The indispensable role of time in autonomous development.

Advances in single-cell analysis have led to a picture of development largely in agreement with Waddington's eponymous epigenetic landscape, in which a cell's fate is determined by its basin of attraction on a high-dimensional gene-expression space. Yet conceptual gaps remain as to how a single progenitor can simultaneously generate multiple endpoints, and why time should be required of the process at all. We propose a theoretical model based on the Hamiltonian mechanics of n-dimensional rotational motion which resolves these paradoxes. We derive the result that systems which become different from themselves over time must initially move in a direction not toward their ultimate endpoints, and propose that this process of resolving ambiguity can be quantified (in an information-theoretic sense) by the volume subtended in gene-expression space by the trajectories taken by the system toward its endpoints. We discuss the implications of this theory for the analysis of single-cell gene-expression data in studies of development.