The escape problem for mortal walkers

Grebenkov, Denis S.; Rupprecht, Jean-François

doi:10.1063/1.4976522

Cited by 64 publications

(53 citation statements)

References 62 publications

(124 reference statements)

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“…The solution of (5) satisfying the boundary condition (∂g n /∂r) r=R = 0 is a linear combination of modified Bessel functions I n (z) and K n (z) of first and second kind, g n (r) = I 0 (α n r/L)K 1 (α n R/L) + K 0 (α n r/L)I 1 (α n R/L), (6) with α n = π 2 n 2 + sL 2 /R 2 . The solution of the inhomogeneous problem with Dirichlet boundary condition at r = ρ reads u 0 (r) = [1 − g 0 (r)/g 0 (ρ)]/s [3,46]. The coefficients a n are determined in the Supplemental Information (SI), section I, and we obtain the final result for the FPT densitỹ…”

Section: Resultsmentioning

confidence: 99%

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

Grebenkov

Metzler

Oshanin

2018

Phys. Chem. Chem. Phys.

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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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Section: Resultsmentioning

confidence: 99%

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

Grebenkov

Metzler

Oshanin

2018

Phys. Chem. Chem. Phys.

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114

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“…One can also easily treat partially absorbing boundaries [52][53][54][55][56][57][58][59][60] by allowing nonzero leakage probability from the sink site x * . If a particle can disappear or loose its activity during diffusion, FPT problems for such "mortal" walkers [61][62][63][64][65][66][67][68] can be treated by introducing two sink sites, x * 1 and x * 2 , that represent an absorbing boundary and a reactive bulk. Using the exchange time distributions ψ xx * 2 (t) depending on x, one can model space-dependent bulk reaction rates.…”

Section: Discussionmentioning

confidence: 99%

Heterogeneous continuous-time random walks

Grebenkov

Tupikina

2018

Phys. Rev. E

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We introduce a heterogeneous continuous time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatio-temporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.

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“…This setting was later called "diffusion-limited reactions", in contrast to conventional "kinetics-limited reactions" [2]. In numerous following studies, the basic diffusion problem (1) was extended in various directions, in particular, by replacing the exterior of a spherical target by an arbitrary Euclidean domain Ω ⊂ R d [3,4], by considering one or multiple targets on the otherwise inert impenetrable boundary [5,6,7,8], by replacing the Laplace operator (ordinary diffusion) by a general second-order elliptic differential operator [9] or a general Fokker-Planck operator [10], by introducing bulk reactivity [11,12,13], trapping events [14,15], or intermittence [16,17,18,19]. However, the focus on the transport step till the first encounter with the target, expressed via Eq.…”

Section: Introductionmentioning

confidence: 99%

Imperfect Diffusion-Controlled Reactions

Grebenkov

2019

Chemical Kinetics

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This chapter aims at emphasizing the crucial role of partial reactivity of a catalytic surface or a target molecule in diffusion-controlled reactions. We discuss various microscopic mechanisms that lead to imperfect reactions, the Robin boundary condition accounting for eventual failed reaction events, and the construction of the underlying stochastic process, the so-called partially reflected Brownian motion. We show that the random path to the reaction event can naturally be separated into the transport step toward the target, and the exploration step near the target surface until reaction. While most studies are focused exclusively on the transport step (describing perfect reactions), the exploration step, consisting is an intricate combination of diffusion-mediated jumps between boundary points, and its consequences for chemical reactions remain poorly understood. We discuss the related mathematical difficulties and recent achievements. In particular, we derive a general representation of the propagator, show its relation to the Dirichlet-to-Neumann operator, and illustrate its properties in the case of a flat surface.

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The escape problem for mortal walkers

Cited by 64 publications

References 62 publications

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

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

Heterogeneous continuous-time random walks

Imperfect Diffusion-Controlled Reactions

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