The narrow escape problem for diffusion in cellular microdomains Z. Schuss†, A. Singer‡, and D. Holcman§¶ʈ †Department of Mathematics, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel; ‡Department of Mathematics, Program in Applied Mathematics, Yale University, 10 Hillhouse Avenue, PO Box 208283, New Haven, CT 05620-8283; §Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel; and ¶De´ partment de Mathe´ matiques et de Biologie, Ecole Normale Supe´ rieure, 46 Rue d’Ulm, 75005 Paris, France Communicated by Robert H. Austin, Princeton University, Princeton, NJ, July 13, 2007 (received for review January 15, 2007) The study of the diffusive motion of ions or molecules in confined molecules involved is assumed sufficiently large, which is not the biological microdomains requires the derivation of the explicit case here. dependence of quantities, such as the decay rate of the population To derive principles and to guess the role of a microdomain, or the forward chemical reaction rate constant on the geometry of we have developed various computations to quantify precisely the domain. Here, we obtain this explicit dependence for a model the role of the geometry in molecular diffusion. The narrow of a Brownian particle (ion, molecule, or protein) confined to a escape problem deals with computing the mean time that a bounded domain (a compartment or a cell) by a reflecting bound- Brownian particle takes to diffuse to a small absorbing portion ary, except for a small window through which it can escape. We call of an otherwise reflecting boundary of a given domain. As the the calculation of the mean escape time the narrow escape prob- absorbing boundary shrinks to zero, the mean time to absorption lem. This time diverges as the window shrinks, thus rendering the diverges to infinity, rendering the narrow escape a singular calculation a singular perturbation problem. Here, we present perturbation problem. Once formulated in terms of boundary asymptotic formulas for the mean escape time in several cases, value problems for partial differential equations, their singular including regular domains in two and three dimensions and in perturbation analysis yields explicit asymptotic expressions for some singular domains in two dimensions. The mean escape time the mean escape time, depending on the diffusion coefficient, comes up in many applications, because it represents the mean the ambient potential, dimensions, and the local and global time it takes for a molecule to hit a target binding site. We present geometrical properties of the domain, and its boundary. several applications in cellular biology: calcium decay in dendritic The narrow escape or the small hole computation leads to the spines, a Markov model of multicomponent chemical reactions in development of Markov models of chemical reactions and microdomains, dynamics of receptor diffusion on the surface of accounts for the small number of molecules involved. After neurons, and vesicle trafficking inside a cell. presenting the modeling framework of the narrow escape, we discuss several applications of the present theory to calcium molecular trafficking ͉ mean first passage time ͉ random motion ͉ dynamics in dendritic spines, the forward binding rate of chem- cellular biology ͉ small hole ical reactions, receptor trafficking on the surface membrane of cells, and an application to vesicle trafficking in neuronal growth. he function of biological microdomains, and specifically Formulation and Mathematical Results Tneurobiological microstructures, such as dendritic spines, is largely unknown, and much effort has been spent in the last 20 We assume that biological particles (e.g., an ion, molecule, or years to unravel the molecular pathways responsible for the receptor) diffuse in a field of force (such as electrical potential) maintenance or modulation of cellular functions and, ultimately, and that their motion can be described by the overdamped Langevin–Smoluchowski (6) equation: to extract fundamental principles (1, 2). The cytoplasm of eukaryotic cells is a complex environment where dynamic or- 1 Ϫ ͑ ͒ ϭ ͱ ganelles, cytoskeletal network, and soluble macromolecules are ˙x ␥ F x 2Dw˙, [1] organized in heterogenous structures and local microdomains (3). These submicrometer domains may contain only a small where number of molecules, of the order between just a few and up to hundreds. This is the case in microdomains like endosomes (4), kBT D ϭ , [2] synapses (5), and the sensory compartments of cells, such as the m␥ outer segment of photoreceptors, but at such low molecular number, the addition of external chemical binding dye mole- ␥ is the friction coefficient per unit of mass, F(x) the force per cules, necessary for experimental purposes, may alter the sig- unit of mass, T is absolute temperature, m is the mass of the naling pathway and thus modify the physiological phenomenol- molecule, kB is Boltzmann’s constant, and w˙ is a vector of ogy. This circumstance calls for physical and mathematical independent ␦-correlated Gaussian white noise, which represent modeling to separate the interfering effects, and ultimately the the effect of the thermal motion. The derivation of the Smolu- physical–mathematical model is expected to be a fundamental chowski equation (Eq. 1) is given in refs. 7 and 8 for the tool for both the quantitative and qualitative study of chemical reactions in microdomains. Because of the small number of molecules involved in chem- Author contributions: Z.S. and D.H. designed research; Z.S., A.S., and D.H. performed ical reactions occurring in extremely confined domains (such as research; D.H. analyzed data; and D.H. wrote the paper. endoplasmic reticulum, caveolae, and mitochondria, to obtain The authors declare no conflict of interest. quantitative information about chemical processes, modeling Freely available online through the PNAS open access option. and simulations seem to be inevitable to reconstruct the mi- Abbreviations: pdf, probability density function; PSD, postsynaptic density. crodomain’s environment and to obtain precise quantitative ʈTo whom correspondence should be addressed. E-mail: [email protected]. information about the molecular dynamics. However, most of This article contains supporting information online at www.pnas.org/cgi/content/full/ the existing models are based on continuum concepts, in which 0706599104/DC1. the medium is assumed homogeneous and the number of © 2007 by The National Academy of Sciences of the USA 16098–16103 ͉ PNAS ͉ October 9, 2007 ͉ vol. 104 ͉ no. 41 www.pnas.org͞cgi͞doi͞10.1073͞pnas.0706599104 where p␧ (x, t͉y) is the pdf conditioned on the initial position, represents the mean conditional sojourn time. It is the solution of the boundary value problem (7) 1 ϭ ⌬ ͑ ͒ ϩ ͑ ͒⅐ٌ ͑ ͒ ϭ Ϫ ʦ ⍀ ͒ ͑ *u␧ y D u␧ y ␥ F y u␧ y 1, for y [10] ͑ ͒ ϭ ʦ Ѩ⍀ u␧ y 0, for y a [11] Ѩu␧͑ y͒ ϭ 0, for y ʦ Ѩ⍀ . [12] Ѩn r Fig. 1. The escape of a Brownian trajectory through a small window. Eq. 12 is the adjoint boundary condition to Eq. 8. The survival probability is three-dimensional motion of a molecule in solution, where ͑ ͒ ϭ ͑ ͒ Einstein’s formula (Eq. 2) can be applied. In two dimensions, S␧ t ͵ p␧ x, t dx, [13] such as for diffusion of a cylinder in the surface of a membrane, ⍀ the diffusion coefficient is given by the Saffman–Delbru¨ck formula (9) where k T ␮h ϭ B ͩ Ϫ ␥ ͪ ͑ ͒ ϭ ͑ ͉ ͒␳ ͑ ͒ D log [3] p␧ x, t ͵ p␧ x, t y y dy. [14] 4␲␮h ␮Јa E 0 ⍀ where ␮ is the viscosity of the solution, ␮Ј is the viscosity of the The density p␧(x, t͉y) can be computed by expanding in eigen- membrane, h the length of the cylinder, a is its radius, and ␥E is Euler’s constant 0.5772. functions A generic problem in cellular biochemistry is to estimate the ϱ mean sojourn time of a Brownian particle in a bounded domain Ϫ␭ ͑␧͒ ͑ ͉ ͒ ϭ ͸ ͑␧͒␺ ͑ ␺ ͒ i t p␧ x, t y ai i,␧ x) i,␧(y e , [15] ⍀ before it escapes through a small absorbing window Ѩ⍀a in its iϭ0 boundary Ѩ⍀. The remaining part of the boundary Ѩ⍀r ϭ Ѩ⍀Ϫ Ѩ⍀a is assumed reflecting for the particle. The reflection may where ␭ (␧) (resp. ␺ ␧) are the eigenvalues (resp. eigenfunctions) represent a high potential barrier on the boundary, or an actual i i, of the Fokker–Planck operator in Eqs. 5, 7, and 8 and the physical obstacle. The opening may represent a narrow corridor coefficients a (␧) depend on the initial function ␳ (y). The mean in the barrier or a defect in the physical obstacle (see Fig. 1). The i 0 biological interpretation of the mean sojourn time is discussed sojourn time can be expressed using the expansion represented below. by Eq. 15 (7), The escape time can be estimated asymptotically in the limit ϱ a ͑␧͒ ͑ ͒ ϭ ͸ i ␺ ͑ ͒ ͉Ѩ⍀ ͉ u␧ y ␧ y . [16] a ␭ i, ␧ϭ ϽϽ i ͉Ѩ⍀͉ 1. [4] iϭ0 When ␧ Ͼ 0, the eigenvalues are strictly positive so that the The probability density function (pdf) p␧ (x, t) of the trajectories of Eq. 1 is the probability per unit volume (area) of finding the steady state is Brownian particle at the point x at time t prior to its escape. The lim p␧͑x, t͉ y͒ ϭ 0 pdf satisfies the Fokker–Planck equation, t3ϱ Ѩp␧͑x, t͒ 1 (ϭ ⌬ ͑ ͒ Ϫ ٌ⅐͓ ͑ ͒ ͑ ͔͒ ϭ ͑ ͒ (the Brownian particle escapes in finite time with probability 1 Ѩ D p␧ x, t ␥ p␧ x, t F x p␧ x, t , [5] t and the mean sojourn time is asymptotically 1/␭0(␧) when ␧ ϽϽ APPLIED 1.