Progenitor Cell Differentiation As a Gradient Flow

Data from single-cell mRNA sequencing are usually represented in a suitable feature space. Multivariate statistics and graph theoretic methods assign to each cell a time label known as pseudo-time and display paths in such space. For example, a process by which progenitors differentiate into one or more types of adult cells is described by a set of paths exhibiting branching points. Herewith the differentiation process is heuristically modelled by introducing a two-dimensional continuous time, autonomous dynamical system of polynomial type. An energy function F of two variables, {σ1,σ2}, is suitably defined; the autonomous dynamical system is obtained from ∇F and gives rise to a gradient flow. The corresponding phase portrait has two attractors, A and B, a saddle point, O, and a separatrix. The orbits of the system interpret experimental data points in feature (≡ state) space. These properties are suggested by data from single cell sequencing. Namely, initial states of the system correspond to progenitors; attractors A and B correspond to the two cell types yielded by progenitor differentiation; the separatrix and the saddle point make sure an orbit asymptotically reaches either A or B. The dynamical system setting replaces paths by orbits, assigns a unique orbit to each cell and enables time ordering as well as compliance with causality, unlike pseudo-time assignment induced by a minimum spanning tree.