Bressloff Paul C
Department of Mathematics, University of Utah 155 South 1400 East, Salt Lake City, Utah 84112, USA.
Phys Rev E. 2022 Sep;106(3-1):034108. doi: 10.1103/PhysRevE.106.034108.
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 ℓ_{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,ℓ_{0},ℓ_{1},t), where P_{n} is the joint probability density for the set (X_{t},ℓ_{0,t},ℓ_{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 ℓ_{0},ℓ_{1} yields an effective system of equations describing diffusion in a bounded domain Ω, in which there is switching between two Robin boundary conditions on ∂Ω. The corresponding constant reactivities are κ_{j}=Dz_{j} and j=0,1, where z_{j} is the Laplace variable corresponding to ℓ_{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. We calculate the flux due to absorption and use this to compute the resulting MFPT in the presence of a renewal-based stochastic resetting protocol. The latter resets the position and conformational state of the particle as well as the corresponding local times. Finally, we indicate how to extend the analysis to higher spatial dimensions using the spectral theory of Dirichlet-to-Neumann operators.
在本文中，我们开发了一种基于相遇的方法的混合版本，用于反应表面上扩散介导的吸收，该方法考虑了扩散粒子构象状态的随机切换。为简单起见，我们考虑一个两态模型，其中表面吸收的概率取决于当前粒子状态以及粒子在每种状态下在表面附近花费的时间量。后者由一对局部时间ℓ_{n,t}，n = 0,1确定，它们是布朗泛函，用于跟踪时间间隔[0,t]内的粒子 - 表面相遇情况。我们通过为一对广义传播子P_{n}(x,ℓ_{0},ℓ_{1},t)构建微分Chapman - Kolmogorov方程来进行，其中P_{n}是当N_{t}=n时集合(X_{t},ℓ_{0,t},ℓ_{1,t})的联合概率密度，其中X_{t}表示粒子位置，N_{t}是相应的构象状态。对ℓ_{0},ℓ_{1}进行双重拉普拉斯变换，得到一个描述有界域Ω中扩散的有效方程组，其中在∂Ω上两个罗宾边界条件之间存在切换。相应的常数反应率为κ_{j}=Dz_{j}，j = 0,1，其中z_{j}是对应于ℓ_{j}的拉普拉斯变量，D是扩散系数。给定拉普拉斯空间中传播子的解，我们构建了一个用于部分吸收的相应概率模型，这需要对z_{0},z_{1}进行拉普拉斯逆变换。我们通过考虑粒子在半直线上的扩散来说明该理论，其边界x = 0有效地在完全反射和部分吸收状态之间切换。我们计算由于吸收产生的通量，并使用此来计算在基于更新的随机重置协议存在下的所得平均首次通过时间。后者会重置粒子的位置和构象状态以及相应的局部时间。最后，我们指出如何使用狄利克雷 - 诺伊曼算子的谱理论将分析扩展到更高空间维度。
