  Gene regulatory network (GRN) refers to the complex network formed by
regulatory interactions between genes in living cells. In this paper, we
consider inferring GRNs in single cells based on single cell RNA sequencing
(scRNA-seq) data. In scRNA-seq, single cells are often profiled from mixed
populations and their cell identities are unknown. A common practice for single
cell GRN analysis is to first cluster the cells and infer GRNs for every
cluster separately. However, this two-step procedure ignores uncertainty in the
clustering step and thus could lead to inaccurate estimation of the networks.
To address this problem, we propose to model scRNA-seq by the mixture
multivariate Poisson log-normal (MPLN) distribution. The precision matrices of
the MPLN are the GRNs of different cell types and can be jointly estimated by
maximizing MPLN's lasso-penalized log-likelihood. We show that the MPLN model
is identifiable and the resulting penalized log-likelihood estimator is
consistent. To avoid the intractable optimization of the MPLN's log-likelihood,
we develop an algorithm called VMPLN based on the variational inference method.
Comprehensive simulation and real scRNA-seq data analyses reveal that VMPLN
performs better than the state-of-the-art single cell GRN methods.
