Markovian robots: Minimal navigation strategies for active particles

Abstract

We explore minimal navigation strategies for active particles in complex, dynamical, external fields, introducing a class of autonomous, self-propelled particles which we call Markovian robots (MR). These machines are equipped with a navigation control system (NCS) that triggers random changes in the direction of self-propulsion of the robots. The internal state of the NCS is described by a Boolean variable that adopts two values. The temporal dynamics of this Boolean variable is dictated by a closed Markov chain-ensuring the absence of fixed points in the dynamics-with transition rates that may depend exclusively on the instantaneous, local value of the external field. Importantly, the NCS does not store past measurements of this value in continuous, internal variables. We show that despite the strong constraints, it is possible to conceive closed Markov chain motifs that lead to nontrivial motility behaviors of the MR in one, two, and three dimensions. By analytically reducing the complexity of the NCS dynamics, we obtain an effective description of the long-time motility behavior of the MR that allows us to identify the minimum requirements in the design of NCS motifs and transition rates to perform complex navigation tasks such as adaptive gradient following, detection of minima or maxima, or selection of a desired value in a dynamical, external field. We put these ideas in practice by assembling a robot that operates by the proposed minimalistic NCS to evaluate the robustness of MR, providing a proof of concept that is possible to navigate through complex information landscapes with such a simple NCS whose internal state can be stored in one bit. These ideas may prove useful for the engineering of miniaturized robots.

Chemotaxis strategies of bacteria with multiple run modes

“…Note in this context that the bias of run times may, in principle, depend on other motion characteristics, such as the speed in the respective run state. This may occur, for example, if the underlying model for the run-time bias relies on a memory kernel ( 16 , 20 – 24 ), but it is also observed in alternative models describing navigation of active particles in concentration gradients ( 25 ).…”

“…In this way, we can reliably predict the long-time chemotactic response even though experimental observations of individual bacteria are restricted to short-time intervals. For this purpose, we generalized the coarse-graining approaches established in ( 18 , 22 , 23 , 25 ) to a theoretical framework that enables us to derive long-term transport properties analytically for arbitrary underlying multimode motility patterns [see also ( 16 , 20 , 21 ) for different theoretical approaches]. By reducing the full dynamics to a Keller-Segel–type equation ( 26 ) for the particle density ρ via a mode expansion we demonstrate that, on large spatial scales and to first order in gradients of the chemoattractant ∇ c , the long-time dynamics of P. putida in an external chemical gradient is given by an upgradient drift (the actual chemotactic response), superimposed by diffusion; as we assumed that run-time distributions decay exponentially for large run times, anomalous diffusion as discussed in ( 27 ) is not expected in this model.…”

“…In this way, we can reliably predict the long-time chemotactic response even though experimental observations of individual bacteria are restricted to short-time intervals. For this purpose, we generalized the coarse-graining approaches established in (18,22,23,25) to a theoretical framework that enables us to derive long-term transport properties analytically for arbitrary underlying multimode motility patterns [see also (16,20,21) for different theoretical approaches]. By reducing the full dynamics to a Keller-Segel-type equation (26) for the particle density  via a mode expansion…”

Chemotaxis in external fields: Simulations for active magnetic biological matter

“…At a coarse-grained level, however, the rates of directional changes (tumbles or reversals) have to depend on both the local oxygen concentration and the local oxygen gradient [12]. The dependence on the oxygen gradient determines the direction of motion (although a recent theoretical study has proposed a mechanism where a dependence on the local oxygen concentration may be sufficient [56]), while a dependence on the oxygen concentration is needed to switch between oxygen as an attractant and a repellent.…”

Chemokinetic Scattering, Trapping, and Avoidance of Active Brownian Particles

“…The original proof can be adapted to inhomogeneous stochastic motion, asserted in [27]. Related results were discussed for position-dependent translational diffusion [6,15,28].…”