Department of Mathematical Sciences, Centre for Mathematical Biology, University of Bath, Claverton Down, Bath, BA2 7AY, UK.
MRC Human Genetics Unit, MRC IGMM, Western General Hospital, University of Edinburgh, Edinburgh, EH4 2XU, UK.
Bull Math Biol. 2017 Dec;79(12):2905-2928. doi: 10.1007/s11538-017-0356-4. Epub 2017 Oct 13.
The stochastic simulation algorithm commonly known as Gillespie's algorithm (originally derived for modelling well-mixed systems of chemical reactions) is now used ubiquitously in the modelling of biological processes in which stochastic effects play an important role. In well-mixed scenarios at the sub-cellular level it is often reasonable to assume that times between successive reaction/interaction events are exponentially distributed and can be appropriately modelled as a Markov process and hence simulated by the Gillespie algorithm. However, Gillespie's algorithm is routinely applied to model biological systems for which it was never intended. In particular, processes in which cell proliferation is important (e.g. embryonic development, cancer formation) should not be simulated naively using the Gillespie algorithm since the history-dependent nature of the cell cycle breaks the Markov process. The variance in experimentally measured cell cycle times is far less than in an exponential cell cycle time distribution with the same mean.Here we suggest a method of modelling the cell cycle that restores the memoryless property to the system and is therefore consistent with simulation via the Gillespie algorithm. By breaking the cell cycle into a number of independent exponentially distributed stages, we can restore the Markov property at the same time as more accurately approximating the appropriate cell cycle time distributions. The consequences of our revised mathematical model are explored analytically as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one non-spatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular proliferation-vital to the accurate modelling of many biological processes-whilst still being able to take advantage of the power and efficiency of the popular Gillespie algorithm.
Gillespie 算法（最初是为模拟混合良好的化学反应系统而推导的）是一种随机模拟算法，现在广泛应用于随机效应起重要作用的生物过程建模。在亚细胞水平的混合情况下，通常可以合理地假设连续反应/相互作用事件之间的时间呈指数分布，可以将其适当建模为马尔可夫过程，并使用 Gillespie 算法进行模拟。然而，Gillespie 算法经常被用于模拟它从未设计过的生物系统。特别是对于细胞增殖很重要的过程（例如胚胎发育、癌症形成），不应使用 Gillespie 算法进行简单模拟，因为细胞周期的历史依赖性破坏了马尔可夫过程。实验测量的细胞周期时间的方差远小于具有相同均值的指数细胞周期时间分布的方差。在这里，我们提出了一种建模细胞周期的方法，该方法使系统恢复无记忆特性，因此与通过 Gillespie 算法进行模拟一致。通过将细胞周期分解为多个独立的指数分布阶段，我们可以在更准确地近似适当的细胞周期时间分布的同时恢复马尔可夫性质。我们尽可能从分析的角度探讨了修正数学模型的后果。我们通过重新生成包含细胞增殖的两个模型（一个是空间的，一个是非空间的）的结果来证明正确使用细胞周期时间分布的重要性，并证明改变细胞周期时间分布会对模型的结果产生定量和定性的差异。我们的改编将使建模者和实验者都能够适当地表示细胞增殖——这对于许多生物过程的准确建模至关重要——同时仍然能够利用流行的 Gillespie 算法的强大功能和效率。
