Moment Fitting for Parameter Inference in Repeatedly and Partially Observed Stochastic Biological Models Author Summary

The inference of reaction rate parameters in biochemical network models for well mixed conditions from time series concentration data is a central task in computational systems biology. The network dynamics usually are described by the chemical master equation, the Fokker Planck equation, the linear noise approximation or the macroscopic rate equation. The inverse problem of estimating the parameters of the underlying differential equation can approached in deterministic and stochastic ways and available methods often involve the difference between individual or mean concentration traces and model predictions when maximizing likelihoods, minimizing regularized least squares functionals, approximating posterior distributions or sequentially processing the data. In this article we assume that the biological reaction network can be at least partially and repeatedly observed over time such that low order statistical moments or central moments at various times for the number of molecules of the chemical species involved can be approximated from the data. Furthermore, we consider closed systems of nonlinear ordinary differential equations that approximatively describe the time evolution of the statistical moments or central moments, can be derived from the chemical master equation or their approximations and depend on the reaction rate parameters. For infering the rate parameters we then suggest to not only consider the distance between the sample mean and the mean prediction of the equations but also to take the error in higher moments explicitly into acount. Cost functions that involve higher statistical moments may form landscapes in the parameter space that have more pronounced curvatures at the minimizer and hence may weaken or even overcome parameter sloppiness and uncertainty. As a consequence both deterministic and stochastic parameter inference algorithms may be improved with respect to accuracy and efficiency. We demonstrate the concept of moment fitting for parameter inference by means of illustrative stochastic biological models from the literature.