Bokes Pavol, Borri Alessandro, Palumbo Pasquale, Singh Abhyudai
Comenius University, Bratislava, Slovakia.
Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia.
J Math Biol. 2020 Jul;81(1):343-367. doi: 10.1007/s00285-020-01512-y. Epub 2020 Jun 24.
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.
基因表达中的噪声会受到产生延迟的实质性影响。在此，我们考虑一个具有蛋白质爆发式产生、一步产生延迟（其通过会激活蛋白质）以及爆发频率反馈的数学模型。我们特别专注于研究该模型在慢激活（即大延迟）状态下的稳态行为。使用一种形式渐近方法，我们推导出了一个适用于慢激活状态的非活性蛋白质的自治常微分方程。如果该微分方程是单稳态的，非活性（活性）蛋白质的稳态分布可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。如果该微分方程是双稳态的（由于协同正反馈），非活性（活性）蛋白质的稳态分布可由位于稳定不动点的高斯（泊松）模式的混合近似；模式的权重由对稳态分布的WKB近似确定。将渐近结果与化学主方程的数值解进行比较。
