The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by
Nico van Kampen
Nicolaas 'Nico' Godfried van Kampen (June 22, 1921 – October 6, 2013) was a Dutch theoretical physicist, who worked mainly on statistical mechanics and non-equilibrium thermodynamics.
Van Kampen was born in Leiden, and was a nephew of Frits Zer ...
[van Kampen, N. G. (2007) "Stochastic Processes in Physics and Chemistry", North-Holland Personal Library] used in the analysis of
stochastic processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
. Specifically, it allows one to find an approximation to the solution of a
master equation
In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determine ...
with nonlinear transition rates. The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...
with linear coefficients determined by the
transition rates and
stoichiometry
Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions.
Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equal ...
of the system.
Less formally, it is normally straightforward to write down a mathematical description of a system where processes happen randomly (for example, radioactive atoms randomly
decay
Decay may refer to:
Science and technology
* Bit decay, in computing
* Software decay, in computing
* Distance decay, in geography
* Decay time (fall time), in electronics
Biology
* Decomposition of organic matter
* Tooth decay (dental caries ...
in a physical system, or genes that are
expressed stochastically in a cell). However, these mathematical descriptions are often too difficult to solve for the study of the systems statistics (for example, the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
and
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of the number of atoms or proteins as a function of time). The system size expansion allows one to obtain an approximate statistical description that can be solved much more easily than the master equation.
Preliminaries
Systems that admit a treatment with the system size expansion may be described by a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, giving the probability of observing the system in state
at time
.
may be, for example, a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
with elements corresponding to the number of molecules of different chemical species in a system. In a system of size
(intuitively interpreted as the volume), we will adopt the following nomenclature:
is a vector of macroscopic copy numbers,
is a vector of concentrations, and
is a vector of deterministic concentrations, as they would appear according to the rate equation in an infinite system.
and
are thus quantities subject to stochastic effects.
A
master equation
In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determine ...
describes the time evolution of this probability.
Henceforth, a system of chemical reactions
[Elf, J. and Ehrenberg, M. (2003) "Fast Evaluation of Fluctuations in Biochemical Networks With the Linear Noise Approximation", ''Genome Research'', 13:2475–2484.] will be discussed to provide a concrete example, although the nomenclature of "species" and "reactions" is generalisable. A system involving
species and
reactions can be described with the master equation:
:
Here,
is the system size,
is an
operator which will be addressed later,
is the stoichiometric matrix for the system (in which element
gives the
stoichiometric coefficient
A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas. The reactant entities are given on the left-hand side and the product entities on the right-hand side with a plus sign between t ...
for species
in reaction
), and
is the rate of reaction
given a state
and system size
.
is a step operator,
removing
from the
th element of its argument. For example,
. This formalism will be useful later.
The above equation can be interpreted as follows. The initial sum on the RHS is over all reactions. For each reaction
, the brackets immediately following the sum give two terms. The term with the simple coefficient −1 gives the probability flux away from a given state
due to reaction
changing the state. The term preceded by the product of step operators gives the probability flux due to reaction
changing a different state
into state
. The product of step operators constructs this state
.
Example
For example, consider the (linear) chemical system involving two chemical species
and
and the reaction
. In this system,
(species),
(reactions). A state of the system is a vector
, where
are the number of molecules of
and
respectively. Let
, so that the rate of reaction 1 (the only reaction) depends on the concentration of
. The stoichiometry matrix is
.
Then the master equation reads:
:
where
is the shift caused by the action of the product of step operators, required to change state
to a precursor state
.
Linear noise approximation
If the master equation possesses
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
transition rates, it may be impossible to solve it analytically. The system size expansion utilises the
ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
that the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of the steady-state probability distribution of constituent numbers in a population scales like the system size. This ansatz is used to expand the master equation in terms of a small parameter given by the inverse system size.
Specifically, let us write the
, the copy number of component
, as a sum of its "deterministic" value (a scaled-up concentration) and a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, scaled by
:
:
The probability distribution of
can then be rewritten in the vector of random variables
:
:
Consider how to write reaction rates
and the step operator
in terms of this new random variable.
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
of the transition rates gives:
:
The step operator has the effect
and hence
:
:
We are now in a position to recast the master equation.
:
This rather frightening expression makes a bit more sense when we gather terms in different powers of
. First, terms of order
give
:
These terms cancel, due to the
macroscopic reaction equation
:
The terms of order
are more interesting:
:
which can be written as
:
where
:
and
:
The time evolution of
is then governed by the linear
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...
with coefficient matrices
and
(in the large-
limit, terms of
may be neglected, termed the linear noise approximation). With knowledge of the reaction rates
and stoichiometry
, the moments of
can then be calculated.
The approximation implies that fluctuations around the mean are
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
distributed. Non-Gaussian features of the distributions can be computed by taking into account higher order terms in the expansion.
Software
The linear noise approximation has become a popular technique for estimating the size of
intrinsic noise
Cellular noise is random variability in quantities arising in cellular biology. For example, cells which are genetically identical, even within the same tissue, are often observed to have different expression levels of proteins, different sizes and ...
in terms of
coefficients of variation and
Fano factor In statistics, the Fano factor, like the coefficient of variation, is a measure of the dispersion of a probability distribution of a Fano noise. It is named after Ugo Fano, an Italian American physicist.
The Fano factor is defined as
:F=\frac, ...
s for molecular species in intracellular pathways. The second moment obtained from the linear noise approximation (on which the noise measures are based) are exact only if the pathway is composed of first-order reactions. However bimolecular reactions such as
enzyme-substrate,
protein-protein and
protein-DNA interactions are ubiquitous elements of all known pathways; for such cases, the linear noise approximation can give estimates which are accurate in the limit of large reaction volumes. Since this limit is taken at constant concentrations, it follows that the linear noise approximation gives accurate results in the limit of large molecule numbers and becomes less reliable for pathways characterized by many species with low copy numbers of molecules.
A number of studies have elucidated cases of the insufficiency of the linear noise approximation in biological contexts by comparison of its predictions with those of stochastic simulations.
[Hayot, F. and Jayaprakash, C. (2004), "The linear noise approximation for molecular fluctuations within cells", ''Physical Biology'', 1:205][Ferm, L. Lötstedt, P. and Hellander, A. (2008), "A Hierarchy of Approximations of the Master Equation Scaled by a Size Parameter", ''Journal of Scientific Computing'', 34:127] This has led to the investigation of higher order terms of the system size expansion that go beyond the linear approximation. These terms have been used to obtain more accurate moment estimates for the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
concentrations and for the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
s of the concentration fluctuations in intracellular pathways. In particular, the leading order corrections to the linear noise approximation yield corrections of the conventional
rate equation
In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reacti ...
s.
[Grima, R. (2010) "An effective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions", ''The Journal of Chemical Physics'', 132:035101] Terms of higher order have also been used to obtain corrections to the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
s and
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
s estimates of the linear noise approximation.
[Grima, R. and Thomas, P. and Straube, A.V. (2011), "How accurate are the nonlinear chemical Fokker-Planck and chemical Langevin equations?", ''The Journal of Chemical Physics'', 135:084103][Grima, R. (2012), "A study of the accuracy of moment-closure approximations for stochastic chemical kinetics", ''The Journal of Chemical Physics'', 136: 154105] The linear noise approximation and corrections to it can be computed using the open source software
intrinsic Noise Analyzer
Within bioinformatics, intrinsic Noise Analyzer (iNA) is an open source software for studying reaction kinetics in living cells. The software analyzes mathematical models of intracellular reaction kinetics such as gene expression, regulatory ne ...
. The corrections have been shown to be particularly considerable for
allosteric
In biochemistry, allosteric regulation (or allosteric control) is the regulation of an enzyme by binding an effector molecule at a site other than the enzyme's active site.
The site to which the effector binds is termed the ''allosteric site ...
and non-allosteric enzyme-mediated reactions in
intracellular compartments
Cellular compartments in cell biology comprise all of the closed parts within the cytosol of a eukaryotic cell, usually surrounded by a single or double lipid layer membrane. These compartments are often, but not always, defined as membrane-b ...
.
References
{{DEFAULTSORT:System Size Expansion
Stochastic processes
Applied mathematics
Chemical kinetics
Stoichiometry
Equations of physics