## Abstract

First passage time (FPT) theory is often used to estimate timescales in cellular and molecular biology. While the overwhelming majority of studies have focused on the time it takes a given single Brownian searcher to reach a target, cellular processes are instead often triggered by the arrival of the first molecule out of many molecules. In these scenarios, the more relevant timescale is the FPT of the first Brownian searcher to reach a target from a large group of independent and identical Brownian searchers. Though the searchers are identically distributed, one searcher will reach the target before the others and will thus have the fastest FPT. This fastest FPT depends on extremely rare events and its mean can be orders of magnitude faster than the mean FPT of a given single searcher. In this paper, we use rigorous probabilistic methods to study this fastest FPT. We determine the asymptotic behavior of all the moments of this fastest FPT in the limit of many searchers in a general class of two- and three-dimensional domains. We establish these results by proving that the fastest searcher takes an almost direct path to the target.

### Similar content being viewed by others

## References

Barlow, P.W.: Why so many sperm cells? Not only a possible means of mitigating the hazards inherent to human reproduction but also an indicator of an exaptation. Commun. Integr. Biol.

**9**(4), e1204499 (2016)Basnayake, K., Holcman, D.: Fastest among equals: a novel paradigm in biology. Phys. Life Rev.

**28**, 96–99 (2019)Basnayake, K., Hubl, A., Schuss, Z., Holcman, D.: Extreme narrow escape: shortest paths for the first particles among \(N\) to reach a target window. Phys. Lett. A

**382**(48), 3449–3454 (2018)Basnayake, K., Schuss, Z., Holcman, D.: Asymptotic formulas for extreme statistics of escape times in 1, 2 and 3-dimensions. J. Nonlinear Sci.

**29**(2), 461–499 (2019a)Basnayake, K., Mazaud, D., Bemelmans, A., Rouach, N., Korkotian, E., Holcman, D.: Fast calcium transients in dendritic spines driven by extreme statistics. PLoS Biol.

**17**(6), e2006202 (2019b)Bénichou, O., Voituriez, R.: Narrow-escape time problem: time needed for a particle to exit a confining domain through a small window. Phys. Rev. Lett.

**100**(16), 168105 (2008)Bernoff, A.J., Lindsay, A.E.: Numerical approximation of diffusive capture rates by planar and spherical surfaces with absorbing pores. SIAM J. Appl. Math.

**78**(1), 266–290 (2018)Bernoff, A.J., Lindsay, A.E., Schmidt, D.D.: Boundary homogenization and capture time distributions of semipermeable membranes with periodic patterns of reactive sites. Multiscale Model. Simul.

**16**(3), 1411–1447 (2018)Bressloff, P.C., Lawley, S.D.: Stochastically gated diffusion-limited reactions for a small target in a bounded domain. Phys. Rev. E

**92**(6), 062117 (2015a). https://doi.org/10.1103/PhysRevE.92.062117Bressloff, P.C., Lawley, S.D.: Escape from subcellular domains with randomly switching boundaries. Multiscale Model. Simul.

**13**(4), 1420–1445 (2015b)Bressloff, P.C., Lawley, S.D.: Escape from a potential well with a randomly switching boundary. J. Phys. A

**48**(22), 225001 (2015c)Bressloff, P.C., Newby, J.M.: Stochastic models of intracellular transport. Rev. Mod. Phys.

**85**(1), 135–196 (2013)Cheviakov, A.F., Ward, M.J., Straube, R.: An asymptotic analysis of the mean first passage time for narrow escape problems: part II: the sphere. Multiscale Model. Simul.

**8**(3), 836–870 (2010)Chou, T., D’Orsogna, M.R.: First passage problems in biology. In: Metzler, R., Oshanin, G., Redner, S. (eds.) First-Passage Phenomena and Their Applications, pp. 306–345. World Scientific, Singapore (2014)

Coles, S.: An Introduction to Statistical Modeling of Extreme Values, vol. 208. Springer, Berlin (2001)

Coombs, D.: First among equals: comment on redundancy principle and the role of extreme statistics in molecular and cellular biology by Z. Schuss, K. Basnayake and D. Holcman. Phys. Life Rev.

**28**, 92–93 (2019)Delgado, M.I., Ward, M.J., Coombs, D.: Conditional mean first passage times to small traps in a 3-d domain with a sticky boundary: applications to t cell searching behavior in lymph nodes. Multiscale Model. Simul.

**13**(4), 1224–1258 (2015)Durrett, R.: Probability: Theory and Examples. Cambridge University Press, Cambridge (2019)

Edwards, L.J., Evavold, B.D.: T cell recognition of weak ligands: roles of signaling, receptor number, and affinity. Immunol. Res.

**50**(1), 39–48 (2011)Eisenbach, M., Giojalas, L.C.: Sperm guidance in mammals—an unpaved road to the egg. Nat. Rev. Mol. Cell Biol.

**7**(4), 276 (2006)Godec, A., Metzler, R.: First passage time distribution in heterogeneity controlled kinetics: going beyond the mean first passage time. Sci. Rep.

**6**, 20349 (2016a)Godec, A., Metzler, R.: Universal proximity effect in target search kinetics in the few-encounter limit. Phys. Rev. X

**6**(4), 041037 (2016b)Grebenkov, D.S.: Searching for partially reactive sites: analytical results for spherical targets. J. Chem. Phys.

**132**(3), 01B608 (2010)Grebenkov, D.S., Oshanin, G.: Diffusive escape through a narrow opening: new insights into a classic problem. Phys. Chem. Chem. Phys.

**19**(4), 2723–2739 (2017)Grebenkov, D.S., Metzler, R., Oshanin, G.: Towards a full quantitative description of single-molecule reaction kinetics in biological cells. Phys. Chem. Chem. Phys.

**20**(24), 16393–16401 (2018a)Grebenkov, D.S., Metzler, R., Oshanin, G.: Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control. Commun. Chem.

**1**(1), 96 (2018b)Guerrier, C., Holcman, D.: The first 100 nm inside the pre-synaptic terminal where calcium diffusion triggers vesicular release. Front. Synaptic Neurosci.

**10**, 23 (2018). https://doi.org/10.3389/fnsyn.2018.00023Gumbel, E.J.: Statistics of Extremes. Columbia University Press, New York City (1962)

Handy, G., Lawley, S.D., Borisyuk, A.: Receptor recharge time drastically reduces the number of captured particles. PLOS Comput. Biol.

**14**(3), e1006015 (2018)Handy, G., Lawley, S.D., Borisyuk, A.: Role of trap recharge time on the statistics of captured particles. Phys. Rev. E

**99**(2), 022420 (2019)Hartich, D., Godec, A.: Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit. J. Phys. A

**52**(24), 244001 (2019)Holcman, D., Schuss, Z.: Time scale of diffusion in molecular and cellular biology. J. Phys. A Math. Theor.

**47**(17), 173001 (2014a)Holcman, D., Schuss, Z.: The narrow escape problem. SIAM Rev.

**56**(2), 213–257 (2014b)Kurella, V., Tzou, J.C., Coombs, D., Ward, M.J.: Asymptotic analysis of first passage time problems inspired by ecology. Bull. Math. Biol.

**77**(1), 83–125 (2015)Larson, D.R., Zenklusen, D., Wu, B., Chao, J.A., Singer, R.H.: Real-time observation of transcription initiation and elongation on an endogenous yeast gene. Science

**332**(6028), 475–478 (2011)Lawley, S.D.: A probabilistic analysis of volume transmission in the brain. SIAM J. Appl. Math.

**78**(2), 942–962 (2018)Lawley, S.D.: Distribution of extreme first passage times of diffusion. arXiv:1910.12170 (2019a)

Lawley, S.D.: Universal formula for extreme first passage statistics of diffusion. arXiv:1909.09883 (2019b)

Lawley, S.D.: Boundary homogenization for trapping patchy particles. Phys. Rev. E

**100**(3), 032601 (2019c)Lawley, S.D., Madrid, J.B.: First passage time distribution of multiple impatient particles with reversible binding. J. Chem. Phys.

**150**, 214113 (2019)Lawley, S.D., Miles, C.E.: Diffusive search for diffusing targets with fluctuating diffusivity and gating. J. Nonlinear Sci. (2019). https://doi.org/10.1007/s00332-019-09564-1

Lindenberg, K., Seshadri, V., Shuler, K.E., Weiss, G.H.: Lattice random walks for sets of random walkers. First passage times. J. Stat. Phys.

**23**(1), 11–25 (1980)Lindsay, A.E., Tzou, J.C., Kolokolnikov, T.: Optimization of first passage times by multiple cooperating mobile traps. Multiscale Model. Simul.

**15**(2), 920–947 (2017a)Lindsay, A.E., Bernoff, A.J., Ward, M.J.: First passage statistics for the capture of a brownian particle by a structured spherical target with multiple surface traps. Multiscale Model. Simul.

**15**(1), 74–109 (2017b)Martyushev, L.M.: Minimal time, Weibull distribution and maximum entropy production principle: comment on redundancy principle and the role of extreme statistics in molecular and cellular biology by Z. Schuss et al. Phys. Life Rev.

**28**, 83–84 (2019)Meerson, B., Redner, S.: Mortality, redundancy, and diversity in stochastic search. Phys. Rev. Lett.

**114**(19), 198101 (2015)Novak, S.Y.: Extreme Value Methods with Applications to Finance. CRC Press, Boca Raton (2011)

Pillay, S., Ward, M.J., Peirce, A., Kolokolnikov, T.: An asymptotic analysis of the mean first passage time for narrow escape problems: part I: two-dimensional domains. Multiscale Model. Simul.

**8**(3), 803–835 (2010)Redner, S., Meerson, B.: Redundancy, extreme statistics and geometrical optics of Brownian motion: comment on redundancy principle and the role of extreme statistics in molecular and cellular biology by Z. Schuss et al. Phys. Life Rev.

**28**, 80–82 (2019)Reynaud, K., Schuss, Z., Rouach, N., Holcman, D.: Why so many sperm cells? Commun. Integr. Biol.

**8**(3), e1017156 (2015)Ro, S., Kim, Y.W.: Parallel random target searches in a confined space. Phys. Rev. E

**96**(1), 012143 (2017)Rusakov, D.A., Savtchenko, L.P.: Extreme statistics may govern avalanche-type biological reactions: comment on “Redundancy principle and the role of extreme statistics in molecular and cellular biology” by Z. Schuss, K. Basnayake, D. Holcman. Phys. Life Rev.

**28**, 85–87 (2019)Schuss, Z., Basnayake, K., Holcman, D.: Redundancy principle and the role of extreme statistics in molecular and cellular biology. Phys. Life Rev.

**28**, 52–79 (2019). https://doi.org/10.1016/j.plrev.2019.01.001Sokolov, I.M.: Extreme fluctuation dominance in biology: On the usefulness of wastefulness: comment on “Redundancy principle and the role of extreme statistics in molecular and cellular biology by Z. Schuss, K. Basnayake and D. Holcman. Phys. Life Rev.

**28**, 88–91 (2019)Tamm, M.V.: Importance of extreme value statistics in biophysical contexts: comment on redundancy principle and the role of extreme statistics in molecular and cellular biology. Phys. Life Rev.

**28**, 94–95 (2019)van Beijeren, H.: The uphill turtle race; on short time nucleation probabilities. J. Stat. Phys.

**110**(3–6), 1397–1410 (2003)Wang, S.S., Alousi, A.A., Thompson, S.H.: The lifetime of inositol 1,4,5-trisphosphate in single cells. J. Gen. Physiol.

**105**(1), 149–171 (1995)Weiss, G.H., Shuler, K.E., Lindenberg, K.: Order statistics for first passage times in diffusion processes. J. Stat. Phys.

**31**(2), 255–278 (1983)Yuste, S.B.: Escape times of \(j\) random walkers from a fractal labyrinth. Phys. Rev. Lett.

**79**(19), 3565 (1997)Yuste, S.B., Acedo, L.: Diffusion of a set of random walkers in Euclidean media. First passage times. J. Phys. A Math. Gen.

**33**(3), 507–512 (2000)Yuste, S.B., Lindenberg, K.: Order statistics for first passage times in one-dimensional diffusion processes. J. Stat. Phys.

**85**(3–4), 501–512 (1996)Yuste, S.B., Lindenberg, K.: Subdiffusive target problem: survival probability. Phys. Rev. E

**76**(5), 051114 (2007)Yuste, S.B., Acedo, L., Lindenberg, K.: Order statistics for \(d\)-dimensional diffusion processes. Phys. Rev. E

**64**(5), 052102 (2001)

## Acknowledgements

SDL was supported by the National Science Foundation (Grant Nos. DMS-1814832 and DMS-1148230). JBM was supported by the National Science Foundation (Grant No. DMS-1814832).

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Communicated by Dr. Paul Newton.

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendix

### Appendix

In this Appendix, we collect the proofs of all the lemmas.

### Proof of Lemma 2

First notice that \(R_{n}(t)\) is a 2d Bessel process. In particular, \(R_{n}(t)\) has the same law as \(\sqrt{2D}\sqrt{W_{1}^{2}(t)+W_{2}^{2}(t)}\), where \(W_{1}(t)\in \mathbb {R}\) and \(W_{2}(t)\in \mathbb {R}\) are independent standard Brownian motions. Next, define the 3d Bessel process,

where \(W_{3}(t)\in \mathbb {R}\) is a third independent standard Brownian motion. If \(\tau \) is the first time that \(R^{\text {3d}}(t)\) hits radius \(a>0\),

then it is immediate that \(\mathbb {P}(\tau _{1,side }\le t)\le \mathbb {P}(\tau \le t)\) for all \(t\ge 0\). Since \(\mathbb {P}(\tau \le t)\) has the explicit formula (see, for example, Bernoff and Lindsay 2018),

we may take \(C_{1}=4a(\pi D)^{-1/2}\) and \(C_{2}=a^{2}/(4D)\) to complete the proof. \(\square \)

### Proof of Lemma 3

Recall that \(\tau _{1,z}\) is the first time \(Z_{1}(t)\) hits \(z=0\), assuming \(Z_{1}(0)=z_{0}>0\) and \(Z_{1}(t)\) reflects from \(z=h\ge z_{0}\). It is immediate that

where \(\tau \) is the first time a standard Brownian motion, \(W(t)\in \mathbb {R}\), hits \(z_{0}/\sqrt{2D}\),

The reflection principle (Durrett 2019) then gives

where \(\text {erf}(x)\) denotes the error function, which has the large *x* behavior,

Hence, we may take \(C_{3}=\frac{\sqrt{D}}{\sqrt{\pi }z_{0}}\) and \(C_{4}=z_{0}^{2}/(4D)\) to complete the proof. \(\square \)

### Proof of Lemma 6

Define the event

and let \(B^{c}\) denote its complement. It is immediate that

since \(\tau _{1}\le \tau _{1}^{+}\) if \(\tau _{1}<\min \{\tau _{1,side },\tau _{1,top }\}\). Next, by the definition of \(\tau _{1}^{+}\), we have that

Therefore

Now, it is an identity that

Therefore, we have that

Therefore, Jensen’s inequality yields

To complete the proof, we need only that \(\mathbb {E}[T_{+}]\) decays slower than \((\mathbb {P}(\tau _{1}>1))^{N}\) as \(N\rightarrow \infty \), which is implied by Lemma 7. \(\square \)

### Proof of Lemma 7

Let \(T_{1d }\) be the first time that any \({\widetilde{Z}}_{n}\) hits \(z=b\),

and let \(n_{f }\in \{1,\dots ,N\}\) denote the random index of this fastest \({\widetilde{Z}}_{n}\) particle

Define the event that the fastest particle first escapes the cylinder through the bottom

Since

we can apply the same argument as in Sect. 2 if we can manage to show that \(\sqrt{\mathbb {P}(A^{c})}(\log N)^{m}\rightarrow 0\) as \(N\rightarrow \infty \).

By De Morgan’s laws, we have that

The first probability can be handled by the same argument as in Sect. 2.

To handle \(\mathbb {P}({\widetilde{\tau }}_{n_{f },top }\le T_{1d })\), note that

As in the proof of Theorem 1, we may bound \(\mathbb {P}(T_{1d }\ge \delta )\) for sufficiently small \(\delta >0\) by

Furthermore, it is immediate that

Now, notice that \({\widetilde{\tau }}_{1,top }\) has the same law as the first time a standard Brownian motion, \(W(t)\in \mathbb {R}\), hits \((h-z_{0})/\sqrt{2D}\). In particular, the reflection principle (Durrett 2019) gives

where \(\text {erf}(x)\) denotes the error function with large *x* behavior given in (5.1). Hence,

Taking \(\delta =(\log N)^{-1/2}\) completes the proof. \(\square \)

### Proof of Lemma 8

Define the first time that the *n*th particle leaves a ball centered at \({{\mathbf {x}}}_{0}\) of radius \(z_{0}>0\),

It is immediate that the *n*th particle cannot reach the target \(\partial \Omega _{T }\) before time \(\tau _{n,ball }\). That is,

Define the radial process

Notice that \(R_{n}\) is not a three-dimensional Bessel process, due to the reflecting boundary \(\partial \Omega \). However, \(R_{n}\) does satisfy the following stochastic differential equation,

where \(R_{n}^{0}\) is a 3d Bessel process. In (5.3), \({{\mathbf {n}}}_{R}\) is the radial component of the inner normal field \({{\mathbf {n}}}:\partial \Omega \mapsto \mathbb {R}^{3}\), and \(L_{n}(t)\) is the local time of \(\mathbf {X}_{n}(t)\) on \(\partial \Omega \). More precisely, \(L_{n}(t)\) is nondecreasing and increases only when \(\mathbf {X}_{n}(t)\) is on \(\partial \Omega \). The significance of the local time term in (5.3) is that it forces \(\mathbf {X}_{n}(t)\) to reflect from \(\partial \Omega \).

By our assumption in (3.7) that *S* in (3.6) is a star domain, we are assured that \({{\mathbf {n}}}_{R}(\mathbf {X}_{n}(t))\le 0\) for all \(t\le \tau _{n,ball }\). Hence, \(R_{n}(t)\le R_{n}^{0}(t)\) for all \(t\le \tau _{n,ball }\). Therefore, \(\tau _{n,ball }^{0}\le \tau _{n,ball }\) almost surely. Hence, \(T_{ball }^{0}\le T_{3d }\) almost surely by (5.2). \(\square \)

## Rights and permissions

## About this article

### Cite this article

Lawley, S.D., Madrid, J.B. A Probabilistic Approach to Extreme Statistics of Brownian Escape Times in Dimensions 1, 2, and 3.
*J Nonlinear Sci* **30**, 1207–1227 (2020). https://doi.org/10.1007/s00332-019-09605-9

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00332-019-09605-9