Optimal search processes in physics and biology

In this thesis, navigation and search strategies are investigated from an optimal control and statistical physics perspective. In Ch.1, a multi-agent decision making problem, a cooperative search game, is treated in the framework of optimal control. It is shown that, surprisingly, phenomenological equations that describe chemotaxis –including perfect adaptation and fold-change response– emerge as the optimal solution, in a mean-field approximation, to this cooperative task. To our knowledge, such an equivalence was never noticed before, and it provides an interesting functional interpretation of chemotaxis. The content of this Chapter is available in Pezzotta et al. Phys.Rev.E 98, 042401 (2018). In Ch.2, the dynamics of a statistical mechanical model (the Conformational Spread model) –that accurately reproduces the dynamics of the flagellar motor switch in Escherichia coli (E.coli)– is studied analytically by means of multi-scale techniques (decimation and averaging), providing a cooperative binding model which effectively describe the locked-state time distribution –ultimately determining the run-and-tumble behaviour of E.coli. Studies of the dynamics of this model were previously limited to numerical simulations, and analytical results were achieved only at equilibrium. This work has been published as a research article in Pezzotta, et al. J.Stat.Mech. 023402 (2017). In Ch.3, we formulate a collective navigation task as an optimal control problem, in which agents have an incentive to align their velocities. A multi-scale analysis (averaging and homogenization) is used for studying the optimally controlled dynamics in the over- damped limit. The analytical solution of the effective equations at the steady state is given in particular instances of the problem (two agents on a torus). In Ch.4, it is shown that the conditioning of Markov processes to lie within a confined region of space can be regarded as an optimal search problem. As a case study, we analysed a jump process conditioned to stay within an infinitely long cylindrical domain, and to go from one end to the other. This example is inspired by the problem of sampling configurations of polymers confined in nanochannels. This work has been published in Adorisio, et al. J.Stat.Phys 170, 79-100 (2017), where more details can be found pertaining to the physics of polymers.