Redundancy principle (biology)

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The redundancy principle in Biology ^{[1] [2] [3] [4] [5] [6] [7] [8] [9]} expresses the need of many copies of the same entity (cell, molecules, ions) to fullfill 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 site 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.

These disproportionate numbers of particles should not be considered wasteful, but rather, they are necessary for generating to perform the function in due time and accelerate the response using the fastest arrival particle. This property seems universal, ranging from the molecular scale to the population level. 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 selection of the fastest in a collective ensemble. Thus the redundancy principle can express in terms of the theory of extreme statistics to determine its laws and quantify how shortest paths are selected.

In most cases, the redundancy principle is used because of the large distance between the source and the target (a small activation site). Had nature use few copies, the activation would have taken a long time as finding a small target by random falls into extremely rare events.

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. 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.

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 ${\displaystyle \tau ^{1}=\min(t_{1},\ldots ,t_{N}),}$ where ${\displaystyle t_{i}}$ are the independent arrival times of the N ions in the medium. The distribution of Pr(\tau^{1}>t) is expressed in terms of a single particle, ${\displaystyle Pr(\tau ^{1}>t)=Pr^{N}(t_{1}>t)}$

Here Pr{t_{1}>t } is the survival probability of a single particle prior to binding at the target. This probability is computed by solving the diffusion equation \cite{Schuss}

Failed to parse (unknown function "\x"): {\displaystyle \frac{\partial p(\x,t)}{\partial t} =&D \Delta p(\x,t)\quad\hbox{\rm for } \x \in Omega,\ t>0\\ p(\x,0)=&p_0(\x)\quad \hbox{\rm for } \x \in\Omega\nonumber\\ \frac{\p p(\x,t)}{\p \n} =&0\quad \hbox{\rm for } \x \in\p\Omega_r\nonumber\\ p(\x,t)=&0 \quad \hbox{\rm for } \x \in \p\Omega_a,\nonumber } where the boundary $\p\Omega$ contains $N_R$ binding sites $\p\Omega_i\subset\p \Omega\ (\p\Omega_a=\bigcup\limits_{i=1}^{N_R}\p\Omega_i,\ \p\Omega_r=\p\Omega-\p\Omega_a)$. The single particle survival probability is

\beq \label{surv} \Pr\{t_{1}>t \} =\int\limits_{\Omega} p(\x,t)\,d\x, \eeq so that

$ \Pr\{\tau^{1}=t \} = \frac{d}{dt}\Pr\{\tau^{1}<t \}=N(\Pr\{t_{1}>t \})^{N-1}\Pr\limits\{t_{1}=t \},$ where $\Pr\{t_{1}=t \}= {\oint_{\p\Omega_a}} \frac{\p p(\x,t)}{\p \n}\, dS_{\x}.$ %and $\Pr\{t_{1}=t \}= N_R {\oint_{\p \Omega_1}} % \frac{\p p(\x,t)}{\p \n}\,dS_{\x}$. The probability density function (pdf) of the arrival time is \beq \label{arrv1} \Pr\{\tau^{1}=t \} =N N_R \left[\int\limits_{\Omega} p(\x,t)d\x \right]^{N-1}\oint\limits_{\p \Omega_1} \frac{\p p(\x,t)}{\p \n} dS_{\x}, \eeq which gives the MFPT \beq\label{mfptmin} \bar{\tau}^{1}=\int\limits\limits_0 ^{\infty}\Pr\{\tau^{1}>t\} dt = \int\limits_0 ^{\infty} \left[ \Pr\{t_{1}>t\} \right]^N dt. \eeq

They are expressed in terms of the shortest distance from the source $S$ to the absorbing window $A$, measured by the distance $\delta_{min}=d(S,A)$, where $d$ is the associated Euclidean distance here. Interestingly, the 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. {\color{red} For a diffusion coefficient $D$ and a window 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 \cite{extreme4,extreme5,Kanishka2}: Failed to parse (syntax error): {\displaystyle \bar\tau^{d1} \approx& \frac{\delta^2_{min}}{4D\ln\left(\frac{N}{\sqrt{\pi}}\right)}, \hspace{2cm}\hbox{ in dim } 1, \hbox{ valid for } N\gg1\\ \bar{\tau}^{d2}\approx& \ds \frac{\ds \delta^2_{min}}{\ds 4 D \log\left(\frac{\pi \sqrt{2}N}{8\log\left(\frac{1}{a}\right)}\right)}, \quad\hbox{ in dim } 2, \,\,\,\,\hbox{ for } \frac{N}{\log (\frac{1}{\eps})}\gg1 \label{finalform2}\\ \bar\tau^{d3} \approx& \frac{\delta^2_{min}}{2D\sqrt{\log\left(\ds N\frac{4a^2}{\pi^{1/2}\delta^2_{min}}\right)}}, \hbox{ in dim } 3, \hbox{ for } \frac{Na^2}{\delta^2_{min}}\gg1. }

These formulas show that the expected arrival time of the fastest particle is in dimension 1 and 2, $O(1/\log(N))$ (see Fig. \ref{fig2}). They should be used instead of the classical forward rate in models of activation in biochemical reactions. The method to derive formulas \ref{finalform} is based on short-time asymptotic and the Green's function representation of the Helmholtz equation \cite{Kanishka2}. These formula can be generalized to any density distribution $\rho$ of initial particles such that $ \int_{\Omega} \rho(A) dS_A=N$ . Indeed, if we consider the case of dimension 2, the mean arrival time for the fastest is computed by averaging over the density $\rho$ in the domain $\Omega$ with surface $|\Omega|$, leading to

\langle \bar\tau^{d2}_{\rho} \rangle = \int_{\Omega} \ds \frac{\ds d^2(S,A)}{\ds 4 D \log\left(\frac{\pi \sqrt{2}\rho(A)|\Omega|}{8\log\left(\frac{1}{a}\right)}\right)} \rho(A) dS_A.

The mathematical analysis of large numbers of molecules, which are obviously redundant in the traditional activation theory, define the in vivo time scale of chemical reactions. The computation relies on asymptotics or probabilistic approach to estimate the time of the fastest in various geometries.^{[10] [11] [12]}