Predicting stochastic gene expression dynamics in single cells

Abstract

Fluctuations in protein numbers (noise) due to inherent stochastic effects in single cells can have large effects on the dynamic behavior of gene regulatory networks. Although deterministic models can predict the average network behavior, they fail to incorporate the stochasticity characteristic of gene expression, thereby limiting their relevance when single cell behaviors deviate from the population average. Recently, stochastic models have been used to predict distributions of steady-state protein levels within a population but not to predict the dynamic, presteadystate distributions. In the present work, we experimentally examine a system whose dynamics are heavily influenced by stochastic effects. We measure population distributions of protein numbers as a function of time in the Escherichia coli lactose uptake network (lac operon). We then introduce a dynamic stochastic model and show that prediction of dynamic distributions requires only a few noise parameters in addition to the rates that characterize a deterministic model. Whereas the deterministic model cannot fully capture the observed behavior, our stochastic model correctly predicts the experimental dynamics without any fit parameters. Our results provide a proof of principle for the possibility of faithfully predicting dynamic population distributions from deterministic models supplemented by a stochastic component that captures the major noise sources.gene networks ͉ systems biology O ne of the central goals of systems biology is to predict the dynamic behavior of a cell's genetic and metabolic networks. These predictions traditionally stem from models in which discrete molecular events, such as transcription and translation, are represented by continuous and deterministic differential equations. Such equations are valid when behavior of individual cells is very similar to the average behavior of the population. However, in many cases, the inherently stochastic nature of biological systems leads to significant cell-to-cell variability (1-3), and previous studies indicate that individual cells often behave very differently from the population average in response to external stimuli. For example, studies of bacterial persistence indicate that the population survival rate can be fundamentally different from the average cell's survival rate in response to environmental stress (4, 5). The impact of noise-induced population heterogeneity is also relevant when studying cellular memory, where fluctuations in protein numbers can cause the stability of epigenetic memory to degrade by effectively causing cells to forget their original states (6, 7). In these cases, stochastic modeling techniques must be used to describe the large cell-tocell variability. Although deterministic models can describe dynamic network behavior and analytical stochastic models can faithfully predict steady-state population distributions (8), little work has been done to connect dynamic cellular behavior with noise models. It is important that stochastic models of biological systems...

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