Delayed production can substantially alter the qualitative behaviour of feedback systems. Motivated by stochastic mechanisms in gene expression, we consider a protein molecule which is produced in randomly timed bursts, requires an exponentially distributed time to activate and then partakes in positive regulation of its burst frequency. Asymptotically analysing the underlying master equation in the large-delay regime, we provide tractable approximations to time-dependent probability distributions of molecular copy numbers. Importantly, the presented analysis demonstrates that positive feedback systems with large production delays can constitute a stable toggle switch even if they operate with low copy numbers of active molecules.

中文翻译：

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推迟生产以指数方式增强随机开关的分子记忆

延迟生产可以大大改变反馈系统的定性行为。受基因表达中的随机机制的启发，我们考虑了一种蛋白质分子，它以随机定时的爆发产生，需要指数分布的时间来激活，然后参与对其爆发频率的正调节。渐近地分析大延迟状态下的基本主方程，我们提供了分子拷贝数的时间相关概率分布的易处理近似值。重要的是，所提出的分析表明，具有大生产延迟的正反馈系统可以构成稳定的拨动开关，即使它们以低拷贝数的活性分子运行。