Chemokinetic Scattering, Trapping, and Avoidance of Active Brownian Particles

Justus A. Kromer, Noelia de la Cruz, and Benjamin M. Friedrich

Phys. Rev. Lett. 124, 118101 – Published 17 March 2020

Abstract

We present a theory of chemokinetic search agents that regulate directional fluctuations according to distance from a target. A dynamic scattering effect reduces the probability to penetrate regions with high fluctuations and thus reduces search success for agents that respond instantaneously to positional cues. In contrast, agents with internal states that initially suppress chemokinesis can exploit scattering to increase their probability to find the target. Using matched asymptotics between the case of diffusive and ballistic search, we obtain analytic results beyond Fox colored noise approximation.

Received 18 April 2019
Revised 3 December 2019
Accepted 21 February 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.118101

Physics Subject Headings (PhySH)

Brownian motion Cell migration Cell signaling Chemotaxis Diffusion Random walks Stochastic processes

Nonlinear DynamicsStatistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Justus A. Kromer¹, Noelia de la Cruz², and Benjamin M. Friedrich ^3,4,5,*

¹Department of Neurosurgery, Stanford University, Palo Alto, California 94304, USA
²Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada
³cfaed, TU Dresden, 01069 Dresden, Germany
⁴Institute of Theoretical Physics, TU Dresden, 01069 Dresden, Germany
⁵Cluster of Excellence Physics of Life, TU Dresden, 01307 Dresden, Germany

^*benjamin.m.friedrich@tu-dresden.de

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Vol. 124, Iss. 11 — 20 March 2020

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Images

Figure 1
Chemokinesis with position-dependent directional fluctuations as consequence of chemotaxis at low concentrations. (a) Radial concentration field $c (x)$ established by diffusion from a source (using parameters for sperm chemotaxis [9]): source (red dot), isoconcentration lines (dashed). (b) Computed signal-to-noise ratio (SNR) of chemotaxis decreases as function of radial distance $R$ from the source. (c) The corresponding effective rotational diffusion coefficient $D_{rot}$ displays a maximum at a characteristic distance (dashed), where $c (x)$ is still large, but $SNR$ is low. (d) Simulated trajectory [labeled (I)] of a chemokinetic ABP subject to $D_{rot} (x)$ from (c), corresponding to instantaneous chemokinesis. A second ABP [labeled (S)] with two-state chemokinesis initially moves ballistically (state 0), but switches to chemokinesis (state 1) as in (I) once it reaches a threshold distance (yellow) for the first time. Shown are two-dimensional reconstructions of three-dimensional trajectories obtained from numerical integration of Eqs. (1) and (2). (e) Penetration probability $p (R | R_{2})$ that an ABP starting at distance $R_{2}$ reaches distance $R$ before returning to $R_{2}$ . For instantaneous chemokinesis (I), this probability is lower than for ballistic motion (dashed). For two-state chemokinesis (S), $p (R | R_{2})$ is higher. Simulation results (black) compare favorably to our analytic theory (red). For parameters, see SM [11].
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Figure 2
Dynamic scattering of chemokinetic agents. (a) Trajectories entering zone 1 (blue) are scattered back to zone 2 due to a high rotational diffusion coefficient $D_{1} > 0$ in zone 1. ( $a^{'}$ ) Penetration probability $p (R | R_{2})$ that an ABP starting at distance $R_{2}$ reaches distance $R$ before returning to $R_{2}$ , analogous to Fig. 1: simulation (black), theory (red), ballistic motion with $D_{1} = D_{2} = 0$ (dashed). ( $a^{''}$ ) Residence time $T (x, t_{0})$ at space position $x$ before time $t_{0}$ as function of radial distance $R = | x |$ . By a mean-chord-length theorem, $\lim_{t_{0} \to \infty} T (x, t_{0}) = T^{\infty} = (π R_{2}^{2} v)^{- 1}$ . (b) Same as (a), but with $D_{1} = 0$ , $D_{2} > 0$ : trajectories can become trapped in zone 1 due to inward scattering of outgoing trajectories at $R_{1}$ . (c) Same as (b), but for two-state chemokinesis: ABPs initially move ballistically with $D_{rot} = 0$ (state 0), and switch to chemokinesis with $D_{rot} = D_{rot} (x)$ as in (b) after crossing a threshold distance (yellow) at $R_{1}$ (state 1). ( $c^{'}$ ) The penetration probability is higher compared to ballistic motion. ( $c^{''}$ ) The mean residence time is now spatially inhomogeneous. Parameters, $R_{1} = R_{2} / 2$ : (a) $D_{1} = 10 v / R_{2}$ , $D_{2} = 0$ ; (b) $D_{1} = 0$ , $D_{2} = 10 v / R_{2}$ ; reference case (dashed) in ( $b^{'}$ ) $D_{1} = D_{2} = 10 v / R_{2}$ .
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Figure 3
Analytic theory by matched asymptotics. (a) The probabilities $p_{1}$ and $p_{2}$ to pass zone 1 and 2, respectively, and the return probability $p_{ret}$ at the boundary between zone 1 and 2, jointly determine the probability $p (R_{0} | R_{2})$ to reach a target of radius $R_{0}$ for ABPs starting at $R_{2}$ ; see Eq. (4). (b) Trade-off between $p_{2}$ (black) and $p_{ret}$ (red) as function of $D_{2}$ . Simulation (dots), limit of high $D_{rot}$ (dotted), ballistic motion [Eq. (5)] (dashed), matched asymptotics [Eq. (6)] (solid).
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