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Preview Mean first-passage time of surface-mediated diffusion in spherical domains

Mean first-passage time of surface-mediated diffusion in spherical domains O. B´enichou,1 D. S. Grebenkov,2,3 P. E. Levitz,2 C. Loverdo,1 and R. Voituriez1 1Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee (UMR 7600), case courrier 121, Universit´e Paris 6, 4 Place Jussieu, 75255 Paris Cedex 2Laboratoire de Physique de la Mati`ere Condens´ee (UMR7643), CNRS – Ecole Polytechnique, F-91128 Palaiseau Cedex France 3Laboratoire Poncelet (UMI 2615), CNRS – Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia 1 (Dated: January 27, 2011) 1 0 We present an exact calculation of the mean first-passage time to a target on the surface of a 2 2D or 3D spherical domain, for a molecule alternating phases of surface diffusion on the domain n boundary and phases of bulk diffusion. The presented approach is based on an integral equation a which can be solved analytically. Numerically validated approximation schemes, which provide J more tractable expressions of the mean first-passage time are also proposed. In the framework of 6 thisminimalmodelofsurface-mediatedreactions,weshowanalyticallythatthemeanreactiontime 2 can be minimized as a function of thedesorption rate from thesurface. ] PACSnumbers: h c e m I. INTRODUCTION - t The kinetics ofmany chemicalreactionsis influencedby the transportproperties ofthe reactantsthatthey involve a t [1, 2]. In fact, schematically, any chemical reaction requires first that a given reactant A meets a second reactant B. s This first reaction step can be rephrased as a search process involving a searcher A looking for a target B. In a very . t dilute regime,exemplifiedbybiochemicalreactionsincells [3]whichsometimesinvolveonlyafew copiesofreactants, a m the targets B are sparse and therefore hard to find in this search process language. In such reactions, the first step of searchfor reactants B is therefore a limiting factor of the globalreaction kinetics. In the generalaim of enhancing - d the reactivity of chemical systems, it is therefore needed to optimize the efficiency of this first step of search. n Recently, it has been shown that intermittent processes, combining slow diffusion phases with a faster transport, o can significantly increase reactions rates [4, 5]. A minimal model demonstrating the efficiency of this type of search, c introduced to account for the fast search of target sequences on DNA by proteins [6] is as follows (see also [7–10]). [ The pathway followed by the protein, considered as a point-like particle, is a succession of 1D diffusions along the 1 DNA strand (called sliding phases) with diffusion coefficient D and 3D excursions in the surrounding solution. The 1 v timespentbytheproteinonDNAduringeachslidingphaseisassumedtofollowanexponentiallawwithdissociation 3 rate λ. In this minimal model, the 3D excursionsare uncorrelatedin space, which means that after dissociation from 4 DNA, the proteinwillrebindthe DNA ata randompositionindependently of its startingposition. Assuming further 0 5 that the mean duration of such 3D excursions τ2 is finite, it has been shown that the mean first-passage time at the 1. targetcanbe minimized asa function of τ1 =λ−1, assoonas the meantime spent inbulk excursionsis not too long. Quantitatively, this condition writes in orders of magnitude as τ L2/D , and the minimum of the search time is 0 2 ≤ 1 obtained for τ τ in the large L limit. Note that in this minimal model, where the time τ is supposed to be a 1 1 2 2 ≃ 1 fixed exterior parameter, bulk phases are always beneficial in the large L limit (i.e. allow one to decrease the search : time with respect to the situation corresponding to 1D diffusion only). v In many practicalsituations however,the duration of the fast bulk excursionsstrongly depends on the geometrical i X propertiesofthe system[11–14]andcannotbe treatedasanindependent variableasassumedinthe mean-field(MF) r model introduced above. An important generic situation concerns the case of confined systems [15–17], involving a transport of reactive molecules both in the bulk of a confining domain and on its boundary, referred to as surface- mediated diffusion in what follows. This type of problems is met in situations as varied as heterogeneous catalysis [18,19], orreactionsinporousmediaandinvesicularsystems [15,16, 20]. Inallthese examples,the durationofbulk excursions is controlled by the return statistics of the molecule to the confining surface, which crucially depends on the volume of the system. This naturally induces strong correlations between the starting and ending points of bulk excursions, and makes the above MF assumption of uncorrelated excursions largely inapplicable in these examples. At the theoretical level, the question of determining mean first-passage times in confinement has attracted a lot of attention in recent years for discrete random walks [21–25] and continuous processes [26–29]. More precisely, the surface-mediated diffusion problem considered here generalizes the so-called narrow escape problem, which refers to the time needed for a simple Brownian motion in absence of surface diffusion to escape through a small window of an otherwise reflecting domain. This problem has been investigated both in the mathematical [20, 30–32] and physical[33–36]literature,partlydueto the challengeoftakingintoaccountmixedboundaryconditions. The caseof 2 surface-mediateddiffusionbringstheadditionalquestionofminimizingthesearchtimewithrespecttothetimespent in adsorption, in the same spirit as done for intermittent processes introduced above. The answer to this question is a priori not clear, since the mean time spent in bulk excursions diverges for large confining domains, so that the condition of minimization mentioned previously cannot be taken as granted, even in the large system limit. In this context, first results have been obtained in [37] where, surprisingly enough, it has been found that, even for bulk and surface diffusion coefficients of the same order of magnitude, the reaction time can be minimized, whereas MF treatments (see for instance [34]) predict a monotonic behavior. Here, we extend the perturbative results of [37] obtained in the small target size limit. Relying on an integral equation approach, we provide an exact solution for the mean FPT, both for 2D an 3D spherical domains, and for anysphericaltargetsize. We alsodevelopapproximationschemes,numericallyvalidated,thatprovidemoretractable expressions of the mean FPT. II. THE MODEL The surface-mediated process under study is defined as follows. We consider a molecule diffusing in a spherical confining domain of radius R (see figure 1), alternating phases of boundary diffusion (with diffusion coefficient D ) 1 and phases of bulk diffusion (with diffusion coefficient D ). The time spent during each one-dimensional phase is 2 assumed to follow an exponential law with dissociation rate λ. At each desorption event, the molecule is assumed to be ejectedatadistancea fromthe frontier(otherwiseitisinstantaneouslyreadsorbed). Althoughformulatedforany value of this parameter a smaller than R, in most physical situations of real interest a R. The target is perfectly ≪ absorbing and defined in 2D by the arc θ [ ǫ,ǫ], and in 3D by the region of the sphere such that θ [0,ǫ] where ∈ − ∈ θ is in this case the elevation angle in standard spherical coordinates. Note that as soon as ǫ = 0, the target can be 6 reachedeither by surface or bulk diffusion. In what follows we calculate the mean first-passagetime at the target for an arbitrary initial condition of the molecule. FIG. 1: Model III. 2D CASE In this section, the confining domain is a disk of radius R and the target is defined by the arc θ [ ǫ,ǫ]. ∈ − 3 A. Basic equations For the process defined above, the mean first-passage time (MFPT) at the target satisfies the following backward equations D 1t′′(θ)+λ[t (R a,θ) t (θ)] = 1 forθ [ǫ,2π ǫ], (1) R2 1 2 − − 1 − ∈ − ∂2 1 ∂ 1 ∂2 D + + t (r,θ) = 1, (2) 2 ∂r2 r∂r r2∂θ2 2 − (cid:18) (cid:19) where t stands for the the MFPT starting from the circle at a position defined on the circumference by the angle 1 θ, and t for the MFPT starting from the point (r,θ) within the disk. In these two equations, the first term of the 2 lhs accounts for the diffusion respectively on the circumference and in the bulk, while the second term of Eq. (1) describes desorption events. They have to be completed by two boundary conditions t (R,θ) = t (θ), (3) 2 1 t (θ) = 0 forθ [0,ǫ] [2π ǫ,2π], (4) 1 ∈ ∪ − which describe the adsorption events and the absorbing target respectively. Eq.(2) is easily shown to be satisfied by the following Fourier series r2 ∞ t (r,θ)=α + α rncos(nθ), (5) 2 0 n − 4D 2 n=1 X with unknown coefficients α to be determined. In particular,we aim at determining the searchtime t , defined as n 1 h i the MFPT, with aninitial position uniformly distributed onthe boundary of the confining domain. Taking Eq.(5) at r =R, we have R2 ∞ t (θ) if θ [ǫ,2π ǫ], α0 + αnRncos(nθ)= 1 ∈ − (6) − 4D2 n=1 (0 if θ ∈[0,ǫ]∪[2π−ǫ,2π], X so that R2 1 2π−ǫ α = t (θ)dθ t , 0 1 1 − 4D 2π ≡h i 2 Zǫ (7) 1 2π−ǫ Rnα = t (θ)cos(nθ)dθ (n 1). n 1 π ≥ Zǫ In what follows we will make use of the following quantities: ω R λ/D , (8) 1 ≡ p a x 1 , (9) ≡ − R and 1 R2 (R a)2 T + − − . (10) ≡ λ 4D 2 As we proceed to show, two different approaches can be used to solve this problem. (i) The first approach, whose mainresultshavebeenpublishedin[37],usestheexplicitformoftheGreenfunctionforthetwo-dimensionalproblem andreliesonasmalltargetsizeǫexpansion. Werecalltheseperturbativeresultsbelowforthesakeofself-consistency and give details of the derivation in Appendix A. (ii) The second approach presented next relies on an integral equation which can be derived for t , and leads to an exact non-perturbative solution. 1 4 B. Perturbative approach It is shown in Appendix that the Fourier coefficients of t (r,θ) as defined in Eq.(5) satisfy an infinite hierarchy of 2 linear equations, which lead to the following small ǫ expansion: R2 ∞ 1 ∞ 1 xm α = +ω2T 2 πǫ+ 1+2ω2 − ǫ2 +..., 0 4D2 ( m=1ω2(1−xm)+m2!− m=1ω2(1−xm)+m2! ) (11) X X ω2T α = 2+n2ǫ2+... . n Rn(ω2(1 xn)+n2) − − (cid:8) (cid:9) Note that Eq.(11) gives in particular the first terms of the perturbative expansion of the search time t defined in 1 (7)andgivenin[37]. Itshouldbestressedthatsincethecoefficientsofǫk ofthisexpansiondivergewithhω,iinpractice one finds that the range of applicability in ǫ of this expansion is wider for ω small. C. Integral equation for t1 Inthis section,wefirstshowthattheresolutionofthe coupledPDEs(1,2)amountstosolvinganintegralequation for t only. As we proceed to show, this integral equation can be solved exactly. Writing Eq. (1) as 1 ∂2t R2 1 = ω2[t (R a,θ) t (R,θ)], (12) ∂θ2 −D − 2 − − 2 1 and expanding its right-hand side into a Taylor series leads to ∂2t R2 ∞ ( a)k ∂kt 1 = ω2 − 2 . (13) ∂θ2 −D − k! ∂rk 1 k=1 (cid:18) (cid:19)R,θ X Substituting the Fourier representation (5) for t into this equation yields 2 ∂2t R2 aR a2 ∞ ( a)k ∞ ∂θ21 =−D −ω2 2D − 4D −ω2 −k! αnn(n−1)...(n−k+1)Rn−kcos(nθ). (14) 1 (cid:18) 2 2(cid:19) k=1 n=k X X Changing the order of summations over n and k, using the binomial formula and the expression (7) for α give n ∂2t R2 aR a2 ω2 ∞ 2π−ǫ 1 = ω2 (xn 1)cos(nθ) cos(nθ′)t (θ′)dθ′. (15) ∂θ2 −D − 2D − 4D − π − 1 1 (cid:18) 2 2(cid:19) n=1 Zǫ X This integro-differential equation for t can actually easily be transformed into an integral equation for t , by inte- 1 1 grating successively two times, which leads to 1 R2 aR a2 t (θ)= +ω2 (θ ǫ)(2π ǫ θ) 1 2 D 2D − 4D − − − (cid:18) 1 (cid:18) 2 2(cid:19)(cid:19) (16) ω2 ∞ cos(nθ) cos(nǫ) 2π−ǫ + (xn 1) − cos(nθ′)t (θ′)dθ′, π − n2 1 n=1 Zǫ X or equivalently to ∞ cos(nθ) cos(nǫ) 2π−ǫ ψ(θ)=(θ ǫ)(2π ǫ θ)+Ω (xn 1) − cos(nθ′)ψ(θ′)dθ′, (17) − − − − n2 n=1 Zǫ X where 2t (θ) 1 ψ(θ) , (18) ≡ ω2T 5 with T defined in Eq. (10) and Ω ω2. Note that Eq. (17) holds for θ [ǫ,2π ǫ]. When there is no desorption ≡ π ∈ − (i.e., λ=0), only the first term in Eq. (18) survives, yielding the classical result [26] R2 t (θ)= (θ ǫ)(2π ǫ θ). (19) 1 D − − − 1 The same result is obtained for a = 0, since xn 1 = (1 a/R)n 1 = 0. The limit a = 0 is in fact equivalent to − − − the limit λ = 0 because, after desorption, the particle immediately returns onto the circle (a = 0) as if it was never desorbed (λ=0). D. Exact solution Iterating the integral equation (17) shows that the solution ψ(θ) writes for θ [ǫ,2π ǫ]: ∈ − ∞ ψ(θ)=(θ ǫ)(2π ǫ θ)+ d cos(nθ) cos(nǫ) , (20) n − − − − n=1 X (cid:2) (cid:3) with the coefficients d which satisfy n ∞ ∞ ∞ dn cos(nθ) cos(nǫ) =Ω Un+ Qn,n′dn′ cos(nθ) cos(nǫ) , (21) − − n=1 n=1(cid:18) n′=1 (cid:19) X (cid:2) (cid:3) X X (cid:2) (cid:3) where we introduced 2π−ǫ xn 1 1 xn Un ≡ n−2 dθ′cos(nθ′)(θ′−ǫ)(2π−ǫ−θ′)=4 −n4 ξn, Zǫ (22) sin(nǫ) ξ (π ǫ)cos(nǫ)+ (n=1,2,...), n ≡ − n and 1 xn Qn,n′ ≡− −n2 Iǫ(n,n′) (n,n′ =1,2,...), (23) with 2π−ǫ I (n,n′) cos(nθ)(cos(n′θ) cos(n′ǫ))dθ ǫ ≡ − Zǫ cos(n′ǫ)sin(nǫ) sin((n′+n)ǫ) sin((n′ n)ǫ) =(1−δn,n′) 2 n − n′+n − n′ −n (cid:18) − (cid:19) (24) sin(2nǫ) +δn,n′ π ǫ+ − 2n (cid:18) (cid:19) cos(nǫ)sin(n′ǫ) cos(n′ǫ)sin(nǫ) sin(2nǫ) =2(1−δn,n′) n′n2 −n′2 n n′2+δn,n′ π−ǫ+ 2n . − (cid:18) (cid:19) Since Eq. (21) should be satisfied for any θ [ǫ,2π ǫ], one gets d=Ω(U +Qd), from which ∈ − d =Ω (I ΩQ)−1U (n=1,2,...). (25) n − n Since (cid:2) (cid:3) ∞ (I ΩQ)−1 = (ΩQ)i, (26) − i=0 X Eq. (20)withthed givenbyEq. (25)canbeseenasaseriesinpowersofΩ,whosen-thordercoefficientisexplicitly n written in terms of the n-th power of the matrix Q. 6 Note that the first term in Eq. (20) can also be expended in a Fourier series ∞ (θ ǫ)(2π ǫ θ), ǫ<θ <2π ǫ, en cos(nθ) cos(nǫ) = − − − − (27) − ( 0, otherwise, n=1 X (cid:2) (cid:3) where the coefficients e are obtained by multiplying this equation by cosmθ and integrating from 0 to 2π: n 4 e = ξ (n=1,2,...). (28) n −πn2 n Once the d determined, the search time t is n 1 h i 2π 2π−ǫ 1 ω2T ω2T 2 ∞ t t (θ)dθ = ψ(θ)dθ = (π ǫ)3 d ξ . (29) 1 1 n n h i≡ 2π 4π 2π 3 − − Z0 Zǫ (cid:26) nX=1 (cid:27) E. Approximate solution While the previous expression of t is exact, it is not fully explicit, since it requires either the inversion of the 1 matrixI ΩQor the calculationofallthe powersofQ. We givehere anapproximationof(I ΩQ)−1, whichinturn − − provides a convenient and fully explicit representation of t . As shown numerically (see Figs. 2, 3, 4 and section V 1 for more details about numerical methods), this approximationof t proves to be in quantitative agreement with the 1 exact expression for a wide range of parameters. This approximation relies on the fact that, in the small target size limit ǫ 0, the matrix Q is diagonal, which → mirrors the orthogonality of the cos(nθ) on [0,2π]. More precisely, one has from Eqs. (23,24): n { } Q =δ Q + (ǫ3), (30) m,n m,n n,n O and keeping only the leading term of this expansion yields d Ω(1 ΩQ )−1U , (31) n n,n n ≈ − from which we obtain the desired approximation: ∞ n(π ǫ)cos(nǫ)+sin(nǫ) 1 xn ψ(θ) (θ ǫ)(2π ǫ θ)+4Ω (cos(nθ) cos(nǫ)) − − . (32) ≈ − − − − n3 n2+Ω(1 xn)I (n,n) ǫ n=1 − X This yields an approximation for the search time: ω2T 2 ∞ 1 xn (π ǫ)cos(nǫ)+ sin(nǫ) 2 t (π ǫ)3 4Ω − − n . (33) h 1i≈ 2π (3 − − nX=1 n2 n2(cid:0)+Ω(1−xn) π−ǫ+ sin2((cid:1)2nnǫ) ) (cid:0) (cid:1) F. Variations of the search time ht1i with the desorption rate λ In this section, we answer two important questions. When are bulk excursions favorable, meaning enabling to reduce the search time (with respect to the situation with no bulk excursion corresponding to λ=0)? If so, is there an optimal value of the desorption rate λ minimizing the search time? 1. When are bulk excursions beneficial to the search? This questioncanbe investigatedby studying the signofthe derivative ∂ht1i atλ=0. The meansearchtime from ∂λ Eq. (29) can also be written as R4 2πD t = 1+λη 1(π ǫ)3 λ ξ (I +λQ˜)−1U , (34) h 1i 2π2D2 3R2 − − · 1 (cid:20) (cid:21) (cid:0) (cid:1) (cid:0) (cid:1) 7 5 10−1 eapxpeaprtcruotrxbiamtiavtee 10−2 appeprtruorxbiamtiavtee ssoolluuttiioonn 4 10−3 3 10−4 θt()1 2 error10−5 1 10−6 0 10−7 −10 1 2 3 4 5 6 10−80 1 2 3 4 5 6 θ θ 6 3 exact approximate perturbative 5 2.5 4 2 θt()13 error1.5 2 1 1 0.5 00 1 2 3 4 5 6 00 1 2 3 4 5 6 θ θ 3 0.8 exact approximate 2.5 appeprtruorxbiamtiavtee 0.6 perturbative 2 0.4 1.5 0.2 θt()1 1 error 0 0.5 −0.2 0 −0.5 −0.4 −1 −0.6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 θ θ 1.5 exact approximate 1 0.5 θ) t(1 0 −0.5 −1 0 1 2 3 4 5 6 θ FIG.2: Comparison betweenthreeapproachesfor computingt1(θ) in2D:theexact solution (20,25),theapproximation (32) and the perturbative formula (A15), with D2 = 1, a = 0.01. In the first row, the other parameters are: ǫ = 0.1, λ = 1, and the series are truncated to N = 100. On the right, the absolute error between the exact solution and the approximation (dashedbluecurve)andbetweentheexactsolution andtheperturbativeformula(solid redcurve). Theapproximationisvery accurate indeed. In the second row, the parameters are: ǫ=0.1, λ=1000, and the series are truncated to N =100 for the exact and approximate solutions, and to N = 1000 for the perturbative solution. One can see that the perturbative solution is inaccurate for large values of λ, while the maximal relative error of the approximate solution is below 2%. In the third row, the parameters are: ǫ = 1, λ = 1, and the series are truncated to N = 100. The perturbative solution is evidently not applicable. In the last row, the parameters are: ǫ=1, λ=1000, and the series are truncated to N =100. In this case, the approximate solution significantly deviates from the exact one (providing mostly negative values). The perturbative solution is completely invalid (not shown). 8 10 0.1 D = 0.5 D = 0.5 2 2 D = 1 D = 1 8 2 2 D = 5 0.05 D = 5 2 2 6 1D λd >1 >/ 0 <t 4 <t1 d −0.05 2 0 −0.1 0 100 200 300 400 500 0 100 200 300 400 500 λ λ FIG. 3: Left: In 2D, the mean time ht1i computed through Eq. (25, 29) with N = 100 as a function of the desorption rate λ for three values of D2: D2 =0.5 (dot-dashed blue line), D2 =1 (dashed green line), and D2 =5 (solid red line). The other parameters are: a = 0.1 and ǫ = 0.01. When D2 < D2,crit ≈ 0.6348... (the first case), ht1i monotonously increases with λ so that there is no optimal value. In two other cases, D2 > D2,crit, and ht1i starts first to decrease with λ, passes through a minimum (the optimal value) and monotonously increases. Symbols show the approximate mean time computed through Eq. (31, 29). Onecan see that the approximation accurate enough even for large values of λ. Right: The derivative dht1i defined dλ by Eq. (35) for thesame parameters. 3.5 10 exact exact approximate approximate 3 perturbative 8 perturbative 2.5 6 2 > > <t1 <t1 4 1.5 2 1 0.5 0 0 −2 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 ε ε FIG. 4: In 2D, the mean time ht1i as a function of ǫ, with D2 =1, a=0.01, and λ=1 (left) or λ=1000 (right). The exact computation through Eq. (25, 29) is compared to the approximation (31, 29) and to the perturbative approach. In all cases, the series are truncated to N = 100. For small λ (λ = 1), the approximate solution is very close to the exact one, while the perturbativesolutionisrelativelycloseforǫupto1. Inturn,forlargeλ(λ=1000),theapproximatesolutionshowssignificant deviations for theintermediate values of ǫ, while theperturbativesolution is not applicable at all. where Q˜ = Q R2 , η = 2aR−a2. The derivative of t with respect to λ is then − πD1 4D2 h 1i ∂ t R4η 2πD (η−1+2λ)I+λ2Q˜ h 1i = 1(π ǫ)3 ξ U . (35) ∂λ 2π2D2 3R2 − − · (I +λQ˜)2 1(cid:20) (cid:18) (cid:19)(cid:21) If the derivative is negative at λ=0, i.e. 2πD η 1(π ǫ)3 < ξ U , (36) 3R2 − · (cid:0) (cid:1) bulk excursions are beneficial to the search. This inequality determines the critical value for the bulk diffusion coefficient D (which enters through η), above which bulk excursions are beneficial: 2,crit D 6R2(ξ U) 24 ∞ 1 xn sin(nǫ) 2 1 = · = − (π ǫ)cos(nǫ)+ . (37) D π(π ǫ)3(2aR a2) π(π ǫ)3(1 x2) n4 − n 2,crit − − − − n=1 (cid:20) (cid:21) X Two comments are in order: 9 1 a = 0.01 a = 0.1 a = 1 0.8 1 D /crit0.6 2, D 0.4 0.2 0 0.5 1 1.5 2 ε FIG.5: D2,crit asafunctionofǫiscomputedfromEq. (37)in2Dforthreevaluesofa/R: 0.01,0.1,and1. Whenǫapproaches π(thewholesurfacebecomesabsorbing),D2,crit diverges(notshown). Infact,inthislimit,thereisnoneedforabulkexcursion because thetarget will befound immediately by thesurface diffusion. (i) Interestingly, this ratio depends only on a/R and ǫ. In the limit of ǫ 0, one gets → D 24 ∞ 1 xn 1 − . (38) D ≈ π2(1 x2) n4 2,crit − n=1 X Taking next the limit a/R 0 finally yields: → D 12ζ(3) 1 1.4615, (39) D ≈ π2 ≈ 2,crit where ζ stands for the Riemann ζ-function. (ii) The dependence ofthe rhsofEq.(37)with ǫ isnottrivial(Fig. 5). Indeedit canbe provedto haveamaximum with respect to ǫ, which can be understood intuitively as follows: in the vicinity of ǫ = 0, increasing ǫ makes the constraint less stringent since the target can be reached directly from the bulk; in the opposite limit ǫ π, the → constraint on D /D has to tend to 0 since the target is found immediately from the surface. Quantitatively, in the 1 2 physical limit a 0, one finds that, as soon as D /D >(D /D ) 0.68..., bulk excursions can be beneficial. 2 1 2,crit 1 → ≈ 2. When is there an optimal value of the desorption rate λ minimizing the search time? If the reaction time t is a decreasing function of the desorption rate λ, the bulk excursions are ”too favorable”, 1 h i and the best search strategy is obtained for λ (purely bulk search). For the reaction time to be an optimizable →∞ functionofλ,thederivative dht1i hastobepositiveatsomeλ. Thisnecessaryandsufficientconditionremainsformal dλ andrequiresnumericalanalysisofEq. (35). Asimple sufficient conditioncanbe usedinsteadby demanding that the search time at zero desorption rate is less than the search time at infinite desorption rate: t (λ=0) < t (λ ) . (40) 1 1 h i h →∞ i This writes in the physically relevant limit a R (using the result of [30]): ≪ D (π ǫ)3 1 π−ǫ usin(u/2) 1 > − , withc(ǫ) du. D2 3πc(ǫ) ≡ π√2Z0 cos(u)+cos(ǫ) (41) p Finally, combining Eqs. (37, 41), the search time is found to be an optimizable function of λ in the limit a R if ≪ (π ǫ)3 D 12 ∞ 1 sin(nǫ) 2 1 − < < (π ǫ)cos(nǫ)+ . (42) 3πc(ǫ) D π(π ǫ)3 n3 − n 2 − n=1 (cid:20) (cid:21) X 10 Knowing that c(ǫ)=ln(2/ǫ)+ (ǫ), Eq. (42) writes in the small ǫ limit: O π2 D 12ζ(3) 1 < < , (43) 3ln(2/ǫ) D π2 2 which summarizes the conditions for the search time to be an optimizable function of λ. This case is illustrated in Fig. 10. IV. 3D CASE In this section, the confining domain is a sphere of radius R and the target is the region on the boundary defined by θ [0,ǫ], where θ is the elevation angle. ∈ A. Basic equations The 3D analogs of Eqs. (1, 2) read as D ∂2t 1 ∂t 1 1 1 + +λ[t (R a,θ) t (θ)] = 1 forθ [ǫ,π], (44) R2 ∂θ2 tanθ ∂θ 2 − − 1 − ∈ (cid:18) (cid:19) ∂2 2 ∂ 1 ∂2 D + + t (r,θ) = 1. (45) 2 ∂r2 r∂r r2∂θ2 2 − (cid:18) (cid:19) These equations have to be completed by two boundary conditions: t (R,θ) = t (θ), (46) 2 1 t (θ) = 0 forθ [0,ǫ], (47) 1 ∈ which respectively describe the adsorption events and express that the target is an absorbing region of the sphere. B. Integral equation for t1 One can search for a solution in the following form r2 ∞ t (r,θ)=α + α rnP (cosθ), (48) 2 0 n n − 6D 2 n=1 X where P stands for the Legendre polynomial of order n. Using the orthonormality of Legendre polynomials, the n projection of t (R,θ) on P writes 2 m π R2 2α Rm m sinθP (cosθ)t (R,θ)dθ =2 α δ + . (49) m 2 0 m,0 − 6D 2m+1 Z0 (cid:18) 2(cid:19) Knowing that t (θ) if θ [ǫ,π], t (R,θ)= 1 ∈ (50) 2 (0 if θ [0,ǫ], ∈ the α can be written in terms of t (θ) as n 1 R2 1 π α = sinθ t (θ)dθ, 0 1 − 6D 2 2 Zǫ (51) 2n+1 π α Rn = sinθ P (cosθ)t (θ)dθ if n 1. n n 1 2 ≥ Zǫ

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