 (Mean and standard deviation of the amount of genome in the three species model : stochastic simulation (black), Gaussian approximation (red), diffusion approximation (green), ODE (blue)) |
Species numbers are modeled as Markov processes and ordinary differential equations are obtained as their limits in case the numbers of molecules of all species are large. In this project, we suggest how to capture the fluctuations of the Markov processes around their deterministic limits using a central limit theorem. The central limit theorem is used to derive an appropriate Gaussian approximation and a diffusion (Langevin) approximation. |
 (Stochastic simulation of the number of molecules of Species 8 in the full model (black) and that of approximation using the limiting model (red) when the time is of order 10000 sec) |
A heat shock response model of Escherichia coli developed by Srivastava, Peterson, and Bentley (2001) has multiscale nature due to its species numbers and reaction rate constants varying over wide ranges. The goal of this project is to apply the multiscale approximation method to the biological model with significant complexity. We scale the species numbers and the rate constants by powers of the scaling parameter with different values of exponents, and the scaling exponents are determined based on several criteria including balance conditions. In the three time scales of interest, we derive limiting models involving a subset of species and reactions, which are used to approximate the full model. |