LinearNoiseApproximation.jl

Purpose

This package provides a numerical method to automatically deriving the linear noise approximation (LNA) for a given stochastic reaction network (SRN) using Catalyst.jl. SRNs have proven their flexibility beyond chemical systems and across many different fields. These include biochemistry with gene expression models, epidemiology with compartmental models, ecology with the Lotka-Volterra equations, condensed matter and semiconductor physics with diffusion processes, and even sociology. The derived expanded system can be solved using DifferentialEquations.jl.

Installation

LinearNoiseApproximation.jl can be installed using following commands inside the Julia REPL

using Pkg
Pkg.add("LinearNoiseApproximation")

It can then be used with

using LinearNoiseApproximation

Usage

While LNA has applications in many different fields, it was first derived as a moment-based approximation method for stochastic chemical kinetics. For brevity and historical consistency, we will therefore stick to the naming conventions from this field.

Reaction network with system size Ω

Suppose we have a reaction network, defined with Catalyst.jl

rn = @reaction_network TwoStage begin
    @parameters  v0 v1 d0 d1 Ω
    @species M(t) P(t)
    v0*Ω, ∅ --> M
    v1, M --> M + P
    d0, M --> ∅
    d1, P --> ∅
end

Here the system size Ω is integrated into the system (propensities).

With LinearNoiseApproximation.jl, you can now simply obtain the LNA system by calling

LNAsys = LNASystem(rn)

This system can be solved with DifferentialEquations.jl

@variables v0, v1, d0, d1, Ω
rates = [v0=>4.0,v1=>10.0,d0=>1.0,d1=>1.0,Ω=>5.0]
tspan = (0.0, 20.0)
u0 =[0, 0] # initial condition for M and P
prob = ODEProblem(LNAsys, u0, tspan, rates)
sol = solve(prob,Vern7(),abstol=1e-7, saveat=1.0)

A complete working version of this example is in the examples folder.

Brief introduction to linear noise approximation

LNA is a moment-based approximation method for stochastic chemical kinetics [1]. A general chemical kinetics reaction can be described as follows: given a set of chemical species $X_{i}, i = 1, \ldots, N$, we define $R$ chemical reactions by the notation

$$\begin{equation} k_r:\sum_{i=1}^{N}s_{ir}X_{i} \rightarrow \sum_{i=1}^{N}s'_{ir}X_{i}, \tag{1} \end{equation}$$

where the stoichiometric coefficients $s_{ir}$ and $s'_{ir}$ are non-negative integer numbers denoting the numbers of reactant and product molecules, respectively. The quantity $k_r$ in Equation (1) is called the reaction rate constant of the $r$-th reaction. Classically, the dynamics of a chemical reaction system as in Equation (1) is modelled by the law of mass action. The law of mass action states that the rate of a reaction is proportional to the product of the concentrations of reactant molecules, which lead to the following rate equations as:

$$\begin{equation} \frac{\mathrm{d}}{\mathrm{dt}} \phi_i = \sum_{r=1}^{R} S_{ir} g_r(\boldsymbol{\phi}), \tag{2} \end{equation}$$

where $\phi_i$ is the concentration of species $X_i$,

$$\boldsymbol{S} = \{S_{ir}\}_{N\times R},\; S_{ir}=s'_{ir} - s_{ir},\; i=1,\ldots,N,\; r=1,\ldots,R \notag$$

is the stoichiometric matrix, and

$$g_r(\boldsymbol{\phi}) = k_r \prod_{i=1}^{N} \phi_i^{s_{ir}},$$

is the rate of the $r$-th reaction.

However, the law of mass action is only valid when the number of molecules is large. When the number of molecules is small, System (1) can instead be modelled by a continuous-time Markov jump process to study the probability of the system being in a particular state at a given time. The dynamics of such a system can be described by the chemical master equation (CME) [2]:

$$\begin{equation} \begin{aligned} \frac{\mathrm{d}}{\mathrm{dt}} P(\boldsymbol{n}, t) = \sum_{r=1}^{R} f_r(\boldsymbol{n} - \mathbf{S}_r, t) P(\boldsymbol{n} - \mathbf{S}_r, t) - \sum_{r=1}^{R} f_r(\boldsymbol{n}, t) P(\boldsymbol{n}, t), \notag \end{aligned} \end{equation}$$

where $P(\boldsymbol{n}, t)$ is the probability of the system being in state $\boldsymbol{n}$ at time $t$,

$$f_r(\boldsymbol{n}, t) = k_r\Omega \prod_{i=1}^{N} \frac{n_i!}{(n_i-s_{ir})!\Omega^{s_{ir}}}$$

is the propensity function of reaction $r$ at state $\boldsymbol{n}$, and $\mathbf{S}_r$ is the stoichiometric vector of reaction $r$, $\Omega$ is the system size (or volume of the system).

The chemical master equation is written directly from the rate constants and stoichiometries of all the elementary reactions of a chemical system, but neither analytical nor numerical solutions are in general available. Fortunately, the chemical master equation can often be simplified in a linear noise approximation. Linear noise approximation is an expansion of the CME taking the inverse system size $1/\Omega$ as the perturbed variable, which is originally developed by [3]. The idea is to separate concentrations into a deterministic part, given by the solution of the deterministic rate equations, and a part describing the fluctuations about the deterministic mean

$$\frac{n_i}{\Omega} = \phi_i + \Omega^{-1/2} \epsilon_i,$$

where $\phi_i$ is the solution of the deterministic rate equations (2), and $\epsilon_i$ represents fluctuations about the deterministic mean. Define $\boldsymbol{\Sigma}=\left(\epsilon_{ij}\right)_{N\times N}$ to be the covariance matrix of the fluctuations, the linear noise approximation is given by:

$$\begin{equation} \partial_t \boldsymbol{\Sigma} = \mathbf{A} \boldsymbol{\Sigma} + \boldsymbol{\Sigma} \mathbf{A}^T + \Omega^{-1} \mathbf{B}, \tag{3} \end{equation}$$

where $\mathbf{A}=\mathbf{A}(\boldsymbol{\phi}),, \mathbf{B}=\mathbf{B}(\boldsymbol{\phi})$ both depend on the solution $\boldsymbol{\phi}$ of the rate equation \eqref{eq:rate_equations}, which are defined by

$$\mathbf{A} = \left(A_{ij}\right)_{N\times N}, A_{ij}=\sum_{r=1}^{R} S_{ir}\partial_{\phi_j}g_r(\boldsymbol{\phi}),$$

as the Jacobian matrix of the deterministic rate equations, and

$$\mathbf{B} =\left(B_{ij}\right)_{N\times N}, B_{ij} = \sum_{r=1}^{R} S_{ir}S_{jr}g_r(\boldsymbol{\phi}).$$

In this formulation, the LNA allows for analytical solutions that are locally valid close to macroscopic trajectories (solution of the rate equations) of the system. We refer to the review paper [2] for more details on the CME and LNA.

References

[1] Nicolaas Godfried van Kampen. The Expansion of the Master Equation. John Wiley & Sons, Inc., 2007.

[2] David Schnoerr, Guido Sanguinetti, and Ramon Grima. Approximation and inference methods for stochastic biochemical kinetics — a tutorial review (2017). Journal of Physics A: Mathematical and Theoretical, 50(9):093001.

[3] Nicolaas Godfried van Kampen. Stochastic Processes in Physics and Chemistry, volume 1. Elsevier, 1992.