Estimating the Stochastic Bifurcation Structure of Cellular Networks

High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.Author Summary: Decades ago, Waddington, and later Kauffman, likened the dynamics of a differentiating cell to a marble rolling downhill on bumpy terrain—the epigenetic landscape. In this metaphor, the valleys of the landscape represent the paths that cells can follow towards a stable cell type, and the fate of the cell is determined by the constant modulation of the epigenetic landscape by internal and external signals. With new technologies for measuring single-cell gene expression, it is increasingly feasible to map out these valleys and how external variables influence cellular responses. Moreover, it is possible to quantify population level effects, such as what fraction of a population of cells arrives at one valley or another, and variability at the cellular level, such as how individual cells bounce around within, and possibly between, valleys due to the stochasticity of cellular biochemistry. In this paper, we discuss which characteristics of the epigenetic landscape can readily be extracted from single-cell gene expression data, and describe computational methods for doing so.