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The epigenetic landscape was introduced by Conrad Waddington as a metaphor of cellular development. Like a ball rolling down a hillside is channelled through a succession of valleys until it reaches the bottom, cells follow specific trajectories from a pluripotent state to a committed state. Transcription factors (TFs) interacting as a network (the gene regulatory network (GRN)) orchestrate this developmental process within each cell. Here, we quantitatively model the epigenetic landscape using a kind of artificial neural network called the Hopfield network (HN). An HN is composed of nodes (genes/TFs) and weighted undirected edges, resulting in a weight matrix ( W ) that stores interactions among the nodes over the entire network. We used gene co-expression to compute the edge weights. Through  W , we then associate an energy score ( E ) to each input pattern (pattern of co-expression for a specific developmental stage) such that each pattern has a specific  E.  We propose that, based on the co-expression values stored in  W , HN associates lower  E  values to stable phenotypic states and higher  E  to transient states. We validate our model using time course gene-expression data sets representing stages of development across 12 biological processes including differentiation of human embryonic stem cells into specialized cells, differentiation of THP1 monocytes to macrophages during immune response and trans-differentiation of epithelial to mesenchymal cells in cancer. We observe that transient states have higher energy than the stable phenotypic states, yielding an arc-shaped trajectory. This relationship was confirmed by perturbation analysis. HNs offer an attractive framework for quantitative modelling of cell differentiation (as a landscape) from empirical data. Using HNs, we identify genes and TFs that drive cell-fate transitions, and gain insight into the global dynamics of GRNs.

Not just a colourful metaphor: modelling the landscape of cellular development using Hopfield networks