The narrow escape problem revisited
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov The narrow escape problem revisited
O. B´enichou and R. Voituriez
Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee (UMR 7600),Universit´e Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex (Dated: November 19, 2018)The time needed for a particle to exit a confining domain through a small window, called thenarrow escape time (NET), is a limiting factor of various processes, such as some biochemicalreactions in cells. Obtaining an estimate of the mean NET for a given geometric environment istherefore a requisite step to quantify the reaction rate constant of such processes, which has raised agrowing interest in the last few years. In this Letter, we determine explicitly the scaling dependenceof the mean NET on both the volume of the confining domain and the starting point to aperturedistance. We show that this analytical approach is applicable to a very wide range of stochasticprocesses, including anomalous diffusion or diffusion in the presence of an external force field, whichcover situations of biological relevance.
PACS numbers: 87.10.+e,05.40.Fb,05.40.Jc
The first-passage time (FPT), namely the time it takesa random walker to reach a given target site is knownto be a key quantity to quantify the dynamics of vari-ous processes of practical interest [1, 2, 3]. Indeed, bio-chemical reactions [4, 5, 6, 7], foraging strategies of an-imals [8, 9, 10], the spread of sexually transmitted dis-eases in a human social network or of viruses through theworld wide web [11] are often controlled by first encounterevents.Among first-passage processes, the case where the tar-get is a small window on the boundary of a confining do-main, defined as the narrow escape problem, has provedvery recently to be of particular importance [12]. Thenarrow escape time (NET) gives the time needed for arandom walker trapped in a confining domain with a sin-gle narrow opening to exit the domain for the first time(see fig.1). The relevance of the NET is striking in cel-lular biology, since it gives for instance the time neededfor a reactive particle released from a specific organelleto activate a given protein on the cell membrane [13].Further examples are given by biochemical reactions incellular microdomains, like dentritic spines, synapses ormicrovesicles to name a few [12, 13]. These submicrom-eter domains often contain a small amount of particleswhich must first exit the domain in order to fulfill theirbiological function. In these examples, the NET is there-fore a limiting quantity whose quantization is a first stepin the modeling of the process.An important theoretical advance has been made re-cently by different groups [12, 14, 15], which obtained theleading term of the mean NET in the limit of small aper-ture in the case of a brownian particle. However, thesedifferent approaches lose track of the dependence of themean NET on the starting point. Obtaining such infor-mation is not only an important theoretical issue, butalso a biologically relevant question. As a matter of fact,biomolecules like membrane signalling proteins or tran-scription factor proteins are generated at specific sitesin the cell [13], whose localization plays an importantrole in the very function of the biomolecules, as under- a Ω r s r r T x S a FIG. 1: The narrow escape problem : the particle starts from r S and evolves in a domain Ω with reflecting walls, except asmall aperture S a of typical radius a centered at r T . lined recently in [7]. In addition, the above mentionedtechniques to estimate mean NETs have been limited sofar to normal brownian diffusion, whereas many experi-mental studies have shown that cellular transport oftendeparts from thermal diffusion due to the complexity ofthe cellular environment. In particular, crowding effectshave proved to induce subdiffusive behavior in many sit-uations [16, 17], while the interaction of a tracer particlewith molecular motors induces a biased motion [13].In this Letter we propose a general theory which (i)provides explicitly the scaling dependence of the meanNET on both the volume of the confining domain andthe source-target distance, (ii) applies to a wide range oftransport processes, including anomalous diffusion, (iii)encompasses the case of transport in the presence of aforce field. Our formalism partially relies on the methodrecently proposed in [2], and considerably broadens itsfield of applications.We start by extending the theory developed in [2] tocompute the mean FPT of a continuous random motionto a closed surface S a ( r T ) of typical radius a containing r T , starting from a source point r S . We will next giveexplicit results when S a ( r T ) is a sphere, and show thatthis approach allows to solve the narrow escape prob-lem. The volume delimited by S a ( r T ) will be denoted by B a ( r T ). We consider that the random walker evolves ina bounded domain Ω of volume V of the d –dimensionalspace R d . Let P ( r , t | r ′ ) be the propagator, i.e. the den-sity probability to be at r at time t , starting from r ′ attime 0, which satisfies the backward equation [18]: ∂∂t P ( r , t | r ′ ) = ∆ r ′ P ( r , t | r ′ ) , (1)where ∆ r denotes the Laplace operator and the diffusioncoefficient has been set to 1. Let F ( r ′ , t | r ) dS ( r ′ ) be theprobability that the first-passage time at the infinitesi-mal surface dS ( r ′ ) located at r ′ , starting from r , is t .By partitioning over the first arrival time t ′ at a surfaceelement dS ( r ′ ) of the sphere S a ( r T ), one obtains P ( r T , t | r S ) = Z r ∈ S a ( r T ) dS ( r ) Z t P ( r T , t − t ′ | r ) F ( r , t ′ | r S ) dt ′ . (2)We next assume a ≪ V /d , so that P ( r T , t | r ∈ S a ( r T )) ≡ P ( r T , t | S a ( r T )) does not depend on r ∈ S a ( r T ). Thiscondition will be fulfilled in the large V limit consid-ered in the following. Denoting b p ( s ) = R ∞ p ( t ) e − st dt theLaplace transform of p ( t ), and setting F ( S a ( r T ) , t | r S ) = Z r ∈ S a ( r T ) dS ( r ) F ( r , t | r S ) , (3)we get from equation (2): b P ( r T , s | r S ) = b P ( r T , s | S a ( r T )) b F ( S a ( r T ) , s | r S ) . (4)We now denote by h T i ( S a ( r T ) | r S ) the mean FPT at S a ( r T ), and write lim t →∞ P ( r T , t | r S ) = P stat ( r T ). Thenone has (cid:26) b F ( S a ( r T ) , s | r S ) = 1 − s h T i ( S a ( r T ) | r S ) + o ( s ) b P ( r T , s | r S ) = P stat ( r T ) /s + H ( r T | r S ) + o (1) (5)where H ( r | r ′ ) = Z ∞ ( P ( r , t | r ′ ) − P stat ( r )) dt. (6)A similar expansion holds for b P ( r T , s | S a ( r T )). From (4),we then get h T i ( S a ( r T ) | r S ) P stat ( r T ) = H ( r T | S a ( r T )) − H ( r T | r S ) , (7)which is an extension of a similar form given in [2, 19].We then consider the large volume limit of equation (7)and define the function φ a ( r T | r S ) aslim V →∞ h T i ( S a ( r T ) | r S ) P stat ( r T ) = φ a ( r T | r S ) . (8) As can be checked directly from definition (6), φ a ( r T | r S )satisfies the following boundary value problem in the in-finite space: ∆ r φ a ( r T | r ) = 0 for r ∈ R d \ B a ( r T ) φ a ( r T | r ) = 0 for r ∈ S a ( r T ) Z r ∈ S a ( r T ) ∂ n φ a ( r T | r ) dS ( r ) = 1 . (9)Equations (8,9) completely define the large volumeasymptotics of the FPT and constitute the central resultof our method. Indeed, rephrased as above, the problemamounts to determining an electrostatic potential outsidea conducting surface S a ( r T ) of charge unity, which hasbeen extensively studied. When S a ( r T ) is a sphere thesolution is straightforward and yields for the mean FPT:lim V →∞ h T i /V = π ln( r/a ) for d = 2Γ( d/ π d/ (cid:18) a d − − r d − (cid:19) for d ≥ r = | r T − r S | . This result is compatible with theform found in [20, 21] using a different method. Besidesthe very useful analogy with potential theory, we willshow in the following that the advantage of the formu-lation (8,9) is threefold. (i) First, it can be adapted toother geometries and in particular to various examplesof extended targets, such as an escape window in thedomain boundary. It therefore extends the main resultof [2] obtained in discrete space for a point-like target.(ii) Second, as the derivation of equations (8,9) is inde-pendent of the operator ∆, it can be reproduced for anydisplacement operator L . In the general case, equations(8,9) still hold, but with ∆ to be substituted by the ad-joint operator L + , and ∂ n φ a to be substituted by theflux of φ a . (iii) Last, and following the previous remark,the formulation (8,9) can be extended to the case of arandom walker experiencing an external force field.We first show that equations (8,9) can be extendedto the example of an escape window W a ( r T ) of smalltypical radius a centered at r T ∈ ∂ Ω, which is preciselythe narrow escape problem. It will be useful to write W a ( r T ) = ∂ Ω ∩ B ǫa ( r T ) where B ǫa ( r T ) is a small volumeof typical thickness ǫ . We then set S ǫa ( r T ) ≡ Ω ∩ ∂B ǫa ( r T ).We now derive the mean NET through S ǫa ( r T ), and wewill use the fact that lim ǫ → S ǫa ( r T ) = W a ( r T ) to obtainthe mean NET through W a ( r T ). Following step by stepthe previous derivation, we obtain: h T i ( S ǫa ( r T ) | r S ) P stat ( r T ) = H ( r T | S ǫa ( r T )) − H ( r T | r S )(11)We now take the infinite volume limit keeping r T − r S fixed, with the prescription that Ω tends to the half space R d + delimited by the hyperplane containing W a ( r T ), anddefine:lim V →∞ h T i ( S ǫa ( r T ) | r S ) P stat ( r T ) = φ ǫa ( r T | r S ) . (12)One can show that φ ǫa ( r T | r ) then satisfies: ∆ r φ ǫa ( r T | r ) = 0 for r ∈ R d + \ B ǫa ( r T ) φ ǫa ( r T | r ) = 0 for r ∈ S ǫa ( r T ) ∂ n φ ǫa ( r T | r ) = 0 for r ∈ ∂ R d + Z r ∈ S ǫa ( r T ) ∂ n φ ǫa ( r T | r ) dS ( r ) = 1 . (13)This shows that φ ǫa ( r T | r ) is the electrostatic potential ina half space delimited by an isolating hyperplane contain-ing a conducting window S ǫa ( r T ) of charge unity. Taking ǫ to 0 gives the mean NET through W a ( r T ). In the caseof a spherical window W a ( r T ), the solution of (13) can beexactly given. For d = 3, we obtain in oblate spheroidalcoordinates [22]:lim V →∞ h T i /V = 12 πa arctan ξ = 14 a − πr + o (cid:18) r (cid:19) (14)where ξ depends on cartesian coordinates according to z a ξ + x + y a ( ξ + 1) = 1 . (15)For d = 2, we use elliptic coordinates and getlim V →∞ h T i /V = µπ ∼ r →∞ π ln( r/a ) (16)where µ depends on cartesian coordinates according to x a cosh µ + y a sinh µ = 1 . (17)We stress that expressions (14,16) of the mean NET areexact for any position of the source point r S . They there-fore extend the results of [12, 14, 15], which give the samesmall a limit, but where the dependence on the sourceposition was not given.We now generalize equations (8,9) to the case of ageneric displacement operator L such that the station-ary distribution is uniform P stat = 1 /V , which actuallyunderlies many models of transport in complex media[23]. We here assume that S a ( r T ) is a sphere, and fol-lowing [2, 24], we further assume that the infinite spacepropagator P satisfies the standard scaling: P ( r , t | r ′ ) ∼ t − d f /d w Π (cid:18) | r − r ′ | t /d w (cid:19) , (18)where the fractal dimension d f characterizes the accessi-ble volume V r ∼ r df within a sphere of radius r , and thewalk dimension d w characterizes the distance r ∼ t /d w covered by a random walker in a given time t . This for-malism in particular covers the case of a random walk ona random fractal like critical percolation clusters, whichgives a representative example of subdiffusive behaviordue to crowding effects [24] and could mimic in a first ap-proximation the cellular environment. Note that we hereimplicitly require that the trajectories and the medium have length scale invariant properties which ensure theexistence of d w and d f . Substituting the scaling (18) inthe definition (6), we obtain from (8) the large V equiv-alence:lim V →∞ h T i /V = α ( a d w − d f − r d w − d f ) for d w < d f α ln( r/a ) for d w = d f α ( r d w − d f − a d w − d f ) for d w > d f (19)Strikingly, the constant α does not depend on the con-fining domain but only on the scaling function Π: α = Z ∞ Π( u − /d w ) u d f /d w du for d w < d f d w Π(0) for d w = d f Z ∞ duu d f /d w (cid:16) Π(0) − Π( u − /d w ) (cid:17) for d w > d f (20)Expressions (19) therefore explicitly elucidate the depen-dence of the mean FPT on the geometrical parameters V and r . Note that for a generic target surface S a ( r T ), the r dependence in (19) still holds for r ≫ a , but the a de-pendence is changed by a numerical factor. Indeed, the r dependence is fully determined by writing the conser-vation of the flux, which does not depend on the windowshape at large r . As previously equations (19) permit toobtain the mean NET: if we assume that the exit win-dow S ǫa ( r T ) is a half sphere of radius a , the mean NETwill be exactly given by two times the mean FPT (19).For a generic window, and in particular in the case of adisk, the r dependence is unchanged for r ≫ a , but the a dependence is modified by a numerical factor.Remarkably, equation (19) highlights two regimes.When the exploration is not compact ( d w < d f ), as in thecase of a brownian particle in the 3–dimensional space,the dependence on the starting point disappears at large r . On the other hand, in the case of compact exploration( d w ≤ d f ), as for 2–dimensional diffusion or subdiffusionon fractals, the mean NET diverges at large r and thestarting point position is crucial.Last, we consider the case of a brownian particle in thepresence of a force field F ( r ) = −∇ r Φ( r ). In the contextof biological cells, such force can be induced by the cou-pling of the particle to molecular motors which perform adirectional motion along cytoskeletal filaments [13]. Weassume that the target is a sphere S a ( r T ), and that theforce field is spherically symmetric and centered at r T ,situation which mimics asters of cytoskeletal filaments,which are ubiquitous in cells [13]. We set the gauge suchthat Φ( r ) = 0 for r ∈ B a ( r T ). Equations (8,9) thenhold with ∆ r to be replaced by the adjoint operator L + governing the evolution of the propagator [18]: L + = F ( r ) ∇ r + ∆ r . (21)We then solve L + φ a ( r T | r ) = 0 with the same boundaryconditions as in (9), and write the stationary distribution P stat ( r ) = e − Φ( r ) R Ω d Ω( r ′ ) e − Φ( r ′ ) . (22)Using equation (8), we finally get the large V equivalenceof the mean FPT: h T i ∼ Γ( d/ π d/ (cid:18)Z Ω d Ω( r ′ ) e − Φ( r ′ ) (cid:19) (cid:18)Z ra u − d e Φ( u ) (cid:19) . (23)As previously, the mean NET through a spherical win-dow is obtained as two times the mean FPT. One shouldnote that the volume dependence entirely lies in the firstintegral factor of (23), which is suitable for a quantitativeanalysis. On the other hand, the r dependence is fullycontained in the second integral factor and will dependon the specific shape of Φ. The volume dependence of(23) agrees with the one found in [25], where howeverthe r dependence was not given.We now give the explicit example of a divergence lessforce, which is a first approximation of the force experi-enced by a particle in an aster of cytoskeletal filaments,if one assumes that the force is proportional to the lo-cal concentration of filaments. This situation also de-scribes a brownian particle advected in an incompress-ible hydrodynamic flow. For d = 3 such force is given by F ( r ) = − γ/r and the equivalent potential governing thedynamics can be taken as Φ( r ) = γ (1 /r − /a ). Applying(23), we getlim V →∞ h T i /V = 14 πγ (cid:16) e γ (1 /a − /r ) − (cid:17) (24)for a generic domain shape. Note that the r dependenceis modified by the hydrodynamic flow, while the V de- pendence at large V is unchanged for any flow intensity γ . For d = 2 the force is given by F ( r ) = − γ/r and theequivalent potential can be taken as Φ( r ) = γ ln( r/a ).Applying (23) in the case of a domain whose boundary isparameterized by R ( θ ) in polar coordinates, we get thelarge volume equivalence h T i ∼ γ (2 − γ ) (cid:18)Z π dθ π R − γ ( θ ) − γ a − γ (cid:19) ( r γ − a γ ) . (25)As opposed to the d = 3 case, the V dependence is nowmodified by the force. Interestingly we find a transitionfor γ = 2. For γ <
2, the mean FPT will scale like V − γ/ at large V , while the V dependence disappears for γ > [1] S. Redner, A guide to first passage time processes (Cam-bridge University Press, Cambridge, England, 2001).[2] S. Condamin, O. B´enichou, V. Tejedor, R. Voituriez, andJ. Klafter, Nature , 77 (2007).[3] M. Shlesinger, Nature , 40 (2007).[4] M. Slutsky and L. A. Mirny, Biophys. J. , 1640 (2004).[5] M. Coppey, O. B´enichou, R. Voituriez, and M. Moreau,Biophys. J. , 1640 (2004).[6] I. Eliazar, T. Koren, and J. Klafter, Journal Of Physics-Condensed Matter , 065140 (2007).[7] G. Kolesov, Z. Wunderlich, O. N. Laikova, M. S. Gelfand,and L. A. Mirny, PNAS , 13948 (2007).[8] G. Viswanathan, S. V. Buldyrev, S. Havlin, M. da Luz,M. Lyra, E. Raposo, and H. E. Stanley, Nature , 911(1999).[9] O. B´enichou, M. Coppey, M. Moreau, P.H. Suet, andR. Voituriez, Phys. Rev. Lett. , 198101 (2005).[10] O. B´enichou, C. Loverdo, M. Moreau, and R. Voituriez,Phys. Rev. E , 020102 (2006).[11] L. Gallos, C. Song, S. Havlin, and H. Makse, PNAS ,7746 (2007).[12] Z. Schuss, A. Singer, and D. Holcman, Proceedings of theNational Academy of Sciences , 16098 (2007).[13] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts,and P. Walter, Molecular Biology of the Cell (GarlandNew York, 2002). [14] I. Grigoriev, Y. Makhnovskii, A. Berezhkovskii, andV. Zitserman, J. Chem. Phys. , 9574 (2002).[15] A. Singer, Z. Schuss, D. Holcman, and R. S. Eisenberg,J. Stat. Phys. , 437 (2006).[16] I. Golding and E.C. Cox, Phys. Rev. Lett. , 098102(2006).[17] M. Platani, I. Goldberg, A. I. Lamond, and J. R. Swed-low, Nature Cell Biology , 502 (2002).[18] N. V. Kampen, Stochastic Processes in Physics andChemistry (North -Holland, 1992).[19] J. D. Noh and H. Rieger, Phys. Rev. Lett. , 118701(2004).[20] S. Condamin, O. B´enichou, and M. Moreau, Phys. Rev.Lett. , 260601 (2005).[21] S. Condamin, O. B´enichou, and M. Moreau, Phys.Rev.E , 21111 (2007).[22] T. L. Hill, Statistical Mechanics (Dover Publ., Inc., NewYork, 1987).[23] J.-P. Bouchaud and A. Georges, Phys.Rep. , 127(1990).[24] D.Ben-Avraham and S.Havlin,
Diffusion and reactionsin fractals and disordered systems (Cambridge UniversityPress, 2000).[25] A. Singer and Z. Schuss, Physical Review E74