AF: Small: Algorithmic Foundations of Hybrid Stochastic Modeling and Simulation Methods with Applications to Cell Cycle Models

许多生化应用中出现的复杂系统通常表现出多尺度特征：这些系统在广泛的尺度上结合了各种物理过程或子系统。典型的多尺度系统可能需要具有宏观、介观和微观动力学、确定性和随机动力学、连续和离散状态空间、快尺度和慢尺度反应以及大种群和小种群物种的尺度。这些复杂的特征给生化系统的建模和仿真带来了巨大的挑战。该项目的目标是通过开发严格的数学理论和创新的混合建模方法数值算法来应对这些挑战。将开发用于将生化系统划分为子系统的混合方法和算法的误差分析的数学基础，并将其应用于不同规模的化学系统。所提出的工作将推进复杂生化系统模拟计算方法的前沿，并实现复杂生化系统中多个尺度之间的自动状态切换。 数值分析还将有助于对离散随机模拟方法中的误差的一般理解。该项目将支持研究生和本科生的教育和培训。该项目开发的仿真方法将被纳入建模和仿真包JigCell和CoPaSi中，所有这些都向公众开放。课程材料和源代码模块也将在大学网站上提供。该项目的动机是对复杂生物控制系统的真实建模和模拟：酵母细胞的生长和分裂周期。该项目侧重于三个具体目标。主要目的是为混合方法的误差分析奠定数学基础。误差将以两种方式进行分析：一种方法基于线性系统化学主方程的近似，另一种方法基于随机模拟轨迹的泊松过程公式。将根据不同的尺度区域及其相互作用来研究不同划分策略和混合求解器引起的误差。错误分析还有助于确定分区策略中的参数。第二个目标是开发生化系统混合随机模拟的算法、自动将反应和状态变量划分到不同尺度区域的机制以及有效模拟多尺度系统的软件。根据反应物群体和反应倾向尺度，生化系统中的反应被划分为四个区域。划分策略是基于对这四个区域和实际规模差异的分析。将仔细研究混合方法的实施细节以实现高效率。该项目的第三个目标是开发一个现实的芽殖酵母细胞周期随机模型，其中包括蛋白质相互作用以及基因和 mRNA 动力学，并将根据野生型芽殖酵母细胞和约 120 个突变菌株的表型进行判断。所开发模型的模拟结果将与湿实验室实验数据进行比较。我们雄心勃勃的目标是建立一个详细的细胞周期模型，反映基因和 mRNA 水平的动态，准确解释酵母细胞中细胞增殖的已知概率特征，并准确预测突变菌株的异常行为。该项目开发的混合方法的算法、理论和软件将在这一复杂细胞周期模型的建模和模拟中得到应用和测试。

Complex systems emerging from many biochemical applications often exhibit multiscale features: the systems incorporate a variety of physical processes or subsystems across a broad range of scales. A typical multiscale system may require scales with macroscopic, mesoscopic, and microscopic kinetics, deterministic and stochastic dynamics, continuous and discrete state space, fast-scale and slow-scale reactions, and species of large and small populations. These complex features present great challenges for the modeling and simulation of biochemical systems. The goal of this project is to face these challenges by developing rigorous mathematical theories and innovative numerical algorithms for hybrid modeling methods. Mathematical foundations for error analysis of hybrid methods and algorithms for partitioning a biochemical system into subsystems will be developed and applied to chemical systems in different scales. The proposed work will advance the frontier of computational methods for the simulation of complex biochemical systems and enable automatic regime switching among multiple scales in complex biochemical systems. The numerical analysis will also contribute to general understanding of errors in discrete stochastic simulation methods. The project will support education and training of graduate and undergraduate students. The simulation methods developed in this project will be incorporated in the modeling and simulation packages JigCell and CoPaSi, all are open for public access. The curriculum material and source code modules will also be available on the university website.This project is motivated by realistic modeling and simulation of a complex biological control system: the cycle of growth and division in yeast cells. The project focuses on three specific aims. The primary aim is to develop mathematical foundations for error analysis of hybrid methods. Errors will be analyzed in two ways: one approach is based on approximation of chemical master equations for linear systems, and the other is based on a Poisson-process formulation of stochastic simulation trajectories. Errors caused by different partitioning strategies and hybrid solvers will be studied corresponding to different scale regions and interactions among them. Error analysis will also help to determine parameters in partitioning strategies. The second aim is to develop algorithms for hybrid stochastic simulation of biochemical systems, mechanisms to automatically partition reactions and state variables into different scale regions, and software to efficiently simulate multiscale systems. Reactions in a biochemical system are partitioned into four regions according to reactant population and reaction propensity scales. The partitioning strategy is based on analysis of these four regions and actual scale differences. Implementation details of hybrid methods will be carefully studied to achieve high efficiency. The third aim of this project is to develop a realistic stochastic model of the budding yeast cell cycle, which will include protein interactions as well as gene and mRNA dynamics and which will be judged with respect to the phenotypes of wild-type budding yeast cells and ~120 mutant strains. Simulation results of the developed model will be compared with wet-lab experimental data. The ambitious goal is to have a detailed cell cycle model that reflects dynamics at gene and mRNA levels, accounts accurately for known probabilistic features of cell proliferation in yeast cells, and accurately predicts the aberrant behaviors of mutant strains. Algorithms, theories, and software of hybrid methods developed in this project will be applied and tested in the modeling and simulation of this complex cell cycle model.