Role of Interactions and Correlations on Collective Dynamics of Molecular Motors Along Parallel Filaments

Abstract

Cytoskeletal motors known as motor proteins are molecules that drive cellular transport along several parallel cytoskeletal filaments and support many biological processes. Experimental evidences suggest that they interact with the nearest molecules of their filament while performing any mechanical work. These interactions modify the microscopic level properties of motor proteins. In this work, a new version of two-channel totally asymmetric simple exclusion process, that incorporates the intra-channel interactions in a thermodynamically consistent way, is proposed. As the existing approaches for multi-channel systems deviate from analyzing the combined effect of inter and intra-channel interactions, a new approach known as modified vertical cluster mean field is developed. The approach along with Monte Carlo simulations successfully encounters some correlations and computes the complex dynamic properties of the system. Role of symmetry of interactions and inter-channel coupling is observed on the phase diagrams, maximal particle current and its corresponding optimal interaction strength. Surprisingly, for all values of coupling rate and most of the interaction splittings, the optimal interaction strength corresponding to maximal current belongs to the case of weak repulsive interactions. Moreover, for weak interaction splittings and with an increase in the coupling rate, the optimal interaction strength tends towards the known experimental results. The effect of coupling as well as interaction energy is also measured for correlations. They are found to be short-range and weaker for repulsive and weak attractive interactions while they are long-range and stronger for large attractions.

Appendices

Appendix A: Expression for bulk current using VCMF approach:

The vertical cluster mean field approach is utilized to compute the bulk current for all eight possible configurations (shown in Fig. 2) contributing to the particle movement in bulk. The total bulk current in the system is an algebraic sum of currents from all of these configurations. We first compute the particle current corresponding to configurations in Fig. 2a–d that doesn’t involve the coupling parameter w i.e. the case when the particle can not hop vertically. We denote total bulk current from these four configurations as

$$\begin{aligned} J_1 = J_a + J_b + J_c + J_d, \end{aligned}$$

(A.1)

where $J_a$, $J_{b}$, $J_c$, and $J_{d}$ are particle currents corresponding to configuration (a), (b), (c), and (d), respectively and are given as

$$\begin{aligned} J_{a}= & {} V_3(V_0 + V_2)^3, \end{aligned}$$

(A.2)

$$\begin{aligned} J_{b}= & {} qV_3(V_1+V_3)(V_0+V_2)^2, \end{aligned}$$

(A.3)

$$\begin{aligned} J_c= & {} rV_3(V_1+V_3)(V_0+V_2)^2, \end{aligned}$$

(A.4)

$$\begin{aligned} \hbox {and} \quad J_d= & {} V_3(V_0+V_2)(V_1+V_3)^2, \end{aligned}$$

(A.5)

thus implying,

$$\begin{aligned} J_1 = V_3(1 - \rho )[(1-\rho )^2 + (q+r)\rho (1-\rho ) + \rho ^2]. \end{aligned}$$

(A.6)

Here $\rho$ = $V_1 + V_3$ denotes the particle bulk density. Similarly particle bulk current for the configurations in Fig. 2e–h which involves the role of parameter w i.e. when lane switching is possible, can be computed as

$$\begin{aligned} J_{e}= & {} (1-w)V_1(V_0 + V_2)^3, \end{aligned}$$

(A.7)

$$\begin{aligned} J_{f}= & {} q(1-w)V_1(V_1+V_3)(V_0+V_2)^2, \end{aligned}$$

(A.8)

$$\begin{aligned} J_g= & {} r(1-w)V_1(V_1+V_3)(V_0+V_2)^2, \end{aligned}$$

(A.9)

$$\begin{aligned} \hbox {and} \quad J_h= & {} (1-w)V_1(V_0+V_2)(V_1+V_3)^2, \end{aligned}$$

(A.10)

where $J_e$, $J_{f}$, $J_g$, and $J_{h}$ denote particle currents corresponding to configuration (e), (f), (g), and (h), respectively. The total bulk current for the above four configurations is expressed as

$$\begin{aligned} J_2= & {} J_e + J_f +J_g +J_h = (1-w)V_1(1 - \rho )[(1-\rho )^2 \nonumber \\&+\;(q+r)\rho (1-\rho ) + \rho ^2]. \end{aligned}$$

(A.11)

The overall bulk current per channel is

$$\begin{aligned} J_{bulk}^{VCMF}= & {} J_1 + J_2 =(V_3 + (1-w)V_1)(1-\rho )[(1-\rho )^2\nonumber \\&+\;(q+r)\rho (1-\rho ) + \rho ^2]. \end{aligned}$$

(A.12)

For any value of $\rho$ between zero and one, the rate of formation of particle cluster, $q = \eta ^{\theta }$, approaches infinity for very large attractive interactions ($\eta \gg 1)$ and $\theta > 0$. This causes bulk current to increase without any limit contradicting to the physical scenario of depletion of particle current under large attractive interactions. Similarly, the rate of breaking from a cluster, $r = \eta ^{(\theta - 1)}$, tends to infinity for $\eta \rightarrow 0$ and $\theta < 1$, for any $\rho \in (0,1)$.

Appendix B: Monte Carlo simulations

Due to approximate nature of our method in calculating the effect of interactions and correlations, we validate the results obtained from the given approximate theoretical method with extensive Monte Carlo (MC) simulations. Random-Sequential update rules are adopted. For a single Monte Carlo step, first a lattice is randomly chosen with equal probability. To avoid any finite-size and boundary effects both lattices are considered to be of size N = 1000 unless otherwise mentioned. The results have been verified by taking large lattice size of $N = 5000$. The simulation starts from a random initial distribution of particles on both the lattices and system evolved for $10^{9}$–$10^{10}$ time steps to ensure the steady state condition. To compute density and particle current at steady state, an average of the last $80\%$ of the steps has been taken. In constructing phase diagrams, density profiles are compared with a precision of 0.01 and for calculating phase boundaries, error estimated in comparing currents is less than $1\%$. Our predicted theoretical results fit well with the simulation results.