[Paper Review] Modelling chemotaxis of microswimmers: from individual to collective behavior

Benno Liebchen, Hartmut Löwen|arXiv (Cornell University)|Feb 22, 2018

Micro and Nano Robotics4 citations

TL;DR

This paper presents a theoretical framework coupling chemical diffusion to microswimmer motion to model chemotaxis across individual, pair, and collective scales. It demonstrates that autochemotaxis leads to self-localization or self-avoidance, predator-prey dynamics emerge from chemotactic coupling, and collective patterns like clusters, spirals, and traveling waves arise from multi-particle chemotactic interactions, with scaling laws and nonlinear diffusion effects revealed via dynamical density functional theory.

ABSTRACT

We discuss recent progress in the theoretical description of chemotaxis by coupling the diffusion equation of a chemical species to equations describing the motion of sensing microorganisms. In particular, we discuss models for autochemotaxis of a single microorganism which senses its own secretion leading to phenomena such as self-localization and self-avoidance. For two heterogeneous particles, chemotactic coupling can lead to predator-prey behavior including chase and escape phenomena, and to the formation of active molecules, where motility spontaneously emerges when the particles approach each other. We close this review with some remarks on the collective behavior of many particles where chemotactic coupling induces patterns involving clusters, spirals or traveling waves.

Motivation & Objective

• To develop a theoretical model linking chemical diffusion to microswimmer motion for chemotaxis.
• To analyze autochemotaxis in single microorganisms that sense their own secreted chemicals, leading to self-localization or self-avoidance.
• To investigate chemotactic coupling between two heterogeneous particles, yielding predator-prey dynamics and active molecule formation.
• To explore collective behavior in many-particle systems, where chemotactic interactions induce clustering, spirals, and traveling waves.
• To extend the model to nonlinear chemical kinetics and strong coupling effects using dynamical density functional theory (DDFT).

Proposed method

• Coupling the diffusion equation with source/sink terms to microswimmer motion via chemotactic response to chemical gradients.
• Using the steady-state solution of the diffusion equation with decay (μ) to model screened chemical fields, analogous to Debye-Hückel theory.
• Applying the upper incomplete Gamma function and modified Bessel functions (K₀) to derive analytical solutions in 1D and 2D for chemical concentration profiles.
• Modeling time-dependent emission rates (λₑ(t)) to study propagating chemical waves and non-equilibrium dynamics.
• Extending the framework to nonlinear diffusion via dynamical density functional theory (DDFT), incorporating particle interactions through free energy functionals.
• Introducing front-rear asymmetry in chemical fields due to moving sources to explain scaling laws in predator-prey systems.

Experimental results

Research questions

• RQ1 How does autochemotaxis—where a microorganism responds to its own secreted chemical—lead to self-localization or self-avoidance?
• RQ2 What emergent dynamics arise from chemotactic coupling between two particles secreting different chemicals, such as in predator-prey systems?
• RQ3 How do chemotactic interactions among many particles give rise to collective patterns like clusters, spirals, and traveling waves?
• RQ4 What are the scaling laws for particle motion in autochemotaxis, and how do they deviate from normal diffusion?
• RQ5 How do nonlinear chemical interactions and strong coupling effects modify the standard diffusion equation in chemotactic systems?

Key findings

• Autochemotaxis leads to non-diffusive scaling laws in mean-square displacement, differing from ordinary Brownian motion.
• In 1D, the steady-state chemical profile decays exponentially as c(x) ∝ exp(−κ|x|) with κ = √(μ/Dc), modeling screened diffusion.
• In 2D, the chemical profile follows a modified Bessel function K₀(κr), yielding a logarithmic decay in the absence of decay (μ=0).
• Chemotactic coupling between two particles can produce chase-and-escape dynamics and spontaneous formation of active molecules when particles approach.
• Many-particle systems exhibit collective patterns such as clusters, spirals, and traveling waves, especially in chiral active systems.
• Nonlinear diffusion effects from strong coupling are captured via dynamical density functional theory (DDFT), generalizing the linear diffusion equation to include particle interactions.