  A key problem in computational biology is discovering the gene expression
changes that regulate cell fate transitions, in which one cell type turns into
another. However, each individual cell cannot be tracked longitudinally, and
cells at the same point in real time may be at different stages of the
transition process. This can be viewed as a problem of learning the behavior of
a dynamical system from observations whose times are unknown. Additionally, a
single progenitor cell type often bifurcates into multiple child cell types,
further complicating the problem of modeling the dynamics. To address this
problem, we developed an approach called variational mixtures of ordinary
differential equations. By using a simple family of ODEs informed by the
biochemistry of gene expression to constrain the likelihood of a deep
generative model, we can simultaneously infer the latent time and latent state
of each cell and predict its future gene expression state. The model can be
interpreted as a mixture of ODEs whose parameters vary continuously across a
latent space of cell states. Our approach dramatically improves data fit,
latent time inference, and future cell state estimation of single-cell gene
expression data compared to previous approaches.
