Transcription rate dynamics and RNA copy number noise: General relations and data-driven predictions

Stochastic models of gene expression typically begin with a microscopic model of transcription and propagate its statistics to the RNA distribution. Here we develop a doubly stochastic framework in which RNA production is Poisson conditional on a time-varying transcription rate $λ(t)$. For RNA lifetimes drawn independently from an arbitrary finite-mean distribution, we derive the RNA Fano factor in terms of the lifetime survival function and the autocovariance of $λ(t)$. For a Poisson degradation process with rate $μ$, the survival kernel provides an exponential temporal filter, and the result depends only on the mean, variance, and normalized autocorrelation of $λ(t)$. These quantities may be calculated from an explicit rate model or, under ergodicity and adequate sampling, estimated from a sufficiently long rate trajectory. We validate the data-driven estimator using simulated transcription-rate trajectories, without supplying the known autocorrelation to the estimator. We also obtain exact analytical results for transcription-rate dynamics modeled by a discrete M/M/1 process and by drift-diffusion with reflecting, periodic, or first-passage-reset boundaries, and verify each result by direct simulation of the coupled transcription-rate and RNA copy-number processes.