Asymptotic Formulas for Extreme Statistics of Escape Times in 1, 2 and 3-Dimensions
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Asymptotic Formulas for Extreme Statistics of Escape Times in 1, 2 and 3-Dimensions K. Basnayake1 · Z. Schuss2 · D. Holcman1 Received: 20 July 2018 / Accepted: 2 September 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract The first of N identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target sets the time scale of activation, which in general is much faster than the arrival to the target of a single trajectory only. Analytical asymptotic expressions for the minimal time is notoriously difficult to compute in general geometries. We derive here asymptotic laws for the probability density function of the first and second arrival times of a large number N of i.i.d. Brownian trajectories to a small target in 1, 2 and 3-dimensions and study their range of validity by stochastic simulations. The results are applied to activation of biochemical pathways in cellular biology. Keywords Short time asymptotics · Diffusion · Narrow escape · Extreme statistics · Transient · Calcium dynamics · Helmoltz · Dendritic spine Mathematics Subject Classification 35K08 · 35J08 · 35J05 · 60G70 · 92C05 · 92C37
1 Introduction Fast activation of biochemical pathways in cell biology is often initiated by the first arrival of a particle to a small target. This is the case of calcium activation in synapses of neuronal cells (Volfovsky et al. 1999; Holcman et al. 2004; Guerrier et al. 2015), fast photoresponse in rods, cones and fly photoreceptors (Katz et al. 2017; Gross et al. 2012; Reingruber et al. 2013), and many more. However, the time scale underlying
Communicated by Paul Newton.
B
D. Holcman [email protected]
1
Applied Mathematics and Computational Biology, Ecole Normale Supérieure, 46 rue d’Ulm, 75005 Paris, France
2
Department of Applied Mathematics, Tel-Aviv University, 69978 Tel Aviv, Israel
123
Journal of Nonlinear Science
these fast activations is not very well understood. We propose here that these generic molecular mechanisms are initiated by the first arrival of one or more of the many identical independently distributed (i.i.d.) Brownian particles to small receptors, such as the influx of many Brownian neurotransmitters inside a synaptic cleft to receptors (Taflia and Holcman 2011; Freche et al. 2011). In general, one or several particles are required to initiate a cascade of chemical reactions, such as the opening of a protein channel (Hille et al. 2001) of a cellular membrane, which amplifies the inflow of ions to an avalanche of thousands or more ions, resulting from the initial binding of few couple of ions or neurotransmitters. The statistic of the minimal arrival times is referred to in the statistical physics literature as extreme statistics (Sokolov et al. 2005). Despite great efforts (Sokolov et al. 2005; Zilman and Bel 2010; Yuste and Lindenberg 1996; Yuste et al. 2001; Majumdar and Pal 2014; Chou and Dorsogna 2014; Schehr 2012; Redner and Meerson 2014), there are no explicit expressions for the probability distributions of arriva
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