Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points

Coombs, Daniel; Straube, Ronny; Ward, Michael J.

doi:10.1137/080733280

Cited by 111 publications

(141 citation statements)

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“…The first term in H is the usual Coulomb singularity in three dimensions, whereas the second term in (2.51b) represents a contribution from surface diffusion on the boundary of the sphere, similar to that studied in [7]. As a remark, for the case of N circular absorbing windows of a common radius ε, the average MFPT,v, is minimized in the limit ε → 0 at the trap configuration {x 1 , .…”

Section: (243)mentioning

confidence: 93%

“…Our results show that, to within the three-term asymptotic approximation, the principal eigenvalue λ(ε) is related to the average MFPTv by λ ∼ 1/(Dv). Related eigenvalue perturbation and optimization problems for the Laplacian in two-dimensional domains with localized interior traps, or with traps on the domain boundary, are studied in [4], [7], [8], [9], and [24] (see also the references therein).…”

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confidence: 99%

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An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Cheviakov¹,

Ward²,

Straube³

2010

Multiscale Model. Simul.

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170

263

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Abstract. The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in R 3 that contains N small nonoverlapping absorbing windows, or traps, on its boundary. For the unit sphere, the method of matched asymptotic expansions is used to derive an explicit three-term asymptotic expansion for the MFPT for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial configuration of the absorbing windows on the boundary of the sphere. The threeterm asymptotic expansion for the average MFPT is shown to be in very close agreement with full numerical results. The average MFPT is shown to be minimized for trap configurations that minimize a certain discrete variational problem. This variational problem is closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling point charges on the unit sphere. Numerical results, based on global optimization methods, are given for both the optimum discrete energy and the arrangements of the centers {x 1 , . . . , x N } of N circular traps on the boundary of the sphere. A scaling law for the optimum discrete energy, valid for N 1, is also derived.Key words. narrow escape, mean first passage time, matched asymptotic expansions, surface Neumann Green's functions, discrete variational problem, logarithmic switchback terms AMS subject classifications. 35B25, 35C20, 35P15, 35J05, 35J08DOI. 10.1137/100782620 1. Introduction. The narrow escape problem concerns the motion of a Brownian particle confined in a bounded domain Ω ∈ R d (d = 2, 3) whose boundary ∂Ω = ∂Ω r ∪ ∂Ω a is almost entirely reflecting (∂Ω r ), except for small absorbing windows, or traps, labeled collectively by ∂Ω a , through which the particle can escape. Denoting the trajectory of the Brownian particle by X(t), the mean first passage time (MFPT) v(x) is defined as the expectation value of the time τ taken for the Brownian particle to become absorbed somewhere in ∂Ω a starting initially fromThe calculation of v(x) becomes a narrow escape problem in the limit when the measure of the absorbing set |∂Ω a | = O(ε d−1 ) is asymptotically small, where 0 < ε 1 measures the dimensionless radius of an absorbing window. Since the MFPT diverges as ε → 0, the calculation of the MFPT v(x) constitutes a singular perturbation problem.The narrow escape problem has many applications in biophysical modeling (see [2], [16], [19], [39] and the references therein). For the case of a two-dimensional domain, the narrow escape problem has been studied with a variety of analytical methods in [19], [42], [43], [20], and Part I of this paper [30]. In this paper, we use

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Section: (243)mentioning

confidence: 93%

mentioning

confidence: 99%

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Cheviakov¹,

Ward²,

Straube³

2010

Multiscale Model. Simul.

Self Cite

170

263

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“…In a different approach, partial differential equations (PDEs) were used on the photoactivation cascade to achieve the same goal of a higher spatial resolution (11)(12)(13). The PDE approach is also useful for deriving general theoretical results, such as first passage times in specific geometries (14)(15)(16)(17). However, PDEs use concentrations of molecules and are therefore impractical when small particle numbers are important.…”

Section: Introductionmentioning

confidence: 98%

Explicit Spatiotemporal Simulation of Receptor-G Protein Coupling in Rod Cell Disk Membranes

Schöneberg

Heck

Hofmann

et al. 2014

Biophysical Journal

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Dim-light vision is mediated by retinal rod cells. Rhodopsin (R), a G-protein-coupled receptor, switches to its active form (R(∗)) in response to absorbing a single photon and activates multiple copies of the G-protein transducin (G) that trigger further downstream reactions of the phototransduction cascade. The classical assumption is that R and G are uniformly distributed and freely diffusing on disk membranes. Recent experimental findings have challenged this view by showing specific R architectures, including RG precomplexes, nonuniform R density, specific R arrangements, and immobile fractions of R. Here, we derive a physical model that describes the first steps of the photoactivation cascade in spatiotemporal detail and single-molecule resolution. The model was implemented in the ReaDDy software for particle-based reaction-diffusion simulations. Detailed kinetic in vitro experiments are used to parametrize the reaction rates and diffusion constants of R and G. Particle diffusion and G activation are then studied under different conditions of R-R interaction. It is found that the classical free-diffusion model is consistent with the available kinetic data. The existence of precomplexes between inactive R and G is only consistent with the data if these precomplexes are weak, with much larger dissociation rates than suggested elsewhere. Microarchitectures of R, such as dimer racks, would effectively immobilize R but have little impact on the diffusivity of G and on the overall amplification of the cascade at the level of the G protein.

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“…The effectiveness of the diffusion mechanism in light of these factors can be understood by studying the first passage time statistics of Brownian walkers to small stationary targets. In many biological settings, the number of individual molecules is typically very large, and so the mean first passage time (MFPT) is an important quantity and the focus of many recent studies [47,24,9,43,1,32,38,37,46,15,10,12,16].…”

Section: Introductionmentioning

confidence: 99%

First Passage Statistics for the Capture of a Brownian Particle by a Structured Spherical Target with Multiple Surface Traps

Lindsay¹,

Bernoff²,

Ward³

2017

Multiscale Model. Simul.

106

153

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Abstract. We study the first passage time problem for a diffusing molecule in an enclosed region to hit a small spherical target whose surface contains many small absorbing traps. This study is motivated by two examples of cellular transport. The first is the intracellular process through which proteins transit from the cytosol to the interior of the nucleus through nuclear pore complexes that are distributed on the nuclear surface. The second is the problem of chemoreception, in which cells sense their surroundings through diffusive contact with receptors distributed on the cell exterior. Using a matched asymptotic analysis in terms of small absorbing pore radius, we derive and numerically verify a high order expansion for the capacitance of the structured target which incorporates surface effects and gives explicit information on interpore interaction through a Coulomb-type discrete energy with additional logarithmic dependencies. In the large N dilute surface trap fraction limit, a single homogenized Robin boundary condition ∂nv + κv = 0 is derived in which κ depends on the total absorbing fraction, the characteristic pore scale, and parameters relating to interpore interactions.

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Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points

Cited by 111 publications

References 46 publications

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Explicit Spatiotemporal Simulation of Receptor-G Protein Coupling in Rod Cell Disk Membranes

First Passage Statistics for the Capture of a Brownian Particle by a Structured Spherical Target with Multiple Surface Traps

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