  In multicellular organisms, cells differentiate into several distinct types
during early development. Determination of each cellular state, along with the
ratio of each cell type, as well as the developmental course during cell
differentiation are highly regulated processes that are robust to noise and
environmental perturbations throughout development. Waddington metaphorically
depicted this robustness as the epigenetic landscape in which the robustness of
each cellular state is represented by each valley in the landscape. This
robustness is now conceptualized as an approach toward an attractor in a
gene-expression dynamical system. However, there is still an incomplete
understanding of the origin of landscape change, which is accompanied by
branching of valleys that corresponds to the differentiation process. Recent
progress in developmental biology has unveiled the molecular processes involved
in epigenetic modification, which will be a key to understanding the nature of
slow landscape change. Nevertheless, the contribution of the interplay between
gene expression and epigenetic modification to robust landscape changes, known
as homeorhesis, remains elusive. Here, we introduce a theoretical model that
combines epigenetic modification with gene expression dynamics driven by a
regulatory network. In this model, epigenetic modification changes the
feasibility of expression, i.e., the threshold for expression dynamics, and a
slow positive-feedback process from expression to the threshold level is
introduced. Under such epigenetic feedback, several fixed-point attractors with
distinct expression patterns are generated hierarchically shaping the
epigenetic landscape with successive branching of valleys. This theory provides
a quantitative framework for explaining homeorhesis in development as
postulated by Waddington, based on dynamical-system theory with slow feedback
reinforcement.
