  Until recently, transcriptomics was limited to bulk RNA sequencing, obscuring
the underlying expression patterns of individual cells in favor of a global
average. Thanks to technological advances, we can now profile gene expression
across thousands or millions of individual cells in parallel. This new type of
data has led to the intriguing discovery that individual cell profiles can
reflect the imprint of time or dynamic processes. However, synthesizing this
information to reconstruct dynamic biological phenomena from data that are
noisy, heterogenous, and sparse---and from processes that may unfold
asynchronously---poses a complex computational and statistical challenge. Here,
we develop a full generative model for probabilistically reconstructing trees
of cellular differentiation from single-cell RNA-seq data. Specifically, we
extend the framework of the classical Dirichlet diffusion tree to
simultaneously infer branch topology and latent cell states along continuous
trajectories over the full tree. In tandem, we construct a novel Markov chain
Monte Carlo sampler that interleaves Metropolis-Hastings and message passing to
leverage model structure for efficient inference. Finally, we demonstrate that
these techniques can recover latent trajectories from simulated single-cell
transcriptomes. While this work is motivated by cellular differentiation, we
derive a tractable model that provides flexible densities for any data (coupled
with an appropriate noise model) that arise from continuous evolution along a
latent nonparametric tree.
