Current-density relation in the exclusion process with dynamic obstacles

J. Szavits-Nossan and B. Waclaw

Phys. Rev. E 102, 042117 – Published 14 October 2020

Abstract

We investigate the totally asymmetric simple exclusion process (TASEP) in the presence of obstacles that dynamically bind and unbind from the lattice. The model is motivated by biological processes such as transcription in the presence of DNA-binding proteins. Similar models have been studied before using the mean-field approximation, but the exact relation between the particle current and density remains elusive. Here, we first show using extensive Monte Carlo simulations that the current-density relation in this model assumes a quasiparabolic form similar to that of the ordinary TASEP without obstacles. We then attempt to explain this relation using exact calculations in the limit of low and high density of particles. Our results suggest that the symmetric, quasiparabolic current-density relation arises through a nontrivial cancellation of higher-order terms, similarly as in the standard TASEP.

Received 25 June 2020
Revised 27 August 2020
Accepted 17 September 2020

DOI:https://doi.org/10.1103/PhysRevE.102.042117

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics

Exclusion processes

Asymmetric simple exclusion process Monte Carlo methods Perturbation theory Series expansions & exact enumeration

Statistical Physics & Thermodynamics

Authors & Affiliations

J. Szavits-Nossan ¹ and B. Waclaw^1,2

¹School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
²Centre for Synthetic and Systems Biology, University of Edinburgh, Edinburgh EH9 3BF, United Kingdom

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Vol. 102, Iss. 4 — October 2020

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Images

Figure 1
TASEP with dynamic obstacles. (a) The model with periodic boundary conditions. (b) The model with open boundaries. (c) A space-time plot of a simulation snapshot for the model with periodic boundaries, for $L = 100, M = 20, k_{+} = 0.002, k_{-} = 0.02$ . Particles are black; obstacles are blue. Particles can be seen moving to the right and stopping when encountering obstacles. Obstacles appear at random sites and disappear after a while.
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Figure 2
(a) Current $J$ in the system with periodic boundaries as a function of the particle density $ρ$ . Different colors represent 235 combinations of the parameters $L, k_{-}, k_{+}$ : $L = 10 \dots 1000$ and $k_{-}, k_{+} = 0.001 \dots 20$ . (b) Difference $(J - J_{par}) / J_{\max}$ between numerically determined $J$ and the inverted parabola $J_{par} = 4 J_{\max} ρ (1 - ρ)$ , for the same data as in the panel (a). (c) Root-mean-square deviation $J - J_{par}$ for $k_{-} = k_{+} = 0.001$ (blue), 0.05 (yellow), and 0.2 (green), for different sizes $L$ . Errors (SEM) are smaller than the point size.
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Figure 3
Root-mean-square (rms) deviation of the current $J (ρ)$ obtained in simulations from the parabola $J (ρ) = 4 J_{max} ρ (1 - ρ)$ , for $k_{-}, k_{+} = 0.001 \dots 20$ and fixed $L = 1000$ .
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Figure 4
(a) Current $J (ρ)$ obtained from exact enumeration of all states of ddTASEP (points), with a parabolic fit (line), for $L = 6$ and $k_{+} = k_{-} = 0.1$ . (b) Root-mean-square deviation from the fitted parabola as a function of $k_{+} = k_{-}$ , for three system sizes $L = 5, 6, 7$ (blue, yellow, green).
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Figure 5
Deviation [calculated as $ln (J_{\max, theor} / J_{\max, sim})$ ] between $J_{\max, sim}$ from simulations and $J_{\max, theor}$ predicted from Eq. (19), for $k_{-}, k_{+} = 0.001 \dots 20$ and a fixed $L = 1000$ .
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Figure 6
Particle density $ρ$ and particle current $J$ in the system with open boundaries. Left-hand panels [(a), (b), (e), (f), (i), and (j)]: $k_{-} = k_{+} = 0.1$ . Right-hand panels [(c), (d), (g), (h), (k), and (l)]: $k_{-} = k_{+} = 5$ . Other parameters common to all figures are $v = 1, L = 100$ . Panels [(a), (c), (e), (g), (i), and (k)] correspond to small- $α$ expansion (fixed $β = 1$ ) and panels [(b), (d), (f), (h), (j), and (l)] to small- $β$ expansion (fixed $α = 1$ ). Points represent Monte Carlo simulations and lines are analytic formulas: Eq. (23a) for panels (a) and (c), Eq. (38) for panels (b) and (d), Eq. (23b) for panels (e) and (g), and Eq. (39) for panels (f) and (h). $J (ρ)$ in panels (i), (j), (k), and (l) has been obtained by inverting the equations for $ρ (α), ρ (β)$ for $α = α (ρ), β = β (ρ)$ and inserting them into the equations for $J (α), J (β)$ .
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