Komarova Natalia L, Wu Lin, Baldi Pierre
Department of Mathematics, University of California, Irvine, CA 92697, United States.
Math Biosci. 2007 Nov;210(1):253-90. doi: 10.1016/j.mbs.2007.04.007. Epub 2007 May 3.
What is the expected number of mutants in a stochastically growing colony once it reaches a given size, N? This is a variant of the famous Luria-Delbruck model which studies the distribution of mutants after a given time-lapse. Instead of fixing the time-lapse, we assume that the colony size is a measurable quantity, which is the case in many in-vivo oncological and other applications. We study the mean number of mutants for an arbitrary cell death rate, and give partial results for the variance. For a restricted set of parameters we provide analytical results; we also design a very efficient computational method to calculate the mean, which works for most of the parameter values, and any colony size, no matter how large. We find that a cellular population with a higher death rate will contain a larger number of mutants than a population of equal size with a smaller death rate. Also, a very large population will contain a larger percentage of mutants; that is, irreversible mutations act like a force of selection, even though here the mutants are assumed to have no selective advantage. Finally, we investigate the applicability of the traditional, 'fixed-time' approach and find that it approximates the 'fixed-size' problem whenever stochastic effects are negligible.
一个随机生长的菌落一旦达到给定大小(N)，其预期的突变体数量是多少？这是著名的卢里亚 - 德尔布吕克模型的一个变体，该模型研究给定时间间隔后突变体的分布。我们不固定时间间隔，而是假设菌落大小是一个可测量的量，这在许多体内肿瘤学和其他应用中都是如此。我们研究了任意细胞死亡率下突变体的平均数量，并给出了方差的部分结果。对于一组受限的参数，我们提供了分析结果；我们还设计了一种非常有效的计算方法来计算平均值，该方法适用于大多数参数值以及任何菌落大小，无论其有多大。我们发现，死亡率较高的细胞群体比大小相同但死亡率较低的群体包含更多的突变体。此外，非常大的群体将包含更大比例的突变体；也就是说，不可逆突变的作用类似于选择力，尽管在这里假设突变体没有选择优势。最后，我们研究了传统的“固定时间”方法的适用性，发现只要随机效应可以忽略不计，它就可以近似“固定大小”问题。
