  A core challenge for modern biology is how to infer the trajectories of
individual cells from population-level time courses of high-dimensional gene
expression data. Birth and death of cells present a particular difficulty:
existing trajectory inference methods cannot distinguish variability in net
proliferation from cell differentiation dynamics, and hence require accurate
prior knowledge of the proliferation rate. Building on Global Waddington-OT
(gWOT), which performs trajectory inference with rigorous theoretical
guarantees when birth and death can be neglected, we show how to use lineage
trees available with recently developed CRISPR-based measurement technologies
to disentangle proliferation and differentiation. In particular, when there is
neither death nor subsampling of cells, we show that we extend gWOT to the case
with proliferation with similar theoretical guarantees and computational cost,
without requiring any prior information. In the case of death and/or
subsampling, our method introduces a bias, that we describe explicitly and
argue to be inherent to these lineage tracing data. We demonstrate in both
cases the ability of this method to reliably reconstruct the landscape of a
branching SDE from time-courses of simulated datasets with lineage tracing,
outperforming even a benchmark using the experimentally unavailable true
branching rates.
