Hierarchical Bayesian Inference of Stochastic Biochemical Processes

Data from population measurements of gene network dynamics have shown that cells exhibit variability even in clonal lines. A reliable mathematical reconstruction of a biological process requires the inference of parameters characterizing this process in a single cell while considering the observed heterogeneity of the population from which data was obtained. Parameter inference, however, is complicated by the fact the outcomes of constituent reactions in a gene circuit are only partially observed in time or are detected indirectly in experiments. One approach is to replace unobserved reactions with time delays, a technique that also simplifies inference through the reduction of model dimension. This simplification, however, results in a non-Markovian model that requires the development of new inference methods. Here, we propose a hierarchical Bayesian inference framework for quantifying the variability of cellular processes within and across cells in a population in a non-Markovian setting, such as a reaction system with delays. We demonstrate our framework using a delayed birth-death process with birth delays which are either fixed or distributed, and show that a model with distributed delays is better when dealing with experimental systems since inference assuming fixed delays lead to underestimation when the true delays are variable. Using synthetic and experimental data, we show that the proposed hierarchical framework is robust and leads to improved estimates as compared to its non-hierarchical analog. We apply our method to data obtained using time-lapse microscopy and infer the parameters that describe the dynamics of fluorescent protein production at the individual cell and population level.