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Crossover from ballistic transport to normal diffusion: a kinetic view

Zhe Xue, Weiran Sun, Zhennan Zhou, Min Tang

TL;DR

This work addresses the mechanism by which a population can transition from ballistic transport to normal diffusion, proposing an intracellularly motivated kinetic description that links run-and-tumble-like dynamics to macroscopic dispersion. By building an individual-based model with internal adaptation dynamics and deriving a corresponding kinetic PDE, the authors prove two distinct macroscopic limits: a self-similar ballistic density with MSD scaling as $\mathrm{MSD}=C_0 t^2$ when $\gamma>\tfrac{1}{2}$ (and $\mu=0$), and a diffusion-dominated density solving $\partial_t \rho - C \Delta_x \rho=0$ for $\gamma\le 0$ (with $\mu=1$). The analysis combines explicit solutions of the transformed kinetic equations with rigorous asymptotic expansions, and is supported by numerical simulations showing ballistic, diffusive, and crossover behaviors across adaptation times. The results illuminate how intracellular noise and adaptation times govern macroscopic transport regimes, offering a mechanistic basis for ballistic-to-diffusive crossovers and a framework applicable to diverse biological and physical systems.

Abstract

The crossover between dispersion patterns has been frequently observed in various systems. Inspired by the pathway-based kinetic model for E. coli chemotaxis that accounts for the intracellular adaptation process and noise, we propose a kinetic model that can exhibit a crossover from ballistic transport to normal diffusion at the population level. At the particle level, this framework aligns with a stochastic individual-based model. Using numerical simulations and rigorous asymptotic analysis, we demonstrate this crossover both analytically and computationally. Notably, under suitable scaling, the model reveals two distinct limits in which the macroscopic densities exhibit either ballistic transport or normal diffusion.

Crossover from ballistic transport to normal diffusion: a kinetic view

TL;DR

This work addresses the mechanism by which a population can transition from ballistic transport to normal diffusion, proposing an intracellularly motivated kinetic description that links run-and-tumble-like dynamics to macroscopic dispersion. By building an individual-based model with internal adaptation dynamics and deriving a corresponding kinetic PDE, the authors prove two distinct macroscopic limits: a self-similar ballistic density with MSD scaling as when (and ), and a diffusion-dominated density solving for (with ). The analysis combines explicit solutions of the transformed kinetic equations with rigorous asymptotic expansions, and is supported by numerical simulations showing ballistic, diffusive, and crossover behaviors across adaptation times. The results illuminate how intracellular noise and adaptation times govern macroscopic transport regimes, offering a mechanistic basis for ballistic-to-diffusive crossovers and a framework applicable to diverse biological and physical systems.

Abstract

The crossover between dispersion patterns has been frequently observed in various systems. Inspired by the pathway-based kinetic model for E. coli chemotaxis that accounts for the intracellular adaptation process and noise, we propose a kinetic model that can exhibit a crossover from ballistic transport to normal diffusion at the population level. At the particle level, this framework aligns with a stochastic individual-based model. Using numerical simulations and rigorous asymptotic analysis, we demonstrate this crossover both analytically and computationally. Notably, under suitable scaling, the model reveals two distinct limits in which the macroscopic densities exhibit either ballistic transport or normal diffusion.
Paper Structure (49 sections, 1 theorem, 144 equations, 10 figures, 2 tables)

This paper contains 49 sections, 1 theorem, 144 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The individual-based model
  3. The two-state model and simplifications
  4. The two-state model
  5. The necessary ingredients for the transition in the CCW duration time
  6. The simplified one-state model
  7. Kinetic model with cell movement
  8. Kinetic model with internal dynamics
  9. The crossover from ballistic transport to normal diffusion
  10. MSD.
  11. The run time distribution (RTD).
  12. PDF of $\Delta x / \Delta t^{\beta}$.
  13. The PDE model and main results
  14. The PDE model and nondimensionalization
  15. The scalings in the numerical simulations
  16. ...and 34 more sections

Key Result

Theorem 3.1

Let $f_{\epsilon}$ satisfy the equation pdemodel with the tumbling rate function $\Lambda$ given by Lambda and adaptation function $g(m)$ given by gm.

Figures (10)

  • Figure 1: The distribution of the CCW duration time with $\alpha_1 = 10$, $\alpha_2 = -2$, $t_0 = 300$, $t_1 = 30$, $\bar{Y} = 5$, $T_m = 6000$ and $\sigma = 0.456$ in the two state model \ref{['TupaperLambda']} and \ref{['TupaperY']}. (a) The log-log plot of $P(T_{ccw} > t)$. The slope $-0.65891$ is fitted in the interval $[30, 10^3]$. (b) The semi-log plot $P(T_{ccw} > t)$. The straight dashed line is fitted in the interval $[5\times 10^3, 1.5\times10^4]$ which indicates the CCW duration time distribution decays exponentially.
  • Figure 2: Comparison of different values of $\alpha_1$ in $\Lambda_1$ in \ref{['TupaperLambda']}. (a) $\Lambda_1(\Delta Y) = \exp \left(\alpha_1 \frac{\Delta Y}{\bar{Y}}\right)$ for $\alpha_1 = 10$ (dash-dot line) and $\alpha_1 = 0.1$ (dotted line). (b) The log-log plot of $P(T_{ccw} > t)$ for $\alpha_1 = 10$ (red stars) and $\alpha_1 = 0.1$ (blue triangles); The slope $-0.65891$ is fitted over the interval $[30, 10^3]$. (c) The semi-log plot for $\alpha_1 = 10$ (red stars) and $\alpha_1 = 0.1$ (blue triangles). The inset shows the CCW duration distribution for $\alpha_1 = 0.1$ on the interval $[0, 60]$. Here $\alpha_2 = -2$, $t_0 = 300$, $t_1 = 30$, $\bar{Y} = 5$, $T_m = 6000$ and $\sigma = 0.456$.
  • Figure 3: The distribution of CCW duration for different values of $T_m$ and $\sigma$ in \ref{['mtequation']}. (a) The log-log plot of $P(T_{ccw} > t)$ for $(T_m, \sigma)=(6000, 0.456)$ (red stars), $(60, 0.456)$ (blue circles) and $(6000, 0.00456)$ (magenta diamonds). (b) The semi-log plot for $(T_m , \sigma)=(6000, 0.456)$ (red stars), $(60, 0.456)$ (blue circles) and $(6000, 0.00456)$ (magenta diamonds). The straight dash-dot line is fitted for the red stars between $5 \times 10^3$ and $1.5 \times 10^4$. The inset shows the CCW duration distribution for $(T_m = 60, \sigma = 0.456)$ and $(T_m = 6000, \sigma = 0.00456)$ on the interval $[0, 1000]$. Here $\alpha_1 = 10$, $\alpha_2 = -2$, $t_0 = 300$, $t_1 = 30$ and $\bar{Y} = 5$.
  • Figure 4: The distribution of CCW duration for two different $g(\Delta Y)$. (a) The log-log plot of $P(T_{ccw} > t)$ for $g(\Delta Y) = \Delta Y$ (red stars) and $g(\Delta Y) = \text{sgn}_0(\Delta Y)$ (blue triangles). (b) The semi-log plot with a logarithmic y-axis for $P(T_{ccw} > t)$ with the two $g(\Delta Y)$. Here, $\alpha_1 = 10$, $\alpha_2 = -2$, $t_0 = 300$, $t_1 = 30$ and $\bar{Y} = 5$$T_m = 6000$ and $\sigma = 0.456$.
  • Figure 5: The distribution of $\Delta Y$ when the cells are at different rotational states. CCW state: orange squares; CW state: cyan squares. Particles from CW to CCW: red stars; from CCW to CW: blue crosses. The two insets display the distributions of red stars and blue crosses near the origin. Here $\alpha_1 = 10$, $\alpha_2 = -2$, $t_0 = 300$, $t_1 = 30$, $\bar{Y} = 5$, $T_m = 6000$ and $\sigma = 0.456$.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Theorem 3.1
  • Remark 3.1