Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a formal asymptotic approach, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable fixed point of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable fixed points; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.

基因表达中的噪音会受到生产延迟的实质性影响。在这里，我们考虑了一个数学模型，其中包含蛋白质的爆发性生产、一步生产延迟（其通过激活蛋白质）和爆发频率的反馈。我们特别专注于检查模型在慢激活（即大延迟）状态下的稳态行为。使用形式渐近方法，我们推导出适用于慢激活方案的非活性蛋白质的自主常微分方程。如果微分方程是单稳态的，则非活性（活性）蛋白质的稳态分布由位于微分方程全局稳定不动点的单个高斯（泊松）模式近似。如果微分方程是双稳态的（由于协同正反馈），则非活性（活性）蛋白质的稳态分布近似为位于稳定不动点的高斯（泊松）模式的混合；模式的权重由 WKB 近似确定到平稳分布。渐近结果与化学主方程的数值解进行比较。