Analysis of controlled biological switches via stochastic motion planning

We consider the problem of regulating a genetic toggle switch by means of an in silico feedback control loop. To achieve this, we first introduce two basic notions of motion planning, and then establish a connection to a class of stochastic optimal control problems concerned with sequential stopping-times. To characterize the desired set of initial conditions, in the context of controlled diffusion processes, we propose a sequence of partial differential equations for which the first one has a known boundary condition, while the boundary conditions of the subsequent ones are determined by the solutions to the preceding steps. We then formulate the control of a bistable system as stochastic motion planning problem, and derive the closed-loop control law that maintains the system inside a prespecified region of its state space. Finally, to provide an autonomous feedback policy, we establish a connection to an eigenvalue problem that describes the asymptotic exit-time of the diffusion process.