Effective Dynamics of Microorganisms That Interact with Their Own Trail

W. Till Kranz, Anatolij Gelimson, Kun Zhao, Gerard C. L. Wong, and Ramin Golestanian

Phys. Rev. Lett. 117, 038101 – Published 11 July 2016

Abstract

Like ants, some microorganisms are known to leave trails on surfaces to communicate. We explore how trail-mediated self-interaction could affect the behavior of individual microorganisms when diffusive spreading of the trail is negligible on the time scale of the microorganism using a simple phenomenological model for an actively moving particle and a finite-width trail. The effective dynamics of each microorganism takes on the form of a stochastic integral equation with the trail interaction appearing in the form of short-term memory. For a moderate coupling strength below an emergent critical value, the dynamics exhibits effective diffusion in both orientation and position after a phase of superdiffusive reorientation. We report experimental verification of a seemingly counterintuitive perpendicular alignment mechanism that emerges from the model.

Received 26 April 2015

DOI:https://doi.org/10.1103/PhysRevLett.117.038101

Physics Subject Headings (PhySH)

Chemotaxis Stochastic processes

Cells

Physics of Living Systems

Authors & Affiliations

W. Till Kranz¹, Anatolij Gelimson¹, Kun Zhao^2,3, Gerard C. L. Wong³, and Ramin Golestanian^1,*

¹Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom
²Key Laboratory of Systems Bioengineering, Ministry of Education, School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, People’s Republic of China
³Bioengineering Department, Chemistry and Biochemistry Department, California Nano Systems Institute, UCLA, Los Angeles, California 90095-1600, USA

^*ramin.golestanian@physics.ox.ac.uk

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Issue

Vol. 117, Iss. 3 — 15 July 2016

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Images

Figure 1
Sample trajectories generated by the effective dynamics [Eqs. (1a), (1c), and (1d)] over a period of time $1 0^{3} τ$ (color coded) for no interaction with the trail, $Ω τ \equiv 0$ (a); weak interaction, $Ω τ = 1.2$ (b); strong interaction, $Ω τ = 1.85$ (c), close to the localization transition, and above it, $Ω τ = 2.15$ (d). The rotational diffusivity is set to $D_{r}^{0} τ = 10^{- 2}$ , and $2 τ$ is the trail-crossing time. (e) shows a magnification of the end of trail (b) with the trail field $ψ (r, t)$ of width $2 R$ in green and the current orientation of the microorganism (ellipse), $\hat{n}$ . (f) Schematic depiction of a microscopic model system such as P. aeruginosa that uses pili for motility and sensing. (g) A schematic for the definition of $Δ θ$ , which is the angle between the current body orientation and the bacterial trajectory (dotted line).
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Figure 2
Angular MSD $δ φ^{2} (t)$ (a) and translational MSD $δ r^{2} (t)$ normalized by the trail width $R$ (b) as a function of time $t$ for several values of the effective turning rate $Ω τ = 0$ , 0.4, 0.7, 1.0, 1.3, 1.6, 1.9. The microscopic diffusivity is set to $D_{r}^{0} τ = 0.1$ . The inset in (a) shows $δ φ^{2} (t)$ for $Ω τ = 1.99$ demonstrating the intermediate superballistic regime. (c) Color-coded translational diffusivity $D$ normalized by the trail width $R$ and the trail-crossing time $τ$ as a function of the control parameters. Note the logarithmic axes. Inset: The translational diffusivity for the same range of parameters normalized by the trail-free ( $Ω τ \equiv 0$ ) value $D^{0}$ on a linear scale. The white contour lines are 0.1 apart.
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Figure 3
Phase diagram of the dynamics of the microorganism with trail-mediated self-interaction, as a function of the dimensionless turning frequency $Ω τ$ . Inset: Enlarged view of Fig. 1.
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Figure 4
(a) The experimentally observed narrowing of the $Δ θ$ distribution with increasing trail deposition [see Fig. 1 for the definition of $Δ θ$ ]. The experiments were carried out with the P. aeruginosa mutants $Δ p s l D$ , which does not secrete Psl (light green) and $Δ P_{p s l} / P_{B A D} − p s l$ , which secretes Psl in response to the arabinose in the environment. For $Δ P_{p s l} / P_{B A D} − p s l$ , the arabinose concentration was varied between 0% (light blue, low Psl deposition) and 1% (light red, high Psl deposition). (b) The corresponding theoretical $Δ θ$ distributions result from the influence of the alignment term. The trail coupling strength is varied between $Ω τ = 0.05$ (green), $Ω τ = 0.12$ (blue), and $Ω τ = 0.35$ (red). The distribution has been sampled using the time interval of $Δ t = 0.2 τ$ , and the rotational diffusion has been set to $D_{r}^{0} τ = 0.015$ . These values roughly correspond to the experimental parameters at which the distribution has been measured. (c) A comparison between experimental (dots) and theoretical (lines) distributions of $Δ θ$ for the $Δ P_{p s l} / P_{B A D} − p s l$ mutant under 0% (blue) and 1% arabinose (red).
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