Redundancy principle for optimal random search in biology
Summary (2 min read)
1 Introduction
- Here, the authors discuss how the many molecules, which are obviously redundant in the traditional activation theory, define the in vivo time scale of chemical reactions.
- Yet, when the number of sperms is reduced by four, infertility ensues [2].
- The forward rate k1 has been used in almost all representations of chemical reactions: it is the basis of the Gillespie algorithm for generating statistics of stochastic simulations with rate k1.
- But, as seen below, its statistics is different from the rate of the fastest arrival.
2 Biochemical reactions in cell biology
- Chemical activation in cell biology starts with the binding of few molecules (Fig 1).
- When there is a separation between the site of the first activation and that of amplification.
- A|, x the positions of injection, and the center of the window is A.
3 Examples of signal transduction
- There are many examples of activation or transduction by the arrival of the first particle at the activation site, which defines the time scale of [1].
- It was shown recently that such a short time scale is generated by an avalanche reaction triggered by the arrival of the fastest calcium ion to a receptor (see Fig. 1).
- This activation is mediated by the release of thousands of neurotransmitters from the pre-synaptic terminal.
- Two binding events on the same receptor are required for activation.
- The number of activated IP3 and the location of IP3 receptors sets the time scale of this transduction pathway.
4 A nonlinear effect of spermatozoa redundancy on
- In the key step of fertilization, sperms have to find the ovule within the short time it is fertile (Fig. 3).
- These are the optimal (bang-bang) solutions of the classical control problem (Fig. 3 and [14]).
- This result suggests that linear trajectories might not be generated by any chemotaxis at a distance of a few centimeters, which is too far away from the source.
- It is thus conceivable that the extreme statistics are responsible for selection of the fastest trajectories determined by sperm dynamics in the uterus and fallopian tubes.
- In addition, it is well documented that reducing their number by a factor of 4 may cause infertility [2].
5 Morale of the story
- Molecular-level simulations of activation processes should avoid the Gillespie algorithm and reaction-diffusion equations especially in the context of cellular fast activation induced by molecular pathways.
- They should use straightforward Brownian paths and extreme statistics, as given in the formulas above.
- The large number of paths produces optimal trajectories that set the observed time scale.
- Disproportionate numbers of particles in natural processes should not be considered wasteful, but rather, they serve a clear purpose: they are necessary for generating the fastest response.
- This property is universal ranging from the molecular scale to the population level.
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Frequently Asked Questions (11)
Q2. what is the diffusion probability of a dendrite?
neuronal connections often occur on a dendritic spine, where the fast calcium increase in dendrites may happen a few milliseconds after initiation in the spine head.
Q3. What is the simplest explanation of the backward rate?
The computation of the backward rate has long history that begins with Arrhenius’ law k−1 = AeE/kT , where A is a constant, and E is activation energy, then Kramers’ rate, derived from the molecular stochastic Langevin equation, which gives the prefactor A.
Q4. What is the speed of the first particle to arrive to a receptor?
When particle move by diffusion, is it sufficient that the first particle arrives to a receptor to open it and lead to an avalanche either through the entry of ions or the opening of the neighboring receptors.
Q5. What is the simplest explanation of the mass-action theory of chemical reactions?
More specifically, mass-action theory of chemical reactions between two reactants insolution, A and B, is expressed asA k1 k−1 B,where k1 and k−1 are the forward and backward reaction rates, respectively.
Q6. What is the purpose of disproportionate numbers of particles in natural processes?
Disproportionate numbers of particles in natural processes should not be considered wasteful, but rather, they serve a clear purpose: they are necessary for generating the fastest response.
Q7. what is the probability of a particle arriving at a certain time?
The single particle survival probability isPr{t1 > t} = ∫ Ω p(x, t) dx, (5)so that Pr{τ 1 = t} = d dt Pr{τ 1 < t} = N(Pr{t1 > t})N−1 Pr{t1 = t}, where Pr{t1 = t} =∮∂Ωa∂p(x,t) ∂n dSx and Pr{t1 = t} = NR ∮ ∂Ω1 ∂p(x,t) ∂n dSx.
Q8. How many spermatozoa are needed to affect the search process?
The number of spermatozoa is thus the main determinant of the selection and since the mean time for the first one to arrive is O(1/logN), a large number of them is necessary to affect the search process.
Q9. What is the role of the first particle to find a small target?
Activation occurs by the first particle to find a small target and the time scale for this activation uses very different geometrical features (optimal paths) than the one described by the traditional mass-action law, reaction-diffusion equations, or Markov-chain representation of stochastic chemical reactions.
Q10. What is the shortest arrival time of a particle?
Brownian trajectories (ions) in a bounded domain Ω to a binding site, the shortest arrival time τ 1 is by definitionτ 1 = min(t1, . . . , tN), (2)where ti are the i.i.d. arrival times of the N ions in the medium.
Q11. what is the probability of a particle arriving at a target?
The asymptotic laws for the expected first arrival time ofBrownian particles to a target for large N , are,τ̄ d1 ≈ δ 2 min4D ln( N√ π ) , in dim 1 (8)τ̄ d2 ≈ ≈ s 2 24D log( π √ 2N8 log ( 1 ε)) , in dim 2 (9)τ̄ d3 ≈ δ 22D √ log ( N 4a2π1/2δ2) , in dim 3. (10)These formulas show that expected arrival time of the fastest particles is O(1/ log(N)) (see Fig. 2).