LNA Reference

LNA

This is a structure which will hold the LNA: the mean of the trajectories and the covariance between the species.

Arguments

• traj_means: A (number of species) x (number of time points) array which holds the mean trajectory for each species on each row of the array.
• traj_covars: An array which holds the covariance matrix of the species at each time point.
• time_points: The timepoints the system was solved for.

LNAInput

This is a structure which holds the inputs needed for computing the Linear Noise Approximation (LNA). This structure will hold the stochastic system as provided by the user; uniquely defined through kinetic parameters, the rates of the system and the stoichiometry matrix.

Arguments

• params::AbstractArray{Float64,1}: The rate parameters of the stochastic model.
• S::AbstractArray{Float64,2}: the stoichiometry matrix of the system. Size: number of reactions x number of species.
• reaction_rate_function::Function,: This is a function f(x, parameters) which should return an array of the reaction rates of the system, i.e. S*f would describe the ODE representation of the system.
• volume::Float64: The volume of the reactants of the system.

compute_LNA(input::LNAInput,
    x0::Tuple{AbstractArray{Float64,1},AbstractArray{Float64,2}},
    Tspan::Tuple{Float64,Float64},
    saveat::Float64,
    solver::OrdinaryDiffEq.OrdinaryDiffEqAlgorithm=RK4();
    kwargs...)

The function computes the linear noise approximation to system through construction of two ODEs: one describing the trajectories of the mean of the LNA and the other describing the change the covariance between the variables. These outputs are held in a LNA structure.

Arguments

• x0::Tuple{AbstractArray{Float64,2},AbstractArray{Float64,2}}: The initial conditions of the system. In the form of (the initial conditions of the species, the initial covariance matrix of the system).
• Tspan::Tuple{Float64,Float64}: The start and end times of the simulation.
• saveat::Float64: The number of time points the use wishes to solve the system for.
• solver::DEAlgorithm: The ODE solver the user wishes to use, for example OrdinaryDiffEq.RK4().

#Returns

• LNA

get_LNA_trajectories(input::LNAInput, n_samples::Int64,
    x0::Tuple{AbstractArray{Float64,1},AbstractArray{Float64,2}},
    Tspan::Tuple{Float64,Float64},
    saveat::Float64,
    solver::OrdinaryDiffEq.OrdinaryDiffEqAlgorithm=RK4();
    kwargs...)

A function which computes the LNA and then samples from the it to output sampled trajectories. The user can also sample more than one trajectory; which are then averaged.

Arguments

• input::LNAInput: LNAInput structure.
• n_samples::Int64: The number of sampled trajectories to be sampled and then averaged.
• x0::Tuple{AbstractArray{Float64,2},AbstractArray{Float64,2}}: The initial conditions of the system. In the form of (the initial conditions of the species, the initial covariance matrix of the system).
• Tspan::Tuple{Float64,Float64}: The start and end times of the simulation.
• saveat::Float64: The number of time points the use wishes to solve the system for.
• solver::DEAlgorithm: The ODE solver the user wishes to use, for example OrdinaryDiffEq.RK4() .

#Returns

• A (number of species) x (number of time points) array which holds the averaged trajectory for each species on each row of the array.

sample_LNA_trajectories(lna::LNA, n_samples::Int64)

A function which samples from the LNA to output sampled trajectories. The LNA gives the mean of the trajectories and the covariance between them; hence a single trajectory can be sampled from a Multivariate Normal distribution. The user can also sample more than one trajectory; which are then averaged.

Arguments

• lna::LNA: LNA structure.
• n_samples::Int64: The number of sampled trajectories to be sampled and then averaged.

#Returns

• A (number of species) x (number of time points) array which holds the averaged trajectory for each species on each row of the array.