Comparison of explicit and mean-field models of cytoskeletal filaments with crosslinking motors

Abstract

In cells, cytoskeletal filament networks are responsible for cell movement, growth, and division. Filaments in the cytoskeleton are driven and organized by crosslinking molecular motors. In reconstituted cytoskeletal systems, motor activity is responsible for far-from-equilibrium phenomena such as active stress, self-organized flow, and spontaneous nematic defect generation. How microscopic interactions between motors and filaments lead to larger-scale dynamics remains incompletely understood. To build from motor–filament interactions to predict bulk behavior of cytoskeletal systems, more computationally efficient techniques for modeling motor–filament interactions are needed. Here, we derive a coarse-graining hierarchy of explicit and continuum models for crosslinking motors that bind to and walk on filament pairs. We compare the steady-state motor distribution and motor-induced filament motion for the different models and analyze their computational cost. All three models agree well in the limit of fast motor binding kinetics. Evolving a truncated moment expansion of motor density speeds the computation by $10^3$–$10^6$ compared to the explicit or continuous-density simulations, suggesting an approach for more efficient simulation of large networks. These tools facilitate further study of motor–filament networks on micrometer to millimeter length scales.

Graphic abstract

Appendices

Appendices

A Determining the time-step for binding

Our kinetic Monte Carlo algorithm assumes that multiple binding/unbinding events do not occur in the same time step $\Delta t$. As $\Delta t$ becomes large relative to the kinetic rates, this approximation fails. A time step is appropriate if the maximum probability of two events occurring in $\Delta t$ satisfies

$$\begin{aligned} \max \{P(C(\Delta t) \cup B(t')|A(0)) \} < \delta \end{aligned}$$

(42)

for a tolerance $\delta$, where A, B, and C denote motor bound states (including unbound, single head bound, and crosslinking) at time $\Delta t> t' > 0$. $P(C(\Delta t) \cup B(t')|A(0)) = P(C(\Delta t)|B(t'))P(B(t')|A(0))$ and each individual state change follows a single event Poisson process with $P(B(t)|A(0)) = 1 - \exp [-k_{A \rightarrow B} t]$. The maximum probability for a double event occurs at $t'=t'_\mathrm{{max}}$ found by solving

$$\begin{aligned} \begin{aligned}&\frac{dP(C(\Delta t) \cup B(t')|A(0))}{dt'}\Bigg |_{t'_{max}} = 0\\&k_{A \rightarrow B} \left( e^{k_{B \rightarrow C}(\Delta t-t')} - 1 \right) - k_{B \rightarrow C} \left( e^{k_{A \rightarrow B} t'} - 1 \right) = 0. \end{aligned} \end{aligned}$$

(43)

While no analytic solution exists, $t'_\mathrm{{max}}$ can be numerically computed.

There are four unique processes that must be considered with a two-step binding process with unbound (U), single head bound (S), and crosslinking (C) states: $U \rightarrow S \rightarrow U$, $U \rightarrow S \rightarrow C$, $S \rightarrow C \rightarrow S$, and $C \rightarrow S \rightarrow U$. The process $C \rightarrow S \rightarrow C$ has the same probability as $S \rightarrow C \rightarrow S$, similarly, $S \rightarrow U \rightarrow S$ has the same probability as $U \rightarrow S \rightarrow U$. If modeling filament motion with some force- or energy-dependent unbinding, may be large. This means that in the limit of large unbinding rate the probabilities $P(C \rightarrow S \rightarrow C) \rightarrow P(S \rightarrow C)$ and $P(C \rightarrow S \rightarrow U) \rightarrow P(S \rightarrow U)$.

B Lookup table for kinetic Monte Carlo binding

Equation (11) gives the transition probability of a singly bound motor crosslinking as an integral of a Boltzmann factor. If $h_\mathrm{cl}=0$, $k_{\mathrm{on},C}$ is functionally similar to an error function. However, to model non-zero-length tethers, we numerically integrate Eq. (11). Rather than directly numerically integrating at each time step, we precompute a lookup table.

The cumulative distribution function (CDF) of Eq. (11), is a function $h_{i,j}$. All other variables in the integral are constant for a given motor species. We reduce the CDF dimensionality by considering the lab position of each bound motor head and an infinite carrier line defined by the position and orientation of the unbound filament. Binding is then determined by the minimum distance $r_{\perp }$ between the bound motor head position and the filament ends $[s_-, s_+]$ on the carrier line.

The carrier line CDF is

$$\begin{aligned} {{\,\mathrm{\mathrm{CDF}}\,}}(r_{\perp },s) = \int _{-\infty }^{s} e^{-\beta U(r_{\perp }, s')}ds', \end{aligned}$$

(44)

allowing us to write the crosslinking rate as

(45)

We notice that $e^{-\beta U_{i,j}}$ is symmetric in s, so ${{\,\mathrm{\mathrm{CDF}}\,}}(r_{\perp },s) - {{\,\mathrm{\mathrm{CDF}}\,}}(r_{\perp },0)$ is anti-symmetric. Therefore, instead of integrating from negative infinity, we use

$$\begin{aligned} {{\,\mathrm{\mathrm{CDF}}\,}}'(r_{\perp },s) = {{\,\mathrm{sgn}\,}}(s)\int _{0}^{s} e^{-\beta U(r_{\perp }, s')}ds' \end{aligned}$$

(46)

and (45) to find the crosslinking rate.

We find the values of Eq. (46) by Gauss–Konrad integration. The accuracy desired sets the maximum values for s and $r_{\perp }$. The integrand is always positive for real values of s and $r_{\perp }$, so the CDF asymptotes for large values of either variable. The maximum of the integral is the point when the Boltzmann factor drops to the accuracy limit $\delta$. Therefore, the lookup table domain is

$$\begin{aligned} s, r_{\perp } \in \left[ 0, \sqrt{-\frac{2\ln (\delta )}{\beta k_\mathrm{cl}}} + h_\mathrm{cl}\right] . \end{aligned}$$

(47)

Given a specified grid spacing $\Delta s, \Delta r$, the memory required for the lookup table scales as $(s_{\max }/\Delta s) \times (r_{\perp , \max }/\Delta r)$.

Fig. 7

Visual representation of the lookup table showing CDF values as a function of distance s along the filament for $h_\mathrm{cl}= 32$ nm, $k_\mathrm{cl}= .3$ pN/nm, $\beta = 1./4.11$ (pN$\cdot$nm)$^{-1}$, and $\delta = 10^{-5}$

1.1 B.1 Interpolation of lookup table values

Since the lookup table is not a continuous function, we interpolate values between discrete grid points. The 2D linear interpolation for input values of $r_{\perp }$ and s is

$$\begin{aligned} \begin{aligned} {{\,\mathrm{\mathrm{CDF}}\,}}(r_{\perp }, s) \approx&\left( 1+m-\frac{r_{\perp }}{\Delta r} \right) \left( 1+n-\frac{s}{\Delta s} \right) \\&{{\,\mathrm{\mathrm{CDF}}\,}}_{m,n} + \left( \frac{r_{\perp }}{\Delta r} - m \right) \left( 1+n-\frac{s}{\Delta s} \right) {{\,\mathrm{\mathrm{CDF}}\,}}_{m+1,n}\\&+ \left( 1+m-\frac{r_{\perp }}{\Delta r} \right) \left( \frac{s}{\Delta s} - n \right) {{\,\mathrm{\mathrm{CDF}}\,}}_{m,n+1} \\&+ \left( \frac{r_{\perp }}{\Delta r} - m \right) \left( \frac{s}{\Delta s} - n \right) {{\,\mathrm{\mathrm{CDF}}\,}}_{m+1,n+1}, \end{aligned}\nonumber \\ \end{aligned}$$

(48)

where ${{\,\mathrm{\mathrm{CDF}}\,}}_{m,n}={{\,\mathrm{\mathrm{CDF}}\,}}(m\Delta r, n\Delta s)$ are the lookup table values at m and n if $r_{\perp }$ lies within $m\Delta r$ and $(m+1)\Delta r$ and s lies within $n\Delta s$ and $(n+1)\Delta s$.

1.2 B.2 Reverse lookup algorithm

When a motor head binds to a filament, the binding position probability distribution function (PDF) is defined by the Boltzmann factor. We sample the PDF by using the lookup table. To transform a uniform random variable X to random variable Y with an arbitrary ${{\,\mathrm{\mathrm{PDF}}\,}}_Y$, X is inserted into the inverted CDF of Y

$$\begin{aligned} Y = {{\,\mathrm{\mathrm{CDF}}\,}}_Y^{-1}(X). \end{aligned}$$

(49)

Since the lookup table holds the CDF values and given a random number from a uniform distribution, we apply a combination of search and interpolation to quickly find the corresponding random number from the PDF. The algorithm is as follows

• 1. Sample a uniform random number $X \in [0, {{\,\mathrm{\mathrm{CDF}}\,}}_{\max }]$. Note that the maximum value does not need to be 1.

• 2. Given $r_{\perp }$, locate index m such that $m\Delta r \le r_{\perp } \le (m +1) \Delta r$

• 3. Use m to find the set of indices $\{n_-, n_+\}$ such that ${{\,\mathrm{\mathrm{CDF}}\,}}_{m,n_-} \le X \le {{\,\mathrm{\mathrm{CDF}}\,}}_{m,n_-+1}$ and ${{\,\mathrm{\mathrm{CDF}}\,}}_{m+1,n_+} \le X \le {{\,\mathrm{\mathrm{CDF}}\,}}_{m+1,n_++1}$.

• 4. Use the CDF values to interpolate the binding locations $s_-, s_+$ corresponding to the perpendicular distances $r_- = m\Delta r$ and $r_+ = (m+1) \Delta r$. For example,

  $$\begin{aligned} s_-&= \Delta s \frac{X - {{\,\mathrm{\mathrm{CDF}}\,}}_{m,n_-}}{{{\,\mathrm{\mathrm{CDF}}\,}}_{m,n_-+1} - {{\,\mathrm{\mathrm{CDF}}\,}}_{m,n_-}} + \Delta s n_- \end{aligned}$$

  (50)
  $$\begin{aligned} s_+&= \Delta s \frac{X - {{\,\mathrm{\mathrm{CDF}}\,}}_{m+1,n_+}}{{{\,\mathrm{\mathrm{CDF}}\,}}_{m+1,n_++1} - {{\,\mathrm{\mathrm{CDF}}\,}}_{m+1,n_+}} + \Delta s n_+ \end{aligned}$$

  (51)

  Note that $s_-$ is not necessarily less than $s_+$.

• 5. Find s by interpolating the across the lookup table grid with respect to $r_{\perp }$

  $$\begin{aligned} s \approx (s_+ - s_-)\frac{r_{\perp }-r_-}{\Delta r} + s_- \end{aligned}$$

  (52)

While this algorithm succeeds in most circumstance, the low slope of the CDF at large values of s can cause errors. For example, if the lookup table has the form of Fig. 7 and a protein is located at a perpendicular distance of $r_{\perp }=35$ nm, given a random number of $X=10^3$, no value for $s_-$ will be found since ${{\,\mathrm{\mathrm{CDF}}\,}}(30, s_\mathrm{max}) < 10^3$. To correct for this, we solve for s using a binary search algorithm.

The binary search algorithm is as follows

• 1. Determine if ${{\,\mathrm{\mathrm{CDF}}\,}}_{m,n_\mathrm{max}}$ or ${{\,\mathrm{\mathrm{CDF}}\,}}_{m+1,n_\mathrm{max}}$ is less than X. If ${{\,\mathrm{\mathrm{CDF}}\,}}_{m,n_\mathrm{max}} < X$, set $s_- =s_\mathrm{max}$. If ${{\,\mathrm{\mathrm{CDF}}\,}}_{m+1,n_\mathrm{max}} < X$, set $s_+ =s_\mathrm{max}$.

• 2. Find other $s_\pm$ using the inverted lookup table and Eq. (50) or (51).

• 3. Find the average of $s_-$ and $s_+$.

• 4. Use the lookup table interpolation algorithm to find the ${{\,\mathrm{\mathrm{CDF}}\,}}(r_{\perp }, s_\mathrm{avg})$.

• 5. If ${{\,\mathrm{\mathrm{CDF}}\,}}(r_{\perp }, s_\mathrm{avg}) > X$ set the larger of the two $s_{\pm }$ values to $s_\mathrm{avg}$. Otherwise, set the smaller of the two to $s_\mathrm{avg}$.

• 6. Repeat steps 3-5 until $|{{\,\mathrm{\mathrm{CDF}}\,}}(r_{\perp }, s_\mathrm{avg}) - X| < \delta$ for some desired tolerance $\delta$.

This process converges at a rate $O(\log _2(\delta s_{\max }))$.

C Numerical integration of the MFMD equation

We approximate the solution $\psi _{i,j}(s_i, s_j, t)$ by discretizing the solution in time and space

(53)

for , where $M_i$ is the number of discretized points along filament i. Additional boundary points for $m,n=0$ are added.

We use forward Euler time-stepping so our discrete differential operator for time is

(54)

To solve the hyperbolic FPE (17), we use a first-order accurate upwind method [88]. The differential operator for $s_i$ becomes

(55)

Note this only holds for the indices $0<m$ and $0<n$. The matrix representation for Eq. (55) is

(56)

where $c_m$ and $d_m$ are chosen to satisfy the boundary conditions. We choose the notation $\rhd ^{m,n}$ for this matrix. To differentiate along $s_j$, we use the identity , apply $\rhd ^{m,n}$ on the matrix, and then convert back,

$$\begin{aligned} \frac{\partial \psi _{i,j}}{\partial s_i} \rightarrow \left( \sum _a \rhd ^{n,a} \psi _{j,i}^{a,m} \right) ^T, \end{aligned}$$

(57)

which in index notation is $\psi _{i,j}^{m,a}(\rhd ^T)^{n,a}$. For brevity, we use the notation $\psi _{i,j}^{m,a}(\rhd ^T)^{n,a}=\psi _{i,j}^{m,a}\lhd ^{a,n}$.

The discretized Fokker–Planck Eq. (17) is then

(58)

where $U_{i,j}^{m,n,k}$ and $v_{i,j}^{m,n,k}$ are the discretized potential and velocity at time $k\Delta t$. Note that $U_{i,j}^{m,n,k} = U_{j,i}^{n,m,k}$, but $v_{i,j}^{m,n,k} \ne v_{j,i}^{n,m,k}$.

In cases where the flux of the motors $\frac{\partial \left( v_{i,j}\psi _{i,j}\right) }{\partial s_i}$ is known at the boundaries, we construct $\rhd$ to satisfy the requirements. When filaments are in solution, there is zero flux from the minus ends, so all $c_m = 0$. In our simulations, motors walk of filament ends with out pausing, so $d_{M_i-1}=-1$ and $d_{M_i}=1$ with all other $d_m=0$. Although not modeled in this paper, some biological motors end pause at filament plus ends. To model this, $d_{M_i-1}=-1$ and every other $d_m=0$.

D Conversion of binding parameters from an explicit to mean-field motor density model

To relate binding parameters of the one-step and multi-step binding models, we use that at steady state, the motor distribution $\psi _{i,j}$ should be equivalent for both models. Since we only compare binding kinetics, we simplify the Fokker–Planck equation to keep only the binding terms: in Eq. (17), we set $v_{i,j} = v_{j,i} = 0$,

(59)

The steady-state solution is

$$\begin{aligned} \psi _{i,j}= c e^{-\beta U_{i,j}}, \end{aligned}$$

(60)

which is a Boltzmann factor multiplied by an effective concentration.

The multi-step binding model can be written

(61)

(62)

(63)

where $\chi _i$ is the mean-field density of motors with one head bound to filament i (cf. Eq. 16). We define $K_\mathrm{E}' = K_\mathrm{E}/V_\mathrm{bind}$ and solve for the steady state, giving

(64)

(65)

(66)

The equations for $\chi _i$ and $\chi _j$ have the forms

$$\begin{aligned} X(s)&= C\int _{a}^{b} \left( Y(t)-X(s)\right) K(s,t)dt + D, \end{aligned}$$

(67)

$$\begin{aligned} Y(t)&= C\int _{c}^{d} \left( X(s)-Y(t)\right) K(s,t)ds + D, \end{aligned}$$

(68)

where $t \in [a,b]$ and $s \in [c,d]$. Distributing the integrals, we can rewrite

$$\begin{aligned} X(s)&= C\int _{a}^{b} Y(t)K(s,t)dt - CX(s)F(s) + D, \end{aligned}$$

(69)

$$\begin{aligned} Y(t)&= C\int _{c}^{d} X(s) K(s,t)ds - CY(t)G(t) + D, \end{aligned}$$

(70)

where $F(s) = \int _{a}^{b}K(s,t)dt$ and $G(t) = \int _{c}^{d}K(s,t)ds$. Solving for X(s) and Y(t) gives

$$\begin{aligned} X(s)&= \frac{D}{1+CF(s)} + \frac{C}{1+CF(s)}\int _{a}^{b} Y(t) K(s,t)dt, \end{aligned}$$

(71)

$$\begin{aligned} Y(t)&= \frac{D}{1+CG(t)} + \frac{C}{1+CG(t)}\int _{c}^{d} X(s) K(s,t)ds, \end{aligned}$$

(72)

After plugging Eq. (71) into (72), we find

$$\begin{aligned} Y(t)= & {} \frac{D}{1+CG(t)} + \frac{CD}{1+CG(t)}\int _{c}^{d}\frac{K(s,t)}{1+CF(s)}ds \nonumber \\&+ C^2\int _{a}^{b}\int _{c}^{d}Y(t') \frac{K(s,t)K(s,t')}{(1+CF(s))(1+CG(t'))} ds dt'.\nonumber \\ \end{aligned}$$

(73)

This can be rearranged into the form

$$\begin{aligned} Y(t) = A(t) + \int _{a}^{b}Y(t') B(t,t') dt', \end{aligned}$$

(74)

which implies that Y(t) and X(s) each satisfy a Fredholm equation of the second kind. Both A and B are continuous given $K(s,t)= e^{-\beta U_{i,j}(s,t)}$, so the Fredholm equations of the second kind have unique solutions. By inspection, the solution to Eqs. (67) and (68) is $X(s)=Y(t)=D$. When we substitute this solution in Eqs. (65) and (66), we find $\chi _i = \chi _j = \epsilon K_a c_o$ and

$$\begin{aligned} \psi _{i,j}= \epsilon ^2 K_a K_\mathrm{E}'c_o e^{-\beta U_{i,j}}. \end{aligned}$$

(75)

Setting Eq. (60) equal to (75) gives

$$\begin{aligned} c = \frac{\epsilon ^2 K_a K_\mathrm{E}}{V_\mathrm{bind}}c_o. \end{aligned}$$

(76)

E Calculating binding parameters from experiments

The experimental parameters for motor binding are not always independently measured. If all but one binding parameters are known, then the unknown parameter can be found from Eq. (18) and the ratio of the number of motors with one head bound and number of motors crosslinking.

As an example, suppose we wish to find $K_\mathrm{E}$. The number of motors with one head bound is $N_S = c_o K_a \epsilon L$, where L is the filament length. In vitro experiments [86] can measure the crosslinking motors number $N_d$. Integrating Eq. (75), we obtain the model prediction for the number of crosslinking motors as

$$\begin{aligned} N_C = c_o \epsilon ^2 K_a K_\mathrm{E}' \int _{L_{i}} \int _{L_{j}} e^{-\beta U_{i,j}} ds_i ds_j. \end{aligned}$$

(77)

For fully parallel or antiparallel filaments of the same length with adjacent centers, the total number of motors in Eq. (77) is proportional to L. If $L \gg \sqrt{2/\beta k_\mathrm{cl}}$, the Gaussian integral $\approx L\sqrt{2\pi /\beta k_\mathrm{cl}}e^{-\beta k_\mathrm{cl}r_{\perp }^2}$, where $r_{\perp }$ is the center-to-center separation between filaments. The ratio of the number of crosslinking motors relative to the number motors with one head bound is

$$\begin{aligned} \rho = \frac{N_C}{N_S} = \epsilon K_\mathrm{E}' \sqrt{\frac{2\pi }{\beta k_\mathrm{cl}}}e^{-\beta k_\mathrm{cl}r_{\perp }^2}, \end{aligned}$$

(78)

allowing us to estimate $K_\mathrm{E}' = \frac{\rho }{\epsilon }\sqrt{\frac{\beta k_\mathrm{cl}}{2\pi }}e^{\beta k_\mathrm{cl}r_{\perp }^2}$.

F Gaussian integrals in the moment expansion

The source terms in the moment expansion require a double integral over two filaments. To lower the numerical integration’s computational cost, we find an analytic solution for either the semi-integrated term $Q_j^l(s_i)$ or the fully integrated term $q_{i,j}^{k,l}$.

The integrated source terms are

(79)

where $\alpha = \sqrt{\frac{2}{\beta k_\mathrm{cl}}}$. We define the quantity so that the integral over $s_j$ becomes

$$\begin{aligned} {\bar{Q}}^l_j(s_i) = \int _{L_j} s_j^l e^{-{\left( \frac{s_j+A}{\alpha }\right) }^2}ds_j. \end{aligned}$$

(80)

This integral has an analytic form in terms of error functions, which can be rapidly computed. For $l={0,1,2,3}$, we find

$$\begin{aligned} {\bar{Q}}_j^0(s_i)= & {} \frac{\alpha \sqrt{\pi }}{2} \left[ {{\,\mathrm{erf}\,}}\left( \frac{s_j+A}{\alpha } \right) \right] _{\partial L_j} \end{aligned}$$

(81)

$$\begin{aligned} {\bar{Q}}_j^1(s_i)= & {} -\frac{\alpha }{2}\left[ \alpha e^{-\left( \frac{s_j+A}{\alpha }\right) ^2} + A\sqrt{\pi }{{\,\mathrm{erf}\,}}\left( \frac{s_j+A}{\alpha } \right) \right] _{\partial L_j} \end{aligned}$$

(82)

$$\begin{aligned} {\bar{Q}}_j^2(s_i)= & {} \frac{\alpha }{4}\left[ 2\alpha (A-s_j) e^{-\left( \frac{s_j+A}{\alpha }\right) ^2} \right. \nonumber \\&\left. + (2A^2+\alpha ^2)\sqrt{\pi }{{\,\mathrm{erf}\,}}\left( \frac{s_j+A}{\alpha }\right) \right] _{\partial L_j} \end{aligned}$$

(83)

$$\begin{aligned} {\bar{Q}}_j^3(s_i)= & {} \frac{-\alpha }{4}\left[ 2\alpha (A^2 - A s_j + s_j^2 + \alpha ^2) e^{-\left( \frac{s_j+A}{\alpha }\right) ^2} \right. \nonumber \\&\left. + (2A^2 + 3\alpha ^2)A\sqrt{\pi }{{\,\mathrm{erf}\,}}\left( \frac{s_j+A}{\alpha }\right) \right] _{\partial L_j} \end{aligned}$$

(84)

G Moment expansion boundary terms

To generally define boundary conditions, instead of integrating over both $s_i$ and $s_i$, we integrate over just one variable. This makes the boundary condition a function of a single filament attachment position. For example, the boundary terms for the first filament are

(85)

These boundary terms are evaluated at $-L_i/2$ and $L_i/2$. We derive a recursion relation by integrating Eq. (85) over $s_j$ and using the definition in Eq. (37)

(86)

Solving this equation requires finding the time evolution of the boundary term spatial derivatives, which solve

(87)

This shows that the boundary terms do not close. However, if the higher-order terms or their coefficients are small compared to the moments $\mu _{i,j}^{k,l}$, we may take a zeroth-order approximation. We consider this approximation in Sect. 6.