Parameter inference for discretely observed stochastic kinetic models using stochastic gradient descent

Abstract

Background: Stochastic effects can be important for the behavior of processes involving small population numbers, so the study of stochastic models has become an important topic in the burgeoning field of computational systems biology. However analysis techniques for stochastic models have tended to lag behind their deterministic cousins due to the heavier computational demands of the statistical approaches for fitting the models to experimental data. There is a continuing need for more effective and efficient algorithms. In this article we focus on the parameter inference problem for stochastic kinetic models of biochemical reactions given discrete time-course observations of either some or all of the molecular species.

Results: We propose an algorithm for inference of kinetic rate parameters based upon maximum likelihood using stochastic gradient descent (SGD). We derive a general formula for the gradient of the likelihood function given discrete time-course observations. The formula applies to any explicit functional form of the kinetic rate laws such as mass-action, Michaelis-Menten, etc. Our algorithm estimates the gradient of the likelihood function by reversible jump Markov chain Monte Carlo sampling (RJMCMC), and then gradient descent method is employed to obtain the maximum likelihood estimation of parameter values. Furthermore, we utilize flux balance analysis and show how to automatically construct reversible jump samplers for arbitrary biochemical reaction models. We provide RJMCMC sampling algorithms for both fully observed and partially observed time-course observation data. Our methods are illustrated with two examples: a birth-death model and an auto-regulatory gene network. We find good agreement of the inferred parameters with the actual parameters in both models.

Conclusions: The SGD method proposed in the paper presents a general framework of inferring parameters for stochastic kinetic models. The method is computationally efficient and is effective for both partially and fully observed systems. Automatic construction of reversible jump samplers and general formulation of the likelihood gradient function makes our method applicable to a wide range of stochastic models. Furthermore our derivations can be useful for other purposes such as using the gradient information for parametric sensitivity analysis or using the reversible jump samplers for full Bayesian inference. The software implementing the algorithms is publicly available at http://cbcl.ics.uci.edu/sgd.

Figure 1

Property of the RJMCMC sampler for the birth-death model. Panel a, c, and e plot the total number of reactions in each of the samples generated by RJMCMC, corresponding to three different observation datasets with Δt = 2, 5 and 10 respectively. Panel b, d, and f shows the autocorrelation on the number of reactions corresponding to panel a, c, and e respectively. The rate parameters used are (k₁, k₂) = (0.03, 0.6).

Figure 2

Property of the RJMCMC sampler for the fully observed case of the auto-regulatory gene network. Panel a, c, and e plot the total number of reactions in each of the samples generated by RJMCMC, corresponding to three different observation datasets with Δt = 0.1, 0.5 and 1 respectively. Panel b, d, and f shows the autocorrelation on the number of reactions corresponding to panel a, c, and e respectively. The rate parameters used are (k₁,...k₈) = (0.1, 0.7, 0.143, 0.35, 0.3, 0.1, 0.9, 0.11, 0.2, 0.1).

Figure 3

Property of the RJMCMC sampler for the partially observed case of the auto-regulatory gene network. Panel a, c, and e plot the total number of reactions in each of the samples generated by RJMCMC, corresponding to three different observation datasets with Δt = 0.1, 0.5 and 1 respectively. Panel b, d, and f shows the autocorrelation on the number of reactions corresponding to panel a, c, and e respectively. The rate parameters used are (k₁,...k₈) = (0.1, 0.7, 0.143, 0.35, 0.3, 0.1, 0.9, 0.11, 0.2, 0.1). Only three species, mRNA, P and P₂, are observed.