Redundancy principle (biology)

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.

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.[15][16][17]

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 survival distribution of arrival time of the fastest ${\displaystyle Pr(\tau ^{1}>t)}$ is expressed in terms of a single particle, ${\displaystyle Pr(\tau ^{1}>t)=Pr^{N}(t_{1}>t)}$. Here ${\displaystyle Pr\{t_{1}>t\}}$ 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 ${\displaystyle \Omega }$:

${\displaystyle {\frac {\partial p(x,t)}{\partial t}}=D\Delta p(x,t){\hbox{ for }}x\in \Omega ,t>0}$

${\displaystyle {\begin{aligned}p(x,0)=&p_{0}(x){\hbox{ for }}x\in \Omega \\{\frac {\partial p}{\partial n}}(x,t)&=0{\hbox{ for }}x\in \partial \Omega _{r}\\p(x,t)&=0{\hbox{ for }}x\in \partial \Omega _{a},\end{aligned}}}$

where the boundary ${\displaystyle \partial \Omega }$ contains NR binding sites ${\displaystyle \partial \Omega _{i}\subset \partial \Omega }$ (${\displaystyle \partial \Omega _{a}=\bigcup \limits _{i=1}^{N_{R}}\partial \Omega _{i},\ \partial \Omega _{r}=\partial \Omega -\partial \Omega _{a}}$). The single particle survival probability is

${\displaystyle \Pr\{t_{1}>t\}=\int \limits _{\Omega }p(x,t)dx,}$ so that ${\displaystyle \Pr\{\tau ^{1}=t\}={\frac {d}{dt}}\Pr\{\tau ^{1}<t\}=N(\Pr\{t_{1}>t\})^{N-1}\Pr \limits \{t_{1}=t\},}$where

${\displaystyle \Pr\{t_{1}=t\}={\oint _{\partial \Omega _{a}}}{\frac {\partial p(x,t)}{\partial n}}\,dS_{x}}$and ${\displaystyle \Pr\{t_{1}=t\}=N_{R}{\oint _{\partial \Omega _{1}}}{\frac {\partial p(x,t)}{\partial n}}\,dS_{x}}$.

The probability density function (pdf) of the arrival time is

${\displaystyle \Pr\{\tau ^{1}=t\}=NN_{R}\left[\int \limits _{\Omega }p(x,t)dx\right]^{N-1}\oint \limits _{\partial \Omega _{1}}{\frac {\partial p(x,t)}{\partial n}}dS_{x},}$ which gives the MFPT

${\displaystyle {\bar {\tau }}^{1}=\int \limits \limits _{0}^{\infty }\Pr\{\tau ^{1}>t\}dt=\int \limits _{0}^{\infty }\left[\Pr\{t_{1}>t\}\right]^{N}dt.}$ The probability ${\displaystyle \Pr\{t_{1}>t\}}$ 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 an semi-interval ${\displaystyle [0,\infty [}$, the survival pdf is solution of

${\displaystyle {\begin{aligned}{\frac {\partial (x,t)}{\partial t}}&=D{\frac {\partial ^{2}p(x,t)}{\partial x^{2}}}\quad {\mbox{ for }}x>0,\ t>0\\p(x,0)&=\delta (x-a)\quad {\mbox{ for }}\ x>0,\quad p(0,t)=0\quad {\mbox{ for }}t>0,\end{aligned}}}$

that is${\displaystyle p(x,t)={\frac {1}{\sqrt {4D\pi t}}}\left[\exp \left\{-{\frac {(x-a)^{2}}{4Dt}}\right\}-\exp \left\{-{\frac {(x+a)^{2}}{4Dt}}\right\}\right].}$

The survival probability with D=1 is ${\displaystyle \Pr\{t_{1}>t\}=\int \limits \limits _{0}^{\infty }p(x,t)\,dx=1-{\frac {2}{\sqrt {\pi }}}\int \limits \limits _{a/{\sqrt {4t}}}^{\infty }e^{-u^{2}}\,du}$. To compute the MFPT, we expand the complementary error function

${\displaystyle {\frac {2}{\sqrt {\pi }}}\int \limits \limits _{x}^{\infty }e^{-u^{2}}\,du={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\left(1-{\frac {1}{2x^{2}}}+O(x^{-4})\right)\quad {\mbox{for}}\ x\gg 1,}$ which gives${\displaystyle {\bar {\tau }}^{1}=\int \limits \limits _{0}^{\infty }\left[\Pr\{t_{1}>t\}\right]^{N}dt\approx \int \limits \limits _{0}^{\infty }\exp \left\{N\ln \left(1-{\frac {e^{-(a/{\sqrt {4t}})^{2}}}{(a/{\sqrt {4t}}){\sqrt {\pi }}}}\right)\right\}\,dt\approx {\frac {a^{2}}{4}}\int \limits \limits _{0}^{\infty }\exp \left\{-N{\frac {{\sqrt {u}}e^{-{\frac {1}{u}}}}{\sqrt {\pi }}}\right\}du}$,

leading (the main contribution of the integral is near 0) to ${\displaystyle {\bar {\tau }}^{1}\approx {\frac {a^{2}}{4D\ln {\frac {N}{\sqrt {\pi }}}}}\quad {\mbox{for}}\ N\gg 1.}$

This result is reminiscent of using the Gumbel's law. Similarly, escape from the interval [0,a] is computed from the infinite sum

${\displaystyle p(x,t\,|\,y)={\frac {1}{\sqrt {4D\pi t}}}\sum \limits _{n=-\infty }^{\infty }\left[\exp \left\{-{\frac {(x-y+2na)^{2}}{4t}}\right\}-\exp \left\{-{\frac {(x+y+2na)^{2}}{4t}}\right\}\right]}$.The conditional survival probability is approximated by[1]

${\displaystyle \Pr\{t_{1}>t\,|\,y\}=\int \limits \limits _{0}^{a}p(x,t\,|\,y)\,dxds\sim 1-\max {\frac {2{\sqrt {t}}}{\sqrt {\pi }}}\left[{\frac {e^{-y^{2}/4t}}{y}},{\frac {e^{-(a-y)^{2}/4t}}{a-y}}\right]\quad {\mbox{as}}\ t\to 0}$, where the maximum occurs at ${\displaystyle \delta =}$ min[y,a-y] for 0<y<a (the shortest ray from y to the boundary). All other integrals can be computed explicitly, leading to

${\displaystyle {\bar {\tau }}^{1}=\int \limits \limits _{0}^{\infty }\left[\Pr\{t_{1}>t\}\right]^{N}dt\approx \int \limits \limits _{0}^{\infty }\exp \left\{N\ln \left(1-{\frac {8{\sqrt {t}}}{\delta {\sqrt {\pi }}}}e^{-\delta ^{2}/16t}\right)\right\}dt\approx {\frac {\delta ^{2}}{16D\ln {\frac {2N}{\sqrt {\pi }}}}}\quad {\mbox{for}}\ N\gg 1.}$

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 ${\displaystyle \delta _{min}=d(S,A),}$where 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 :

${\displaystyle {\bar {\tau }}^{d1}\approx {\frac {\delta _{min}^{2}}{4D\ln \left({\frac {N}{\sqrt {\pi }}}\right)}},{\hbox{in dim 1, valid for}}N\gg 1,}$

${\textstyle {\bar {\tau }}^{d2}\approx {\frac {\delta _{min}^{2}}{4D\log \left({\frac {\pi {\sqrt {2}}N}{8\log \left({\frac {1}{a}}\right)}}\right)}},{\hbox{ in dim 2 for }}{\frac {N}{\log({\frac {1}{\epsilon }})}}\gg 1,}$

${\displaystyle {\bar {\tau }}^{d3}\approx {\frac {\delta _{min}^{2}}{4D{\log \left(N{\frac {4a^{2}}{\pi ^{1/2}\delta _{min}^{2}}}\right)}}},{\hbox{ in dim }}3,{\hbox{ for }}{\frac {Na^{2}}{\delta _{min}^{2}}}\gg 1.}$

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.

How nature sets the disproportionate numbers of particles remain unclear, but can be found using the theory of diffusion. One example is the number of neurotransmitters around 2000 to 3000 released during synaptic transmission, that are set to compensate the low copy number of receptors, so the probability of activation is restored to one.[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. This property is universal, ranging from the molecular scale to the population level.[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.[21]