Physique de la Matière Condensée, Ecole Polytechnique, Palaiseau, France

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1 Denis Grebenkov a and Gleb Oshanin b a Physique de la Matière Condensée, Ecole Polytechnique, Palaiseau, France b Physique Théorique de la Matière Condensée (UMR CNRS) Sorbonne Universités Université Pierre et Marie Curie Paris, France Buenos Aires, March 2-22, 217 1

2 ! Narrow escape problem (NEP) Bounded domain (e.g., a sphere, a circle, whatever) enclosed by an impenetrable wall with a tiny hole to go out an escape window (EW) (or chemically active or binding site) linear size of EW/linear size of the domain = ε << 1 (hence, such a name) domain - cell, microvesicle, compartment, endosome, caviola, etc A particle (ion, chemically active molecule, protein, receptor, ligand, etc) starts from a random or some prescribed location and moves diffusively (with diffusion coefficient D bulk ) until it finds the EW (or binding site) The goal is to determine the mean first passage time to the EW esirably, the whole pdf) 2

3 ! What is known about NEP? No interactions beyond the hard-wall Exact asymptotic result for a sphere T = πr2 1 ε + ln + O( 1) ε Good review: Bressloff & Newby, Rev. Mod. Phys. 85, 135 (213) Exact result for a circle (Singer, Schuss & Holcman) T = R2 D 1 8 ln sin ε = R2 2 D ln + O 1 ε Impenetrable wall + attraction to the wall upon a contact Intermittent motion : particle diffuses freely in the bulk (D bulk ), encounters the surface, performs a «surface diffusion» tour (D surf ), desorbs and goes diffusively to the bulk, encounters the surface again, adsorbs, and etc. Some exact results (requiring numerical inversion) and different mean-field descriptions e.g. G. Oshanin, M. Tamm and O. Vasilyev, JCP 21; O. Bénichou, D. Grebenkov, P. Levitz, C. Loverdo, R. Voituriez, PRL 21; F. Rojo, H. S. Wio, C. E. Budde, PRE 212; A. Berezhkovski, L. Dagdug, JCP 212 For some rates of adsorption/desorption, T can have an optimum be minimal. (side remark 1: contradicts to Adam-Delbrück scenario of «dimensionality reduction» for search processes in systems with «complex» geometry) [side remark 2: IM tells that there is no optimum for D surf = D bulk ] ( ) 3

4 ! Motivations & Goals 1. For applications, it is not only important to appear at the EW, but also to pass through. In reality, there is an energy (ΔE) and/or an entropy (ΔS) barrier at the EW (finite binding rate) giving rise to some finite rate κ exp(- βδe(or ΔS)) of barrier-crossing events. We focus on the mean first EXIT time and take κ into account. 2. Intermittent modelling supposes that interactions are δ-functions and presumes that adsorbing/desorption rates are «free» parameters. In reality, they are linked on a microscopic level by interactions with the wall. It may turn out that an «optimum» corresponds to a physically non-plausible choice. To avoid having «free» parameters we introduce explicit potential interactions with the wall beyond the hard-wall ones. 3. Physical interactions are typically long-ranged (e.g., electrostatic, van der Waals). We focus on the effects of the «range» of the potential on the mean first exit time D. Grebenkov & G. Oshanin, Phys. Chem. Chem. Phys. 19, 2723 (217) *G. Oshanin, M. Popescu & S. Dietrich, Active colloids in the context of chemical kinetics, J Phys A 5 (217); special issue on Smoluchowski s paper, Eds. E. Gudowska-Nowak, K. Lindenberg, and R. Metzler 4

5 ! Mathematical formulation t(r,θ) mean first exit time through the EW for a diffusive particle started at (r,θ). It obeys the backward Fokker-Planck equation for a sphere for a circle 2 t r r β du t dr r + 1 r 2 sin θ ( ) 2 t r r β du t dr r t r 2 θ = 1 2 D θ sin θ ( ) t = 1 θ D with mixed boundary conditions: D t r r=r t = κ t r=r for θ ε and r r=r = for ε < θ π Global mean first exit time T ε = 1 V dv t( r,θ) We do not know how to solve the problem exactly. We will resort to some self-consistent approximation (SCA) 5

6 ! Shoup-Lipari-Szabo approach (Biophys. J. 36, ) Impenetrable sphere with a small specific site on its surface (e.g., a receptor in the model of Berg & Purcell) Reactants (e.g., ligands) diffuse outside of the sphere and react at a rate κ when they appear at the specific site No potential interactions beyond the hard-wall ones The approach consists in replacement of the mixed bc-s by an inhomogeneous Neumann bc: D t r r=r = Q Heaviside( ε θ) where Q is a trial parameter to be determined self-consistently, by requiring that the reactivecondition holds on average over the surface of the specific site. This replacements implies that the result is determined up to an additive constant, but is known to work fairly well. We suitably generalize this approach by adding U(r) and check our predictions against MC simulation and numerical solution with «true» bc-s. By-product: rate constant for Berg-Purcell model with arbitrary U(r) 6

7 ! Main result in (sphere) T ε =3) = 2RL (3) U 3κ (1 cosε) + R2 L U (3) Ψ (3) ε + R2 1 x 4 dx L (3) U (xr) barrier-crossing contribution MFPT to the EW MFPT to reach any point on the surface from a random point r L (3) U (r) = 3 eβu(r) ρ 2 dρ e βu (ρ ) Ψ (3) r 3 ε = n =1 g n Rg n ' ( P n 1 (cosε) P n +1 (cosε)) 2 (2n +1) g n (r) are eigenfunctions of the FP equation, which encode all info about U(r), P n (cos ε) are Legendre polynomials - An analogue of the «Collins-Kimball» relation for the apparent rate constant in classical chemical kinetics - Contributions due to barrier-crossing and due to diffusive search for the EW are additive, which permits to study them separately - A very similar result for 2D (circle) 7

8 ! Barrier-crossing contribution for ε << 1 T =3) barrier = 2RL (3) U 1 3κ ε 2 In case of repulsive interactions For attractive interactions L U exp( βu ) 1 L U β U For an energy barrier, κ is independent of ε. Hence, T barrier 1/ε 2 For an entropy barrier, βδs = ln(1/ε) (Malgaretti, Pagonabarraga & Rubi, JCP 216). Hence, T barrier 1/ε 3 and the entropy barrier is most important factor for the NEP (but not for chemical reactions with a binding site when this barrier is absent). 2D T =2) barrier = πrl (2) U 1 2κ ε Behavior of L (2) is analogous to L (3) In 2D, for an energy barrier, T barrier 1/ε For an entropy barrier, T barrier 1/ε 2 8

9 ! Barrier-crossing contribution: U(r)= D T ε /R SCA (ε =.1) FEM (ε =.1) SCA (ε =.2) FEM (ε =.2) SCA (ε = π/4) FEM (ε = π/4) D T ε /R SCA (ε =.1) FEM (ε =.1) SCA (ε =.2) FEM (ε =.2) SCA (ε = π/4) FEM (ε = π/4) (a) κ R/D (b) κ R/D 2D Comparison of our predictions against numerical solution (by the finite elements method, symbols) of the exact mixed boundary problem for different values of ε. 9

10 ! Diffusive search for the EW T =3) diff = R2 L (3) U Ψ (3) ε + R2 1 x 4 dx L (3) U (xr) L (3) U (r) = 3 eβu(r) ρ 2 dρ e r 3 does not depend on ε r βu (ρ ) Ψ ε (3) = n =1 g n Rg n ' ( P n 1 (cosε) P n +1 (cosε)) 2 (2n +1) To calculate T diff we have - to specify U(r) (if U(r)= ψ can be calculated exactly) - solve the differential equation for g n (r) - sum infinite series Impossible mission, except for very few choices of U(r) 1

11 ! Divine revelation Ψ ε (3) = n =1 g n Rg n ' ( P n 1 (cosε) P n +1 (cosε)) 2 (2n +1) For potentials U(r) with bounded first derivative, for n >> 1, lim ε ( P n 1 (cosε) P n +1 (cosε)) 2 = (2n +1) 2 g n g n ' 1 n The sum for small ε is supported by the behavior of g n with large «n»! d 2 g n dr r β du dg n dr dr n(n +1) r 2 g n = big parameter We seek solution g n = f (r) 1 and in fact, we need to know only f 1 and f n + f (r) 2 + f (r) n 2 n 3 2 Ψ (3) ε π ε + ( 1 βru ' )ln ε + ln π 2 12 βru ' + ( 2n +1) General expression (holds for U(r) such that U and U < ) lim ε Ψ ε (3) = The leading behavior is independent of the potential, subleading (logarithmic) depends on force at the boundary. We need to know g n only for calculation of the ε-independent term. And we do not care about it (recall that all results come up to a constant). n =1 g n 1 n + βr 2n 2 U' + O(ε) Rg n ' 11

12 ! Diffusive search for the EW: explicit results for ε << 1 T =3) diff = 32R2 L (3) U 1 9π D ε + R2 L U (3) ( 1 βru' )ln ε + O(1) L (3) U (r) = 3 eβu(r) ρ 2 dρ e r 3 r βu (ρ ) The leading term diverges as 1/ε contribution of trajectories, diffusion in the bulk. This term depends on interactions only via the amplitude «L». The subleading term diverges as ln(1/ε) contribution of 2D-like trajectories which predominantly go along the wall. Depends on interactions via the amplitude «L» and also on the force at the boundary. 2D T =2) diff = R2 L (2) U D ln ε + O(1) L U (2) (r) = 2 eβu(r) r 2 r ρ dρ e βu (ρ ) The leading term diverges as ln(1/ε). This term depends on interactions only via the amplitude «L». 12

13 ! Check against exact results for hard-wall interactions T =3) diff πr2 1 ε + ln + O( 1) ε T =3) diff = 32R2 1 9π D ε + R2 ln + O(1) ε Numerical factor is 1% off T =2) diff R2 D 1 8 ln sin ε = R2 2 D ln + O 1 ε T =2) diff = R2 D ln + O(1) ε Numerical factor is exact ( ) (Singer et al) Beyond small-ε limit (ψ can be calculated in this case, g n (r) = 1/r n ) D T ε /R SCA asympt FEM MC D T ε /R D SCA exact FEM MC 1 1 (a) ε 1 (b) ε 13

14 ! Case study: triangular-well potential U(r) U(r) = r r U R r for r r for r < r R U o r o R r DT ε (3) /R U = 1 U 1 = 2 U = 5 (a) r /R DT ε (2) /R r /R (b) «Optimum» exists for our case D bulk =D surf, at odds with intermittent modelling «Optimal» interactions are neither too short-ranged, nor extend too deeply into the bulk Adam-Delbruck scenario is not optimal: optimal paths combination of excursions in the bulk and along the surface 14

15 ! Barrier-crossing vs Diffusive search: Dominant factor? =3) = 4RL U 3κ =3) T barrier T diff (3) ε 2 32R 2 L U (3) 9πD ε 1 =1.2 D 1 (κ R) ε 2D =2) = πrl U 2κ =2) T barrier T diff (2) ε 1 R 2 L U (2) D ln(1/ε) = π 2 D (κ R) 1 ε ln(1/ε) Recall, that for an entropic barrier κ is proportional to ε Formally, the rhs of these two expressions diverge when ε -> (even for κ independent of ε). Hence, in the true NEP limit, the barrier-crossing becomes to be a bigger problem than the diffusive search meaning that for ε -> the NEP is «kinetic-controlled» rather than «diffusion-controlled». For arbitrarily small but finite ε, precise values of D, R and κ do matter (estimates for «typical» systems with an entropy barrier give D/(κ R) ε the ratio is large). Previous work neglected a non-negligible hindering effect of the barrier, it is not justified and does not permit for an interpretation of experimental data, despite such claims 15

16 Thank you! 16