Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Bobrowski, Adam

doi:10.1007/s00028-020-00617-7

Cited by 7 publications

(5 citation statements)

References 75 publications

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“…A similar result has been established in [10], where the case of diffusion in 3D layers separated by a flat membrane was studied in detail. In particular, it was shown there that the principle just mentioned, i.e., that permeability coefficients of the membrane become integral parts of the limit equation, is robust to a change in the way the particles filter through the membrane (i.e., the membrane might be partly sticky): while such a change influences the initial condition of the limit Cauchy problem, it does not alter the limit master equation itself.…”

Section: \Biggr) supporting

confidence: 78%

“…The main results: Three scenarios. As already mentioned, in the present paper we deal with different geometry than in [10]: we focus on two dimensions and choose the example of a circular membrane as a case study. A particular goal of this study is to establish the fact that in the thin layer approximation transmission conditions become integral parts of the limit equation in three natural scenarios similar to the scenario described in section 1.2.…”

Section: \Biggr) mentioning

confidence: 99%

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Modeling Diffusion in Thin 2D Layers Separated by a Semipermeable Membrane

Bobrowski¹

2020

SIAM J. Math. Anal.

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Motivated by models of signaling pathways in B-lymphocytes, which have extremely large nuclei, we study the question of how reaction-diffusion equations in thin 2D domains may be approximated by diffusion equations in regions of smaller dimensions. In particular, we study how transmission conditions featuring in the approximating equations become integral parts of the limit master equation. We devise a scheme which, by appropriate rescaling of coefficients and finding a common reference space for all Feller semigroups involved, allows deriving the form of the limit equation formally. The results obtained, expressed as convergence theorems for the Feller semigroups, may also be interpreted as a weak convergence of underlying stochastic processes.

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Section: \Biggr) supporting

confidence: 78%

Section: \Biggr) mentioning

confidence: 99%

Modeling Diffusion in Thin 2D Layers Separated by a Semipermeable Membrane

Bobrowski¹

2020

SIAM J. Math. Anal.

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“…The papers [26,27] extend the result just described to the case where there are two thin layers lying on two sides of a semi-permeable membrane. The role of boundary conditions is then played by various sorts of transmission conditions, but the effect is the same: transmission conditions become integral parts of the limit master equation, and describe jumps of a limit process from one side of the membrane to the other.…”

Section: Thin Layer Approximation In Modelling Of Signalling Pathwaysmentioning

confidence: 92%

On convergence and asymptotic behaviour of semigroups of operators

Bobrowski

Rudnicki

2020

Phil. Trans. R. Soc. A.

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“…However, incorporating the microscopic analog of the permeable boundary condition is non-trivial. A rigorous probabilistic formulation of one-dimensional (1D) Brownian motion (BM) in the presence of a semipermeable barrier has recently been introduced by Lejay [38,39], see also [1,14]. This is based on so-called snapping out BM, which sews together successive rounds of partially reflecting BM that are restricted to either the left-hand or righthand side of the barrier.…”

Section: Introductionmentioning

confidence: 99%

Diffusion with stochastic resetting screened by a semipermeable interface

Bressloff¹

2023

J. Phys. A: Math. Theor.

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In this paper we consider the diffusive search for a bounded target $\Omega \in \R^d$ with its boundary $\partial \Omega$ totally absorbing. We assume that the target is surrounded by a semipermeable interface given by the closed surface $\partial \calM$ with $\Omega \subset \calM\subset \R^d$. That is, the interface totally surrounds the target and thus partially screens the diffusive search process. We also assume that the position of the diffusing particle (searcher) randomly resets to its initial position $\x_0$ according to a Poisson process with a resetting rate $r$. The location $\x_0$ is taken to be outside the interface, $\x_0\in \calM^c$, which means that resetting does not occur when the particle is within the interior of $\partial \calM$. (Otherwise, the particle would have to cross the interface in order to reset to $\x_0$.) Hence, the semipermeable interface also screens out the effects of resetting. We illustrate the theory by explicitly solving the boundary value problem (BVP) for a three-dimensional (3D) spherically symmetric interface and a concentric spherical target. We calculate the mean first passage time (MFPT) to find (be absorbed by) the target and explore its behavior as a function of the permeability $\kappa_0$ of the interface and the radius of the interface $R$. In particular, we find that increasing $R$ for a fixed target size reduces the MFPT and increases the optimal resetting rate at which the MFPT is minimized. We also find that the sensitivity of the MFPT to changes in $\kappa_0$ is a decreasing function of $R$. Finally, we introduce a stochastic single-particle realization of the search process based on a generalization of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of reflecting Brownian motion on either side of the interface. The main challenge is establishing that the probability density generated by the snapping out BM satisfies the permeable boundary conditions at the interface. We show how this can be achieved using renewal theory.

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Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Cited by 7 publications

References 75 publications

Modeling Diffusion in Thin 2D Layers Separated by a Semipermeable Membrane

Modeling Diffusion in Thin 2D Layers Separated by a Semipermeable Membrane

On convergence and asymptotic behaviour of semigroups of operators

Diffusion with stochastic resetting screened by a semipermeable interface

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