate, the capping rate is assumed to be slowed exponentially by opposing force, if the filament is in contact with the obstacle. Then

$${k}_{\text{cap}}(x,\theta )=\{\begin{array}{cc}{k}_{\mathrm{c}}{u}_{\mathrm{c}}\hfill & \text{if}\phantom{\rule{thinmathspace}{0ex}}x<0,\hfill \\ {k}_{\mathrm{c}}{u}_{\mathrm{c}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\phantom{\rule{thinmathspace}{0ex}}(-{f}_{}\text{if}\phantom{\rule{thinmathspace}{0ex}}x\ge 0,\hfill \hfill \\ \phantom{\}}\end{array}$$

(8)

where *k _{c}* is the barbed capping rate without depletion effects, and

Free and attached filaments can inter-convert by attaching to or detaching from the obstacle. We assume that such conversions leave *x* and *θ* unchanged. The conversion rates for the free and attached filaments are

$${\phantom{\frac{{P}_{\mathrm{f}}t\mathrm{conv}}{=}}}_{}$$

(9)

$${\phantom{\frac{{P}_{\mathrm{a}}t\mathrm{conv}}{=}}}_{}$$

(10)

where *k*_{att} is the free filament attachment rate and *k*_{det} is the detachment rate of attached filaments. We assume that *k*_{att} is a constant *k*_{a} for the free filaments that are in contact with the obstacle:

$${k}_{\mathrm{att}}\left(x\right)={k}_{\mathrm{a}}H\left(x\right),$$

(11)

where *H*(*x*) is the Heaviside step function. We also assume that *k*_{det} depends on the force in the *X*–direction (see Ref. [2]):

$${k}_{\mathrm{det}}(x,\theta )=\{\begin{array}{cc}{k}_{\mathrm{d}}\hfill & \text{if}\phantom{\rule{thinmathspace}{0ex}}x>0,\hfill \\ {k}_{\mathrm{d}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\phantom{\rule{thinmathspace}{0ex}}(-{f}_{x}/{f}_{0})\hfill & \text{if}\phantom{\rule{thinmathspace}{0ex}}x\le 0,\hfill \end{array}\phantom{\}}$$

(12)

where *k*_{d} is the filament detachment rate in the absence of force and *f _{x}* ≤ 0 is the

The rates of change of *P*_{f} and *P*_{a} due to the motion of filament tips relative to the obstacle are

$${\phantom{\frac{{P}_{\mathrm{f},\mathrm{a}}t\text{motion}}{=}}}_{}$$

(13)

where ${v}_{\mathrm{f},\mathrm{a}}^{\mathrm{rel}}$ are the velocities of the attached and free filaments. These are measured in the *X*-direction relative to the obstacle. In order to calculate ${v}_{\mathrm{f},\mathrm{a}}^{\mathrm{rel}}$, we first calculate the force exerted on a filament with filament deformation *x*. We assume that the force in the *X*-direction follows Hooke’s law

$${f}_{x}={k}_{\mathrm{el}}x,$$

(14)

where *k*_{el} (see Appendix A) is the effective spring constant for the base of an actin filament. Then, if *x* > 0, *f _{x}* > 0 (pushing force); if

The magnitude of the force that can be sustained by a filament will be limited by its bending. We thus define a mechanical stall force *f*_{mech}(*θ*) as the minimum force that will bend a filament with orientation *θ* to be parallel to the obstacle surface. Since elongation of a filament parallel to the obstacle surface does not contribute to force generation in the *X*-direction, the pushing force of the filament cannot exceed *f*_{mech}. We assume that *f*_{mech} also depends on the typical length of filament from the tip to the first anchoring or cross-linking point. We take this length to be the same as the crosslink distance *l _{c}* ≈ 0.1–0.2

$${f}_{\mathrm{mech}}\left(\theta \right)\approx \frac{\alpha {l}_{\mathrm{p}}{k}_{\mathrm{B}}T}{{l}_{c}^{2}}(1-\frac{2\theta \pi}{)},$$

(15)

where *α* = 3.4 and *l _{p}* ≈ 10

$${f}_{x}=\mathrm{min}\phantom{\rule{thinmathspace}{0ex}}[{k}_{\mathrm{el}}x,{f}_{\mathrm{mech}}(\theta \left)\right]$$

(16)

for *x* > 0. The force component parallel to the filament, which enters its growth rate, is

$${f}_{}$$

(17)

The growth velocities of free and attached filaments in the *X*-direction are calculated from these forces as follows. For all three single-filament mechanisms, the free filament velocity has the same form [3]:

$${v}_{\mathrm{f}}=\left[{v}_{0}{u}_{\mathrm{v}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\phantom{\rule{thinmathspace}{0ex}}\right(-{f}_{}$$

(18)

where *v*_{0} is the free filament barbed end growth velocity, *u*_{v} is the depletion factor for the actin polymerization rate (see Appendix B), and *v*_{d} is the depolymerization velocity. The thermodynamic stall force of each free filament is on the scale of *f*_{0}, which is consistent with the measured value of a few piconewtons [31], if one assumes that in this experiment only one or two filament are pushing at a time. For the attached filaments, the growth velocity is given by

$${v}_{\mathrm{a}}=\{\begin{array}{cc}0\hfill & \mathrm{TR},\hfill \\ {\scriptstyle \frac{1}{2}}({v}_{0}{u}_{\mathrm{v}}-{v}_{\mathrm{d}})\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\theta [1-\mathrm{tanh}({f}_{}\mathrm{PP},\hfill \hfill & {\scriptstyle \frac{1}{2}}({v}_{0}{u}_{\mathrm{v}}-{v}_{\mathrm{d}})\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\theta [1-\mathrm{tanh}(({f}_{}\mathrm{ET}.\hfill \hfill \\ \phantom{\}}\end{array}$$

(19)

Note that in the ET mechanism, we have used a *tanh* function and two parameters *f _{w}* = 1.5 pN and

The velocities of the free and attached filaments in the *X*-direction, relative to the obstacle, are

$${v}_{\mathrm{f},\mathrm{a}}^{\mathrm{rel}}={v}_{\mathrm{f},\mathrm{a}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\theta -{v}_{\mathrm{b}}.$$

(20)

Using Eqs. 1, 7, 9, 10 and 13, we write a time evolution equation for the two filament populations:

$$\frac{{P}_{\mathrm{f},\mathrm{a}}t={\phantom{\frac{{P}_{\mathrm{f},\mathrm{a}}t\mathrm{nuc}}{+}}}_{}}{}$$

(21)

so that

$$\frac{{P}_{\mathrm{f}}t=-\frac{x\left({v}_{\mathrm{f}}^{\mathrm{rel}}{P}_{\mathrm{f}}\right)-({k}_{\mathrm{att}}+{k}_{\text{cap}}){P}_{\mathrm{f}}+{k}_{\mathrm{det}}{P}_{\mathrm{a}},}{}}{}$$

(22)

$$\frac{{P}_{\mathrm{a}}t=-\frac{x\left({v}_{\mathrm{a}}^{\mathrm{rel}}{P}_{\mathrm{a}}\right)+{k}_{\mathrm{att}}{P}_{\mathrm{f}}-{k}_{\mathrm{det}}{P}_{\mathrm{a}}+{k}_{\mathrm{nuc}}\delta \left(x\right).}{}}{}$$

(23)

Eqs. 22 and 23 are the basis of our simulations. The parameters are given in Table I, and the justification of their values is given in Appendix D.

Eqs. 22 and 23 are solved numerically with a firstorder upwind scheme [32]. We use a two-dimensional mesh in the *x-θ* plane, where *θ* ranges from 0 to *π*/2, and the maximum and minimum values of *x* are functions of *θ*. The filament distribution in the range of −*π*/2 ≤ *θ* < 0 can be obtained from symmetry. Defining ${x}_{\mathrm{f}}^{\mathrm{max}}\phantom{\rule{thinmathspace}{0ex}}\left(\theta \right)$ and ${x}_{\mathrm{a}}^{\mathrm{max}}\phantom{\rule{thinmathspace}{0ex}}\left(\theta \right)$ to be the solutions to the equations ${v}_{\mathrm{f}}^{\mathrm{rel}}\left[{x}_{\mathrm{f}}^{\mathrm{max}}\phantom{\rule{thinmathspace}{0ex}}\left(\theta \right)\right]={v}_{\mathrm{a}}^{\mathrm{rel}}\left[{x}_{\mathrm{a}}^{\mathrm{max}}\phantom{\rule{thinmathspace}{0ex}}\left(\theta \right)\right]=0$, the maximum value for *x* at a given *θ* is ${x}_{\mathrm{max}}=\mathrm{max}\left[{x}_{\mathrm{f}}^{\mathrm{max}}\phantom{\rule{thinmathspace}{0ex}}\left(\theta \right),{x}_{\mathrm{a}}^{\mathrm{max}}\phantom{\rule{thinmathspace}{0ex}}\left(\theta \right),0\right]$. The minimum value of *x* is chosen such that *P*_{f}(*x,θ*) and *P*_{a}(*x,θ*) are negligible. For *x* that is beyond this range, we take *P*_{f}(*x,θ*) = *P*_{a}(*x,θ*) = 0.

To calculate the force-velocity and force-filament number relations, we fix the obstacle velocity and start with distributions *P*_{f} and *P*_{a} that are concentrated at *x* = 0 but uniformly spread out as a function of *θ*. We then let *P*_{f} and *P*_{a} evolve until a steady state is reached. The steady state in the simulation is defined as follows: over the time it takes a filament to traverse the entire range of deformation distribution *t* ≈ *x*_{max}/*v*^{rel}, the variations in *F*_{tot}, *N*_{f} and *N*_{a} are smaller than 1%. At steady state, the external force balances the total force that the filament network exerts on the obstacle in the *X*-direction:

$${F}_{\mathrm{ext}}={F}_{\mathrm{tot}}={\int}_{-\infty}^{\infty}{f}_{x}{n}_{\mathrm{t}}\left(x\right)\mathrm{d}x,$$

(24)

where ${n}_{\mathrm{t}}\left(x\right)={\int}_{-\pi /2}^{\pi /2}\left[{P}_{\mathrm{f}}\right(x,\theta \left)H\right(x)+{P}_{\mathrm{a}}(x,\theta \left)\right]\mathrm{d}\theta $ is the total number of filaments with deformation *x* that are in touch with the obstacle and *f _{x}* is given by Eq. 16. Here we assume that the force generation is cooperative among filaments, because in our model filaments have various orientations and do not form parallel bundles. Therefore the exchange of load-bearing duty [31] is neglected. The numbers of free and attached filaments in the network are

$${N}_{\mathrm{f}}={\int}_{-\infty}^{\infty}{\int}_{-\pi /2}^{\pi /2}{P}_{\mathrm{f}}(x,\theta )\mathrm{d}\theta \mathrm{d}x,$$

(25)

$${N}_{\mathrm{a}}={\int}_{-\infty}^{\infty}{\int}_{-\pi /2}^{\pi /2}{P}_{\mathrm{a}}(x,\theta )\mathrm{d}\theta \mathrm{d}x,$$

(26)

and we define

$${N}_{\mathrm{tot}}={N}_{\mathrm{f}}+{N}_{\mathrm{a}}.$$

(27)

To calculate the zero-force velocity, we use a root finding routine to find the value of *v*_{b} at which the total force produced by the actin network at steady state is zero.

We have checked the simulation results against a previous three-dimensional stochastic simulation [9]. As in the stochastic-simulation studies, we take all the filaments to be free and created via autocatalytic branching, and we use the same parameters as in Ref. [9]. We have compared the force-velocity relation and the branching- and capping-rate dependence of the zero-force velocity to those of the stochastic simulations, and the difference between the two approaches is always less than 10%. Thus our continuum methodology describes the stochastic growth process accurately.

The main assumptions and approximations in our calculations are the following:

Gardel *et al*. [33] have shown that the actin network *in vitro* is homogeneous, isotropic and elastic over a large range of actin and cross-linker concentrations, in the absence of bundling proteins. Given the rapid kinetics of actin *in vivo*, it is reasonable to make the same assumption in the straight region of a lamellipod. Since we only aim to treat cases where bundling is absent, and the size of the obstacle is much larger than the mesh size of the actin network, it is legitimate to treat the actin network as a homogeneous and isotropic elastic medium. A similar assumption has been made in Ref. [25]. If the mesh size of the actin network were comparable to the size of the obstacle, this approximation would no longer be valid.

We also assume that *k*_{el} is independent of *F*_{ext}. This is clearly a simplification, because a higher *F*_{ext} induces a larger *N*_{tot}, which should lead to a smaller *l _{c}* and thus to a higher

We note that in our elastic-gel model, the possibility of filament polymerization and nucleation being blocked by steric interactions is ignored. The simulations in Ref. [9] included these effects and found them to affect the growth velocity by 10% or less, even at high F-actin densities of 1 *mM*.

In a real actin network, filaments have varying distances from the tips to the first cross-linking points. The forces required to stall these filament sections via mechanical bending are therefore different. But the filaments are cross-linked to the network, and the cross-linking points can yield under external forces. The strongest forces will be felt by the shortest filaments, so this effect will reduce the variations in *f*_{mech}. Thus we feel that using a uniform *l _{c}* to characterize

We also assume that *f*_{mech} is independent of *F*_{ext}. Since *F*_{ext} probably decreases *l _{c}*, as mentioned above, it should increase

The deflection angle at a filament’s tip should increase continuously with the force on the filament, until the filament is parallel to the obstacle’s surface or detached. This increase is slow and linear at low forces, but is fast and nonlinear at high forces near *f*_{mech}. In our calculations, we assume that the filaments keep their orientations at the tip up to the mechanical stall point. This approximation tends to overestimate *f*_{} on the pushing filaments, and underestimate *f*_{} on pulling filaments. Therefore actual filaments could grow faster, and then the actual *v*_{b} at a given *F*_{ext} could be higher than our prediction.

We assume that filaments do not change their pointed-end orientations when forces are exerted on the barbed ends. This is because a single filament can have multiple crosslinking points (with an average spacing of *l _{c}*). When a filament is bent at the barbed end, the torque on the filament is distributed among multiple crosslinking points. Therefore the change of filament orientation itself is small, if

The actin network is reported to act like a viscoelastic gel under external forces [34, 35]. The effects of the resulting deformation on the *v-F* relation are not included in our model. For a given external load, the gel’s elasticity affects the gel length by allowing compression, and the gel’s viscosity affects its growth velocity by allowing compressional flows. One would expect both the extent of compression of the gel, and the flow velocity, to be proportional to *FL*/*A*, where *F* is the force, *L* is the length of the gel, and *A* is its cross-sectional area. Therefore to experimentally reduce the gel flow and compression effects, one could use a geometry in which the gel length is much smaller than its width. Then it should be legitimate to compare our predicted actin gel growth velocity with experimental observations.

If the elastic compression is not negligible, one way to relate the experimental results to the present theory is to measure the compression ratio of the actin network Δ*L*/*L* by suddenly releasing the external load. Then our predicted *v* would correspond to the measured *v* divided by the factor (1 − Δ*L*/*L*). To account for viscous flow, one could adjust the measured velocity by subtracting off the compression-flow velocity, and then compare this adjusted velocity to our predictions. Using a parallel-dashpot model of the gel, the compression-flow velocity has been estimated to be *v*_{flow} ~ *FL*/*η*_{gel}*A*, where *η*_{gel} is the gel’s viscosity [36]. However, these compression effects are probably small in the experiment of Ref. [20], because otherwise the measured velocity would be dropping more rapidly at small forces.

One would expect that if filaments are laterally pinned to the obstacle at their attachment points, preventing lateral motion, the resulting forces could impact the growth of the filaments. Therefore we have performed additional calculations in which filaments are laterally pinned to the obstacle and experience lateral forces *f _{y}*. In this case, we assume that a pushing filament can detach only in the

At the molecular level, the processes of filament creation, elongation, capping, and detachment are stochastic. Such stochastic effects are not included in our calculations, because we study the network growth properties at a much larger scale, on the basis of averaged properties. As mentioned in the previous subsection, we have compared our results to those from stochastic simulations [9], and find close agreement. Therefore the neglect of stochastic effects does not have a major impact on the conclusions.

Filament uncapping is not included in our calculations. However, the properties of interest here are mainly determined by the average capping state of the filaments. Therefore the effects of uncapping would be similar to those of reducing the capping rate.

The main outputs of our simulations which could be compared to experiments are the force-velocity relation, the force-filament number relation, and the zero-force velocity’s dependence on the detachment and capping rates.

Figure 3a shows the force-velocity relations for the six network growth models with a crosslink distance of *l _{c}* =0.2

The scale of the obstacle velocity also depends on the single-filament growth mechanism. Unfortunately, this difference cannot be a used to distinguish between the mechanisms, because it is sensitive to the choice of parameter values. For example, if most of the filaments in the TR mechanism are free, which corresponds to a high *k*_{d} and low *k*_{a}, the zero-force velocity can be as high as (*v*_{0} − *v*_{d}) ≈ 0.95*v*_{0}, close to the value from the ET mechanism. Note also that monomer depletion effects have a large effect on the results for the branching nucleation mode (solid curves). In their absence, *v*_{b} is independent of *F*_{ext}, as found in Ref. [9].

The force-velocity curves for the ET mechanism do not have the same plateau region at low force as found for the single-filament ET mechanism [6]. There are at least two effects that cause this difference. 1) Since *F*_{tot} contains contributions from a range of filament orientations, there exists a critical angle *θ _{c}* = cos

Figure 3a also compares our simulation results to the experimental data in Ref. [20]. We found that none of the six models are able to reproduce the plateau region in the force-velocity curve observed experimentally. We have tried different parameters in our simulations but found that the shapes of the force-velocity curve remain fairly similar. As an extreme case, we have tried using a step-function as the single-filament’s force-velocity relation, but we still fail to reproduce the observed behavior. A possible reason is that the actin network in the experiment undergoes a cooperative mechanical instability, as suggested by the observed loading-history dependence [20], which is beyond the scope of our modeling. An alternate possibility is that liquid-like effects of the actin gel are important. A recent model based on these effects [37] reproduced some of the qualitative features seen in the force-velocity curve of Ref. [20]

Figure 3b illustrates the effect of mechanical stalling on the force-velocity relation, for the ET mechanism, by varying *l _{c}*. At the smaller value of

Figure 4a compares the *F*_{ext}-*N*_{tot} relations of the six network models with *l _{c}* = 0.2

Force-filament number relation for branching nucleation mode (solid lines) and spontaneous nucleation mode (dashed lines). a) TR, PP, and ET mechanisms with *l*_{c} = 0.2 *μ*m. b) ET mechanism with *l*_{c} = 0.1 *μ*m and 0.2 *μ*m.

Fig. 4b shows that varying *l _{c}* also has a moderate effect on the force-filament number relation in the ET model. For the larger value of

In order to explore further avenues for distinguishing between the network models, we evaluate the *k*_{d}- and *k*_{c}-dependence of the zero-force velocity.

The *k*_{d}-dependence displayed in Figure 5a for *l _{c}* = 0.2

Zero-force velocity as a function of a) *k*_{d} and b) *k*_{c}, with *k*_{a} = 1 s^{−1}. TR, PP, and ET mechanisms with both branching nucleation (solid lines) and spontaneous nucleation (dashed lines) are compared.

Figure 5b shows corresponding results for the *k*_{c}-dependence of the obstacle’s zero-force velocity. The velocity decreases with increased *k*_{c} in all six models. However, the extent of the decrease differs strongly between the single-filament growth mechanisms. The TR mechanism has the largest change in velocity; the velocity drops to 0 at large *k*_{c}. The ET mechanism has the smallest change. For both the *k*_{d}- and the *k*_{c}-dependence of the zero-force velocity, the branching and spontaneous nucleation modes give very similar results, except that the branching nucleation curves for the PP and TR models terminate earlier as a function of *k*_{d} because the filament number becomes very low (*N*_{tot} < 100).

The above results show that some of the pronounced qualitative differences between the single-filament force-velocity relations of competing growth mechanisms are “washed out” when these mechanisms are embedded in more complete network models. However, observable differences persist in the force-velocity relations and other experimentally accessible quantities. Large values of the thermodynamic stall force *f*_{therm} in the single-filament growth mechanism tend to produce a sharp, concave-down, drop in the force-velocity relation. On the other hand, differences in the force-filament number relation are mainly related to differences in filament nucleation mechanism, with branching nucleation giving a sharper climb in the filament number. The capping and detachment-rate dependence of the velocity depend mainly on whether free filaments push more effectively (TR and PP mechanisms) or less effectively (ET mechanism) than attached filaments.

The factor affecting the shape of the force-velocity relation most is the magnitude of the thermodynamic stall force *f*_{therm} in comparison with the mechanical stall force *f*_{mech}. If *f*_{therm} > *f*_{mech}, the force-velocity curve drops rapidly near a critical force per filament around *f*_{mech}, where many of the filaments begin to be bent. The drop causes a concave-down shape. This effect may be related to the observed sudden drop of obstacle velocity as reported in Refs. [20] and [21]. However, none of our force-velocity curves reproduces the low-force plateau seen in the experimental data. On the other hand, for *f*_{mech} > *f*_{therm}, filaments can produce strong pushing forces without being significantly bent. This gives the force-velocity curve a concave-up shape.

This prediction could be tested experimentally by varying the crosslinking distance *l _{c}*. With decreasing cross-linking protein concentration,

In making the transition from single-filament models to our network model, the ET mechanism undergoes the most significant change in the shape of the force-velocity relation. The reason is that in the ET mechanism, attached filaments with large angles turn into pulling filaments, which slows down the obstacle especially when the obstacle is moving at high speed. This makes the force-velocity curve decrease more rapidly with external force near the high velocity region.

For the branching nucleation mode, our model does not have the force-independent protrusion velocity predicted by Ref. [9], because we find that the branching rate is limited by protein concentrations which are limited by diffusion. We have checked that if the branching rate is kept constant, our network model does give a force-independent protrusion velocity, although it does not reproduce the subsequent drop seen in Ref. [20].

Schaus and Borisy [18] proposed a network model in which a force-velocity curve with a plateau at low forces and a concave-up decay at high forces is produced. The plateau is believed to occur because the velocity is limited by the detachment rate of tethers. The discrepancy between these results and the present ones, however, cannot definitely establish the fundamental difference between their model and ours.

For all six network models, the filament number increases with external force. This occurs because an increased load slows down the obstacle, which in turn decreases the filament capping, detachment, and leaving rates, which increases the number of filaments in contact with the obstacle. Our results also show that the filament number increases more rapidly for the branching nucleation mode. This occurs because the filament creation rate increases with the filament number [11]. Such positive-feedback is absent in spontaneous nucleation. As mentioned above, the force-filament number relation could be evaluated by measuring the fluorescence intensity of appropriately labeled actin as a function of force. Analogous measurements of the amount of polymerized actin as a function of time have been made for Listeria and actin-propelled beads [19, 22, 23].

For the detachment and capping rate dependence of the zero-force velocity, there are no significant differences between the branching and spontaneous nucleation modes. However, substantial differences between the three single-filament growth mechanisms exist. Under increases in the detachment rate, the velocity in the TR mechanism changes most, and the velocity in the ET mechanism changes least. Under increases in the capping rate, the velocity in the TR mechanism has the largest change and drops to zero at high capping rates, while the velocity in the ET mechanism has the smallest change and approaches a constant value at high capping rate. We have checked the above trends by assuming that new filaments are created free, and found similar dependencies of the velocity.

A possible way of distinguishing the three single-filament mechanisms experimentally would be to measure the dependence of the zero-force velocity on the detachment rate. One could, for example, add the protein VASP, which appears to increase the filament detachment rate [39-41], to the protein mix used in *in vitro* experiments. Such experiments could use a cantilever setup similar to that of Ref. [20], or freely moving disks as described in Ref. [42]. Existing experiments on beads, to which the present theory is not directly applicable, show that the zero-force speed is enhanced by high VASP levels [39, 43-45].

In the types of cells where the flat-lamellipodium approximation is most appropriate, such as migrating keratocytes, VASP can be up- or down-regulated and the effect on lamellipodial protrusion velocities measured. Our two-dimensional calculations show trends very similar to those for the three-dimensional ones. Thus in either case, one would expect a strong increase of velocity with VASP concentration or activity for the TR mechanism, a modest increase for the PP mechanism, and little variation for the ET mechanism. Measurements of this type have been performed on keratocytes and they give an increase of the velocity with overexpression of VASP [46]. Tentatively these results would speak in favor of the TR or PP mechanisms for keratocytes. Additional experiments with a broader range of cell types would help to establish the generality of this result.

Measurements of the effects of the capping rate on the zero-force velocity are harder to interpret because capping protein affects the free-actin concentration. Therefore it is difficult to separate the effect of capping by itself. However, the very strong downturn seen in the TR results should be present regardless of the effects on the free-actin concentration. Thus if the TR mechanism holds, very large increases of capping protein activity in either *in vivo* or *in vitro* should lead to a slowdown in velocity. Such experiments have not been performed for the geometry treated here, but experiments with *Listeria* [43] indicated that the speed first increases at low capping protein concentrations (presumably because more monomeric actin becomes available for propulsion), and subsequently decreases at higher concentrations.

We appreciate valuable discussions with David Sept’s research group, Philip Bayly, and Frank Brooks. This work was supported by the National Institutes of Health under Grant R01-GM086882.

By dimensional analysis, one can estimate the spring constant for the displacement of a filament base (which has units of energy/length^{2}) as *k*_{el} ≈ *El _{c}*, where

The recruitment rates of actin monomers, capping protein, and Arp2/3 complex are limited by the depletion of free proteins in the polymerization zone. Therefore as the rate of actin monomer consumption or the density of actin filaments increase, the growth, capping, and nucleation rates may decrease. Dickinson and Purich [28] estimated that near a biomimetic bead, the slowing of polymerization due to monomer depletion satisfies

$${u}_{\mathrm{v}}=\left[\mathrm{G}\right]/{\left[\mathrm{G}\right]}_{\infty}\approx {(1+N{k}_{B}^{+}/4\pi DR)}^{-1},$$

(B1)

where *N* is the number of growing filaments & (*N* = *N*_{f} for the TR model; *N* = *N*_{tot} for the ET and PP models), ${k}_{\mathrm{B}}^{+}$ is the barbed end monomer on-rate constant as mentioned before, *D* is the monomer diffusion constant, and *R* is the bead’s radius; the first equality holds since the on-rate is proportional to the monomer concentration. The geometry treated in the cantilever experiments [20] is a flat surface, roughly rectangular, of area approximately 180 *μ*m^{2} (visually estimated from Fig. 1 of the paper). The diffusion calculation is considerably more complex for this geometry than for a sphere. For this reason, and because our calculations do not aim for quantitative accuracy, we obtain our depletion coefficients by performing a calculation for a sphere of area 180 *μ*m^{2}, so that in Eq. B1 *R* = 3.8 *μ*m. In addition, we take *D* = 4 *μ*m^{2}/s [49, 50].

To calculate *u*_{c}, we note that the intracellular concentration of capping protein is typically about one to two orders of magnitude lower than the G-actin concentration [51]. On the other hand, for a typical filament length of 0.1 to 0.3 *μ*m, there are thirty to a hundred subunits per filament. If we assume that all filaments eventually become capped, this means that the consumption rate of capping protein will be lower than that of actin by a factor equal to the filament length. Therefore the depletion factors should be similar in magnitude, and we take *u*_{c} = *u*_{v}.

For branching nucleation, *k*_{b} depends on both the free actin monomer concentration [G] and the Arp2/3 concentration [Arp2/3]. Carlsson *et al*. [27] estimated *k*_{b} [G]^{2}[Arp2/3] from a fit to polymerization dynamics, which corresponds to a critical nucleus of one Arp2/3 complex and two actin monomers. In cells, the ratio of free Arp2/3 complex to G-actin is usually on the same order of magnitude as that of capping protein [51]. If we assume, as for capping protein, that one Arp2/3 complex is used for every filament, it is then reasonable to assume that the depletion factor for Arp2/3 complex is the same as that for G-actin. Therefore, since the branching rate contains one factor of the Arp2/3 complex concentration and two factors of the G-actin concentration, we take

$${u}_{\mathrm{b}}={{u}_{\mathrm{v}}}^{3}\approx {(1+N{k}_{B}^{+}/4\pi DR)}^{-3}.$$

(B2)

Assuming that spontaneous nucleation involves a critical nucleus of size similar to that for branching, we take *u*_{s} = *u*_{b}.

The geometry of a filament pushing against a surface is shown in Figure 2. We define the angle between the filament and the *X*-direction at the pointed end to be *θ*, and the angle at the barbed end to be *θ*_{tip}. We assume that the pointed end of the filament is clamped, and that the barbed end can move freely along the surface. Therefore *θ* is independent of force, and *θ*_{tip} increases with the pushing force *f _{x}*. Dickinson

$${f}_{x}=\frac{{l}_{\mathrm{p}}{k}_{\mathrm{B}}T}{{l}^{2}}{\left[K\left(\mathrm{sin}\frac{{\theta}_{\text{tip}}}{2}\right)-F\left({\varphi}_{0},\mathrm{sin}\frac{{\theta}_{\text{tip}}}{2}\right)\right]}^{2},$$

(C1)

where *ϕ*_{0} = sin^{−1}[sin(*θ*/2)/sin(*θ*_{tip}/2)]; *K*(*k*) and *F*(*z*, *k*) are the complete and incomplete elliptic integrals of the first kind, respectively. We define *f*_{mech}(*θ*) to be the force in the *X*-direction that gives *θ*_{tip} = *π*/2, at which filament can no longer produce pushing forces. We have evaluated *f*_{mech}(*θ*) numerically and find that a linear relation between ${f}_{\mathrm{mech}}\left(0\right)={l}_{\mathrm{p}}{k}_{\mathrm{B}}T{K}^{2}(1/\sqrt{2})/{l}^{2}\approx 3.4{l}_{\mathrm{p}}{k}_{\mathrm{B}}T/{l}^{2}$ and *f*_{mech}(*π*/2) = 0 gives a very good fit. Therefore, we use this linear approximation to calculate *f*_{mech}(*θ*) in Eq. 15.

To our knowledge, there are no definitive experimental measurements of the attachment rate *k*_{a} or the detachment rate *k*_{d}. To obtain a very rough estimate of *k*_{a}, we calculate the first-passage time 1/*k*_{a} for a filament to reach a binding site. We take the spacing between the binding sites to be *d* = 17 nm [48], and the radius of a binding site to be *r*_{b} = 0.5 nm. This radius represents not the size of the protein to which the tip binds, but rather the displacement from the optimal binding position which is required to reduce binding substantially. The filament tip is treated as a freely diusing particle moving on the obstacle surface. Once it touches a binding site, it becomes attached. To estimate the first-passage time, we evaluate the first-passage time *w*(*r*) for a particle diusing from an outer circle with radius *d*/2 to an inner circle with radius *r*_{b}. This time satisfies *w*″(*r*) + *w*′(*r*)/*r* = −1/*D*_{tip}, with boundary condition *w*(*r*_{b}) = *w*′(*d*/2) = 0, where *D*_{tip} is the diffusion constant of the filament tip [52]. Then a simple calculation shows that $w(d/2)={d}^{2}\phantom{\rule{thinmathspace}{0ex}}\mathrm{ln}(d/2{r}_{b})/8{D}_{\mathrm{tip}}-({d}^{2}/4-{r}_{\mathrm{b}}^{2})/4{D}_{\mathrm{tip}}$. The value of *D*_{tip} has been calculated using a normal mode analysis [53]. For a filament with length *l* = 200 nm, a lowest-mode approximation gives *D*_{tip} = 4*Dδ*/*l* ≈ 0.2 *μ*m^{2}/s, where *D* ≈ 4 *μ*m^{2}/s is the diffusion constant of an actin monomer [49, 50] and *δ* = 2.7 nm is the filament length increment per actin subunit. Then *k*_{a} ≈ 1/*w*(*d*/2) ≈ 6000 s^{−1}.

Literature estimates of *k*_{d} vary widely. In a study of actin propelled beads [54], a few filaments about 1 *μ*m in length were observed to be attached to sub-micron beads. Evaluation of the time that it would take a filament to grow to this length suggests a detachment rate on the order of 1 s^{−1} or less. On the other hand, Vavylonis *et al*. [55] estimated the profilin-barbed end dissociation rate to be 2500 s^{−1}. If the interaction between the filaments and the obstacle in our study is similar to that between filaments and profilin, then the value of *k*_{d} could be as high. Since there are no accurate measurements of *k*_{a} or *k*_{d}, we have evaluated the effects of a broad range of variation of these parameters on our results. Varying *k*_{d} from 1 s^{−1} to 2500 s^{−1}, we find that the results are determined mainly by the ratio of *k*_{a} to *k*_{d}. Therefore our strategy for fixing *k*_{a} and *k*_{d} is to use baseline values of 1 s^{−1} which lead to numerically tractable calculations, and subsequently to evaluate the effects of deviations from these baseline values.

We take the maximum polymerization velocity to be *v*_{0} = 70 nm/s, which, using an on-rate constant of 11.6 *μ*M^{−1}s^{−1} [56], corresponds to a free monomer concentration of *G*=2-3 *μ*M, similar to the typical *in vitro* concentrations [57]. The actin filament depolymerization velocity is ${v}_{\mathrm{d}}={k}_{\mathrm{B}}^{-}\delta \approx 3.8\phantom{\rule{thickmathspace}{0ex}}\mathrm{nm}/\mathrm{s}$, where ${k}_{\mathrm{B}}^{-}\approx 1.4\phantom{\rule{thickmathspace}{0ex}}{\mathrm{s}}^{-1}$ is the barbed-end depolymerization rate of ATP-actin [56].

The on-rate of capping protein to the barbed end of actin filaments has been estimated to be ${k}_{\mathrm{cap}}^{\mathrm{B},+}=3-8\phantom{\rule{thickmathspace}{0ex}}\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$ [27, 58]. Here we take ${k}_{\mathrm{cap}}^{\mathrm{B},+}=5\phantom{\rule{thickmathspace}{0ex}}\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, and assume the concentration of capping protein to be [CP]=0.1 *μ*M, an intermediate value between *in vivo* [51] and *in vitro* [43, 57] estimates. Then the capping rate in the absence of depletion effects is ${k}_{\mathrm{c}}={k}_{\mathrm{cap}}^{\mathrm{B},+}\left[\mathrm{CP}\right]\approx 0.5\phantom{\rule{thickmathspace}{0ex}}{\mathrm{s}}^{-1}$.

The spontaneous nucleation rate *k*_{s} is estimated as follows. At steady state, the rate of filament creation should be equal to the rate of filament extinction. Since extinction in our model results only from capping, *k*_{s} satisfies

$${k}_{\mathrm{s}}={\int}_{-\infty}^{\infty}{\int}_{-\pi /2}^{\pi /2}{k}_{\text{cap}}(x,\theta ){P}_{\mathrm{f}}(x,\theta )\mathrm{d}\theta \mathrm{d}x\le {k}_{\mathrm{c}}{N}_{\mathrm{f}}\le {k}_{\mathrm{c}}{N}_{\mathrm{tot}},$$

(D1)

where *N*_{f} is the total number of free filaments and *N*_{tot} is the total number of free and attached filaments. For concreteness we take the area to be 180 *μ*m^{2} (see Appendix B). We thus estimate the maximum value of *N*_{tot}, corresponding to a density of 1000 *μ*m^{−2} [59], to be ${N}_{\mathrm{tot}}^{\mathrm{max}}=1.8\times {10}^{5}$. Then the upper limit of *k*_{s} is ${k}_{\mathrm{s}}^{\mathrm{max}}={k}_{\mathrm{c}}{N}_{\mathrm{tot}}^{\mathrm{max}}\approx 9\times {10}^{4}\phantom{\rule{thickmathspace}{0ex}}{\mathrm{s}}^{-1}$, and we use an intermediate value of *k*_{s} = 4.5 × 10^{4} s^{−1}. The branching nucleation rate *k*_{b} is estimated from a filament branch spacing of *l*_{b} ≈ 50 nm [29]. Using the maximum polymerization velocity given above, we obtain *k*_{b} = *v*_{0}/*l*_{b} ≈ 1.4s^{−1}.

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