异质细胞群体内竞争的随机多尺度模型：模拟方法和平均场分析
Stochastic multi-scale models of competition within heterogeneous cellular populations: simulation methods and mean-field analysis
摘要：
我们提出了一个建模框架来分析异质性，多尺度细胞群体的随机行为。我们用一个特定的例子来说明我们的方法，在这个例子中，我们研究了一个具有氧调节增殖率的群体。我们的配方是基于一个年龄相关的随机过程。群体中的细胞以其年龄为特征。年龄依赖性（氧调节）出生率由氧依赖性细胞周期进程的随机模型给出。然后，我们制定了一个年龄依赖的出生和死亡过程，这决定了细胞群的时间演变。种群处于控制其稳态大小的反馈回路下：细胞消耗氧气，这反过来又促进细胞增殖。我们表明，我们的细胞周期进程的随机模型允许由随机效应引起的细胞群体内的异质性。这种异质性行为反映在增殖率的变化中。在这一框架内，我们取得了三个主要成果。首先，我们已经证明，年龄的G1/S过渡，这基本上决定了出生率，表现出一个非常简单的标度行为。这使得我们的数值方法大大简化。进一步的结果是观察到异质种群经历了准中性竞争的内部过程。最后，我们研究了细胞周期时相依赖性治疗（如放射治疗）对异质性人群的影响。特别是，我们研究了人口中包含一个静态的子人口的情况。我们的平均场分析和数值模拟证实，如果生存分数的治疗是太高，救援的静止人口发生。这导致出现对治疗的抗性，因为获救的人口对治疗不太敏感。
英文摘要：
We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age. The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. We then formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size: cells consume oxygen which in turns fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. This allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.
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