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 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. When a large distance separates 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 event 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
Stochastic process The motion of a particle located at position X t {\displaystyle X_{t}} can be described by the Smoluchowski's limit of the Langevin equation:
d X t = 2 D d B t + 1 γ F ( x ) d t , {\displaystyle dX_{t}={\sqrt {2D}}\,dB_{t}+{\frac {1}{\gamma }}F(x)dt,}
where D {\displaystyle D} is the diffusion coefficient of the particle, γ {\displaystyle \gamma } is the friction coefficient per unit of mass, F ( x ) {\displaystyle F(x)} the force per unit of mass, and B t {\displaystyle B_{t}} is a Brownian motion. This model is classically used in molecular dynamics simulations.
Jump processes
x n + 1 = { x n − a , with probability l ( x n ) x n + b , with probability r ( x n ) {\displaystyle {\begin{aligned}x_{n+1}={\begin{cases}x_{n}-a,&{\text{with probability }}l(x_{n})\\x_{n}+b,&{\text{ with probability }}r(x_{n})\end{cases}}\end{aligned}}} , which is for example a model of telomere length dynamics. Here r ( x ) = 1 1 + β x , {\displaystyle r(x)={\frac {1}{1+\beta x}},} , with r ( x ) + l ( x ) = 1 {\displaystyle r(x)+l(x)=1} .
Directed motion process
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