Clonal populations of microbial and cancer cells are often driven into a drug-tolerant persister state in response to drug therapy, and these persisters can subsequently adapt to the new drug environment via genetic and epigenetic mechanisms. Estimating the frequency with which drug-tolerance states arise, and its transition to drug-resistance, is critical for designing efficient treatment schedules. Here we study a stochastic model of cell proliferation where drug-tolerant persister cells transform into a drug-resistant state with a certain adaptation rate, and the resistant cells can then proliferate in the presence of the drug. Assuming a random number of persisters to begin with, we derive an exact analytical expression for the statistical moments and the distribution of the total cell count (i.e., colony size) over time. Interestingly, for Poisson initial conditions the noise in the colony size (as quantified by the Fano factor) becomes independent of the initial condition and only depends on the adaptation rate. Thus, experimentally quantifying the fluctuations in the colony sizes provides an estimate of the adaptation rate, which then can be used to infer the starting persister numbers from the mean colony size. Overall, our analysis introduces a modification of the classical Luria--Delbruck experiment, also called the ``Fluctuation Test", providing a valuable tool to quantify the emergence of drug resistance in cell populations.

中文翻译：

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改良的波动试验，用于阐明微生物和癌细胞的耐药性

微生物和癌细胞的克隆种群通常会响应药物治疗而进入耐受药物的持久性状态，这些持久性随后可以通过遗传和表观遗传机制适应新的药物环境。估计药物耐受状态出现的频率及其向耐药性的转变，对于设计有效的治疗方案至关重要。在这里，我们研究了细胞增殖的随机模型，其中耐药性持久性细胞以一定的适应率转化为耐药性状态，然后耐药性细胞可以在存在药物的情况下增殖。假设开始时有随机数的持久性，我们就统计矩和总细胞数（即菌落大小）随时间的分布得出精确的解析表达式。有趣的是 对于泊松初始条件，菌落大小的噪声（由Fano因子量化）变得与初始条件无关，并且仅取决于适应率。因此，实验上量化菌落大小的波动可提供适应率的估计值，然后可将其用于从平均菌落大小中推断出起始持续性数量。总体而言，我们的分析引入了对经典Luria-Delbruck实验的改进，也称为``波动测试''，为量化细胞群体中耐药性的出现提供了有价值的工具。实验上量化菌落大小的波动可提供适应率的估计值，然后可将其用于从平均菌落大小中推断出起始持续性数量。总体而言，我们的分析引入了对经典Luria-Delbruck实验的改进，也称为``波动测试''，为量化细胞群体中耐药性的出现提供了有价值的工具。实验上量化菌落大小的波动可提供适应率的估计值，然后可将其用于从平均菌落大小中推断出起始持续性数量。总体而言，我们的分析引入了对经典Luria-Delbruck实验的改进，也称为``波动测试''，为量化细胞群体中耐药性的出现提供了有价值的工具。