Complex dynamics of interacting populations of intrinsically stochastic individuals can be mathematically represented by a discrete-state, continuous-time Markov jump process.  T. G. Kurtz's theorem establishes a relation between this stochastic process, in the limit of V tending infinity, and the traditional dynamical systems based on ODEs.  We apply this theory to several problems in current cell biology in terms of the biochemical constituents, and illustrate the emergent notions of epigenetic phenotypes and their switching, and relation to the classical idea of phase transition.  The mathematical origin of the latter in non-uniform convergence when time goes to infinity with respect to the parameter V is discussed, and a large deviations result based on WKB ansatz will be presented. We suggest some open questions and a set of computational issues as well.

钱紘，一九六零年生于上海。北京大学天体物理学本科毕业，圣路易斯华盛顿大学生物化学博士，现为西雅图华盛顿大学应用数学系教授。早年的研究工作包括单分子生物物理的实验与数据分析，蛋白质折叠的物理化学，和分子马达的数学模型。近年来主要研究方向为用随机动态系统的数学语言和统计热力学的思想来刻画细胞层面的生命过程：包括细胞内生物化学反应动力学作为开系统及其定态的理论，首创化学自由能耗散与细胞内生物化学信号传导定量关系的研究。他从具体生物化学建模经验中发展出来的一套介观随机热力学的数学理论现为国际前沿课题之一。