Actin Cross-Linkers and the Shape of Stereocilia

Abstract

Stereocilia are actin-based cellular protrusions essential for hearing. We propose that they are shaped by the detachment dynamics of actin cross-linkers, in particular espin. We account for experimentally observed stereocilium shapes, treadmilling velocity to length relationship, espin 1 localization profile, and microvillus length to espin level relationship. If the cross-linkers are allowed to reattach, our model yields a dynamical phase transition toward unbounded growth. Considering the simplified case of a noninteracting, one-filament system, we calculate the length probability distribution in the growing phase and its stationary form in a continuum approximation of the finite-length phase. Numerical simulations of interacting filaments suggest an anomalous power-law divergence of the protrusion length at the growth transition, which could be a universal feature of cross-linked depolymerizing systems.

Introduction

The exquisite frequency selectivity of our hearing can be tracked back to the intricate and remarkably well-regulated internal structure of the ear. At the heart of this mechanotransduction machinery are stereocilia, which are present in reptiles, birds, and mammals. They are 1–120 μm-long rodlike protrusions of so-called hair cells that pivot around their ankle upon mechanical stimulation (1). This motion causes the opening of ion channels, which induces a depolarization of the membrane that results in the propagation of a nervous signal. Stereocilia are primarily made of a paracrystal of up to 200 densely packed (2), parallel actin filaments enclosed by the cell membrane (3). They are roughly cylindrical over most of their length, but taper at their base. This indicates that some filaments do not extend all the way to the cell body, although some others do penetrate far into the underlying cuticular plate (see Fig. 1 b). Within the stereocilium the filaments are in register, meaning that their helical periods are perfectly aligned in the vertical direction. Their barbed (polymerizing) ends point toward the stereocilium tip while their depolymerizing ends point toward the cell body. Although stereocilia are maintained throughout the life span of an individual, they are dynamic structures that are constantly renewed by actin treadmilling. During this process, actin is continuously incorporated at the tip of the stereocilium and depolymerized at its base (4). Interestingly, the actin treadmilling velocity is proportional to the stereocilium height, so that the time necessary to fully renew any auditory stereocilium is independent of the stereocilium height (it has been shown to be ≃48 h in rats (5)).

Figure 1.

Model stereocilium and comparison with experimental shapes. (a) Cross-linked actin is produced in ℓ = 0 and treadmills down with a velocity v. Meanwhile, espins are exchanged with the surrounding solution with rates k_on and k_off. An actin filament not held by a cross-linker at its pointed end immediately depolymerizes to the next espin. (b) Comparison between our predictions (Eq. 4, plotted as thick black lines, the top ends of which indicate ℓ = 0, the polymerization front), and three guinea pig stereocilia from the same hair cell (micrographs taken from Fig. 3 a of (25)). Note the stereocilia's long cylindrical top section, tapered base, and the fact that they insert into the cuticular plate (the top part of cell body). The diameter of the tallest stereocilium is ∼250 nm.

Stereocilia have recently been the focus of theoretical attention (6,7), and two of us have suggested that this renewal time is essentially an intrinsic timescale associated with the actin bundle's depolymerization dynamics (8). However, the origin of the timescale proposed in this previous work yields a strong sensitivity of the stereocilium shape on the model's parameters. In addition, this model does not account well for the long quasicylindrical section observed in healthy stereocilia. Here we improve the notion of an intrinsic timescale put forward in Prost et al. (8) by suggesting that it originates in the binding-unbinding dynamics of actin cross-linkers, which were mentioned but not treated explicitly in this work. Our description of this experimentally well-characterized mechanism allows us to reproduce stereocilium shapes faithfully with few adjustable parameters and to quantitatively account for experimental results previously only considered from a qualitative point of view. Finally, it yields robust structures, which is very significant because the frequency sensitivity of the ear requires a delicate regulation of the stereocilia's mechanical properties, which are in turn determined by their shape.

Actin cross-linking was described early in the study of stereocilia (9) and could be responsible for the filaments being in register (10). Although cross-linkers of two types, espin and fimbrin, have been identified in stereocilia (11,12), we hereafter focus on espin, which is thought to provide sturdier cross-linking than fimbrin (13,14). Note, however, that our study is general enough to apply to any cross-linker, and could be extended to account for the simultaneous presence of several cross-linker species. Espin slows actin depolymerization down in vitro (15), and could thus play an important role in stereocilia, as the actin depolymerization rate there (≃0.002–0.04 s⁻¹ over the whole stereocilium, meaning ≈10⁻⁴ s⁻¹ for each filament (5)) is much smaller than that of F-actin in vitro (≃1 s⁻¹ (16)). Several in vivo experiments indeed support the notion that cross-linking plays a major role in the length regulation of stereocilia and related cellular protrusions. When transfected with espin, LLC-PK1-CL4 epithelial cells (referred to as CL4 cells in the following) undergo a dramatic lengthening of one such type of protrusions, microvilli, which could be due to espin preventing their disassembly (17). Other actin cross-linkers are also known to inhibit the disassembly of actin bundles in Drosophila bristle (18). Espin is incorporated at the stereocilium tip and treadmills down simultaneously with actin (5). Its overexpression (but not that of actin) induces the lengthening of stereocilia and a mutation resulting in espin underexpression causes their shortening (19). Under normal in vivo conditions, the variability in stereociliar length is correlated with the espin expression level (15,20) and isoform expression pattern (21). Finally, two recessive and four dominant mutations of espin are responsible for deafness in humans (13). Out of the four dominant ones, three induce less microvillus lengthening than wild-type espin when transfected into CL4 cells (17).

This article is organized as follows. In Model for the Actin and Cross-Linker Dynamics, we present a model for the coupled dynamics of espin cross-linking and actin depolymerization. Solving the simple case where espin is incorporated into the actin bundle only at the tip of stereocilia, we show in Stereocilium Shape without Espin Reattachment that our formalism yields robust stereocilia shapes with only one adjustable parameter and accounts for experimental results not previously discussed in the theoretical literature. In Single Filament with Reattachment, we show the modifications induced by espin reattachment during the course of treadmilling by discussing a simplified situation involving only one filament. Coupling Between Filaments then focuses on the lateral correlations that espin reattachment induces in a multifilament bundle, and we discuss our results in the last section, Discussion and Conclusions.

Model for the Actin and Cross-Linker Dynamics

Our model is presented in Fig. 1 a. Completely cross-linked actin is continuously produced at a location ℓ = 0 with an externally imposed treadmilling velocity v, the regulation of which is discussed in Prost et al. (8). The polymerization dynamics of the actin bundle is thus assumed to be deterministic. In practice, this polymerization is highly regulated by several proteins comprised in the electron-dense tip complex located at the stereocilium tip (19,22–24). Because the filaments across the bundle are cross-linked, they move together at a velocity equal to the average polymerization rate of the filaments. As there are many filaments in the bundle, the fluctuations of this average should be small.

As actin moves down, espin is exchanged with the surrounding medium. Considering that the typical time for the depolymerization dynamics in stereocilia is ≃1000 s (the time required to depolymerize one helical period of the actin filament according to (25)) and assuming a diffusion constant of 60 μm².s⁻¹ (estimated from the Stokes radius of espin (26)), we estimate that the unbound espin concentration is homogeneous over length scales of order at least 250 μm, i.e., larger than the size of the stereocilium. We thus consider that the espin attachment and detachment rates k_on and k_off are constant throughout the stereocilium (k_off also accounts for espin degradation). Note that this reasoning would not hold if espin were actively localized in some regions of the stereocilia, or if the diffusion of espin were slowed down considerably, for instance by crowding effects. It is, however, not known how much the actin bundle slows the diffusion of espin down, and we assume throughout this article that this effect is not sufficient to induce significant espin density gradients. The opposite hypothesis is considered in Naoz et al. (7), which we further discuss in the last section. A similar argument applies to the supply of actin to the stereocilium tip, which we consider to always be sufficient to maintain the treadmilling velocity v. Finally, espin attachment at the altitude ℓ is only possible between two neighboring filaments of length equal to or larger than ℓ, as espin cannot reattach if there are no actin filaments.

We formulate the simplifying hypothesis that actin filaments can only depolymerize from their pointed ends. In agreement with the experimental results presented in the previous section, we assume that espin prevents the depolymerization of the actin filaments that it cross-links. Furthermore, we assume that the depolymerization of actin alone happens on much shorter timescales (≃1 s) than the espin detachment dynamics (≃1000 s). Hence, on the timescales relevant for the morphogenesis of stereocilia, actin filaments depolymerize instantaneously up to the next point where they are cross-linked, and are then stalled until the detachment of the cross-linker, which occurs at a rate k_off. We denote by a the vertical spacing between two actin cross-linkers (see Fig. 1 a). A filament cannot depolymerize beyond ℓ = 0 (this description is justified if, for instance, a filament of vanishing length is immediately renucleated by the tip complex so that the total number of filaments is conserved).

From the model described here, we expect the lower end of the actin bundle to have a very irregular shape due to the stochastic character of the espin detachment and subsequent actin depolymerization (as in Fig. 1 a, for example). However, we show in the Supporting Material that membrane tension pushes the filaments together, so that they are always in close contact (see Fig. S1 in the Supporting Material).

Unless otherwise specified, in the following we express lengths in units of the distance a between espin sites and times in units of the average cross-linker lifetime k_off⁻¹. We denote the dimensionless polymerization velocity v/(ak_off) by v, and define k = k_on/k_off.

Stereocilium Shape without Espin Reattachment

In this section we solve our model in the case where espin is incorporated in the actin bundle only at the stereocilium tip (k = 0). In this situation, an espin attachment site located at a distance ℓ from the polymerization front is occupied if and only if an espin has been incorporated when this site was located at the polymerization front and has then survived detachment for a time ℓ/v. Because the detachment process is analogous to a radioactive decay-like stochastic process with rate 1, the site in question is occupied with probability

$$P_{\text{on}}\left(\ell \right)=P_{\text{0}}{e}^{-\ell /\nu },$$ (1)

where P_on(0) = P₀ is the probability with which an espin cross-linker is incorporated at ℓ = 0. For a maximally cross-linked bundle, P₀ = 1. Now considering not one espin site, but a full espin column (defined in Fig. 1 a), we ask for the probability that the lowermost espin of the column is located at a distance ℓ or smaller from the polymerization front. This probability is given by the infinite product

$$P_c^{\le }\left(\ell \right)=\left[1-P_{\text{on}}\left(\ell +1\right)\right]\times \left[1-P_{\text{on}}\left(\ell +2\right)\right]\times \left[1-P_{\text{on}}\left(\ell +3\right)\right]\times \mathrm{\dots },$$ (2)

where ℓ ≥ 0. Now turning to the actin filaments, we see that an actin filament has a length smaller or equal to ℓ if and only if all neighboring espin columns have their lowermost espin at a location ℓ′ ≤ ℓ. Denoting by n the number of neighbors of an actin filament (filaments are hexagonally packed in mammalian and bird stereocilia so that n = 6 (10); n = 2 in Fig. 1 a), the probability for a filament to have a length smaller than or equal to ℓ in the absence of espin reattachment reads

$$P_f^{\le }\left(\ell \right)=P_c^{\le }{\left(\ell \right)}^n={\prod _{i=1}^{+\mathrm{\infty }}\left[1-P_0{e}^{-\left(\ell +i\right)/v}\right]}^n.$$ (3)

We now discuss this result and compare it to experimental data. For the sake of clarity, in the remainder of this section we go back to nonscaled units. Qualitatively, P^≤_f(ℓ) is equal to 0 for small ℓ-values, and to 1 for large ℓ-values. If a large number of filaments are present, the number of filaments of length larger than ℓ is proportional to

$$P_f^{>}\left(\ell \right)=1-P_f^{\le }\left(\ell \right).$$

Because the filaments are closely packed as discussed in Model for the Actin and Cross-Linker Dynamics and in Section S1 in the Supporting Material, the section π[r(ℓ)]² of the stereocilium at position ℓ is proportional to the number of filaments longer than ℓ, so that

$$r\left(\ell \right)=r\left(0\right)\sqrt{1-P_f^{\le }\left(\ell \right)}.$$ (4)

Here we do not specify the physical processes imposing r(0), the radius at the polymerizing end of the actin bundle. For relatively short-lived actin-based protrusion, r(0) could be fixed by dynamical processes operating during the initial actin bundling phase (27). In stereocilia, mechanical effects within the tip complex might lead to its continuous regulation (8). Because the length of the stereocilia (≃5 μm) is much larger than that the distance between two cross-linking sites (≃10 nm), we can use the continuum limit of Eq. 3,

$$P_f^{\le }\left(\ell \right)\underset{\upsilon /\left(a{k}_{\mathrm{off}}\right)\gg 1}{\mathbf{\sim }}\exp \left[-e^{-\left(\ell -{\ell }_s\right)k_{\text{off}}/v}\right],$$ (5)

where

$${\ell }_s=\frac{v}{k_{\text{off}}}\ln \left(\frac{nv{P}_0}{a{k}_{\text{off}}}\right)$$ (6)

(see Sec. S3.3 in the Supporting Material for a rigorous discussion of this limit). For small values of ℓ, this equation yields a cylindrical profile with a characteristic length ℓ_s given by Eq. 6. The cylinder then tapers over a length v/k_off. These predictions are plotted and compared to actual stereocilia shapes in Fig. 1 b. Several parameters involved in our theoretical shapes are well-known experimentally. Up to six espins can bind to each helical period of the actin filament, which yields a/n = (37/6) μm (10). The actin of the part of the stereocilium that sticks out of the cell is completely renewed by treadmilling in 48 h = ℓ_s/v (5), which imposes a different value of v depending on the length of the stereocilium. In agreement with electron microscopy studies, we assume that the actin bundle is heavily cross-linked by espin, so that P₀ = 1. This leaves only one free parameter k_off. Because the three stereocilia of Fig. 1 b belong to the same cell, we furthermore impose that they are all described by the same value of k_off. Taking k_off = 0.14 h⁻¹ yields a good fit for all three stereocilia.

More quantitative experimental results are also accounted for by our model. First, the relationship Eq. 6 between ℓ_s and v is almost linear, and we show in Fig. 2 a that it is compatible with the observation that the stereocilium's treadmilling velocity is roughly proportional to its length (5). Here the value of k_off is the same as the one determined in Fig. 1 b, meaning that no adjustable parameter is used in Fig. 2 a. In Fig. 2 b, we compare the experimentally measured (24) density profile of one specific type of espin, espin 1, along three stereocilia belonging to the same vestibular hair cell to an exponential, because the espin density is expected to be proportional to the probability P_on defined in Eq. 1 (note that the actin bundle renewal time in vestibular hair cells is 72 h (5)). The decay length of the experimental curves increases with stereocilium length (and therefore treadmilling velocity) as predicted by this equation. Consequently, three different stereocilia of the same cell are again well described by using one common value of k_off. Note, however, that although espin 1 does bind actin, its main role could be the regulation of actin polymerization, while other espins might be responsible for most of the cross-linking (B. Kachar, National Institute of Health, private communication, 2009). Another interesting result is presented in Loomis et al. (15). In this study, CL4 cells are transfected with espin, which causes the elongation of the cells' microvilli. The average elongation is measured and correlated to the espin expression level. Assuming that espin is incorporated at the tip of the protrusion at a rate proportional to its expression level c_e, we can consider that P₀ is proportional to c_e. Following this, Eq. 6 yields a prediction for the dependence of ℓ_s on c_e, which we show in Fig. 2 c. We use two new adjustable parameters there, as these experiments deal with a different cell type and with other protrusions than stereocilia (in particular, the renewal time of microvilli is much shorter than that of stereocilia). The best fit is found for v/k_off = 1.5 μm. The value of the other parameter,

$$\text{d}\left(\frac{nv{P}_0}{a{k}_{\text{off}}}\right)/\text{d}{c}_e,$$

does not contain any exploitable information because only relative values of c_e are known experimentally.

Figure 2.

Dependence of the protrusion length on various parameters predicted by Eq. 6 and determined from experiments. (a) Measured treadmilling velocity versus length in the stereocilia of the rat cochlea. In mammals, cochlear stereocilia are arranged into three rows of graded height (black circles, experimental data for the tallest row; gray circles, middle row; and open circles, shortest row). (Line) Plot of Eq. 6, using the same value k_off = 0.14 h⁻¹ as in Fig. 1b. Experimental data taken from Rzadzinska et al. (5). (b) Espin 1 density as a function of ℓ in the vestibular stereocilia of guinea pigs. The three curves correspond to three stereocilia of the same hair cell with different lengths (T ≃ 35 μm, M ≃ 20 μm, and S ≃ 10 μm). Agreement with Eq. 1 is found for k_off = 0.35 h⁻¹, which is of the same order of magnitude as the value deduced from the fit of Fig. 1b. Experimental data taken from Salles et al. (24). (c) Dependence of microvilli length in CL4 cells on the espin overexpression level. Experimental data taken from Loomis et al. (15).

Overall, we find that the simple case where espin does not reattach to actin yields good agreement with experimental data, while relying on only one adjustable parameter. Note also that the stereocilium length given by Eq. 6 has a smooth dependence on both $v/k_{\text{off}}$ and $nv{P}_0/a{k}_{\text{off}}$, as illustrated by Fig. 2, a and c. This makes the stereocilium robust with respect to perturbations of the cellular conditions, which is expected for such a well-regulated structure.

Single Filament with Reattachment

Although the results presented above give a good description of the shape of experimentally observed stereocilia, it is interesting to study the effects of espin reattachment in our model. We might indeed have to take this effect into account in more detailed studies of stereocilia or when interested in other types of cellular protrusions. In such protrusions, cross-linkers detaching from the actin filaments might diffuse for a while, and then reattach elsewhere in the actin bundle. If diffusion is considered fast in the sense of Model for the Actin and Cross-Linker Dynamics, this is equivalent to putting the filament in contact with a reservoir of cross-linkers, represented by the attachment rate k. In this configuration, the espin dynamics influences actin depolymerization in the same way as above, but unlike in Stereocilium Shape without Espin Reattachment, actin depolymerization now also influences the espin dynamics. Indeed, espin can reattach at a given site only if this site is surrounded by two actin filaments. Therefore, in contrast to the previous section, actin is no longer slaved to espin.

In this section, we consider only the simplified case of a single filament cross-linked to a wall, as shown in Fig. 3 a. We furthermore assume that P₀ = 1, i.e., that the actin bundle is completely cross-linked at the polymerization front. In Discrete Master Equation and Solution Far from the Polymerization Front, we write a master equation for the dynamics of the filament's depolymerizing end and solve it far from the polymerization front. We then consider the case where the depolymerizing end comes close to the polymerization front and discuss the resulting treadmilling steady state in Growth Transition and Stationary State.

Figure 3.

Schematics of the single-filament problem. (a) Single filament bound to a single wall and the coordinate system used in Single Filament with Reattachment. (b) Single filament bound to n = 3 walls.

Discrete master equation and solution far from the polymerization front

Unlike in the previous section, in the following we consider the altitude in the reference frame of the filament, not of the polymerization front. We assume that the polymerization front is at altitude zero at time t = 0. Because it moves with a velocity v in the reference frame of the filament, it is at altitude vt at time t. Thus the altitude z = vt – ℓ of the pointed end of the filament is an integer smaller than or equal to the altitude vt of the polymerization front (Fig. 3 a).

Let us define the quantity

$$\delta \left(i,t\right)=\exp \left[-\left(1+k\right)\left(t-\frac{i}{v}\right)\right].$$ (7)

In Sec. S2.1 in the Supporting Material, we write a master equation for the model described in Model for the Actin and Cross-Linker Dynamics and show that the probability P(Z, t) for the filament's depolymerizing end to be at altitude Z such that 0 ≤ Z < vt at time t obeys the simplified master equation

$${\partial }_{t}P\left(Z,t\right)=-P\left(Z,t\right)+\frac{k+\delta \left(Z,t\right)}{1+k}\sum _{Z^{\prime }=-\mathrm{\infty }}^{Z-1}\left[\prod _{i=Z^{\prime }+1}^{Z-1}\frac{1-\delta \left(i,t\right)}{1+k}\right]P\left(Z^{\prime },t\right),$$ (8)

with the boundary condition at the polymerization front

$${\partial }_{t}P\left(\mathrm{\lfloor }vt\mathrm{\rfloor },t\right)=\sum _{Z^{\prime }=-\mathrm{\infty }}^{\mathrm{\lfloor }vt\mathrm{\rfloor }-1}\left[\prod _{i=Z^{\prime }+1}^{\mathrm{\lfloor }vt\mathrm{\rfloor }-1}\frac{1-\delta \left(i,t\right)}{1+k}\right]P\left(Z^{\prime },t\right),$$ (9)

and where we assume that the filament has a vanishing length at t = 0:

$$P\left(Z,t=0\right)={\delta }_{Z,0}.$$ (10)

Here ⌊x⌋ denotes the integral part (or floor) of real number x, and δ_i,j is the Krönecker delta. Note that the probability distribution from Eq. 3 is a solution of this problem in the special case n = 1, k = 0, P₀ = 1 (see Sec. S2.2 in the Supporting Material).

We now consider the altitude i located strictly above the depolymerizing end of the filament and strictly below the polymerization front (i.e., Z < i < vt). We show in Sec. S2.1.3 in the Supporting Material that the probability for the espin site located at altitude i to be occupied is

$$\frac{k+\delta \left(i,t\right)}{1+k}.$$

The function δ(i, t) can thus be interpreted as the deviation of the espin density at site i from the steady-state density $k/(1+k)$ corresponding to a situation where site i is in equilibrium with the espin reservoir. This imbalance originates in the fact that espin sites are always occupied at the polymerization front (they are incorporated into the actin bundle with a probability of 1). With time, however, espin sites lose the memory of their initial conditions, and relax back to an equilibrium with the espin reservoir. This is reflected by the fact that δ(i, t) vanishes far away from the polymerization front, i.e., in the region where $vt-i\gg v/(1+k)$. Let us assume that the filament's depolymerizing end is at the altitude

$$\overline{Z}\gg \frac{v}{1+k}$$

at time $\overline{t}$. We solve this problem exactly in Sec. S2.3 in the Supporting Material. We then show that on long timescales the dynamics of the depolymerizing end is well approximated by the Gaussian distribution

$$P\left(Z,t\right)\underset{t\to +\mathrm{\infty }}{\propto }\exp \left\{-\frac{k^2}{2\left(1+k\right)\left(2+k\right)\left(t-\overline{t}\right)}{\left[Z-\overline{Z}-\frac{\left(1+k\right)\left(t-\overline{t}\right)}{k}\right]}^2\right\}.$$ (11)

This is characteristic of a biased diffusion with diffusion coefficient

$$D_d=\frac{\left(1+k\right)\left(2+k\right)}{2k^2}$$

and average depolymerization velocity

$$v_d=\frac{1+k}{k}.$$

The depolymerization velocity can be recovered from the following very simple argument: consider a filament cross-linked to the wall at its pointed end. Because the cross-link detaches with a rate 1, the average waiting time for the filament to unpin is τ = 1. Once the filament is released, it quickly depolymerizes to the next cross-linker, and then becomes pinned again. Because the espins are at equilibrium with the reservoir, the average cross-linker density is

$$\rho =\frac{k}{1+k},$$

meaning that the filament depolymerizes over an average distance d = 1/ρ before becoming pinned again. Therefore, the average depolymerization velocity of the filament is

$$v_d=d/\tau =\frac{1+k}{k}.$$

Growth transition and stationary state

If the depolymerization velocity v_d is smaller than the polymerization velocity (v_d < v), then the pointed end never catches up on the polymerization front, and Eq. 11 is a good approximation of its dynamics. In this case, the filament length—which is equal to the distance between polymerization front and pointed end—grows indefinitely at velocity v – v_d and the filament has no stationary state. Heavy cross-linking of the actin favors this regime, because it has the effect of slowing depolymerization down. However, v_d cannot be smaller than 1, which corresponds to a maximally cross-linked situation (i.e., to jumps of size 1 at a rate 1). Therefore, if v < 1, the growth regime described here does not exist. Conversely, if the depolymerization velocity is larger than the polymerization velocity (v_d > v), the pointed end moves closer and closer to the polymerization front. Thus, the length of the filament is bounded in this regime. This is the situation considered in this section.

We hereafter call the threshold v = v_d the growth transition. As it comes closer to the polymerization front, the pointed end penetrates into regions where the cross-links have not yet lost the memory of their incorporation into the bundle, and are therefore denser than at equilibrium. More specifically, their average density is given by

$$\rho \left(\ell \right)=\frac{k+e^{-\frac{1+k}{v}\ell }}{1+k},$$ (12)

where ℓ = vt – z is the length of the filament. Using the same argument as in the previous section, the depolymerization velocity of a filament of length ℓ is equal to 1/ρ(ℓ). A stationary filament length is obtained when this velocity matches the polymerization velocity. This reasoning yields an estimate for the stationary length ℓ_s,

$$v=\frac{1}{\rho \left({\ell }_s\right)}\iff {\ell }_s=\frac{v}{1+k}\ln \left[\frac{1}{\left(1+k\right)\left(\frac{1}{v}-\frac{1}{v_d}\right)}\right],$$ (13)

where

$$v_d=\frac{1+k}{k}.$$

Equation 13 matches Eq. 6 for k = 0, P₀ = 1, and n = 1. In vivo, stereocilia are much longer than the spacing between two cross-linkers, meaning that we are interested in the regime ${\ell }_s\gg 1$. There are two ways to enter this regime. One is for the logarithm in Eq. 13 to be very large, which can only be achieved if

$$\frac{1}{v}-\frac{1}{v_d}\ll 1.$$

This happens when the polymerization and equilibrium depolymerization velocities are very well matched. Because we expect the stereocilium shape to be robust under perturbations of the model parameters, this is not reasonable from a biological point of view. Therefore, we discard this first way of obtaining ${\ell }_s\gg 1$ and turn to the second one, which is

$$\frac{v}{1+k}\gg 1.$$

In this case, because v < v_d,

$$1\ll \frac{v}{1+k}<\frac{v_d}{1+k}=\frac{1}{k}.$$ (14)

This implies k ≪ 1, meaning that we do not need to consider the depolymerization problem in all its generality, but only its small-k, large-v limit. Let α = kv. Multiplying Eq. 14 by k, we note that below the growth transition, 0 ≤ α < 1. Therefore, in the limit of large v, the growth transition occurs for α = 1 (or equivalently $v=v_d=(1+k)/k$, which is its definition). The interesting regimes to consider are therefore those where α is of order 1, and in the following we take the v → +∞ limit at fixed, finite α.

In Sec. S3.2 in the Supporting Material, we generalize our approach to a filament bound to a number n of walls, as exampled in Fig. 3 b. Defining the coordinate ξ by

$$\ell =vt\mathrm{-}Z=v\ln v+v\xi ,$$ (15)

we expect from Eq. 13 that the interesting part of the dynamics takes place in the scaling region ξ ≈ 1. Indeed, we show in Sec. S3.3 in the Supporting Material that the master equation has the following continuum limit:

$$\frac{\text{d}P}{\text{d}\xi }\left(\xi \right)=-P\left(\xi \right)+n\left(\alpha +e^{-\xi }\right)\exp \left(n\alpha \xi -n{e}^{-\xi }\right){\int }_{\xi }^{+\mathrm{\infty }}\frac{P\left({\xi }^{\prime }\right)}{\exp \left(n\alpha {\xi }^{\prime }-n{e}^{-{\xi }^{\prime }}\right)}\text{d}{\xi }^{\prime }.$$ (16)

The stationary profile of the filament length probability distribution is the only normalized stationary solution of this equation. The corresponding cumulative distribution reads (see Sec. S3.4 in the Supporting Material)

$$P^{\le }\left(\xi \right)=\frac{\Gamma \left(1-n\alpha ,n{e}^{-\xi }\right)}{\Gamma \left(1-n\alpha \right)}.$$ (17)

Here Γ(b) = Γ(b, 0) is the usual γ-function, where the incomplete γ-function is defined as

$$\Gamma \left(b,x\right)={\int }_x^{+\mathrm{\infty }}\left(u^{b-1}{e}^{-u}\right)\text{d}u.$$ (18)

Plots of P^≤ as a function of ℓ are presented in Fig. 5. Equation 17 implies that the average filament length diverges as

$$\langle \ell \rangle \underset{k\to k_c^{-}}{\sim }\frac{v}{1-nkv}\propto \frac{1}{|k-k_c|},$$ (19)

when k approaches the critical value

$$k_c=\frac{1}{nv}.$$ (20)

Therefore, for a large enough espin reattachment rate, a stationary filament profile ceases to exist. This is the n-walls generalization of the growth transition discussed at the beginning of this section. Indeed, for k ≥ k_c, espin slows the depolymerization down so much that the pointed end can never catch up on the polymerization front.

Figure 5.

(Color online) Profiles of multifilament bundles for various values of k. (Red lines; leftmost line in (a) and rightmost line in (b-d)) P^> = 1 – P^≤, with P^≤ given by the single-filament theory equation (Eq. 17) scaled to the number of filaments in the bundle with n = 2.5. (Black lines) Average number of filaments longer than ℓ for numerical 8 × 8 bundles. (Gray area) Standard deviation of the steady-state fluctuations around this average. The cyan (gray) lines have different meanings depending on the figure considered. (a) n = 4 single filament theory, equivalent to the fit of Fig. 1b; note that the representation used here does not reflect the aspect ratio of the predicted stereocilia shapes. (b) Average number of filaments longer than ℓ and fluctuations for a 16 × 16 bundle (data normalized to match the black line in ℓ = 0). (c) Average number of filaments longer than ℓ and fluctuations for a circular bundle of 32 filaments (data normalized to match the black line in ℓ = 0). Note the contracted ℓ scale in panel d, as compared to panels a–c.

Coupling between Filaments

In this section we use Monte Carlo simulations of a square (n = 4) lattice of filaments (described in Sec. S4.1 in the Supporting Material) to study the effect of espin reattachment in the biologically relevant situation of a stereocilium composed of several filaments. In the following, we focus on long stereocilia, for which we expect the continuum approach introduced in Growth Transition and Stationary State to apply. This approach is valid for v ≫ 1. Because simulating long bundles is time-consuming, we use v = 20 throughout, which represents a good compromise. Unlike in the previous section, filaments are bound to each other and not to walls. Their espin environment thus depends on both their altitude and on the state of their neighbors. In the next subsection, we study how this modifies the growth transition. Then, in the following subsection, Multifilament Stereocilium Profiles, we compare the stereocilium shapes obtained from numerical simulations to those derived from a one-filament calculation.

Couplings modify the growth transition

To investigate whether multifilament bundles have a growth transition, we simulate several 8 × 8 periodic filament bundles for various value of the espin reattachment rate k.

We first focus on the values of k where stationary stereocilium profiles exist and monitor the average filament length, as shown in Fig. 4 a. At k = k_c = 0.02, the average filament length diverges, showing that coupled filaments do undergo a growth transition. This value of k_c matches the threshold of Eq. 20 if n is set to n^eff = 2.5. This effective n can approximately be viewed as the average number of neighbors available for each filament to cross-link at each given instant, i.e., the number of neighbors as long as or longer than the filament. We give an argument for its numerical value in Sec. S4.2 in the Supporting Material. In Fig. 4 a, we fit a power law to the divergence of the stereocilium length and show that

$$\langle \ell \rangle \underset{k\to k_c^{-}}{\propto }\frac{1}{|k-k_c{|}^{0.33}},$$ (21)

which is an anomalous divergence compared to the case of Eq. 19. This is likely to be related to the build-up of long-ranged correlations across the actin bundle, as discussed in Sec. S4.3 in the Supporting Material.

Figure 4.

Growth transition for multifilament bundles with reattachment. (a) Average length as a function of k below the transition and comparison with the average length calculated from Eq. 17 for n = 2.5 (line). (Open circles) 8 × 8 periodic arrays. (Crosses) 16 × 16 periodic arrays, showing that the lengths do not depend much on the simulation size. Error bars represent the root-mean-square height fluctuations in the steady state. (Inset) Log-log representation of the same 〈ℓ〉 data from 8 × 8 arrays as a function of the distance k_c – k to the growth transition threshold. (Solid line) Prediction from Eq. 17 as in the main figure. (Dotted line) Power law fit as in Eq. 21. (b) Growth velocity of the bundle as a function of k above the growth transition for 8 × 8 periodic arrays (open circles) and comparison to the generalization of the one-filament theory given in Eqs. 22 and 23 for n = 2.5 (line).

For values of k above the growth transition, the stereocilium grows indefinitely and at constant velocity. In Fig. 4 b, we plot the stereocilium's growth velocity as a function of k. As k is reduced, the pointed ends depolymerize faster and faster and catch up to the polymerization front for k_c = 0.02, which is consistent with the threshold determined in Fig. 4 a. At steady state, the stereocilium lengthening velocity is the difference between its polymerization velocity and its depolymerization velocity far from the polymerization front:

$$\frac{\text{d}\ell }{\text{d}t}=v-v_d.$$ (22)

This growth velocity vanishes at the growth transition. While v is imposed in our simulations, v_d depends on k and n. We now discuss our theoretical predictions for this dependence. Far away from the polymerization front, the probability for an espin to be on is $k/(1+k)$. In the cases considered here, k ≪ 1, meaning that espins are scarce far from the polymerization front: the probability for a given pointed end to be bound to more than one cross-linker is negligible. Thus, the interesting part of the filament is bound to cross-linkers with an average density ρ, and is very unlikely to be bound to more than one cross-linker at any given altitude. The discussion at the end of Discrete Master Equation and Solution Far From the Polymerization Front thus applies, although the density of the cross-linkers in the case considered here is n times larger, because there are n walls instead of one. To lowest order in k, this yields

$$v_d=\frac{1}{nk}.$$ (23)

This single-filament result is compared to the multifilament simulations in Fig. 4 b using n = n^eff, and the two are found to be in very good agreement. Note that we expect the function v_d(k) to diverge as k goes to 0, but to be a smooth function of k for k > 0. In particular, v_d(k) has no reason to have a singularity in k = k_c: indeed, k_c is defined by v_d(k_c) = v, and v_d does not depend on v. Thus, k = k_c is a generic point of the function v_d(k). Therefore, at the transition, the following generic crossing scenario applies, whether or not the filaments are coupled:

$$\frac{\text{d}\ell }{\text{d}t}\left(k\right)\underset{k\to k_c^{+}}{\propto }\left(k-k_c\right).$$ (24)

Multifilament stereocilium profiles

We now return to the question of the shape of stereocilia. In Fig. 5, we compare the shapes obtained from the simulations with theoretical expectations from the single-filament theory. For each value of k, the theoretical curve Eq. 17 is plotted using the effective number of neighbors n^eff = 2.5 from the previous subsection. As k is increased, the description of the bundle by the single-filament theory becomes worse and worse, as expected from Fig. 4 a.

Another theoretical result our simulations should be compared with is Eq. 17 using the actual number of neighbors n = 4. Note, however, that this is only possible for k smaller than 0.0125, which is the growth transition threshold for n = 4. Consistent with this, we plot the n = 4 theoretical curve only in Fig. 5 a, where k = 0. Excellent agreement with the numerical simulations is found. This is expected, because when espins are not allowed to reattach, Eq. 17 is identical to Eq. 5—which is the exact solution of the multifilament problem for k = 0.

In Fig. 5 b, we illustrate the dependence of the bundle shape on the number of filaments included in the simulations. No change in the shape is observed when multiplying the number of filaments by four, but the amplitude of the fluctuations is reduced. This suggests that in this regime at least, the average profile given by our 8 × 8 simulations is a good assessment of the infinite bundle limit.

In Fig. 5 c, we illustrate the dependence of the bundle shape on the boundary conditions of the bundle. It is found that a circular bundle (see illustration in Fig. S3 a in Supporting Material) is markedly shorter than a bundle with periodic boundary conditions. This is because the filaments close to the rim of the circular bundle tend to depolymerize faster, due to the fact that they have fewer neighbors. In the parameter regime presented here, this is sufficient to reduce the average length of the bundle significantly. This effect becomes negligible for small k and for large bundle radii, i.e., if the filaments are correlated over a length much shorter than the radius of the bundle.

Finally, in Fig. 5 d we note that as the growth transition is approached, the amplitude of the bundle's fluctuations increases dramatically. Indeed, as the depolymerization velocity becomes very close to the polymerization velocity, the filaments are more and more loosely confined to a finite length. Similarly to what happens, e.g., for a Brownian particle in a harmonic potential, a looser confinement leads to fluctuations of a larger amplitude.

Discussion

In this article we present a simple physical model for the morphogenesis of stereocilia, whose very well-regulated shapes are crucial for the frequency selectivity of hearing in a wide range of animals. Our model is to be understood in the framework of Prost et al. (8), where the shape of stereocilia is attributed to an “internal clock” of the actin bundle. Here we propose that the internal clock is provided by the stochastic attachment-detachment dynamics of the well-characterized protein espin, or some other actin cross-linker.

Although the emphasis of this article is on stereocilia, the simplicity of our model makes it general enough to describe several other biological length-regulation processes (28). The most obvious of these are of course other cellular protrusions, such as filopodia, microvilli, and Drosophila bristles, where actin filaments are also coupled by cross-linkers. More specifically, in filopodia the ratio of the actin treadmilling velocity (29) and detachment rate of the cross-linker fascin (30) is ≈(1 μm/min)/(0.12 s⁻¹) ≈ 1 μm, which is commensurate with the length of this type of protrusion. This suggests that the mechanism described here could be relevant in filopodia. In addition, the study presented in Single Filament with Reattachment is relevant to single-filament problems where each monomer stochastically switches between two states, such as the phosphorylation-dependent depolymerization of a single actin filament (16) or microtubule (31), or association with proteins making the filament more susceptible to depolymerization (32).

Stereocilia models have been previously proposed in the literature that yield good agreement with electron micrographs of stereocilia. This article is based on Prost et al. (8), which analyzes the forces at play in stereocilium treadmilling and the interaction of the actin bundle with the membrane and the cytoplasm. Here we improve on this work by proposing a refined description of the dynamics of the actin bundle itself based on the role of cross-linkers. This leads to improvements in three directions, which evidences the importance of espin in shaping stereocilia:

First, the model of Prost et al. (8) depends partly on a hypothetical actin pointed end-capping protein, whereas we only assume well-identified proteins. Note that Prost et al. (8) suggests that espin could be described as such a capping protein. We show here, however, that there are important differences:

• 1.Our model cross-links interact with the actin all along the filament, whereas capping proteins only bind to its end.
• 2.The probability for the pointed end to be cross-linked depends on the filament length.
• 3.Capping proteins introduce no interfilament correlations.

Second, the stereocilia shapes calculated in Prost et al. (8) resemble those of deaf Shaker 2J mutants, while we account for those of healthy animals.

Third, the shapes of Prost et al. (8) are highly sensitive on the fine tuning of actin's polymerization and depolymerization rates, which is not compatible with the biological robustness of the well-controlled stereocilia shapes.

Another quite different model is proposed in Naoz et al. (7). It is based on the fact that actin-associated proteins could be actively localized at the stereocilium base, e.g., by molecular motors. For instance, this work suggests that actin-severing proteins localized at its base could drive the narrowing of the actin bundle there. This model offers an interesting insight into the possible roles of the experimentally observed active transport within the stereocilium. It is, however, difficult to assess its validity quantitatively, because it hypothesizes several experimentally uncharacterized protein-protein interactions, and has an accordingly large number of adjustable parameters. Note also that this model does not address the issue of the stereocilium height regulation.

The model presented in this article is in agreement with several experiments showing the importance of espin in stereocilium length regulation. We predict that the actin bundle can only reach a stationary profile if the attachment rate of espin to actin is much smaller then its detachment rate (k ≈ v⁻¹ ≪ 1). This is consistent with the observation made in Rzadzinska et al. (5) that espin in the stereocilium seems to treadmill along with actin—in other words, that espin is essentially incorporated at the tip of the stereocilium and not so much exchanged with the solution in the bulk of the actin bundle. More quantitatively, we are able to reproduce the shape of several stereocilia within the same hair bundle with only one adjustable parameter. We also account for the apparent proportionality between stereocilium length and turnover time, as well as for espin 1 localization at the stereocilium tip. Finally, our approach faithfully captures the quantitatively measured relationship between microvillus length and espin expression. A possible extension of our model as applied to stereocilia would be to consider that the espin detachment dynamics might be different in the bulk of the actin bundle and at its lateral surface. For instance, in the presence of preferential espin detachment at the surface, actin filament termination would happen more rarely in the bulk. This could account for the fact that filament bending as pictured in Fig. S1 in the Supporting Material is not clearly observed in electron micrographs.

On a broader level, the dynamics of the cytoskeleton involves many out-of-equilibrium surface growth processes. In addition to actin bundle-based protrusions, one could quote the dynamics of the cell cortex, which undergoes polymerization and depolymerization as well as transient cross-linking, similarly to the system studied here. In addition, its dynamics involves actin filament branching and barbed end capping, as well as molecular motors binding which makes it contractile. Another similarly complicated system is the lamellipodium, a thin sheet of actin that some cell types (e.g., keratocytes) extend in front of them while moving.

The precise interplay between all the sources of activity in these processes is not well understood. The more formal aspects of our study of the novel, nontrivial growth model introduced here reveal interesting directions to pursue in order to characterize those processes. Indeed, in the sections Single Filament with Reattachment and Coupling between Filaments, we discuss what we expect to be two very robust features that might be universal across a large range of cross-linking-limited disassembly models: the growth transition and an anomalous length divergence exponent in the presence of local interactions between filaments. By identifying those features and recognizing them in actual cellular systems, one might be able to use them as signatures of the underlying interface-shaping phenomena, and therefore show which mechanism dominates which type of interface.