The question of how cell-to-cell differences in transcription rate affect RNA count distributions is fundamental for understanding biological processes underlying transcription. Answering this question requires quantitative models that are both interpretable (describing concrete biophysical phenomena) and tractable (amenable to mathematical analysis). This enables the identification of experiments which best discriminate between competing hypotheses. As a proof of principle, we introduce a simple but flexible class of models involving a continuous stochastic transcription rate driving a discrete RNA transcription and splicing process, and compare and contrast two biologically plausible hypotheses about transcription rate variation. One assumes variation is due to DNA experiencing mechanical strain, while the other assumes it is due to regulator number fluctuations. We introduce a framework for numerically and analytically studying such models, and apply Bayesian model selection to identify candidate genes that show signatures of each model in single-cell transcriptomic data from mouse glutamatergic neurons.

To what extent do cell-to-cell differences in transcription rate affect RNA copy number distributions, and what can this variation tell us about biological processes underlying transcription? We argue that successfully answering such questions requires quantitative models that are both interpretable (describing concrete biophysical phenomena) and tractable (amenable to mathematical analysis); in particular, such models enable the identification of experiments which best discriminate between competing hypotheses. As a proof of principle, we introduce a simple but flexible class of models involving a stochastic transcription rate (governed by a stochastic differential equation) coupled to a discrete stochastic RNA transcription and splicing process, and compare and contrast two biologically plausible hypotheses about observed transcription rate variation. One hypothesis assumes transcription rate variation is due to DNA experiencing mechanical strain and relaxation, while the other assumes that variation is due to fluctuations in the number of an abundant regulator. Through a thorough mathematical analysis, we show that these two models are challenging to distinguish: properties like first- and second-order moments, autocorrelations, and several limiting distributions are shared. However, our analysis also points to the experiments which best discriminate between them. Our work illustrates the importance of theory-guided data collection in general, and multimodal single-molecule data in particular for distinguishing between competing hypotheses. We use this theoretical case study to introduce and motivate a general framework for constructing and solving such nontrivial continuous-discrete models.

Stochastic mechanics-the study of classical stochastic systems governed by things like master equations and Fokker-Planck equations-exhibits striking mathematical parallels to quantum mechanics. In this article, we make those parallels more transparent by presenting a quantum mechanics-like formalism for deriving a path integral description of systems described by stochastic differential equations. Our formalism expediently recovers the usual path integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup forms) and is flexible enough to account for different variable domains (e.g. real line versus compact interval), stochastic interpretations, arbitrary numbers of variables, explicit time-dependence, dimensionful control parameters, and more.We discuss the implications of our formalism for stochastic biology.

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