Our graphvelo paper just appeared online (Link)
This work is a continuation of our efforts to place single-cell data analyses under rigorous mathematical and theoretical physics foundations, which I stress to be important. It shows how one infers velocities from a subset of genes with reliable velocity estimation based on that the data points form a low-dimensional manifold. Existence of such a manifold is typical for a dynamical system due to time-scale separation. Theoretically one can link to Einstein’s work on the Brownian motion, and a formal derivation by Zwanzig using the projection operator in the Hilbert space–most physicists learn the technique in quantum mechanics (e.g., von neumann’s mathematical foundations of quantum mechanics) and statistical physics (Zwanzig’s nonequilibrium statistical physics). Later I was pulled by my LLNL colleague Ken Kim to the problem and spent 4-5 years to figure out how to apply to non-Hamiltonian systems to get a generalized Langevin equation, which in some limit reduces to a set of Langevin equations (see Zwanzig’s book or Kubo’s statistical physics II). This set of equations form the foundation of our framework. This graphvelo work is on preprocessing data, which feeds into our Dynamo framework to reconstruct a set of ODEs describing “whole cell” dynamics– which can be viewed as biophysical virtual cell models.
