Dynamical stability and accuracy of moment closure approximations for bursty gene expression with nonlinear degradation

Moment-closure methods provide a widely used approach for approximating low-order statistics of stochastic biochemical reaction networks, but the resulting closed equations may exhibit physically meaningless dynamics. Here, we systematically investigate the stability of second-order moment-closure approximations for a bursty gene expression model with nonlinear protein degradation. We consider the Gaussian, log-normal, and negative-binomial closures. By analyzing the resulting dynamical systems, we show that the Gaussian closure fails the proposed stability criterion for all parameter values, whereas both the log-normal and negative-binomial closures possess a unique physically accessible fixed point and are stable if and only if kB(1 + 2B) [≥] 1, where B is the mean burst size and k is the dimensionless protein synthesis rate. Moreover, this condition is automatically satisfied when the steady-state mean protein copy number is at least one, suggesting that it is mild in biologically relevant regimes. Numerical comparisons over a broad parameter range further show that the negative-binomial closure is the most accurate, while the Gaussian closure is the least accurate. We further develop higher-order negative-binomial moment-closure schemes and demonstrate that increasing the closure order substantially improves predictive accuracy. These results highlight the importance of considering dynamical stability, in addition to numerical accuracy, when assessing moment-closure approximations.