Driving directed differentiation: Bridging microscopic biological processes and macroscopic Hill function modeling

Cell differentiation emerges as an irreversible biological transition governed by the expression of core gene circuit. While Hill functions provide phenomenological descriptions of gene network behaviors, when addressing the issue of cell differentiation, it is often inevitable to introduce a time-correlated driving term, which undermines the theoretical closure. Utilizing a prototypical gene circuit, we develop a microscopic theory for the Hill functions based on the detailed biochemical reactions and the principles of statistical physics. This approach establishes equivalence mappings between different biological scales: the complete microscopic reaction network model, reduced microscopic model, and effective mesoscopic model. Under this equivalence, without introducing external information, we identify the differentiation-driving forces, while the remaining components of the model are exactly represented by Hill functions. Our theoretical results also demonstrate that the macroscopic force described by Hill functions maintains cellular stability over small time scales, whereas intrinsic driving forces propel directed differentiation of cells across large time scales. Furthermore, numerical simulations conducted using equivalent methods agrees with our theoretical findings. The derived relationships between reaction kinetic constants and phenomenological parameters establish a physical basis for bridging genotype-phenotype mapping in developmental systems.