![Figure 3: (A) Schematic representation of the uterus and the diffusive trajectory of the sperm that reaches an ovule. The possibility to form a zygote is determined by whether the time of the fastest sperm to reach the ovule is within the fertile period. (B) In a Brownian simulation in a schematic uterus, the trajectories with short arrival time are concentrated along two symmetric optimal paths (white dashed line) [14].](/figures/figure-3-a-schematic-representation-of-the-uterus-and-the-1vcj3wep.webp)
Redundancy principle for optimal random search in
biology
Z. Schuss
∗
, K. Basnayake
†
, D. Holcman
†
Abstract
Chemical activation rate is traditionally determined by the diffusion flux into
an absorbing ball, as computed by Smoluchowski in 1916. Thus the rate is set
by the mean first passage time (MFPT) of a Brownian particle to a small target.
This paradigm is shifted in this manuscript to set the time scale of activation in
cellular biology to the mean time of the first among many arrivals of particles at
the activation site. This rate is very different from the MFPT and depends on
different geometrical parameters. The shift calls for the reconsideration of physi-
cal modeling such as deterministic and stochastic chemical reactions based on the
traditional forward rate, especially for fast activation processes occurring in living
cells. Consequently, the biological activation time is not necessarily exponential.
The new paradigm clarifies the role of population redundancy in accelerating search
processes and in defining cellular-activation time scales. This is the case, for ex-
ample, in cellular transduction or in the nonlinear dependence of fertilization rate
on the number of spermatozoa. We conclude that statistics of the extreme set the
new laws of biology, which can be very different from the physical laws derived for
individuals.
1 Introduction
Why are specialized sensory cells so sensitive and what determines their efficiency? For
example, rod photoreceptors can detect a single photon in few tens of milliseconds [1],
olfactory cells sense few odorant molecules on a similar time scale, calcium ions can
induce calcium release in few milliseconds in neuronal synapses, which is a key process
in triggering synaptic plasticity that underlies learning and memory. Decades of research
have revealed the molecular processes underlying these cellular responses, that identify
molecules and their rates, but in most cases, the underlying physical scenario remains
unclear.
∗
Department of Applied Mathematics, Tel-Aviv University, Tel-Aviv 69978, Israel.
†
Computational Bioplogy and Applied Mathematics, IBENS, Ecole Normale Sup´erieure, Paris, France.
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Here, we discuss how the many molecules, which are obviously redundant in the tra-
ditional activation theory, define the in vivo time scale of chemical reactions. This redun-
dancy is particulary 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 molecular pathways. 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.
We also discuss the role of the fastest particle to arrive at a small target in the context
of fertilization, where the enormous and disproportionate number of spermatozoa, relative
to the single ovule, remains an enigma. Yet, when the number of sperms is reduced by
four, infertility ensues [2]. The analysis of extreme statistics can explain the waste of
resources in so many natural systems. So yes, numbers matter and wasting resources
serve an optimal purpose: selecting the most fitted or the fastest.
More specifically, mass-action theory of chemical reactions between two reactants in
solution, A and B, is expressed as
A
k
1
k
−1
B,
where k
1
and k
−1
are the forward and backward reaction rates, respectively. The compu-
tation of the backward rate has long history that begins with Arrhenius’ law k
−1
= Ae
E/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. For the past sixty
years, chemical physicists computed the activation energy E and clarified the role of the
energy landscape, with extensions to applications in chemistry, signal processing (time
to loss of lock in phase trackers [3]), finance (time for binary option price to reach a
threshold), and many more [4].
In contrast, the forward rate k
1
represents the flux of three-dimensional Brownian par-
ticles arriving at a small ball or radius a. Smoluchowski’s 1916 forward rate computation
reveals that
k
1
= 4πDca, (1)
where D is the diffusion coefficient, when the concentration c is maintained constant
far away from the reaction site. When the window is in a smooth surface or inside a
hidden cusp, the forward rate k
1
is the reciprocal of the MFPT of a Brownian particle
to the window. The precise geometry of the activating small windows has been captured
by general asymptotics of the mean first arrival time at high activation energy. This
mean time is, indeed, sufficient to characterize the rate, because the binding process is
2
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Poissonian and the rate is precisely the reciprocal of the MFPT. These computations are
summarized in the narrow escape theory [5–9]. For example, when the small absorbing
window represents binding at a surface, the forward rate is given by
k
−1
1
=
|Ω|
4aD
1 +
L(0) + N(0)
2π
a log a + o(a log a)
,
with |Ω| the volume of the domain of Brownian motion, a is the radius of the absorbing
window [5], and L(0) and N(0) are the principal mean curvatures of the surface at the
small absorbing window.
The forward rate k
1
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 k
1
. It has also been used in coarse-graining stochastic chemical-reactions into a
Markov chain. However, the rate k
1
is not used in simulations of Brownian trajectories.
In the latter case, the statistics of arrival times can be computed directly. Obviously,
diffusion theory computes k
1
from the mean arrival rate of a single particle. However,
in cell biology, biochemical processes are often activated by the first (fastest) particle
that reaches a small binding target, so that the average time for a single particle do es
not necessarily represent the time scale of activation. Thus even the Gillespie algorithm
would give a rate, sampled with mean k
1
. But, as seen below, its statistics is different
from the rate of the fastest arrival. This difference is the key to the determination of
the time scale of cellular activation that can be computed from full Brownian simulations
and/or asymptotics of the fastest particle. This is an important shift from the traditional
paradigm.
2 Biochemical reactions in cell biology
Chemical activation in cell biology starts with the binding of few molecules (Fig 1). The
signal is often amplified so that a molecular event is transformed into a cellular signal.
How fast is this activation? What defines its time scale? when there is a separation
between the site of the first activation and that of amplification. 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.
Thus the time scale of activation is not given by the reciprocal of the forward rate, but
rather by the extreme statistics, that is, by the mean arrival time of the first particle to
the activation site (the target).
The statistics of the first particle to arrive to a target can be computed from the
statistics of a single particle when they are all independent and identically distributed
[10–13]. With N non-interacting i.i.d. Brownian trajectories (ions) in a bounded domain
Ω to a binding site, the shortest arrival time τ
1
is by definition
τ
1
= min(t
1
, . . . , t
N
), (2)
3
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RyR
First Ca
to arrive
2+
Activation of
an avalanche
mGluR
ER
IP3
to arrive
IP3
receptor
Ca
2+
Release
First four
Avalanche
arrive
AMPA/
First
by input currents
Neuro-
transmitter
release
Lost
receptor
(A)
(B)
(C)
Amplification
two to
molecules
Cell membrane
NMDA
Spine
Apparatus
Figure 1: (A) Calcium-induced-calcium-release in a dendritic spine. The first Ryanodine Re-
ceptor (RyR) at the base of the spine apparatus that opens is triggered by the fastest calcium
ion. An avalanche of calcium release ensues by opening the neighboring receptors. This leads
to rapid amplification at a much shorter time than the MFPT of the diffusing calcium ions.
(B) Activation of calcium release by IP3 receptors, which are calcium channels gated by IP3
molecules, which function as secondary messengers. When the first IP3 molecules arrives at the
first IP3R, its calcium release induces an avalanche due to the opening of subsequent IP3 recep-
tors. (C) In the post-synaptic terminal, the influx of ions due to the opening of NMDA/AMPA
receptors is the amplification process. The signal of the pre-synaptic signal transmitted by the
neurotransmitter molecules diffusing in the synaptic cleft and the time scale of the amplification
are determined by the fastest molecules that arrive at the receptor targets and open them.
4
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where t
i
are the i.i.d. arrival times of the N ions in the medium. The narrow escape
problem (NEP) is to find the PDF and the MFPT of τ
1
. The complementary PDF of τ
1
,
Pr{τ
1
> t} = Pr
N
{t
1
> t}. (3)
Here Pr{t
1
> t} is the survival probability of a single particle prior to binding at the
target. This probability can be computed by solving the diffusion equation
∂p(x, t)
∂t
=D∆p(x, t) for x ∈ Ω, t > 0 (4)
p(x, 0) =p
0
(x) for x ∈ Ω
∂p(x, t)
∂n
=0 for x ∈ ∂Ω
r
p(x, t) =0 for x ∈ ∂Ω
a
,
where the boundary ∂Ω contains N
R
binding sites ∂Ω
i
⊂ ∂Ω (∂Ω
a
=
N
R
i=1
∂Ω
i
, ∂Ω
r
=
∂Ω − ∂Ω
a
). The single particle survival probability is
Pr{t
1
> t} =
Ω
p(x, t) dx, (5)
so that Pr{τ
1
= t} =
d
dt
Pr{τ
1
< t} = N(Pr{t
1
> t})
N−1
Pr{t
1
= t}, where Pr{t
1
= t} =
∂Ω
a
∂p(x,t)
∂n
dS
x
and Pr{t
1
= t} = N
R
∂Ω
1
∂p(x,t)
∂n
dS
x
. The pdf of the arrival time is
Pr{τ
1
= t} = NN
R
Ω
p(x, t)dx
N−1
∂Ω
1
∂p(x, t)
∂n
dS
x
, (6)
which gives the MFPT
¯τ
1
=
∞
0
Pr{τ
1
> t}dt =
∞
0
[Pr{t
1
> t}]
N
dt. (7)
The shortest ray from the source to the absorbing window δ
min
plays a key role, because
the fastest trajectory is as close as possible to that ray. The diffusion coefficient is D,
the number of Brownian particles is N, s
2
= |x − A|, x the positions of injection, and
the center of the window is A. The asymptotic laws for the expected first arrival time of
5
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