Stochastically switching diffusion with partially reactive surfaces

In this paper we develop a hybrid version of the encounter-based approach to diffusion-mediated absorption at a reactive surface, which takes into account stochastic switching of a diffusing particle's conformational state. For simplicity, we consider a two-state model in which the probability of surface absorption depends on the current particle state and the amount of time the particle has spent in a neighborhood of the surface in each state. The latter is determined by a pair of local times $\ell_{n,t}$, $n=0,1$, which are Brownian functionals that keep track of particle-surface encounters over the time interval $[0,t]$. We proceed by constructing a differential Chapman-Kolmogorov equation for a pair of generalized propagators $P_n(\x,\ell_0,\ell_1,t)$, where $P_n$ is the joint probability density for the set $(\X_t,\ell_{0,t},\ell_{1,t})$ when $N_t=n$, where $\X_t$ denotes the particle position and $N_t$ is the corresponding conformational state. Performing a double Laplace transform with respect to $\ell_0,\ell_1$ yields an effective system of equations describing diffusion in a bounded domain $\Omega$, in which there is switching between two Robin boundary conditions on $\partial \Omega$. The corresponding constant reactivities are $\kappa_j=D z_j$, $j=0,1$, where $z_j$ is the Laplace variable corresponding to $\ell_j$ and $D$ is the diffusivity. Given the solution for the propagators in Laplace space, we construct a corresponding probabilistic model for partial absorption, which requires finding the inverse Laplace transform with respect to $z_0,z_1$. We illustrate the theory by considering diffusion of a particle on the half-line with the boundary at $x=0$ effectively switching between a totally reflecting and a partially absorbing state. Finally, we indicate how to extend the analysis to higher spatial dimensions using the spectral theory of Dirichlet-to-Neumann operators.