A Mechanistic View of Collective Filament Motion in Active Nematic Networks

Abstract

Protein filament networks are structures crucial for force generation and cell shape. A central open question is how collective filament dynamics emerges from interactions between individual network constituents. To address this question, we study a minimal but generic model for a nematic network in which filament sliding is driven by the action of motor proteins. Our theoretical analysis shows how the interplay between viscous drag on filaments and motor-induced forces governs force propagation through such interconnected filament networks. We find that the ratio between these antagonistic forces establishes the range of filament interaction, which determines how the local filament velocity depends on the polarity of the surrounding network. This force-propagation mechanism implies that the polarity-independent sliding observed in Xenopus egg extracts and in vitro experiments with purified components is a consequence of a large force-propagation length. We suggest how our predictions can be tested by tangible in vitro experiments whose feasibility is assessed with the help of simulations and an accompanying theoretical analysis.

Significance

Cells perform a variety of vital tasks ranging from cell division to motion and force generation. These abilities are intrinsically dynamic and rely on active network structures consisting of cytoskeletal filaments and cross-linking motor proteins. How does collective dynamics at the macroscopic level emerge from interactions of individual filaments and motor proteins? We address this open question through a conceptual model for motor-induced motion in networks of interconnected filaments. A prominent representative of this class of structures is the mitotic spindle, in which motor-driven filament flux is essential to maintain shape and functionality. Through theoretical and numerical analysis, we identify a mechanism that qualitatively accounts for experimental observations of both the spindle and of systems with purified components.

Introduction

Living cells have the remarkable ability to actively change their shape and to generate forces and motion. A key component enabling cells to exhibit these stunning mechanical properties is the cytoskeleton. This structure is built out of various proteins and forms diverse functional networks consisting of polymer filaments such as actin and microtubules, motor proteins, and associated proteins (1,2). The motor proteins expend chemical energy to generate forces that act on the cytoskeletal filaments (3, 4, 5). In particular, motors that have two binding domains, e.g., kinesin-5, can walk along two filaments at once, causing filaments of opposite polarity to slide past one another (6).

To understand the nonequilibrium physics underlying the dynamics of motor-filament systems, it has proven fruitful to study reconstituted systems of purified components in vitro (7, 8, 9, 10). Despite their reduced complexity, these systems still self-organize into intricate patterns and structures reminiscent of those found in living cells. But how is their collective behavior at the macroscopic level linked to the interactions between individual filaments and motors? What are the underlying mechanisms? To provide an answer, we focus here on a generic class of systems in which filaments exhibit nematic order and motors drive relative sliding of filaments. A prominent representative of this class is the poleward flux of microtubules in Xenopus mitotic spindles (11, 12, 13). This process has been attributed to antiparallel, motor-driven interactions between filaments, especially if the motor protein dynein is inhibited (11,14,15). A quite puzzling observation made in these systems was the correlation—or rather, the lack of correlation—between filament speed and network polarity, i.e., the ratio of parallel/antiparallel filaments. Although filament motion is induced by sliding antiparallel filaments past each other, polarity was observed to have barely any influence on the filament speed (14,16,17). This surprising behavior was recently replicated in a system of purified components composed of the kinesin-14 XCTK2 and microtubules and interpreted in terms of a hydrodynamic theory for heavily cross-linked filament networks (18). These observations are at variance with previous predictions for dilute filament networks, in which filament motion depends linearly on the local polarity (19, 20, 21). How can these conflicting results be reconciled? What are the biophysical mechanisms determining the relation between filament speed and network polarity?

To gain insight into these important questions, we study a minimal but generic model consisting of nematically ordered cytoskeletal filaments (such as microtubules) and molecular motors (such as kinesin-5) that are capable of cross-linking and sliding antiparallel filaments apart. Our mathematical analysis of this theoretical model shows that the interplay between motor-induced forces and viscous drag acting on the filaments determines the relation between filament velocity and the polarity of filaments. Depending on the relative strengths of these forces, we find that the velocity-polarity relation varies continuously between a local and a global law. Our theory reveals the mechanism that underlies this relation between filament velocity and network polarity: for high motor-induced forces and small fluid drag, local forces on the filaments propagate through the strongly interconnected network without dissipation and thereby influence the overall network dynamics. In contrast, for small motor-induced forces or high fluid drag, local forces are quickly damped and only influence the local dynamics. This mechanism provides a deeper understanding of the link between collective filament dynamics and molecular interactions. Moreover, it reconciles previously conflicting results for the velocity-polarity relation in the limit of dilute (19,20) and heavily cross-linked systems (14,18,22). Strikingly, our theoretical analysis shows that the insensitivity of filament velocities to changes in the network polarity, which was reported for the spindle (14,16) and in vitro systems (18), occurs in a biologically relevant parameter range. In addition, our theory predicts how the ratio between the spectrum of measured polarities and filament speeds depends on the ratio of drag/motor-induced forces in the system. We suggest an in vitro experiment to validate those predictions. The feasibility of this experiment is assessed with the help of computer simulations and an accompanying theory.

Methods

Biophysical agent-based model of motor-induced filament movement

We are interested in understanding how the interplay between viscous drag and molecular forces between cytoskeletal filaments, mediated by molecular motors, drives the internal dynamics of filament networks. Specifically, we focus on reconstituted in vitro systems consisting of microtubules and motors capable of cross-linking neighboring filaments and sliding them apart; see Fig. 1, A and B. Such motor proteins can walk on both filaments simultaneously, so that the forces generated between filaments depend on their relative orientation (Fig. 1 A). In vitro, such microtubule-motor mixtures were observed to self-organize into a nematic network, in which neighboring filaments may be disposed approximately parallel or antiparallel (18).

Figure 1.

Biophysical model for motor-driven filament motion. (A) Microscopic, motor-mediated interactions between microtubules are shown. Neighboring microtubules are connected by motors (red) that walk toward the microtubule’s (green) plus end with velocity V_m. A motor exerts zero force if filament motion is such that the motor is not stretched. (A, left) A motor connecting two parallel microtubules counteracts relative motion between the filaments. (A, right) In contrast, two antiparallel microtubules connected by a motor are slid apart. The force falls to zero once their relative velocity equals twice the motor velocity (−2V_m). (B) A sketch of a microtubule-motor mixture in a nematically aligned state is shown. The springs denote motors cross-linking neighboring filaments. The highlighted region includes all interactions of the center microtubule. (C) The one-dimensional model system is shown. Possible interaction partners of the microtubule in the center (dark blue) are in the highlighted region. To account for the reduced number of interaction partners in the experimental filament network, we draw on average N out of all possible partners to interact (interaction partners are highlighted in color, parallel interaction partners in light blue, and antiparallel interaction partners in light red). To see this figure in color, go online.

Motivated by the nematic order of these filament networks, we set up a biophysical agent-based model, which is effectively one-dimensional. We consider a system of size S in which the filaments (microtubules) are assumed to be rigid polar rods of fixed length L, oriented with their plus end either to the left (+) or right (−); see Fig. 1 C. Hence, the dynamics of each polar filament i is determined (solely) by its velocity v_i^(±). Relative motion between filaments is caused by molecular motors that walk on these cross-linked filaments and thereby exert forces. In vitro assays involving pairs of isolated microtubules cross-linked by kinesin-5 motors reveal that 1) kinesin-5 has the ability to walk simultaneously on both microtubules with approximately the zero-load velocity V_m, 2) antiparallel microtubules are pushed apart with a relative velocity of ∼2V_m, and 3) parallel microtubules remain static (6). Integrating this information with experiments showing a linear force-velocity relation for kinesin motors (23, 24, 25, 26), we assume that the forces between two cross-linked parallel (±±) and antiparallel (±$\mp$) filaments per motor are given by

$$F_{ij}^{\left(\pm \pm \right)}=F_{\text{m}}\left(\frac{v_j^{\left(\pm \right)}-v_i^{\left(\pm \right)}}{2V_{\text{m}}}\right),F_{ij}^{\left(+-\right)}=-F_{ji}^{\left(-+\right)}=F_{\text{m}}\left(1+\frac{v_j^{\left(-\right)}-v_i^{\left(+\right)}}{2V_{\text{m}}}\right).$$ (1)

Here, F_ij denotes the force that filament j exerts on filament i, with F_m signifying the motor stall force; because of force balance, F_ij = −F_ij. These forces vanish if the relative motion of the filaments does not induce strain in the cross-linking motors. Although for parallel filaments, this is the case if the filaments move at the same speed, a motor walking on antiparallel filaments is not strained if these slide apart with relative velocity 2V_m, i.e., v_i⁽⁺⁾ − v_j⁽⁻⁾ = 2V_m. On the other hand, the maximal force between two filaments corresponds to the stall force, F_m, which is defined as the force between two antiparallel filaments fixed at their relative position (v_i⁽⁻⁾ = v_j⁽⁺⁾). In that case, the motor heads move apart until the motor stalls and exerts its maximal force on the filaments. An analogous situation occurs if a motor is attached to two parallel filaments that move with a relative speed v_j^(±) − v_i^(±) = 2V_m. So, the corresponding force is also F_m.

The velocity v_i^(±) of a specific microtubule i in the network is determined by the force balance equation

$$\gamma v_i^{\left(\pm \right)}=\sum _j{n}_{ij}{F}_{ij}^{(\pm \pm )}+\sum _k{n}_{ik}{F}_{ik}^{(\pm \mp )},$$ (2)

where γ denotes the fluid drag coefficient and n_ij the number of motors cross-linking microtubule i and j. The sums run over all parallel and antiparallel interaction partners of microtubule i, respectively. In general, the number of interaction partners, as well as the strength of their interaction, can depend on a variety of factors. For example, the interactions are influenced by the density of motors in the cytosolic volume, as well as along the filament, or the local structure of the filament network. Inclusion of all these factors would lead to a microscopic description with many unknown parameters. Focusing on the mechanistic basis of filament motion here, we make the following assumptions (see Fig. 1 C): first, we consider a homogeneous motor density in the cytosolic volume and along the microtubules. Thus, we describe motors effectively by a constant density with, on average, N_m motors per filament. Second, we assume that all filaments have, on average, N interaction partners that are drawn randomly. This, on average, accounts for the limited number of neighbors in the three-dimensional network structure. Finally, note that we neglect hydrodynamic interactions between the filaments. A priori, it is not clear whether such interactions would not change the dynamics. However, in a recent study on a motor-filament system, it turned out that the experimental results are well described by a theory neglecting hydrodynamic interactions (27). Presumably, this is due to “hydrodynamic screening” in dense systems (28).

Results

A local mean-field approximation predicts strong velocity-polarity sensitivity

To gain initial insight into the dynamics of microtubules, we simplify the system even further using a local, continuum mean-field approximation that neglects any lateral displacement between cross-linked filaments. In the continuum description, each microtubule i is identified by its midpoint position x_i. As a crude simplification, we assume that all cross-linked, equally oriented filaments passing through position x move at (roughly) the same velocity v^(±)(x). This entails that the forces between all parallel filaments, F^(±±)(x), vanish. Denoting the fraction of filaments at position x oriented in the (±) direction by φ^(±)(x), Eq. 2 then simplifies to γv⁽⁺⁾(x) = N_mφ⁽⁻⁾(x)F⁽⁺⁻⁾(x) and γv⁽⁻⁾(x) = N_mφ⁽⁺⁾(x)F⁽⁻⁺⁾(x), with N_m denoting the number of motors per filament as above. Inserting the force-velocity equation, Eq. 1, and solving for the velocity yields v⁽⁺⁾(x) ∝ 1 − P(x) and v⁽⁻⁾(x) ∝ 1 + P(x), where we defined P(x) = φ⁽⁺⁾(x) − φ⁽⁻⁾(x) as the local network polarity at position x. Hence, the central result of this local mean-field analysis, which will ultimately turn out to be oversimplified, is a linear dependence of the local velocities on the local polarity. This result corresponds to the intuition that forces between filaments—and their relative motions—strongly depend on their relative orientation. In particular, whereas antiparallel interactions between two filaments introduce motion of both filaments, parallel filaments remain static. As a consequence, filaments with a higher number of antiparallel interactions are expected to exhibit an enhanced speed. However, as we will see next, this intuition is in conflict with numerical simulations (in the biologically relevant parameter regime) as well as with experimental findings for heavily cross-linked filament gels (18).

The agent-based model can describe the weak velocity-polarity sensitivity

To test whether our model is capable of describing the observations in heavily cross-linked filament networks, we solved the full set of coupled linear equations (Eq. 2) for a one-dimensional network numerically. To compare our results to experimental data, we assessed the model parameters as follows: first, we determined the mean number of interaction partners per filament. The typical maximal distance between two microtubules connected by a sliding motor is estimated to be on the order of the tail length of kinesin-5, ∼0.1 μm (5), plus two times the microtubule radius, ∼0.024 μm (29). Together with the typical microtubule length, estimated to be ∼6–7 μm, these values yield an interaction volume of approximately 1/3 μm³. Fürthauer and collaborators argue that the number density of filaments in their experimental setup is approximately 17/μm³ (18). So, all in all, we estimate that there are N ≈ 5.5 interaction partners per filament. In an analogous manner, we assessed the number of microtubules in our one-dimensional representation of the experimental chamber of length 400 μm to be ∼400. Those filaments are placed randomly as described below (In Silico Study: Random Polarity Field) and experience a drag coefficient of γ = 0.5 pN s/μm (30, 31, 32). As motor parameters, we use V_m = 20 nm/s (6), F_m ∼ 1 pN (5), and N_m = 25 as the average number of motors per filament (18).

Using these parameters, we performed numerical simulations and found good agreement with experimental results (compare Fig. 2 and Fig. 2 in (18)). In particular, the average filament speed (solid black circles in Fig. 2) is found to be independent of the local polarity. This clearly contradicts the local mean-field theory as discussed above (see A Local Mean-Field Approximation Predicts Strong Velocity-Polarity Sensitivity). To assess why this simplified local view is misleading, we next give a comprehensive mathematical analysis of the full agent-based model.

Figure 2.

Local microtubule speed versus local polarity obtained by numerically solving the full set of coupled linear equations (Eq. 2) for a one-dimensional microtubule network. The microtubule network is generated as described in In Silico Study: Random Polarity Field. Gray dots represent individual measurements, and black dots show the average speed binned for local polarities (bin size ΔP = 0.1). In contrast to the oversimplified discussion (dashed-dotted lines), the velocity does not depend linearly on the local polarity. Instead, the average speed is mostly independent of the local polarity. Note that the vertical stripes are artifacts arising from the discrete nature of the agent-based simulation: because of the finite number of filaments in an interval [x, x + Δx], the polarity can only take on discrete values.

Nonlocal continuum theory

It is evident that in the simplified local mean-field analysis discussed above, we neglected the finite extension of filaments. Actually, two filaments that pass through the same location do not necessarily have the same midpoint position. Although they share some overlap, they will interact with different neighbors at different positions. If all filaments have the same length L, a filament with midpoint at position x can interact with filaments whose midpoints lie in the interval [x − L, x + L] (see Fig. 1 C). In this way, the velocities of filaments located at different spatial positions are coupled, leading to nonlocal correlation effects that could explain the weak dependence of filament speed on local polarity.

Motivated by this heuristic argument, we set out to formulate a continuum theory that quantifies the nonlocal coupling between the filament velocities (v^±(x)) and densities (ρ^(±)(x)). To this end, we rewrote the local balance equation, Eq. 2, assuming a continuum limit:

$$\gamma v^{\left(\pm \right)}\left(x\right)=\frac{1}{L}{\int }_{x-L}^{x+L}\text{d}y\left\{f_{\text{parallel}}^{\left(\pm \pm \right)}\left(x,y\right)+f_{\text{antiparallel}}^{\left(\pm \mp \right)}\left(x,y\right)\right\},$$ (3)

where the local forces are given by

$$f_{\text{parallel}}^{\left(\pm \pm \right)}\left(x,y\right)={\hat{N}}_{\text{m}}\left(x,y\right)\times N{\varphi }^{\left(\pm \right)}\left(y\right)\times F^{\left(\pm \pm \right)}\left(x,y\right),$$ (4a)

$$f_{\text{antiparallel}}^{\left(\pm \mp \right)}\left(x,y\right)={\hat{N}}_{\text{m}}\left(x,y\right)\times N{\varphi }^{\left(\mp \right)}\left(y\right)\times F^{\left(\pm \mp \right)}\left(x,y\right).$$ (4b)

Here, the force a motor exerts on the filaments it cross-links is simply given by the continuum version of Eq. 1, e.g., F^(±±)(x, y) = F_m[v^(±)(y) − v^(±)(x)]/(2V_m). The second factor in (4a), (4b) accounts for the expected number of interaction partners at position y, given by the number fraction of filaments with the respective polarity φ^(±)(y) multiplied by the average number N of interaction partners: Nφ^(±)(y). For this functional form to apply, we implicitly assumed that the filament network is not sparse, i.e., that there is always a sufficient number of interaction partners, namely more than N, available. The number fraction can be written in terms of the filament densities as φ^(±) = ρ^(±)/(ρ⁽⁺⁾ + ρ⁽⁻⁾). The first factor in (4a), (4b), ${\hat{N}}_{\text{m}}$(x, y), specifies the average number of motor proteins mediating the interaction between a pair of filaments located at positions x and y. This number is determined by the size of the overlapping region, L_ov = L − |x − y|, and the number of motors per filament, N_m. Because all the available motors on a filament have to be shared among all of its N interaction partners, only N_m/N are available for the interaction with any specific filament. Hence, assuming a uniform motor distribution along each microtubule, the effective number of motors cross-linking a filament pair is on average given by ${\hat{N}}_{\text{m}}$(x, y) = N_m/N × L_ov/L.

Based on this nonlocal continuum representation of our agent-based model, we seek a quantitative understanding of how the opposing forces in the filament network give rise to collective (uniform) motion. Ultimately, our goal is to provide an explicit expression relating the polarity and velocity fields.

Analytic solution for motor-induced filament movement

In this section, we present an analytic solution to our nonlocal continuum description (Eq. 3). We restrict our analysis to the limit in which the system size is large compared to the filament length L and to all other intrinsic length scales of the system we might encounter in the course of the mathematical analysis. Making use of complex calculus, in this limit it is possible to find an explicit expression for the velocity field v^(±)(x) in terms of the polarity field P(x). This expression thus constitutes a velocity-polarity relation that quantifies how the polarity field affects the velocities.

In an experimentally reasonable parameter regime, one finds an approximate expression that reads (for a detailed analysis, see Supporting Materials and Methods)

$$v^{\left(\pm \right)}\left(x\right)=\pm V_{\text{m}}\left(1-\alpha \right)\left(1\mp \text{Π}\left(x\right)\right),$$ (5a)

$$\text{Π}\left(x\right)=\frac{1}{2l_c}{\int }_{-\infty }^{\infty }\text{d}y{\text{e}}^{-\left|x-y\right|/l_c}P\left(y\right),$$ (5b)

where 1/α := 1 + 12(l_c/L)²; for biologically plausible parameter values, one has $\alpha \ll 1$. Importantly, Eq. 5a shows that the motion of filaments is neither solely dependent on the local polarity nor fully independent of the polarity field. Instead, the local filament velocities, v^(±)(x), now depend in a nonlocal way on the polarity, P(y), as specified by the convolution integral (weighted average), Π(x), with an exponential kernel (weight) $\sim {\text{e}}^{-\left|x-y\right|/l_c}$. To emphasize this nonlocal dependence of the velocities on the polarity, we refer to Π(x) as the ambient polarity in the following. The characteristic interaction range l_c, over which the polarity field is averaged, is given by

$$l_c=L\sqrt{\frac{F_{\text{m}}{N}_{\text{m}}}{24\gamma V_{\text{m}}}}.$$ (6)

It is set by the ratio of the total force exerted by motors between microtubules, F_mN_m, to the drag imposed on the microtubule by the surrounding fluid, γV_m. Furthermore, it can be interpreted as the length scale over which motion generated by antiparallel filament sliding is propagated by parallel and antiparallel filament interactions through the network. As a result, the interaction range l_c reflects the antagonism between motion-propagating forces (parallel and antiparallel interactions) and the attenuation of force propagation in the filament network mediated by viscous drag. (An insightful intuitive explanation for the characteristic length l_c can be found in “Extension of the analysis to systems with several different types of cross-linking motors” in the Supporting Materials and Methods. It relies on an analysis of systems with several types of motors that proposes a clear separation of the forces acting on microtubules: active forces that determine the magnitude of the filament speed and passive friction forces between microtubules that determine the force propagation in the network.) This antagonism is captured by the spatial average of the polarity field, which effectively corresponds to a low-pass filter. Because of averaging over local polarities, high-frequency fluctuations in the spatial polarity profile are filtered out and hence do not contribute to the velocity. Explicitly, by Fourier transforming Eq. 5b, we find a Lorentzian Fourier weight

$${\Pi }_k=P_k\frac{1}{1+{\left(k{l}_c\right)}^2},$$ (7)

where k denotes the wave number. Hence, the characteristic frequency of the low-pass filter is proportional to the reciprocal of the characteristic length, 1/l_c, implying that the larger the l_c, the stronger the filter and the less relevant local fluctuations in the polarity. To put it another way, the speed of a filament at position x depends only on the local “view” of the polarity field within a range defined by l_c (Fig. 3).

Figure 3.

Typical polarity field, P(x), and two choices of interaction kernel, exp(−|x − y|/l_c), characterizing global and local polarity dependence, respectively. Filaments positioned in a range of l_c around x contribute to the motion of microtubules at x. Depending on the ratio of average motor force exerted on a microtubule/attenuation (drag of microtubules in the fluid), the characteristic propagation length l_c takes different values. (A) For large l_c and polarity fields that vary randomly on length scales smaller than l_c, this averaging yields a roughly constant ambient polarity profile, Π(x), and hence a roughly constant velocity profile. On a microscopic level, this corresponds to a heavily cross-linked filament network (inset). (B) In the limit of small l_c, only the local environment, i.e., the direct interaction partners, has an influence on the microtubule motion. The ambient polarity field (velocity field) varies as the polarity field varies. To see this figure in color, go online.

To gain an impression of how the interplay between the different forces in the network leads to the nonlocal effects, it is helpful to consider the limiting cases of large and small l_c, respectively. For large l_c, motor forces dominate viscous drag (F_mN_m $\gg$ γV_m). Then, because of either weak dissipation or strong motor-mediated filament coupling, parallel and antiparallel cross-linked filaments translate the motion, generated by interactions between antiparallel filaments, over long distances (∼l_c). As a result, motion generated at one position in the network propagates through the entire network. In the asymptotic limit l_c → ∞, the velocity-polarity relation (Eq. 5a) reduces to v^(±) = ±V_m (for zero overall polarity), confirming recently published findings for a heavily cross-linked network (18). In contrast, for small l_c (F_mN_m $\ll$ γV_m) force generated at a certain position in the network has only a local effect. Forces generated by antiparallel interactions cannot propagate through the network because of either strong dissipation or a lack of filament interactions. In this limit, the velocity-polarity relation reduces to the result obtained with the local mean-field theory discussed in A Local Mean-Field Approximation Predicts Strong Velocity-Polarity Sensitivity. This relation agrees with the velocity-polarity relation found for dilute filament networks in which only local bundles of filaments are considered (20,21). (Care has to be taken when comparing our result to the dilute limit: we restricted our discussion to the case of a sufficient number of interaction partners, whereas for dilute systems, disconnected patches of filaments are usually considered. For a more comprehensive discussion on how our results are related to results for dilute systems, we refer the reader to the Supporting Materials and Methods.)

Interpretation of the velocity-polarity relation

With regard to previous results, our considerations offer a solution to the seemingly contradictory behavior of dilute and heavily cross-linked networks. More specifically, our results identify a common mechanism for collective filament dynamics: because of the finite extension of the microtubules, one microtubule can be cross-linked with several others whose center positions are spread over a region up to twice the microtubule length (Fig. 1 C). As a result, although microtubules at different positions might in fact not be directly linked by a motor, an interaction between them can be mediated by successive cross-links through a chain of microtubules. In this way, the velocity of microtubules at one position influences the velocity at a different position and information on the local polarities propagates through the system. How far this information propagates (l_c) depends on how “effectively” movement at one position is translated into movement at a different position. The greater the efficiency, the smaller the ratio between the passive drag on microtubules in the fluid (and thus the attenuation) and the average maximal force exerted on one microtubule by all motors linking it to other microtubules.

Taken together, our results shed light on the question of what determines the local speed of microtubules in a nematic network: generally, it is neither the local polarity, P(x), that determines the velocity of microtubules at a certain position nor the overall polarity in the system, P_glob. Instead, the ambient polarity, Π(x), is informative. The ambient polarity corresponds to an average of the polarity with a weight that decays exponentially with the distance from the position of interest (see Eq. 5b). The characteristic decay length, l_c, is proportional to the filament length L and increases with the ratio of the motor force on a microtubule, F_mN_m, to the fluid drag, γV_m. In general, for a finite decay length and a spatially varying polarity profile, the ambient polarity also varies in space. As can be inferred from Eq. 5b, for larger values of l_c, a larger region of space contributes to the ambient polarity (see also Fig. 3). Accordingly, the ambient polarity then corresponds to an average of the local polarity over more positions. As a result, for a fixed spatial polarity profile, the range of values of the ambient polarity decreases with increasing characteristic propagation length l_c. Because of the linear relationship between the velocities and the ambient polarity, Eq. 5a, the same holds true for the range of velocities.

In the following, we illustrate these predictions with the help of two examples. First, we consider a spatially linear polarity profile. Besides being an instructive case, this polarity profile is of biological relevance. It resembles the measured, approximately linear polarity profiles in the mitotic spindle (see Discussion). As a complement, the setup of the second example is designed to mimic typical in vitro experiments. To make testable predictions, we analyze the suggested (idealized) experiment in detail and focus on quantities that we believe to be accessible in experiments.

A simple example: The linear polarity profile

Our theory predicts that the range of velocities decreases with increasing characteristic propagation length, l_c. To demonstrate this correlation, we consider a linear polarity profile P(x) = a(x − S/2) in a finite interval x ∈ [0, S] (for details, see Supporting Materials and Methods). As motivated above, we describe the local polarity profile in terms of its Fourier coefficients ${\hat{P}}_k$. The wave numbers are now discrete, $k\in \mathbb{N}$, because the system is finite. The Fourier coefficients of the ambient polarity, ${\hat{\Pi }}_k$, are given by the Fourier coefficient of the local polarity, ${\hat{P}}_k$, times a k-dependent weighting factor: ${\hat{\Pi }}_k={\hat{P}}_k$/[1 + (2πkl_c/S)²] (see Supporting Materials and Methods). Correspondingly, the ratio between the range of the local polarity 2P_max = aS and the range of the ambient polarity 2Π_max can be approximated as (see Supporting Materials and Methods)

$$\frac{{\text{Π}}_{\text{max}}}{P_{\text{max}}}\approx \frac{1}{1+{\left(\pi l_c/S\right)}^2}.$$ (8)

This finding confirms the intuitive expectation that with increasing characteristic length l_c, the ambient polarity range 2Π_max (or analogously, the velocity range 2Π_maxV_m(1 − α)) should decrease relative to the local polarity range, 2P_max, and the spatial profile gets “squeezed” (Fig. 4). From the approximate expression, Eq. 8, we infer that for characteristic lengths of the same order as the system size, l_c/S ∼ $\mathcal{O}$(1), the range of the ambient polarity 2Π_max is only a tenth of the range of the local polarity 2P_max. Because of the linear relationship between the velocity and ambient polarity, Eq. 5a, this small range of ambient polarities implies that also the velocity range for equally oriented microtubules is small. As a result, for l_c ≥ S, all equally oriented microtubules move as a collective with approximately uniform velocity. In particular, there is also movement in regions where locally the polarity is P(x) = ±1, corresponding to stretches populated only by parallel microtubules.

Figure 4.

“Squeezing” of the ambient polarity in a finite system with reflecting boundary conditions. (A) A sketch of the linear spatial polarity profile, P(x) = a(x − S/2), x ∈ [0, S], together with the ambient polarity profile, Π(x, l_c), is shown for two different values of the characteristic length l_c/S (normalized by the system size S). The solid and dashed lines indicate the solutions relevant for (+) and (−) filaments, respectively. For larger l_c, the range of the ambient polarity, 2Π_max, becomes more restricted. (B) Ratio between the range of the ambient polarity 2Π_max and the range of the local polarity 2P_max = aS plotted against l_c/S is given. The curve is well approximated by a Lorentzian decay 1/(1 + (πl_c/S)²) (estimate). For the exact expression, please refer to the Supporting Materials and Methods. For larger l_c/S, the range of the ambient polarity relative to that of the local polarity falls off rapidly. To see this figure in color, go online.

For in vitro experiments with filament gels or reconstituted systems, it might not be feasible to get information on the entire spatial polarity and velocity fields. Instead, in typical experiments, the local polarity and velocity are recorded only at single points in the filament gel (17,18,33). Data obtained in this way are similar to those shown in Fig. 2, in which one data point corresponds to a polarity-velocity pair measured at one location in the gel. In the next section, we thus perform an in silico study in which we make single velocity and polarity measurements only and do not measure the entire spatial field. Nevertheless, the key idea motivating the setup of the in silico study is the expectation that the spectrum of measured velocities is squeezed compared to the spectrum of local polarities: because of the filtering of short-wavelength modes, extreme values of the local polarity are averaged out, and the velocity profile is smoother than the local polarity profile. In the following, we thus focus on deriving a relation between the measured distribution of local polarities and velocities.

In silico study: Random polarity field

The goal of this section is to suggest an experimental setup that should permit the antagonism between the different forces in the system due to drag and motor-mediated interactions to be explored. To this end, we performed an in silico study intended to closely emulate the situation in experiments with in vitro filament gels. Photobleaching experiments have proven to be a feasible option to simultaneously determine sliding velocities and local gel polarity in filament gels (17,18,33). In these experiments, the fluorescently labeled microtubules in the gel are photobleached along a line by laser light. Because of the motion of the filaments in the gel, the bleached line splits into two lines that move to the right (left) and correspond to left-oriented (right-oriented) microtubules, respectively. From the motion of the two lines, the local velocity and the local polarity can be inferred simultaneously: the local velocity of the left-oriented (right-oriented) microtubules is directly obtained from the velocity of the respective line. Furthermore, the local polarity is determined from the ratio of the bleach intensities of the two lines. The data so obtained only contain local information about the velocity and polarity, but no spatially resolved information. To make experimentally testable predictions, our goal is, therefore, to derive a relationship between the distribution of measured local polarities and the distribution of measured velocities for which spatial resolution is not necessary.

Setup of the in silico study

To illustrate how a given polarity distribution affects the velocity distribution, we consider a specific example, namely a polarity “environment” resulting from random filament assemblies; for details, please refer to the Supporting Materials and Methods. We assume that the filament network is nematically ordered and filaments are randomly oriented to the left or to the right, and therefore neglect the possibility that in the experimental system, the spontaneous self-organization into the nematic state might involve some polarity sorting. More specifically, filaments are randomly placed in a chamber of size S $\gg$ L with periodic boundary conditions. Because for random filament assemblies, there is no reason why the average number of left- and right-pointing filaments should differ, we choose the number density for both left- and right-pointing microtubules to be identical: μ⁽⁺⁾ = μ⁽⁻⁾ = μ. Importantly, because of the finite extension of the microtubules, the polarity at different positions is not independent. Instead, one finds a positive covariance for the polarities at distances less than one microtubule length L apart (see Supporting Materials and Methods). As a result, the polarity profile is not completely random, but correlated on lengths smaller than the microtubule length L (for a typical profile, please refer to the Supporting Materials and Methods).

Signature of the ambient polarity in the velocity distribution

Based on our theoretical understanding, we expect that depending on the characteristic length l_c, the distribution of velocities is squeezed compared to the polarity distribution. This is because, depending on the ratio of the antagonistic forces, filament motion arises from averaging the polarity over longer (large l_c) or shorter (small l_c) distances. Because we expect the degree of averaging to be reflected in the distribution of velocities, the standard deviation (SD) of the microtubule velocities should be an interesting quantity to look at in experiments.

To predict the variance of the velocities (ambient polarities) analytically, we describe the local polarity field resulting from the random placement and orientation of filaments in the “experimental” chamber by a set of correlated random variables (see Supporting Materials and Methods). Using their correlation structure, we average the local polarity according to the expression for the ambient polarity (Eq. 5b) and find (see Supporting Materials and Methods)

$$\frac{\text{Var}\left[v/V_m\right]}{\text{Var}\left[P\right]}={\left(1-\alpha \right)}^2\left[1-\frac{3l_c}{2L}\left(1-{\text{e}}^{-L/l_c}\right)+\frac{1}{2}{\text{e}}^{-L/l_c}\right].$$ (9)

Here, Var[P] = Var[P(x)] = $\langle P{\left(x\right)}^2\rangle -{\langle P\left(x\right)\rangle }^2$ denotes the variance of the local polarity and Var[v/V_m] = Var[v(x)/V_m] the variance of the (normalized) velocity v/V_m measured in units of the motor velocity. The above equation implies that the variance of the normalized velocity can be considerably smaller than the variance of the spatial polarity profile; see Fig. 5 B. The ratio between the two only depends on the characteristic length l_c/L and quickly decays with respect to it. For larger l_c/L, the ambient polarity corresponds to an average over a larger region in space. Therefore, its variance decreases. Because of the linear relationship between the velocity and the ambient polarity, the variance of the velocity decreases to an equal extent.

Figure 5.

Results for the in silico study. (A) A density plot is given, displaying the probability distribution for all combinations of local polarity, P(x), and speed, |v(x)/V_m|, as measured in the in silico study described in In Silico Study: Random Polarity Field. The histograms at the top and on the right are projections of the density plot on the respective axis. In both cases, the solid line is the corresponding analytic prediction that was obtained by approximating the distributions by a normal distribution with the respective predicted mean and variance. In comparison to the local polarity (top), the velocity distribution (right) is less broad, i.e., it exhibits a smaller but nonzero SD. The parameters are chosen to match the stochastic agent-based simulation described in The Agent-Based Model Can Describe the Weak Velocity-Polarity Sensitivity, namely μ = 3 and l_c/L = 10. (B) The ratio between the SDs of the normalized velocity, σ[v/V_m], and of the local polarity, σ[P], is plotted against the normalized characteristic length, l_c/L. The results of the numerical solution of the continuum equation, Eq. 5b (symbols) for μ = {3, 10, 17, 24} (red stars, blue circles, yellow squares, green crosses) collapse onto one master curve. The solid line corresponds to the analytic prediction of the master curve. For larger characteristic length, the SD of the velocity decreases relative to the SD of the local polarity. Note that for small μ = 3, there is a slight deviation from the master curve. In this case, the variance of the local polarity is so high that the corresponding, approximate normal distribution has not decayed to zero at P = ±1 (see histogram at top of A). To see this figure in color, go online.

To compare our results to in vitro experiments, we assessed the values for both the one-dimensional number density of filaments 2μ and the characteristic length l_c. From recent experimental data (18), we estimated 2μ = 6 and l_c/L ≈ 10; see also The Agent-Based Model Can Describe the Weak Velocity-Polarity Sensitivity. Given these estimates, our theory yields an SD of the polarity distribution $\sigma \left[P\right]=\sqrt{\text{Var}\left[P\right]}\approx 0.46$, corresponding to a broad range of observable polarities similar to what is seen in experiments. Using our theoretical results, we predict the ratio between the SDs of the local polarity and the normalized velocity to be approximately $\sigma \left[P\right]/\sigma \left[v/V_m\right]=\sqrt{\text{Var}\left[P\right]/\text{Var}\left[v/V_m\right]}\approx 6.3$. Thus, we expect the mismatch between the widths of the two distributions to be clearly visible in experiments.

Polarity and velocity distribution in the in silico study

Fig. 5 A shows a comparison of the distribution of the local polarity and velocity, as measured in the in silico study (density plot and histograms) and as predicted analytically (black lines). The density plot shows the measured probability distribution for all combinations of local polarity and velocity. The histograms for both quantities were obtained as projections of the density plot onto the respective axis. Whereas the local polarity takes values in a broad range between ±1 (histogram at top of Fig. 5 A), the distribution of the velocity is squeezed to values of approximately (1 ± 0.2)V_m (histogram at right of Fig. 5 A).

The disparity between the two distributions nicely illustrates the filtering of high-frequency modes discussed in the Analytic Solution for Motor-Induced Filament Movement. This filtering is due to long-range interactions induced by the averaging of the polarity field over a length l_c. Because the filtering strongly depends on the characteristic length l_c, the ratio between the SDs of the local polarity and velocity distributions, σ[P]/σ[v/V_m], decreases as l_c increases (Fig. 5 B). It would be interesting to test this prediction experimentally by changing, for instance, the concentration of the molecular motors or the drag in the fluid.

Fig. 5 B shows how the SD of the velocity distribution, normalized to the SD of the local polarity distribution, depends on the characteristic length l_c. For small l_c ∼ L, the effective interaction range of microtubules l_c is small, and the microtubule dynamics is predominantly determined by the local polarity at their respective position. Conversely, in the limit of large l_c, the dynamics of all microtubules is determined by the same average global polarity. Consequently, all microtubules then exhibit the same velocity, and the SD of the velocity σ[v/V_m] decays to zero. Notably, the normalized curves for different values of the microtubule density, μ, collapse onto one master curve when plotted against l_c/L (see Supporting Materials and Methods). Thus, in our thought experiment, in which we make a certain assumption with regard to the spatial polarity profile, knowledge of the microtubule density is not necessary.

Experimental relevance

In an experimental filament gel, other factors also influence filament dynamics. For instance, as molecular motors randomly attach and detach from microtubules, even microtubules at the same position can interact with a different set of microtubules and thus experience different environments. As a result, different microtubules at the same position might actually have a (slightly) different velocity. Correspondingly, for two experimental realizations with an identical polarity profile, the respective average filament speeds at one position x might indeed differ. This effect is not captured by our continuum description, which assumes deterministic velocity profiles v^(±)(x). Thus, we expect a broader distribution of velocities for in vitro measurements compared to our theoretical prediction. To gauge the strength of this effect, we compared our theoretical predictions with the results from stochastic agent-based simulations of the system (for details, see Supporting Materials and Methods). We find that the specific value of the width of the velocity distribution depends on details of the velocity measurement in the experiments. Nevertheless, irrespective of these details, the velocity distribution is significantly smaller than the width of the polarity distribution. Similarly, we expect that fluctuations in the concentration of motors lead to a slight broadening of the velocity distribution but will not change the behavior qualitatively.

The in silico study considered here clearly simulates an idealized system insofar as we have assumed that there is no overall spatial structure. The analysis can be readily extended to a broader class of systems, in which knowledge of the covariance structure of the polarity field (Cov[P](x, y)) is sufficient to predict the covariance structure of the velocity field (Cov[v/V_m](x, y)) (see Supporting Materials and Methods). Because this signature of our results is strongly dependent on the characteristic length, l_c, we expect such measurements to provide insight into network parameters. Actually, even low-resolution information on the spatial variation of the polarity field could be helpful to test our predictions. As we have seen above, the Fourier coefficients are suppressed by 1/(1 + (2πl_ck/S)²), $k\in \mathbb{Z}$, in a finite system of size S (or, equivalently, by 1/(1 + (l_ck)²), $k\in \mathbb{R}$, in the infinite system). So, for large l_c, the velocity modes with wave vector k ≥ 1/l_c should not be visible in experiments.

Discussion

In this work, we have considered a mesoscopic model for microtubule dynamics in a nematic, motor-cross-linked network. So far, research has focused on either the dilute or heavily cross-linked limit. Strikingly, the observed behavior in these two cases is qualitatively different: whereas in the dilute case, the microtubule velocities strongly depend on the local network polarity (20,21), in the heavily cross-linked case, the velocity has been found to be independent of the polarity (14,16,18)). These distinct phenomenologies are puzzling because the underlying microscopic motor-mediated microtubule interactions are presumably the same in both cases. Starting from these filament interactions, we have shown how the interplay between movement resulting from motor cross-linking and the countervailing effects of fluid drag determines the sensitivity of the local filament dynamics to the network polarity. Thereby, we provide a better understanding of the essential physical principles that lead to such diverse dynamics.

To this end, we derived a nonlocal mean-field theory of our system from the microscopic interactions. This theory enabled us to obtain an explicit analytic expression relating the local microtubule velocity to the spatial polarity profile. Our key result is that the local velocity depends on the local ambient polarity, which is given by the averaged polarity a microtubule senses in its environment. More specifically, the local velocity is given by the convolution of the polarity and an exponentially decaying interaction kernel with characteristic propagation length, l_c. Hence, it is not the local polarity at the position of a microtubule that determines its motion, but rather the entire polarity profile in an environment of length l_c. This finding implies that a one-to-one mapping between the local velocities of microtubules and the local polarity as shown in Fig. 2 is not the whole story. Instead, to predict the velocity at a specific location, knowledge of the spatially varying polarity profile in the entire vicinity is needed. In general, such detailed spatial information appears to be inaccessible with current experimental techniques. Fortunately, to infer the distribution of velocities from the distribution of local polarities, such detailed information is not essential. For example, in a gel where microtubules are randomly placed in an experimental chamber and stochastically oriented, our theory predicts how the variances of the local polarity and of the velocity are related.

The relationship between the velocity and polarity distributions strongly depends on the characteristic propagation length l_c, which is an important emerging length scale in the system. It can be interpreted as a nonlocal interaction range of filaments and is determined by the ratio between the average motor-driven force on a microtubule and the microtubule’s drag in the fluid. Thus, this intrinsic length reflects how effectively motion generated at one position is propagated through the interconnected network of filaments. It strongly depends on the network properties.

We have identified a common mechanism explaining the microscopic origin of both uniform filament motion in percolated nematic networks and the strong polarity dependence of microtubule motion in dilute systems: because of their finite extension, microtubules directly interact with several parallel and antiparallel neighbors within a spatial range equal to twice their filament length. Motors between parallel microtubules induce a resistance against relative motion and thus promote uniform motion of cross-linked microtubules. Thereby, motion generated by antiparallel interactions translates through the percolated network of microtubules even into regions with only parallel and no antiparallel interactions, where a priori no motion is expected. The degree of efficiency of this propagation of motion is quantified by the characteristic propagation length l_c. Hence, it is influenced by the average number of motors per interaction and the drag of filaments in the fluid, among other factors. Filaments at distances larger than l_c apart can be considered to be part of disconnected patches. That is, for small l_c, only motor cross-links between nearest neighbor filaments are relevant for filament motion, as in the dilute limit. For this case, we recover the linear relationship between local polarity and filament velocity (19, 20, 21). On the other hand, in the limit of large l_c, which corresponds to systems in which the patch size exceeds the system size, we find a dependence of the velocity on the global polarity only. Here, the velocity for equally oriented microtubules is the same everywhere in space. In particular, our results explain the weak sensitivity of the filament velocities to the local polarity observed in recent experiments (18) and in the spindle apparatus (11,14,16).

In particular, we predict a strong dependency of the velocity distribution on the characteristic propagation length. To test this prediction, we suggest a practicable in vitro experiment whose feasibility we assessed with the help of an in silico study intended to mimic the suggested in vitro experiment. Intriguingly, it is not necessary to determine the entire spatial polarity and velocity profile to check the validity of our theory. Instead, it suffices to determine the polarity and velocity distributions by measuring the local velocity and polarity at random positions in the filament gel. When plotting the ratio of the SDs of the polarity and velocity distribution against the characteristic length l_c, we expect the data to collapse onto a master curve, irrespective of the explicit number of filaments in the experimental chamber (Fig. 5). Furthermore, the ratio of the SDs of the polarity and velocity distributions for a specific experimental setup could be used to identify the characteristic propagation length l_c and allow one to draw conclusions regarding network features (Eq. 6).

Microtubule motion in mitotic spindles formed in Xenopus egg extract is a prominent example for polarity-independent sliding. The polarity profile in these spindles is approximately linear, ranging from zero polarity in the center to highly polar regions at the spindle poles (17). Nonetheless, microtubules drift with roughly constant velocity toward the spindle poles, especially if dynein is inhibited. Our theory can account for this behavior. In particular, the individual velocities deviate only slightly from the mean velocity if motor cross-linking is strong, i.e., if the characteristic length exceeds the system size (see A Simple Example: The Linear Polarity Profile). Interestingly, for biologically plausible parameters, the interaction range is on the same order as the length of the spindles formed in Xenopus egg extracts, l_c ∝ 30–80 μm. Correspondingly, as seen in Fig. 4 A, the velocity of the poleward moving microtubules is expected to be slightly smaller close to the pole than in the center of the spindle. This variation is due to the dependence of the velocity on the ambient polarity (the local polarity environment). Taken together, our results suggest that depending on the value of the characteristic length compared to the spindle size, the spatial polarity profile, and, in particular, the fact that the poles are highly polar, could be significant for the velocity profile as well. To examine this behavior experimentally, it would be instructive to investigate the velocity distribution of microtubules in a dynein-depleted, unfocused spindle as a function of the distance from the spindle boundary.

More generally, there is not only one type of motor present in vivo. Instead, several types of motors can, in principle, cross-link and exert forces on filaments. Extending our analysis to a broader class of cross-linking proteins does not change our results qualitatively (see Supporting Materials and Methods). The velocity still depends on the ambient polarity with a characteristic length that is determined by the ratio between the absolute “friction” between microtubules and the drag in the fluid.

From a broader perspective, it would be interesting to extend our work on nematic networks to a more general description of filament gels. To this end, it could be promising to start from recent work on heavily cross-linked filament gels, for which a sophisticated hydrodynamic framework has been established from microscopic properties (18). This theoretical framework assumes an infinitely large characteristic length l_c, so that motion generated at one position propagates through the whole network without loss. Our results suggest that incorporating an exponential interaction kernel into this framework can provide a more comprehensive description of filament motion in cross-linked gels. Such a description would also offer the chance to understand the transition from heavily cross-linked to weakly coupled gels.

Author Contributions

M.S., I.R.G., and E.F. designed research, performed research, and wrote the article.

Editor: Anatoly Kolomeisky.