Fastest first-passage time statistics for time-dependent particle injection

A common scenario in a variety of biological systems is that multiple particles are searching in parallel for an immobile target located in a bounded domain, and the fastest among them that arrives to the target first triggers a given desirable or detrimental process. The statistics of such extreme events—the first-passage to the target—is well-understood by now through a series of theoretical analyses, but exclusively under the assumption that all <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math> particles start , i.e., all are introduced into the domain instantly, by <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>δ</b:mi></b:math>-function-like pulses. However, in many practically important situations this is not the case: to start their search, the particles often have to enter first into a bounded domain, e.g., a cell or its nucleus, penetrating through gated channels or nuclear pores. This entrance process has a random duration so that the particles appear in the domain sequentially and with a time delay. Here we focus on the effect of such an extended-in-time injection of multiple particles on the fastest first-passage time (fFPT) and its statistics. We derive the full probability density function <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msub><c:mi>H</c:mi><c:mi>N</c:mi></c:msub><c:mrow><c:mo>(</c:mo><c:mi>t</c:mi><c:mo>)</c:mo></c:mrow></c:mrow></c:math> of the fFPT with an arbitrary time-dependent injection intensity of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>N</d:mi></d:math> particles. Under rather general assumptions on the survival probability of a single particle and on the injection intensity, we derive the large-<e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>N</e:mi></e:math> asymptotic formula for the mean fFPT, which is quite different from that obtained for the instantaneous <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>δ</f:mi></f:math>-pulse injection. The extended injection is also shown to considerably slow down the convergence of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:msub><g:mi>H</g:mi><g:mi>N</g:mi></g:msub><g:mrow><g:mo>(</g:mo><g:mi>t</g:mi><g:mo>)</g:mo></g:mrow></g:mrow></g:math> to its large-<h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>N</h:mi></h:math> limit—the Gumbel distribution—so that the latter may be inapplicable in the most relevant settings with few tens to few thousands of particles.