Stochastic analysis of Chemical Reaction Networks using Linear Noise Approximation

Abstract

Stochastic evolution of Chemical Reactions Networks (CRNs) over time is usually analyzed through solving the Chemical Master Equation (CME) or performing extensive simulations. Analysing stochasticity is often needed, particularly when some molecules occur in low numbers. Unfortunately, both approaches become infeasible if the system is complex and/or it cannot be ensured that initial populations are small. We develop a probabilistic logic for CRNs that enables stochastic analysis of the evolution of populations of molecular species. We present an approximate model checking algorithm based on the Linear Noise Approximation (LNA) of the CME, whose computational complexity is independent of the population size of each species and polynomial in the number of different species. The algorithm requires the solution of first order polynomial differential equations. We prove that our approach is valid for any CRN close enough to the thermodynamical limit. However, we show on four case studies that it can still provide good approximation even for low molecule counts. Our approach enables rigorous analysis of CRNs that are not analyzable by solving the CME, but are far from the deterministic limit. Moreover, it can be used for a fast approximate stochastic characterization of a CRN.

Molecular Filters for Noise Reduction

“…7 is nonlinear, a general and exact analysis as in the linear case cannot be performed, as the moment equations cannot be solved. Consequently, we make use of the LNA (23, 28) and derive an analytical solution for the expectation and Fano factor of B at steady state for such an input process A . We get (see Supporting Materials and Methods, Section D)E[B]∞=r1E[A]∞r2andFB=2r13/2r2kpkd+4r1r2kp−r1kd2−kd38r1r2kp−2kd3,where F B stands for the Fano factor of B at steady state.…”

Peregrine: A Tool for the Analysis of Population Protocols

“…As mentioned in the introduction, population protocols are isomorphic to chemical reaction networks (CRNs), a popular model in natural computing. Cardelli et al have recently developed model checking techniques and analysis algorithms for stochastic CRNs [7][8][9]. The problems studied therein are incomparable to the parameterized questions addressed by Peregrine.…”

On Quantitative Comparison of Chemical Reaction Network Models

“…The recovery of infected individuals is given by the fluxes to the reaction r 2 , that is, the flux edge r 1 displays the normalised flux values in these simulations for the complete simulation as well as for the time intervals 0 to 2 and 2 to 4. As it can be seen in these plots, the fluxes in the shorter intervals (F [0, 2] and F [2,4]) add up to contribute to the fluxes of the complete simulation (F [0, 100]). While a speed up in α results in the infection of the whole population, a smaller γ delays the recovery.…”