MULTISCALE MODEL. SIMUL. c 20XX Society for Industrial and Applied Mathematics Vol. 0, No. 0, pp. 000–000 BROWNIAN MOTION IN DIRE STRAITS∗ D. HOLCMAN† AND Z. SCHUSS‡ Abstract. The passage of Brownian motion through a bottleneck in a bounded domain is a rare event, and as the bottleneck radius shrinks to zero the mean time for such passage increases indefinitely. Its calculation reveals the effect of geometry and smoothness on the flux through the bottleneck. We find new behavior of the narrow escape time through bottlenecks in planar and spatial domains and on a surface. Some applications in cellular biology and neurobiology are discussed. Key words. AUTHOR MUST PROVIDE AMS subject classifications. AUTHOR MUST PROVIDE DOI. 10.1137/110857519 1. Introduction. The narrow escape problem is to calculate the mean first pas- sage time (MFPT) of Brownian motion from a domain with mostly reflecting bound- ary to a small absorbing window. This time is also referred to as the narrow escape time (NET). The problem, which goes back to Lord Rayleigh [1] in the context of the theory of sound, arises in mechanics and continuum theory [2], [3]. The more recent interest in the problem is due to its relevance to molecular biophysics, most significantly to neuroscience. It concerns dendritic spines, which are believed to be the locus of postsynaptic transmission. Recognized more than 100 years ago by Ram´on y Cajal, dendritic spines are small terminal protrusions on neuronal dendrites and are the postsynaptic parts of excitatory synaptic connections. The spine consists of a relatively narrow cylindrical neck connected to a bulky head. The geometrical shape of a spine correlates with its physiological function [4], [5], [6], [7], [8]. Several physiological phenomena are regulated by diffusion in dendritic spines. For example, synaptic plasticity is induced by the transient increase of calcium concentration in the spine, which is regulated by spine geometry, by endogenous buffers, and by the number and rates of exchangers [6], [9], [10], [11], [12]. Another significant function of the spine is the regulation of the number and type of receptors that contribute to the shaping of the synaptic current [13], [14], [15], [16]. Indeed, the neurotransmitter receptors, such as AMPA and NMDA, whose motion on the spine surface is diffusion, mediate the glutamatergic-induced synaptic current. Thus dendritic spines regulate both two-dimensional motion of neurotransmitter receptors on its surface and three- dimensional diffusive motion of ions (e.g., calcium), molecules, proteins (e.g., mRNA), or small vesicles in the bulk. Our results give a quantitative measure of the effect of geometry on regulation of flux. In a biochemical context, the narrow escape time (NET) accounts for the local geometry near an active binding site occluded by the molecular structure of the pro- tein. This is the case for active sites of complex molecules, such as hemoglobin or penicillin-binding proteins, which are hidden inside α-andβ-sheet structures. In that ∗Received by the editors December 1, 2011; accepted for publication (in revised form) June 11, 2012; published electronically DATE. http://www.siam.org/journals/mms/x-x/85751.html †Group of Applied Mathematics and Computational Biology, IBENS, Ecole Normale Sup´erieure, 75005 Paris, France ([email protected]). This author’s research was supported by an ERC Starting Grant. ‡Department of Mathematics, Tel-Aviv University, Tel-Aviv 69978, Israel ([email protected]). 1 2 D. HOLCMAN AND Z. SCHUSS Fig. 1. Narrow straits formed by a partial block (solid disk) of the passage from the head to the neck of the domain enclosed by the black line. Inside the circle the narrow straits can be approximated by the gap between adjacent circles, as in Figure 2 (left). case, a ligand, such as β-lactam antibiotic, has to bind to a small site hidden inside the molecule [17]. Another application is that of the turnaround time of Brownian needle. The mean turnover time provides the time scale associated with double strand DNA break motion confined between two-dimensional membranes [18]. The geometry is the main controller of the flux through narrow passages, an effect that is ubiquitous in biological systems. The common feature of the geometries studied in this paper is the cusp-shaped narrow passage leading to the absorbing boundary. More specific ap- plications of the NET to dendritic spines are given in [33], where composite domains are studied. The NET was calculated in [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] for small absorbing windows in a smooth reflecting boundary. Several more complex cases were considered in [24], [25], [26], such as the NET through a window at a corner or at a cusp in the boundary and the NET on Riemannian manifolds. The calculation of the NET in composite domains with long necks was attempted in [30], [31], [32] and ultimately accomplished in [33]. The NET problem in a planar domain with an absorbing window at the end of a funnel was considered in [34]. The case of planar domains that consist of large compartments interconnected by funnel-shaped bottlenecks was also considered in [34]. In this paper we consider the NET problem for Brownian motion in two- and three-dimensional domains with several types of geometries. First, we consider do- mains whose boundaries are smooth and reflecting, except for a small absorbing win- dow at the end of a cusp-shaped funnel. The cusp can be formed by a partial block of a planar domain, as shown in Figure 1. A mathematical idealization is shown in Figure 2, where the narrow neck can be asymmetric with finite or zero curvature (see Figure 2 (right)). The NET from this type of a domain was calculated in [34] only for the planar case. The NET problem in this type of domain cannot be solved by the methods of [24], [25], [26], [35], because the contribution of the singular part of Neumann’s function to the MFPT is not necessarily dominant. Also the method of matched asymptotic expansions, used in [19], [20], [21], [36] for calculating the MFPT to a small absorbing disc on a smooth boundary, requires major modifications for an interface at the end of a funnel, because the boundary layer problem does not reduce to the classical electri- BROWNIAN MOTION IN DIRE STRAITS 3 Fig. 2. Geometry near a cusp. Left: The planar (dimensional) domain Ω is bounded by a large circular arc connected smoothly to a funnel formed by moving apart two tangent circular arcs of radius Rc ε (i.e., AB = ε). Right: Blowup of the cusp region. The solid, dashed, and dotted necks correspond to ν± =1, 0.4,and5 in (12), respectively. Fig. 3. Composite domains with narrow necks connected to bulky heads with or without a funnel serve as mathematical idealizations of the cross sections of neuronal spine morphologies. Left: The bulky head Ω1 is connected smoothly by an interface ∂Ωi = AB toanarrowneckΩ2.Theentire boundary is ∂Ωr (reflecting), except for a small absorbing part ∂Ωa = CD. Right: The neck is connected to the bulky head without a funnel. fied disk problem [37]. Altogether different boundary or internal layers at absorbing windows located at the end of a cusp-like funnel are needed. Second, we consider the NET problem in composite domains that consist of a head connected with or without a funnel to a narrow cylindrical neck, as shown in Figure 3. Finally, we consider the NET problem and the exit probability when there are N absorbing windows at the ends of narrow necks. These are related to the prin- cipal eigenvalue of the Laplacian in domains that consist of heads interconnected by narrow necks, which, in turn, is related to the effective diffusion in such domains. In particular, we consider dumbbell-shaped domains that consist of two heads intercon- nected, with or without a funnel, by a narrow cylindrical neck. The methods used in [30] and [31] for constructing the MFPT in composite domains of the type shown in Figure 3 (right) are made precise here, and the new method extends to domains of 4 D. HOLCMAN AND Z. SCHUSS the type shown in Figure 3 (left). Summary of results. The results of [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] for small absorbing windows in a smooth reflecting boundary of a domain Ω can be summarized as follows. In the two-dimensional case considered in [25], the absorbing boundary ∂Ωa is a small window in the smooth boundary ∂Ω that is otherwise reflecting to Brownian trajectories. The MFPT from x ∈ Ωtothe absorbing boundary ∂Ωa, denoted τ¯x→∂Ωa , is the NET from the domain Ω to the small window ∂Ωa (of length a), such that ε = π|∂Ωa|/|∂Ω| = πa/|∂Ω|1 (this corrects the definition in [25]). Because the singularity of Neumann’s function in the plane is logarithmic, the MFPT is given by |Ω| |∂Ω| (1) τ¯x→∂Ωa = ln + O(1) for x ∈ Ω outside a boundary layer near ∂Ωa. πD π|∂Ωa| In the three-dimensional case the MFPT to a circular absorbing window ∂Ωa of small radius a centered at 0 on the boundary ∂Ω is given by [35] | | Ω (2) τ¯x→∂Ωa = , aD L(0)+N(0) a a o a a 4 1+ 2π log + ( log ) where L(0)andN(0) are the principal curvatures of the boundary at the center of ∂Ωa. The new results of this paper are as follows. In section 2 we prove that the MFPT to the narrow straits formed by a partial block of a planar domain (see Figures 1 and 2) is given by Rc(Rc + rc) π| Ω| (3) τ¯ = (1 + o(1)) for ε |∂Ω|,Rc,rc, 2rcε 2D where Rc and rc are the curvatures at the neck and ε is the width of the straits.