Doob-WKB Rates in a Bursty Gene-Expression Switch with Catastrophic Loss

Low-copy gene-expression circuits switch through discrete bursts and multi-molecule loss events that are poorly represented by Gaussian diffusion. We study an effective onedimensional continuous-time Markov chain for an autoregulated protein with geometric burst production, linear degradation, and catastrophic binomial thinning. The thinning event is treated as a coarse-grained fixed-volume copy-number loss mechanism, not as a full cell-division model. We combine exact first-passage calculations with semiclassical rate interpretation. Augmented uniformization of the killed generator gives exact first-passage-time densities, survival curves, hitting probabilities, and conditional means. Jump-WKB theory gives the exponential switching scale while preserving the full burst and thinning statistics. A Doob-WKB tilted killed process then provides a calibrationfree semiclassical estimate of the order-one prefactor through its principal eigenvalue. The Doob-WKB rate estimate is benchmarked against exact killed-spectrum rates and stochastic simulation. It tracks the exact rate across representative low-copy bistable regimes, while diffusion and Eyring–Kramers baselines show prefactor errors and miss catastrophe-induced directional asymmetry. Parameter scans separate barrier controls, such as production strength and burst size, from conditioned-flux controls, such as catastrophe rate and severity. The result is a benchmarked one-dimensional rate theory for barrier versus prefactor control in jump-driven switching.