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Diffusive flux into a stochastically gated tube

Sean D Lawley

Abstract

Diffusion-influenced reactions in the presence of gates which randomly open and close have been studied for decades in a variety of biophysical and biochemical scenarios. The diffusive flux from a large bulk reservoir to the end of a narrow tube with a stochastically gated entrance has been previously estimated. In this paper, we extend this gated flux estimate to be valid if (i) the tube is not necessarily narrow and/or (ii) the diffusivity differs in the tube versus the bulk. Extension (i) is challenging because it entails a nontrivial three-dimensional geometry. Extension (ii) is challenging because it introduces multiplicative noise. We derive an explicit flux estimate formula and prove that it is exact in certain parameter regimes. We further use stochastic simulations to show that the estimate remains accurate across a very broad range of parameters. Our results differ from prior work on extensions (i) and (ii).

Diffusive flux into a stochastically gated tube

Abstract

Diffusion-influenced reactions in the presence of gates which randomly open and close have been studied for decades in a variety of biophysical and biochemical scenarios. The diffusive flux from a large bulk reservoir to the end of a narrow tube with a stochastically gated entrance has been previously estimated. In this paper, we extend this gated flux estimate to be valid if (i) the tube is not necessarily narrow and/or (ii) the diffusivity differs in the tube versus the bulk. Extension (i) is challenging because it entails a nontrivial three-dimensional geometry. Extension (ii) is challenging because it introduces multiplicative noise. We derive an explicit flux estimate formula and prove that it is exact in certain parameter regimes. We further use stochastic simulations to show that the estimate remains accurate across a very broad range of parameters. Our results differ from prior work on extensions (i) and (ii).
Paper Structure (24 sections, 123 equations, 5 figures)

This paper contains 24 sections, 123 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A pedagogical model
  3. Always open
  4. Gating
  5. Takeaways from the pedagogical model
  6. Tube model
  7. Always open
  8. Gating
  9. Stochastic simulations
  10. Predictions of flux formula
  11. Slow and fast switching
  12. Large $\rho=(L/a)(D_{\mathrm{b}}/D)^\alpha$
  13. Small $\rho=(L/a)(D_{\mathrm{b}}/D)^\alpha$
  14. Fast and slow switching on opposite sides of the gate
  15. Discussion
  16. ...and 9 more sections

Figures (5)

  • Figure 1: Particles can diffuse from a bulk reservoir into a tube and get absorbed at the right end of the tube or wander back into the bulk. The tube is stochastically gated, meaning that its entrance randomly switches between being open (left panel) and closed (right panel). Particles can freely enter and exit the tube at the left end when the gate is open, whereas particles can neither enter nor exit at the left end when the gate is closed. Particles diffuse with diffusivity $D_{\mathrm{b}}$ in the bulk and $D$ in the tube.
  • Figure 2: Plot of $P(0^-)$ in \ref{['eq:Pgatedsimple']} as a function of the open probability $p_0$ in different parameter regimes. See section \ref{['sec:opposite1D']} for details.
  • Figure 3: Comparison of $\mathcal{P}_{\text{approx}}$ in \ref{['eq:Prho0']} (curves) to stochastic simulations (markers).
  • Figure 4: Comparison of $\mathcal{P}_{\text{approx}}$ in \ref{['eq:Pmain']} (black curves) to stochastic simulations (markers). The red circle markers are for $\alpha=0$. The blue $+$ markers and green $\times$ markers are both for $\alpha=1/2$.
  • Figure 5: Plot of $\mathcal{P}_{\text{approx}}$ in \ref{['eq:Pmain']} as a function of the open probability $p_0$ in different parameter regimes. See section \ref{['sec:opposite3D']} for details.