|
Data from single-cell mRNA sequencing, made available by leading-edge experimental methods, demand proper representation and understanding. Multivariate statistics and graph theoretic methods represent cells in a suitable feature space, assign to each cell a time label known as “pseudo-time” and display “trajectories” (in fact orbits) in such space. Orbits shall describe a process by which progenitors differentiate into one or more types of adult cells: broncho-alveolar progenitors are e.g., found to evolve into two distinct pneumocyte types. This work aims at applying the qualitative theory of dynamical systems to describe the differentiation process. Some notions of qualitative theory are presented (§ 2). The main stages of single-cell data analysis are outlined (§ 3). Next, a two-dimensional continuous time, autonomous dynamical system of polynomial type is looked for, the orbits of which may interpret some sequences of data points in feature (⌘ state) space. Section 4 defines an energy function F of two variables, 1,!2}, and the autonomous dynamical system obtained from rF, which thus generates a gradient flow. Both F and the gradient flow give rise to a phase portrait with two attractors, A and B, a saddle point, O, and a separatrix. These properties are suggested by data from single cell sequencing. 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. Why and how a gradient flow model shall be applied to data from single-cell sequencing is discussed in § 5. The application of dynamical system theory presented herewith relies on a heuristic basis, as all population dynamics models do. Nonetheless, placing a given cell on an orbit of its own enables time ordering and compliance with causality, unlike pseudo-time assignment induced by a minimum spanning tree. An earlier (2009) application in a much simpler context, the evolving morphology of cytoskeletal tubulines, is finally recalled: from cyto-toxicity experiments, epifluorescence images of tubulin filaments were obtained, then analysed and assigned to morphology classes; class centroids formed a sequence in feature (⌘ state) space describing loss of cytoskeletal structure followed by its recovery.
|
