Efficiency, accuracy and robustness of probability generating function based parameter inference method for stochastic biochemical reactions

Biochemical reactions are inherently stochastic, with their kinetics commonly described by chemical master equations (CMEs). However, the discrete nature of molecular states renders likelihood-based parameter inference from CMEs computationally intensive. Here, we introduce an inference method that leverages analytical solutions in the probability generating function (PGF) space and systematically evaluate its efficiency, accuracy, and robustness. Across both steady-state and time-resolved count data, our numerical experiments demonstrate that the PGF-based method consistently outperforms existing approaches in terms of both computational efficiency and inference accuracy, even under data contamination. These favorable properties further enable the extension of the PGF-based framework to model selection—a task typically considered computationally prohibitive. Using time-resolved data, we show that the method can correctly identify complex gene expression models with more than three gene states, a task that cannot be reliably achieved using steady-state data alone.Author summary: Biochemical processes within cells, such as gene expression, are inherently stochastic. To understand these dynamics, researchers use mathematical models like the Chemical Master Equation (CME) to infer kinetic parameters from experimental data. However, traditional inference methods often face a bottleneck: they are either computationally too slow or lack the necessary accuracy when dealing with the complex, noisy data produced by modern single-cell experiments. In this study, we introduce a high-performance inference framework based on the Probability Generating Function (PGF). By leveraging analytical solutions, our method achieves exceptional efficiency and accuracy across both steady-state snapshots and transient, time-resolved data. We demonstrate that the PGF-based approach is highly robust, maintaining reliable performance even when data is corrupted by experimental artifacts such as molecular loss or extreme outliers. Crucially, we extend this framework to the critical task of model selection. Using a cross-validation strategy, our method can accurately distinguish between competing biological hypotheses—for instance, correctly identifying the number of hidden states a gene transitions through before activation. This versatile and scalable tool provides a powerful resource for researchers to decode the hidden mechanisms of life from complex single-cell datasets.