Size effects in reactive circular site interactions

Saddawi, Salma; Strieder, William

doi:10.1063/1.3668312

Cited by 6 publications

(6 citation statements)

References 13 publications

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“…In boundary homogenization approaches [6,7,8], the cluster is replaced by a single circular pore on which the Robin boundary condition (1.3) applies, and functional forms of \kappa are postulated and free parameters estimated either numerically or asymptotically. Asymptotic approaches utilize singular perturbation [12,25] or separable methods [42,50] to obtain series approximations of the flux into the absorbing pores.…”

Section: Refining Berg Andmentioning

confidence: 99%

Boundary Homogenization and Capture Time Distributions of Semipermeable Membranes with Periodic Patterns of Reactive Sites

Bernoff¹,

Lindsay²,

Schmidt³

2018

Multiscale Model. Simul.

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We consider the capture dynamics of a particle undergoing a random walk in a halfspace bounded by a plane with a periodic pattern of absorbing pores. In particular, we numerically measure and asymptotically characterize the distribution of capture times. Numerically we develop a kinetic Monte Carlo (KMC) method that exploits exact solutions to create an efficient particlebased simulation of the capture time that deals with the infinite half-space exactly and has a run time that is independent of how far from the pores one begins. Past researchers have proposed homogenizing the surface boundary conditions, replacing the reflecting (Neumann) and absorbing (Dirichlet) boundary conditions with a mixed (Robin) boundary condition. We extend previous work to asymptotically determine the leakage parameter for the mixed boundary condition for arbitrary periodic pore configurations in the dilute fraction limit. In this asymptotic limit, we pose and solve an optimization problem for the Bravais lattice which maximizes the capture rate of the absorbing pores, finding the hexagonal lattice to be the global maximum.

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Section: Refining Berg Andmentioning

confidence: 99%

Boundary Homogenization and Capture Time Distributions of Semipermeable Membranes with Periodic Patterns of Reactive Sites

Bernoff¹,

Lindsay²,

Schmidt³

2018

Multiscale Model. Simul.

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“…As a check on the validity of (2.21), we compare to an exact solution in the N = 2 case for two unit discs separated by d = |x 1 − x 2 |. A separable solution of (1.1) in bi-polar coordinates [35,41] determined that…”

Section: )mentioning

confidence: 99%

“…For the normalization condition (2.1c), the capacitance is determined from lim |x|→∞ u(x) = −C −1 p . Exact solutions to (2.1) have been developed in the simple cases where the absorbing set Γ a is one [40] or two non-overlapping absorbing pores [35,41]. However, these methods rely heavily on exploiting symmetries of the set Γ a and cannot be easily generalized for larger N .…”

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confidence: 99%

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Numerical Approximation of Diffusive Capture Rates by Planar and Spherical Surfaces with Absorbing Pores

Bernoff¹,

Lindsay²

2018

SIAM J. Appl. Math.

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In 1977 Berg & Purcell published a landmark paper entitled Physics of Chemoreception which examined how a bacterium can sense a chemical attractant in the fluid surrounding it [7]. At small scales the attractant molecules move by Brownian motion and diffusive processes dominate. This example is the archetype of diffusive signaling problems where an agent moves via a random walk until it either strikes or eludes a target. Berg & Purcell modeled the target as a sphere with a set of small circular targets (pores) that can capture a diffusing agent. They argued that, in the limit of small radii and wide spacing, each pore could be modeled independently as a circular pore on an infinite plane. Using a known exact solution, they showed the capture rate to be proportional to the combined perimeter of the pores. In this paper we study how to improve this approximation by including inter-pore competition effects and verify this result numerically for a finite collection of pores on a plane or a sphere. Asymptotically we have found the corrections to the Berg-Purcell formula that account for the enhancement of capture due to the curvature of the spherical target and the inhibition of capture due to the spatial interaction of the pores. Numerically we develop a spectral boundary element method for the exterior mixed Neumann-Dirichlet boundary value problem. Our formulation reduces the problem to a linear integral equation, specifically a Neumann to Dirichlet map, which is supported only on the individual pores. The difficulty is that both the kernel and the flux are singular, a notorious obstacle in such problems. A judicious choice of singular boundary elements allows us to resolve the flux singularity at the edge of the pore. In biological systems there can be thousands of receptors whose radii are 0.1% the radius of the cell. Our numerics can now resolve this realistic limit with an accuracy of roughly one part in 10 8 .

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“…In this article we discuss screening effects in the framework of the simplest model assuming that diffusing molecules are point particles and binding sites are perfectly absorbing circular disks located on the otherwise reflecting flat surface. From a mathematical point of view, this is a complicated many-body problem, which has a highly nontrivial exact solution only in the case of a two-disk cluster (17,18). One can learn about sophisticated formalisms developed to analyze the problem in the literature ((17-23) and references therein).…”

Section: Introductionmentioning

confidence: 99%

Trapping by Clusters of Channels, Receptors, and Transporters: Quantitative Description

Berezhkovskii

Dagdug

Lizunov

et al. 2014

Biophysical Journal

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Various membrane functional units such as receptors, transporters, and channels, whose action necessarily involves capturing diffusing molecules, are often organized into multimeric complexes forming clusters on the cell and organelle membranes. These functional units themselves are usually oligomers of several integral proteins, which have their own symmetry. Depending on the symmetry, they form clusters on different packing lattices. Moreover, local membrane inhomogeneities, e.g., the so-called membrane domains, rafts, stalks, etc., lead to different patterns even within the structures on the same packing lattice. Units in the cluster compete for diffusing molecules and screen each other. Here we propose a general approach that allows one to quantify the screening effects. The approach is used to derive simple approximate formulas giving the trapping rates of diffusing molecules by clusters of absorbers on lattices of different packing symmetries. The obtained results describe smooth variation of the trapping rate from the sum of the rates of individual absorbers forming the cluster to the effective collective rate. The latter shows how the trapping efficiency of an individual absorber decreases as the number of absorbers in the cluster increases and/or the inter-absorber distance decreases. Numerical tests demonstrate good agreement between the rates predicted by the theory and obtained from Brownian dynamics simulations for clusters of different shapes and sizes.

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Size effects in reactive circular site interactions

Cited by 6 publications

References 13 publications

Boundary Homogenization and Capture Time Distributions of Semipermeable Membranes with Periodic Patterns of Reactive Sites

Boundary Homogenization and Capture Time Distributions of Semipermeable Membranes with Periodic Patterns of Reactive Sites

Numerical Approximation of Diffusive Capture Rates by Planar and Spherical Surfaces with Absorbing Pores

Trapping by Clusters of Channels, Receptors, and Transporters: Quantitative Description

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