Modeling the temporal interplay of molecular signaling and gene expression by using dynamic nested effects models

Abstract

Cellular decision making in differentiation, proliferation, or cell death is mediated by molecular signaling processes, which control the regulation and expression of genes. Vice versa, the expression of genes can trigger the activity of signaling pathways. We introduce and describe a statistical method called Dynamic Nested Effects Model (D-NEM) for analyzing the temporal interplay of cell signaling and gene expression. D-NEMs are Bayesian models of signal propagation in a network. They decompose observed time delays of multiple step signaling processes into single steps. Time delays are assumed to be exponentially distributed. Rate constants of signal propagation are model parameters, whose joint posterior distribution is assessed via Gibbs sampling. They hold information on the interplay of different forms of biological signal propagation. Molecular signaling in the cytoplasm acts at high rates, direct signal propagation via transcription and translation act at intermediate rates, while secondary effects operate at low rates. D-NEMs allow the dissection of biological processes into signaling and expression events, and analysis of cellular signal flow. An application of D-NEMs to embryonic stem cell development in mice reveals a feed-forward loop dominated network, which stabilizes the differentiated state of cells and points to Nanog as the key sensitizer of stem cells for differentiation stimuli.

Fig. 1.

Elementary example of a D-NEM. Shown is a network of three S-genes together with binary time series tables for typical E-genes connected to the S-genes. Each table holds three rows corresponding to the three possible perturbation experiments of S-genes. A one in column t_i, row S_j of table E_k represents the observation of a downstream effect in E_k, t_i time units after perturbation of S_j.

Fig. 2.

Stem cell data analysis. (A) Discretized data of the last time point across E-genes (rows) and S-gene perturbations (columns), with black representing downstream effects and white no effects. (B) The transitively closed nested effects model estimated from the data shown in A using static NEM. (C) A histogram of the posterior probabilities for the average time delay associated with the edge from Oct4 to its target E-genes. (D) Heat map of the posterior distribution of average time delays. Rows correspond to edges of the network including those between S- and E-genes, whereas columns refer to average time delays. Marginal posterior probabilities are gray-scale coded. The top row corresponds to the histogram shown above. (E) The final network structure estimated by time delay analysis using D-NEM. Edge colors correspond to estimated average time delays: fast signal propagation (green), intermediate signal propagation (blue), and slow signal propagation (red).