Redundancy principle (biology)

The redundancy principle in biology expresses the need of many copies of the same entity (cells, molecules, ions) to fulfill a biological function. Examples are numerous: disproportionate numbers of spermatozoa during fertilization compared to one egg, large number of neurotransmitters released during neuronal communication compared to the number of receptors, large numbers of released calcium ions during transient in cells and many more in molecular and cellular transduction or gene activation and cell signaling. This redundancy is particularly relevant when the sites of activation is 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 shortest paths are selected. The main goal is to estimate these large numbers from physical principles and mathematical derivations.
When large distance separate the source and the target (a small activation site), the redundancy principle explains that this geometrical gap can be compensated by large number. 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 events and falls into narrow escape problems.
Molecular rate.
The time for the fastest particles to reach a target in the context of redundancy depends on the numbers and the local geometry of the target. In most of the time, it is the rate of activation. This rate should be used instead of the classical Smoluchowski's rate describing the mean arrival time, but not the fastest. The statistics of the minimal time to activation set kinetic laws in biology, which can be quite different from the ones associated to average times.
Physical models.
Stochatic processes.
The motion of a particle located at position formula_1 can be described by the Smoluchowski's limit of the Langevin equation:
formula_2
where formula_3 is the diffusion coefficient of the particle, formula_4 is the friction coefficient per unit of mass, formula_5 the force per unit of mass, and formula_6 is a Brownian motion. This model is classically used in molecular dynamics simulations.
Jump processes.
formula_7, which is for example a model of telomere length dynamics. Here formula_8 , with formula_9.
Directed motion process.
formula_10 where formula_11 is a unit vector chosen from a uniform distribution. Upon hitting an obstacle at a boundary point formula_12, the velocity changes to formula_13 where formula_14 is chosen on the unit sphere in the supporting half space at formula_15 from a uniform distribution, independently of formula_11. This rectilinear with constant velocity is a simplified model of spermatozoon motion in a bounded domain formula_17. Other models can be diffusion on graph, active graph motion.
Mathematical formulation: Computing the rate of arrival time for the fastest.
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.
With N non-interacting i.i.d. Brownian trajectories (ions) in a bounded domain Ω that bind at a site, the shortest arrival time is by definition
formula_18 where formula_19 are the independent arrival times of the N ions in the medium. The survival distribution of arrival time of the fastest formula_20 is expressed in terms of a single particle, formula_21. Here formula_22 is the survival probability of a single particle prior to binding at the target.This probability is computed from the solution of the diffusion equation in a domain formula_23:
formula_24
formula_25
where the boundary formula_26 contains NR binding sites formula_27 (formula_28). The single particle survival probability is
formula_29 so that formula_30where
formula_31and formula_32.
The probability density function (pdf) of the arrival time is
formula_33 which gives the MFPT
formula_34 The probability formula_35 can be computed using short-time asymptotics of the diffusion equation as shown in the next sections.
Explicit computation in dimension 1.
The short-time asymptotic of the diffusion equation is based on the ray method approximation. For an semi-interval formula_36, the survival pdf is solution of
formula_37
that isformula_38
The survival probability with D=1 is formula_39. To compute the MFPT, we expand the complementary error function
formula_40 which givesformula_41,
leading (the main contribution of the integral is near 0) to formula_42
This result is reminiscent of using the Gumbel's law. Similarly, escape from the interval [0,a] is computed from the infinite sum
formula_43.The conditional survival probability is approximated by
formula_44, where the maximum occurs at formula_45 min[y,a-y] for 0<y\bar{\tau}^{1}= \int\limits\limits_0 ^{\infty} \left[ \Pr\{t_{1}>t\} \right]^N dt \approx 
e^{-\delta^2/16t}\right) \right\}dt \approx \frac{\delta^2}{16D\ln\frac{2N}{\sqrt{\pi}}}\quad\mbox{for}\ N\gg1. </math>
Arrival times of the fastest in higher dimensions.
The arrival times of the fastest among many Brownian motions are expressed in terms of the shortest distance from the source S to the absorbing window A, measured by the distance formula_46where d is the associated Euclidean distance. Interestingly, trajectories followed by the fastest are as close as possible from the optimal trajectories. In technical language, the associated trajectories of the fastest among N, concentrate near the optimal trajectory (shortest path) when the number N of particles increases. 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 :
formula_47
formula_48
formula_49
These formulas show that the expected arrival time of the fastest particle is in dimension 1 and 2, O(1/\log(N)). They should be used instead of the classical forward rate in models of activation in biochemical reactions. 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.
Optimal Paths.
Minimizing The optimal path in large 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 formula_50 constant, so that it is given by the probability of a sampled path until it exits at the small window formula_51at the random time T
formula_52
formula_53
where
formula_54 in the product and T is the exit time in the narrow absorbing window formula_55 Finally,
formula_56
where formula_57 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 formula_23. The probabilityformula_59 is used to show that the empirical stochastic trajectories of formula_60 concentrate near the shortest paths starting from y and ending at the small absorbing window formula_61, under the condition that formula_62.  The paths of formula_57 can be approximated using discrete broken lines among a finite number of points and we denote the associated ensemble by formula_64.  Bayes'rule leads toformula_65 where formula_66 is the probability that a path of formula_64  exits in m-discrete time steps. A path made of broken lines (random walk with a time stepformula_68) can be expressed using Wiener path-integral.  The probability of a Brownian path x(s) can be expressed in the limit of a path-integral with the functional:
formula_69
The Survival probability conditioned on starting at y is given by the Wiener representation:
formula_70
where formula_71 is the limit Wiener measure: the exterior integral is taken over all end points x and the path integral is over all paths starting from x(0). When we consider n-independent paths formula_72 (made of points with a time step formula_68 that exit in m-steps, the probability of such an event is
formula_74.Indeed, when there are n paths of m steps, and the fastest one escapes in m-steps, they should all exit in m steps. Using the limit of path integral, we get heuristically the representation
<math>Pr \{ \hbox{ Path }\sigma \in \tilde S_n(y)|m, \tau_{\sigma}=m \Delta t \}= \left(\int\limits_