Systems Biology (Methods in Molecular Biology)

A.9 Stochastic
Resonance

Since several times the system shows a bistability one can consider

the following stochastic equation:

dX¼

∂UðX,tÞ

∂UðXÞ

dtþηdWt ð 45 Þ

wheredWtstands for a Wiener process andηrepresents the noise

level.

Now consider potentials of the formU(X,t)¼Uo(X)þEXcos

(2πt/τ), composed of a stationary partUowith two minima atX

andX+and a periodic forcing with amplitudeEand periodτ.IfEis

small enough,Xwill oscillate around eitherXorX+, without ever

switching to the other.

But what happens if one increases the noise amplitudeη? Then

there is some probability thatXwill jump from one basin to the

other. If the noise level is just right,Xwill follow the periodic

forcing and oscillate betweenXandX+ with periodτ. This is

what we mean by stochastic resonance.

In more general terms, there isstochastic resonancewhenever

adding noise to a system improves its performance or, in the lan-

guage of signal processing, increases its signal-to-noise ratio. Note

that the noise amplitude cannot be too large or the system can

become completely random.

A.10 Stochastic
Solutions

The deterministic dynamics of populations in continuous time are

traditionally described using coupled, first-order ordinary differen-

tial equations. While this approach is accurate for large systems, it is

often inadequate for small systems where key species may be present

in small numbers or where key reactions occur at a low rate. The

Gillespie stochastic simulation algorithm (SSA) [31] is a procedure

for generating time-evolution trajectories of finite populations in

continuous time and has become the standard algorithm for these

types of stochastic models. It is well known that stochasticity in

finite populations can generate dynamics profoundly different from

the predictions of the corresponding deterministic model. For

example, demographic stochasticity can give rise to regular and

persistent population cycles in models that are deterministically

stable and can give rise to molecular noise and noisy gene expres-

sion in genetic and chemical systems where key molecules are

present in small numbers or where key reactions occur at a low

rate. Because analytical solutions to stochastic time-evolution equa-

tions for all but the simplest systems are intractable, while numerical

solutions are often prohibitively difficult, stochastic simulations

have become an invaluable tool for studying the dynamics of finite

biological, chemical, and physical systems.

The Gillespie stochastic simulation algorithm (SSA) is a proce-

dure for generating statistically correct trajectories of finite well-

mixed populations in continuous time. The trajectory that is

Inverse Problems in Systems Biology 91