Biomedical Engineering Reference
In-Depth Information
expression lead to non-genetic individuality: even in the case when two indi-
viduals are genetically identical, protein concentrations between the two indi-
viduals can vary significantly because of the stochastic nature of protein
synthesis. Pioneering experiments by Ko et al. (6) demonstrated that gene ex-
pression levels can vary significantly from cell to cell in an isogenic population.
It is thought that the main source of the noise in gene expression arises from
statistical fluctuations in the concentrations of mRNAs, transcriptional and
translational machinery, and regulatory proteins (see also related chapters 1.2
(by Hofacker and Stadler) and 1.3 (by Wagner), Part III, this volume). In a sin-
gle cell these concentrations can fluctuate widely since many of the molecules
are present at low numbers. For example the lactose repressor protein in
Es-
cherichia coli
is present at only 30 copies on average. Several stochastic models
(7-13) have been proposed that recently have been tested experimentally (14-
17) (see (18) for an excellent review). The goal of this chapter is to review two
powerful analytical modeling techniques that can be used to determine the noise
characteristics of genetic networks: (i) the master equation approach and (ii) the
Langevin approach. These approaches will be applied to calculate the statistical
properties of noisy genetic circuits. A more general treatment of master equa-
tions and the Langevin technique can be found in van Kampen (19).
2.
THE MASTER EQUATION APPROACH
The master equation corresponds to the statement that the probability of
being in a given state changes depending on the probabilities of transition to and
from any other state in the system. It provides the full probability distribution
when it can be directly solved. Unfortunately, this is not often the case, so we
must settle for some of the moments of the distribution. These are easily ob-
tained from the generating function, so we will work with the master equation in
a form in which it depends on the generating function rather than the distribu-
tion.
The genetic network is defined by
N
state variables
n
1
...
n
N
and
M
rate con-
stants
k
1
...
k
M
. The variables denote the number of copies of a certain chemical
species such as mRNAs or proteins. Before applying the master equation ap-
proach to determine the noise properties of a genetic network we will start by
obtaining the master equation in the generating function form for some elemen-
tary chemical equations.
2.1. Synthesis from a Template
In numerous genetic reactions, such as transcription and translation,
mRNAs and proteins are synthesized from a template (DNA and mRNA, respec-
