Recovering nonlinear dynamics from first-passage events: A Markov process inversion approach with application to gene expression variability

Stochastic nonlinear systems, from gene circuits to financial markets, are ubiquitous. A key challenge is inferring their governing parameters from sparse event times rather than continuous trajectories. We introduce a Markov process inversion approach (MPIA) that recovers drift and diffusion coefficients of nonlinear diffusions solely from first-passage times to a fixed boundary. Using arithmetic and geometric Brownian motions-representing constant-coefficient and state-dependent (log-linear) diffusions, respectively-the latter a paradigmatic model of multiplicative noise capturing state-dependent fluctuations-we develop a two-step procedure: fitting the inverse Gaussian distribution to hitting times, then analytically recovering generator parameters. We provide a comprehensive analysis of distribution fitting strategies, including maximum likelihood, Bayesian, and robust methods, and establish their relative merits. Asymptotic normality and consistency are proven, and the method is validated through high-precision simulations involving both direct sampling and full-path simulation. Applied to a synthetic gene circuit, MPIA disentangles effective growth rate from noise strength using only threshold-crossing times-readily accessible in high-throughput microfluidic experiments but invisible to trajectory-based methods. This event-based framework directly addresses the data constraints of modern single-cell technologies, where photobleaching and storage limitations make continuous fluorescence tracking impractical for large populations. Practical guidelines for handling censored data, drift direction verification, and model diagnostics are provided. This work lays a foundational blueprint for inversion strategies tailored to diffusions with analytically tractable first-passage-time distributions-specifically, constant-coefficient, log-linear, and the broader class of Lamperti-reducible diffusions. Extensions to non-reducible models, such as Ornstein-Uhlenbeck, Cox-Ingersoll-Ross, and logistic diffusions, via spectral and moment-based techniques, are discussed as future directions.