Laboratoire Jean Kuntzmann, Bâtiment IMAG, 700 avenue centrale, Saint Martin d'Hères 38401, France.
Math Biosci. 2018 Sep;303:83-100. doi: 10.1016/j.mbs.2018.06.006. Epub 2018 Jun 19.
The classic Luria-Delbrück model can be interpreted as a Poisson compound (number of mutations) of exponential mixtures (developing time of mutant clones) of geometric distributions (size of a clone in a given time). This "three-ingredients" approach is generalized in this paper to the case where the split instant distributions of cells are not i.i.d. : the lifetime of each cell is assumed to depend on its birth date. This model takes also into account cell deaths and non-exponentially distributed lifetimes. Previous results on the convergence of the distribution of mutant counts are recovered. The particular case where the instantaneous division rates of normal and mutant cells are proportional is studied. The classic Luria-Delbrück and Haldane models are recovered. Probability computations and simulation algorithms are provided. Robust estimation methods developed for the classic mutation models are adapted to the new model: their properties of consistency and asymptotic normality remain true; their asymptotic variances are computed. Finally, the estimation biases induced by considering classic mutation models instead of an inhomogeneous model are studied with simulation experiments.
经典的 Luria-Delbrück 模型可以解释为泊松复合（突变数量）的指数混合（突变克隆的发展时间）的几何分布（给定时间内克隆的大小）。本文将这种“三要素”方法推广到细胞分裂瞬间分布不是独立同分布的情况：每个细胞的寿命假设取决于其出生日期。该模型还考虑了细胞死亡和非指数分布的寿命。恢复了关于突变计数分布收敛的先前结果。研究了正常和突变细胞的瞬时分裂率成比例的特殊情况。恢复了经典的 Luria-Delbrück 和 Haldane 模型。提供了概率计算和模拟算法。为经典突变模型开发的稳健估计方法被应用于新模型：它们的一致性和渐近正态性的特性仍然是正确的；计算了它们的渐近方差。最后，通过模拟实验研究了考虑经典突变模型而不是非均匀模型引起的估计偏差。
