Author summaryDecoding biochemical reaction networks-the number of species, reactions connecting them and reaction rates-from snapshot data, such as data drawn from single molecule fluorescence in situ hybridization (smFISH) experiments, is of critical interest. Yet, this task's challenges are currently addressed with network specification heuristics since: 1) network size cannot be specified independently of rates; 2) rates may be separated by orders of magnitude, generating stiff ODEs; 3) uncertainty is not propagated into networks and parameters. We present, inspired by recent computational statistics tools, a method to simultaneously deduce reaction networks and associated rates from snapshot data while propagating error over all unknowns. We achieve this by treating network models as random variables and, within the Bayesian nonparametric paradigm, develop a posterior over models themselves. This multidimensional posterior naturally contains multiple hills and valleys, so we propose a combination of samplers allowing for the first simultaneous and self-consistent inference of networks and their associated rates. Our method's ability to treat arbitrary numbers of states contrasts the current state of the art and may modify previous biological conclusions based on network-determination heuristics. We demonstrate on method to synthetic data mimicking smFISH experiments and demonstrate its improvement over naive MCMC schemes. Accessing information on an underlying network driving a biological process often involves interrupting the process and collecting snapshot data. When snapshot data are stochastic, the data's structure necessitates a probabilistic description to infer underlying reaction networks. As an example, we may imagine wanting to learn gene state networks from the type of data collected in single molecule RNA fluorescence in situ hybridization (RNA-FISH). In the networks we consider, nodes represent network states, and edges represent biochemical reaction rates linking states. Simultaneously estimating the number of nodes and constituent parameters from snapshot data remains a challenging task in part on account of data uncertainty and timescale separations between kinetic parameters mediating the network. While parametric Bayesian methods learn parameters given a network structure (with known node numbers) with rigorously propagated measurement uncertainty, learning the number of nodes and parameters with potentially large timescale separations remain open questions. Here, we propose a Bayesian nonparametric framework and describe a hybrid Bayesian Markov Chain Monte Carlo (MCMC) sampler directly addressing these challenges. In particular, in our hybrid method, Hamiltonian Monte Carlo (HMC) leverages local posterior geometries in inference to explore the parameter space; Adaptive Metropolis Hastings (AMH) learns correlations between plausible parameter sets to efficiently propose probable models; and Parallel Tempering takes into account multiple models simultaneously with tempered information content to augment sampling efficiency. We apply our method to synthetic data mimicking single molecule RNA-FISH, a popular snapshot method in probing transcriptional networks to illustrate the identified challenges inherent to learning dynamical models from these snapshots and how our method addresses them.
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