Cooperative Dynamics in Bidirectional Transport on Flexible Lattice


Several theoretical models based on totally asymmetric simple exclusion process (TASEP) have been extensively utilized to study various non-equilibrium transport phenomena. Inspired by the the role of microtubule-transported vesicles in intracellular transport, we propose a generalized TASEP model, where two distinct particles are directed to hop stochastically in opposite directions on a flexible lattice immersed in a three dimensional pool of diffusing particles. We investigate the interplay between lattice conformation and bidirectional transport by obtaining the stationary phase diagrams and density profiles within the framework of mean field theory. For the case when configuration of flexible lattice is independent of particle density on lattice, the phase diagram only differs quantitatively in comparison to that obtained for bidirectional transport on rigid lattice. However, if the lattice occupancy governs the global conformation of lattice, in addition to the pre-existing phases for bidirectional transport a new asymmetric shock-low density phase originates in the system. We identified that this phase is sensitive to finite size effect and vanishes in the thermodynamic limit.

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Correspondence to Arvind Kumar Gupta.

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Communicated by Shin-ichi Sasa.


A Summary of Expressions Obtained for Effective Entry Rate in Distinct Phases

We determine the effect of 3D environment on the entry rates of two type of particles (\({\bar{\alpha }}_{\pm }\)) in three distinct phases low-density (LD), high-density (HD), maximal current (MC) [10]. To do so utilize the expression of effective entry rate \({\bar{\alpha }}_{\pm }=\alpha _{\infty }+J_{\pm }{\varGamma }\), plug the different forms of current according to phases for simple TASEP model on rigid lattice [2], that yields the effective entry rate for,

LD phase,

$$\begin{aligned} {\bar{\alpha }}_{\pm }^{LD}=\alpha _{\infty }+{\bar{\alpha }}^{LD}_{\pm }(1-{\bar{\alpha }}^{LD}_{\pm }), \end{aligned}$$

which on solving gives,

$$\begin{aligned} {\bar{\alpha }}^{LD}_{\pm }=\dfrac{({\varGamma }-1)\pm \sqrt{(1-{\varGamma })^2+4\alpha _{\infty }{\varGamma }}}{2{\varGamma }}. \end{aligned}$$

HD phase,

$$\begin{aligned} {\bar{\alpha }}_{\pm }^{HD}=\alpha _{\infty }+\beta (1-\beta ). \end{aligned}$$

MC phase,

$$\begin{aligned} {\bar{\alpha }}_{\pm }^{MC}=\alpha _{\infty }+\dfrac{{\varGamma }}{4}. \end{aligned}$$

Further, the bulk density, current and conditions of existence of different phases are elaborated in Table 1. These effective rates are further used to analyse the bidirectional movement in a 3D environment.

Continuum Equations

The continuum version of governing equations obtained in Eq. (36) is discretized using finite-difference scheme where time and space derivative are replaced using forward and central difference formula. Choosing \({\varDelta } x=\frac{1}{N}\) and suitable \({\varDelta } t\) satisfying the stability criteria \(\frac{{\varDelta } t}{{\varDelta } x}\le 1\), the solution is captured in the limit \(j\rightarrow \infty \) (time variable) for \(1<i<N\) (space variable) to ensure the occurrence of steady-state.

$$\begin{aligned} \rho _i^{(j+1)}=&\rho _i^{(j)}+\dfrac{\epsilon }{2}\dfrac{{\varDelta } t}{{\varDelta } x^2}(\rho _{i+1}^{(j)}-2\rho _i^{(j)}+\rho _{i-1}^{(j)}) \nonumber \\&+\dfrac{{\varDelta } t}{2{\varDelta } x}(2\rho _i^{(j-1)}-1)(\rho _{i+1}^{(j)}-\rho _{i-1}^{(j)}),\end{aligned}$$
$$\begin{aligned}&\sigma _i^{(j+1)}=\sigma _i^{(j)}+\dfrac{\epsilon }{2}\dfrac{{\varDelta } t}{{\varDelta } x^2}(\sigma _{i+1}^{(j)}-2\rho _i^{(j)}+\sigma _{i-1}^{(j)})\nonumber \\&+\dfrac{{\varDelta } t}{2{\varDelta } x}(1-2\sigma _i^{(j)})(\sigma _{i+1}^{(j)}-\sigma _{i-1}^{(j)}) \end{aligned}$$

As in the proposed model two type of particles are allowed to interact only at the boundaries, it is not feasible to explicitly calculate the boundary conditions in the continuum limit from Eq. 3. Hence, the boundary conditions given in Eq. 3 are implemented using finite difference scheme given by,

$$\begin{aligned} \dfrac{d\rho _1^{(j+1)}}{dt}&={\bar{\alpha }}_+(1-\rho _1^{(j)} -\sigma _1^{(j)})-\rho _1^{(j)}(1-\rho _2^{(j)}),\nonumber \\ \dfrac{d\sigma _1^{(j+1)} }{dt}&=\sigma _2^{(j)}(1-\rho _1^{(j)})-\beta \sigma _1^{(j)}, \nonumber \\ \dfrac{d\rho _N^{(j+1)}}{dt}&=\rho _{N-1}^{(j)}(1-\rho _N^{(j)})-\beta \rho _N^{(j)},\nonumber \\ \dfrac{d\sigma _N^{(j+1)} }{dt}&={\bar{\alpha }}_-(1-\rho _N^{(j)}-\sigma _N^{(j)})-\sigma _N^{(j)}(1-\sigma _{N-1}^{(j)}). \end{aligned}$$

This reduces the problem to a set of algebraic equations that are solved simultaneously to obtain the steady state solution for density profile in the whole lattice.

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Jindal, A., Verma, A.K. & Gupta, A.K. Cooperative Dynamics in Bidirectional Transport on Flexible Lattice. J Stat Phys 182, 5 (2021).

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  • Bidirectional transport
  • Symmetry breaking
  • Shocks