Memoryless control of boundary concentrations of diffusing particles

Abstract

Flux between regions of different concentration occurs in nearly every device involving diffusion, whether an electrochemical cell, a bipolar transistor, or a protein channel in a biological membrane. Diffusion theory has calculated that flux since the time of Fick (1855), and the flux has been known to arise from the stochastic behavior of Brownian trajectories since the time of Einstein (1905), yet the mathematical description of the behavior of trajectories corresponding to different types of boundaries is not complete. We consider the trajectories of noninteracting particles diffusing in a finite region connecting two baths of fixed concentrations. Inside the region, the trajectories of diffusing particles are governed by the Langevin equation. To maintain average concentrations at the boundaries of the region at their values in the baths, a control mechanism is needed to set the boundary dynamics of the trajectories. Different control mechanisms are used in Langevin and Brownian simulations of such systems. We analyze models of controllers and derive equations for the time evolution and spatial distribution of particles inside the domain. Our analysis shows a distinct difference between the time evolution and the steady state concentrations. While the time evolution of the density is governed by an integral operator, the spatial distribution is governed by the familiar Fokker-Planck operator. The boundary conditions for the time dependent density depend on the model of the controller; however, this dependence disappears in the steady state, if the controller is of a renewal type. Renewal-type controllers, however, produce spurious boundary layers that can be catastrophic in simulations of charged particles, because even a tiny net charge can have global effects. The design of a nonrenewal controller that maintains concentrations of noninteracting particles without creating spurious boundary layers at the interface requires the solution of the time-dependent Fokker-Planck equation with absorption of outgoing trajectories and a source of ingoing trajectories on the boundary (the so called albedo problem).

Unidirectional Flux In Brownian And Langevin Simulations Of Diffusion

“…This shortcoming of refining the time step is remedied by replacing the constant rate sources with time-step-dependent sources, as predicted by eqs. A similar boundary layer arises at the interface in Langevin simulations [13]. Figure 2 describes the concentration profiles for three different values of ∆t and source strengths that are proportional to 1/ √ ∆t.…”

Langevin Trajectories between Fixed Concentrations

“…Without the approximation the limiting distribution of velocities is (4). Note, however, that injecting trajectories by any Markovian scheme, with the limiting distribution (10) and with any time step ∆t, creates a boundary layer [20]. A LD simulation with C L = 0, C R = 0, and the parameters γ = 100, ε = 1, L = 1, ∆t = 10…”

“…Brownian dynamics simulations with different boundary protocols seem to indicate that density fluctuations near the channels are independent of the boundary conditions, if the boundaries are moved sufficiently far away from the channel [19]. However, as shown in [20], many boundary protocols for maintaining fixed concentrations lead to the formation of spurious boundary layers, which in the case of charged particles may produce large long range fluctuations in the electric field that spread throughout the entire simulation volume Ω. The analytic structure of these boundary layers was determined in [21,22], following several numerical investigations (e.g, [23]).…”

“…The boundary of the simulation has to be interfaced with the bath in a manner that does not distort the physics. This means that new LT have to be injected into the simulation at the correct rate and with the correct distribution of displacement and velocity, for otherwise, spurious boundary layers will form [20].…”

A Poisson–Nernst–Planck Model for Biological Ion Channels—An Asymptotic Analysis in a Three-Dimensional Narrow Funnel

“…Furthermore, it is possible to simulate only the region of the channel and its immediate surrounding, rather than the entire baths. Connecting the discrete simulation region to a continuum modeled bath in a self-consistent way is a challenging mathematical problem [16,17,18]. Computer simulations may eventually predict the correct channel function, possibly even in our lifetime; however, they are no substitute for physical and mathematical insight.…”