Putting (single-cell) data into orbit

摘要

Data from single-cell mRNA sequencing, made available by leading-edge experimental methods, demand properrepresentation and understanding. Multivariate statistics and graph theoretic methods represent cells in asuitable feature space, assign to each cell a time label known as "pseudo-time" and display "trajectories" (infact orbits) in such space. Orbits shall describe a process by which progenitors di↵erentiate into one or moretypes 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 di↵erentiation 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 lookedfor, the orbits of which may interpret some sequences of data points in feature (≡ state) space. Section 4defines an energy function F of two variables, {σ_1,σ_2}, and the autonomous dynamical system obtained fromΔF, which thus generates a gradient flow. Both F and the gradient flow give rise to a phase portrait withtwo attractors, A and B, a saddle point, O, and a separatrix. These properties are suggested by data fromsingle cell sequencing. Initial states of the system correspond to progenitors. Attractors A and B correspondto the two cell types yielded by progenitor di↵erentiation. The separatrix and the saddle point make sure anorbit asymptotically reaches either A or B. Why and how a gradient flow model shall be applied to data fromsingle-cell sequencing is discussed in § 5. The application of dynamical system theory presented herewith relieson a heuristic basis, as all population dynamics models do. Nonetheless, placing a given cell on an orbit of itsown enables time ordering and compliance with causality, unlike pseudo-time assignment induced by a minimumspanning tree. An earlier (2009) application in a much simpler context, the evolving morphology of cytoskeletaltubulines, is finally recalled: from cyto-toxicity experiments, epifluorescence images of tubulin filaments wereobtained, 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.