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Derivation of the bacterial run-and-tumble kinetic model : quantitative and strong convergence results

Alain Blaustein

TL;DR

The article rigorously derives a gradient-dependent run-and-tumble kinetic model from a detailed internal-variable framework for bacterial signaling. By introducing a fast-adapting methylation variable $m$ and a rescaled Maxwellian structure, it shows that as the adaptation-to-observation ratio $\varepsilon\to 0$, the microscopic model converges strongly in $L^1$ to the macroscopic, gradient-sensing kinetics with an explicit tumbling kernel $\bar{\Lambda}$. The authors provide explicit convergence rates leveraging a relative-entropy approach and a careful control of the Maxwellian concentration, achieving global-in-time estimates and precise links between the internal dynamics and the emergent chemotactic bias. This work clarifies how intracellular signaling can give rise to gradient sensing in a mathematically controlled way and informs the validity range of gradient-dependent kinetic models in chemotaxis. It also outlines several extensions, including removing the well-preparedness assumption and exploring more general internal dynamics and noise structures with potential nonlinear couplings to the external signal.

Abstract

During the past century, biologists and mathematicians investigated two mechanisms underlying bacteria motion: the run phase during which bacteria move in straight lines and the tumble phase in which they change their orientation. When surrounded by a chemical attractant, experiments show that bacteria increase their run time as moving up concentration gradients, leading to a biased random walk towards favorable regions. This observation raises the following question, which has drawn intense interest from both biological and mathematical communities: what cellular mechanisms enable bacteria to feel concentration gradients\,? In this article, we investigate an asymptotic regime that was proposed to explain this ability thanks to internal mechanisms. More precisely, we derive the run-and-tumble kinetic equation with concentration's gradient dependent tumbling rate from a more comprehensive model, which incorporates internal cellular mechanisms. Our result improves on previous investigations, as we obtain strong convergence towards the gradient dependent kinetic model with quantitative and formally optimal convergence rates. The main ingredient consists in identifying a set of coordinates for the internal cellular dynamics in which concentration gradients arise explicitly. Then, we use relative entropy methods in order to capture quantitative measurement of the distance between the model incorporating cellular mechanisms and the one with concentration gradient dependent tumbling rate.

Derivation of the bacterial run-and-tumble kinetic model : quantitative and strong convergence results

TL;DR

The article rigorously derives a gradient-dependent run-and-tumble kinetic model from a detailed internal-variable framework for bacterial signaling. By introducing a fast-adapting methylation variable and a rescaled Maxwellian structure, it shows that as the adaptation-to-observation ratio , the microscopic model converges strongly in to the macroscopic, gradient-sensing kinetics with an explicit tumbling kernel . The authors provide explicit convergence rates leveraging a relative-entropy approach and a careful control of the Maxwellian concentration, achieving global-in-time estimates and precise links between the internal dynamics and the emergent chemotactic bias. This work clarifies how intracellular signaling can give rise to gradient sensing in a mathematically controlled way and informs the validity range of gradient-dependent kinetic models in chemotaxis. It also outlines several extensions, including removing the well-preparedness assumption and exploring more general internal dynamics and noise structures with potential nonlinear couplings to the external signal.

Abstract

During the past century, biologists and mathematicians investigated two mechanisms underlying bacteria motion: the run phase during which bacteria move in straight lines and the tumble phase in which they change their orientation. When surrounded by a chemical attractant, experiments show that bacteria increase their run time as moving up concentration gradients, leading to a biased random walk towards favorable regions. This observation raises the following question, which has drawn intense interest from both biological and mathematical communities: what cellular mechanisms enable bacteria to feel concentration gradients\,? In this article, we investigate an asymptotic regime that was proposed to explain this ability thanks to internal mechanisms. More precisely, we derive the run-and-tumble kinetic equation with concentration's gradient dependent tumbling rate from a more comprehensive model, which incorporates internal cellular mechanisms. Our result improves on previous investigations, as we obtain strong convergence towards the gradient dependent kinetic model with quantitative and formally optimal convergence rates. The main ingredient consists in identifying a set of coordinates for the internal cellular dynamics in which concentration gradients arise explicitly. Then, we use relative entropy methods in order to capture quantitative measurement of the distance between the model incorporating cellular mechanisms and the one with concentration gradient dependent tumbling rate.
Paper Structure (6 sections, 5 theorems, 139 equations)

This paper contains 6 sections, 5 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main result
  3. A priori estimates
  4. Convergence of the re-scaled distribution
  5. Proof of Theorem \ref{['TH:MAIN']}
  6. Conclusion and perspectives

Key Result

Theorem 2.1

Under assumptions hyp:M on $M$, hyp:Lambda+-hyp:Lambda- on $\Lambda$, hyp1:p:eps-hyp0:p on $\left(p^\varepsilon _0\right)_{\varepsilon >0}$ and hyp1:p on $\bar{p}_0$, consider a sequence of strong solutions $\left(p^\varepsilon \right)_{\varepsilon >0}$ to eq:p-def:tumbling:op with initial data $\le and where $\mathcal{M}_{\varepsilon ,N}$ is defined in def:M:eps and where the solution $N$ to eq:

Theorems & Definitions (9)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof