Department of Physics, Technion, Haifa, Israel.
PLoS One. 2012;7(12):e52105. doi: 10.1371/journal.pone.0052105. Epub 2012 Dec 27.
Biological cells in a population are variable in practically every property. Much is known about how variability of single cells is reflected in the statistical properties of infinitely large populations; however, many biologically relevant situations entail finite times and intermediate-sized populations. The statistical properties of an ensemble of finite populations then come into focus, raising questions concerning inter-population variability and dependence on initial conditions. Recent technologies of microfluidic and microdroplet-based population growth realize these situations and make them immediately relevant for experiments and biotechnological application. We here study the statistical properties, arising from metabolic variability of single cells, in an ensemble of micro-populations grown to saturation in a finite environment such as a micro-droplet. We develop a discrete stochastic model for this growth process, describing the possible histories as a random walk in a phenotypic space with an absorbing boundary. Using a mapping to Polya's Urn, a classic problem of probability theory, we find that distributions approach a limiting inoculum-dependent form after a large number of divisions. Thus, population size and structure are random variables whose mean, variance and in general their distribution can reflect initial conditions after many generations of growth. Implications of our results to experiments and to biotechnology are discussed.
生物群体中的细胞在几乎所有属性上都是可变的。人们已经了解了单细胞的可变性如何反映在无限大群体的统计特性中；然而，许多与生物学相关的情况都需要有限的时间和中等大小的群体。因此，有限群体的集合的统计特性成为焦点，这就提出了关于群体间变异性和对初始条件的依赖性的问题。基于微流控和微滴的群体生长的新技术实现了这些情况，并使其立即与实验和生物技术应用相关。我们在这里研究了在有限环境（例如微滴）中生长至饱和的微群体集合中，由单个细胞的代谢可变性引起的统计特性。我们为这个生长过程开发了一个离散的随机模型，将可能的历史描述为在表型空间中的随机游走，其中有一个吸收边界。通过映射到概率论中的 Polya urn，我们发现，在经过大量分裂后，分布会趋近于一种有限制接种物依赖性的形式。因此，群体大小和结构是随机变量，其均值、方差以及一般分布可以反映经过多代生长后的初始条件。我们的结果对实验和生物技术的影响进行了讨论。
