This redundancy is particularly relevant when the sites of activation are physically separated from the initial position of the molecular messengers.
The redundancy is often generated for the purpose of resolving the time constraint of fast-activating pathways.
It can be expressed in terms of the theory of extreme statistics to determine its laws and quantify how the shortest paths are selected.
The main goal is to estimate these large numbers from physical principles and mathematical derivations.
Had nature used less copies than normal, activation would have taken a much longer time, as finding a small target by chance is a rare event and falls into narrow escape problems.
This rectilinear with constant velocity is a simplified model of spermatozoon motion in a bounded domain
[14] The mathematical analysis of large numbers of molecules, which are obviously redundant in the traditional activation theory, is used to compute the in vivo time scale of stochastic chemical reactions.
The computation relies on asymptotics or probabilistic approaches to estimate the mean time of the fastest to reach a small target in various geometries.
Brownian trajectories (ions) in a bounded domain Ω that bind at a site, the shortest arrival time is by definition
can be computed using short-time asymptotics of the diffusion equation as shown in the next sections.
The short-time asymptotic of the diffusion equation is based on the ray method approximation.
For a diffusion coefficient D and a window of size a, the expected first arrival times of N identically independent distributed Brownian particles initially positioned at the source S are expressed in the following asymptotic formulas :
These formulas show that the expected arrival time of the fastest particle is in dimension 1 and 2, O(1/\log(N)).
The method to derive formulas is based on short-time asymptotic and the Green's function representation of the Helmholtz equation.
Note that other distributions could lead to other decays with respect N. The optimal paths for the fastest can be found using the Wencell-Freidlin functional in the Large-deviation theory.
These paths correspond to the short-time asymptotics of the diffusion equation from a source to a target.
In general, the exact solution is hard to find, especially for a space containing various distribution of obstacles.
The Wiener integral representation of the pdf for a pure Brownian motion is obtained for a zero drift and diffusion tensor
is the ensemble of shortest paths selected among n Brownian trajectories, starting at point y and exiting between time t and t+dt from the domain
concentrate near the shortest paths starting from y and ending at the small absorbing window
can be approximated using discrete broken lines among a finite number of points and we denote the associated ensemble by
A path made of broken lines (random walk with a time step
The probability of a Brownian path x(s) can be expressed in the limit of a path-integral with the functional:
Indeed, the Euler-Lagrange equations for the extremal problem are the classical geodesics between y and a point in the narrow window
in the opening and a curvature R. The diffusion coefficient is D. The shortest arrival time, valid for large n is given by
[18] How nature sets the disproportionate numbers of particles remain unclear, but can be found using the theory of diffusion.
[19][20] In natural processes these large numbers should not be considered wasteful, but are necessary for generating the fastest possible response and make possible rare events that otherwise would never happen.
[21] Nature's strategy for optimizing the response time is not necessarily defined by the physics of the motion of an individual particle, but rather by the extreme statistics, that select the shortest paths.
In addition, the search for a small activation site selects the particle to arrive first: although these trajectories are rare, they are the ones that set the time scale.
We may need to reconsider our estimation toward numbers when punctioning nature in agreement with the redundant principle that quantifies the request to achieve the biological function.