Analytical and Numerical Results for an Escape Problem

Caginalp, Carey; Chen, Xinfu

doi:10.1007/s00205-011-0455-6

Cited by 24 publications

(22 citation statements)

References 6 publications

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“…Integrating (S92) over the area of the circular micro-domain, we arrive at our result in (39). Lastly, we note that the problem of finding the MFET through the fully reactive arc (−ε, ε) of a disk, without LRI potential (U (r) ≡ 0) and without a barrier at the EW (κ = ∞) was solved analytically by Singer et al [60] (see also [61,62]).…”

Section: B 2d Casementioning

confidence: 71%

“…The first term in (46), which is proportional to 1/κ and diverges as 1/ε in the limit ε → 0, is forced to vanish in the idealised case κ ≡ ∞, and the leading small-ε behaviour of T (2) ε in (46) becomes determined by the second term, which exhibits only a slow, logarithmic divergence with ε. At this point it is expedient to recall that the original mixed boundary-value problem in (3) and (4) for the 2D case with κ ≡ ∞ can be solved exactly [60][61][62] and here T (2) ε reads:…”

Section: A Global Mfet For Systems Without Long-range Potentialsmentioning

confidence: 99%

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Diffusive escape through a narrow opening: new insights into a classic problem

Oshanin

2017

Phys. Chem. Chem. Phys.

145

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We study the mean first exit time (T) of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size ε. Focusing on the effects of an energy/entropy barrier at the EW, and of the long-range interactions (LRIs) with the boundary on the diffusive search for the EW, we develop a self-consistent approximation to derive for T a general expression, akin to the celebrated Collins-Kimball relation in chemical kinetics and accounting for both rate-controlling factors in an explicit way. Our analysis reveals that the barrier-induced contribution to T is the dominant one in the limit ε → 0, implying that the narrow escape problem is not "diffusion-limited" but rather "barrier-limited". We present the small-ε expansion for T, in which the coefficients in front of the leading terms are expressed via some integrals and derivatives of the LRI potential. Considering a triangular-well potential as an example, we show that T is non-monotonic with respect to the extent of the attractive LRI, being minimal for the ones having an intermediate extent, neither too concentrated on the boundary nor penetrating too deeply into the bulk. Our analytical predictions are in good agreement with the numerical simulations.

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Section: B 2d Casementioning

confidence: 71%

Section: A Global Mfet For Systems Without Long-range Potentialsmentioning

confidence: 99%

Diffusive escape through a narrow opening: new insights into a classic problem

Oshanin

2017

Phys. Chem. Chem. Phys.

145

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“…(46), in which the matrices K and M are determined by Eqs. (43,48). The Laplacetransformed survival probability is still given by Eq.…”

Section: B Diffusion Outside a Ballmentioning

confidence: 99%

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

Grebenkov

2019

The Journal of Chemical Physics

108

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We propose a general theoretical description of chemical reactions occurring on a catalytic surface with heterogeneous reactivity. The propagator of a diffusion-reaction process with eventual absorption on the heterogeneous partially reactive surface is expressed in terms of a much simpler propagator toward a homogeneous perfectly reactive surface. In other words, the original problem with general Robin boundary condition that includes in particular mixed Robin-Neumann condition, is reduced to that with Dirichlet boundary condition. Chemical kinetics on the surface is incorporated as a matrix representation of the surface reactivity in the eigenbasis of the Dirichlet-to-Neumann operator. New spectral representations of important characteristics of diffusion-controlled reactions, such as the survival probability, the distribution of reaction times, and the reaction rate, are deduced. Theoretical and numerical advantages of this spectral approach are illustrated by solving interior and exterior problems for a spherical surface that may describe either an escape from a ball or hitting its surface from outside. The effect of continuously varying or piecewise constant surface reactivity (describing, e.g., many reactive patches) is analyzed.

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“…Most available analytical results to quantify the firstpassage dynamics were obtained for the mean firstpassage time (MFPT) [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], corresponding to the inverse of the mean rate constant conventionally used in * Electronic address: denis.grebenkov@polytechnique.edu † Electronic address: rmetzler@uni-potsdam.de ‡ Electronic address: oshanin@lptmc.jussieu.fr biochemistry. For a bounded domain the MFPT is typically proportional to the domain volume, and it diverges as the target region shrinks.…”

Section: Introductionmentioning

confidence: 99%

Towards a full quantitative description of single-molecule reaction kinetics in biological cells

Grebenkov

Metzler

Oshanin

2018

Phys. Chem. Chem. Phys.

114

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The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of instances when biomolecules in a biological cell reach their specific binding sites and trigger cellular regulation. Typically, the first-passage properties are given in terms of mean first-passage times. However, modern experiments now monitor single-molecular binding-processes in living cells and thus provide access to the full statistics of the underlying first-passage events, in particular, inherent cell-to-cell fluctuations. We here present a robust explicit approach for obtaining the distribution of FPTs to a small partially reactive target in cylindrical-annulus domains, which represent typical bacterial and neuronal cell shapes. We investigate various asymptotic behaviours of this FPT distribution and show that it is typically very broad in many biological situations, thus, the mean FPT can differ from the most probable FPT by orders of magnitude. The most probable FPT is shown to strongly depend only on the starting position within the geometry and to be almost independent of the target size and reactivity. These findings demonstrate the dramatic relevance of knowing the full distribution of FPTs and thus open new perspectives for a more reliable description of many intracellular processes initiated by the arrival of one or few biomolecules to a small, spatially localised region inside the cell.

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Analytical and Numerical Results for an Escape Problem

Cited by 24 publications

References 6 publications

Diffusive escape through a narrow opening: new insights into a classic problem

Diffusive escape through a narrow opening: new insights into a classic problem

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

Towards a full quantitative description of single-molecule reaction kinetics in biological cells

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