Model reduction in systems biology: the Mori-Zwanzig projection method

生物化学网络的复杂性来自于这样一个事实，即它们是由远离平衡的非线性动力学控制的。近年来，系统生物学已经成为一门从综合系统而不是分离组件的角度来研究生物复杂性的学科（还原论者？观点）。系统方法可能有助于解决生命科学中的许多基本问题，这些问题与系统水平的相互作用有关，而不是与单个生物分子物种有关。为了实现完整的系统级描述，除了新的实验进展外，还需要开发许多新的分析技术和理论。一个主要的挑战涉及微分方程的生物动力学的计算建模。由于任何特定的生化控制系统的复杂性，需要大量的变量和参数来描述其动态。为了具有所需的预测能力，需要以足够的精度确定这些参数的值，但是对于大多数系统来说，所需的实验数据是不可用的。这是许多科学领域的普遍问题，包括物理和化学，其中强大的Mori-Zwanzig投影方法被广泛用于研究哈密顿动力学。该方法首先将复杂动力系统分解为主次子系统。 通过系统的信息收缩，人们可以专注于包含主要感兴趣的变量（特别是实验上可解析的变量）的主要系统。次级子系统没有被明确地处理，但它对初级系统的影响在数学上得到了适当的解释。在这项提案中，研究人员将为没有详细平衡的非哈密顿系统开发一种通用形式和数值算法，重点是细胞调控网络。该方法在以下三种情况下特别有用：1）需要粗粒度模型，或者可用数据阻止更详细的模型; 2）所研究的网络嵌入在更大的网络中; 3）想要执行多尺度建模。所提出的方法将为系统生物学家提供一个强大的工具，以接近理解复杂的生物过程和改善人类健康的最终目标。在许多其他研究领域，人们也会遇到类似的情况，即处理信息不完整的复杂系统，例如：金融交易、供电网络以及流行病或生物恐怖袭击期间病毒的传播和进化。

The complexity of biochemical networks derives from the fact that they are governed by nonlinear kinetics far-from-equilibrium. In recent years, systems biology has emerged as a discipline to examine biological complexity from the point-of-view of integrated systems rather than separate components (the reductionists? point-of-view). A systems approach may help to resolve many fundamental issues in the life sciences that relate to systems-level interactions rather than individual biomolecular species. To achieve a full systems-level description, many new analytical techniques and theories need to be developed, in addition to new experimental advances. One major challenge concerns computational modeling of biological dynamics by differential equations. Due to the complexity of any particular biochemical control system, a large number of variables and parameters are needed to describe its dynamics. To have the desired predictive power, the values of these parameters need to be determined with sufficient precision, but for most systems the requisite experimental data are not available. This is a general problem in many areas of science, including physics and chemistry, where the powerful Mori-Zwanzig projection method is widely used for studying Hamiltonian dynamics. In this method, the complex dynamical system is first separated into primary and secondary subsystems. Through systematic information contraction, one can focus on the primary system, which contains the variables of primary interest (in particular, experimentally resolvable variables). The secondary subsystem is not treated explicitly, but its effect on the primary system is properly accounted for mathematically. In this proposal, the researchers will develop a general formalism and numerical algorithms for non-Hamiltonian systems without detailed balance, with a focus on cellular regulatory networks. The method will be especially useful in three cases: 1) where a coarse-grained model is desirable, or available data prevents a more detailed model; 2) where the network under study is embedded in a larger network; and 3) where one wants to perform multi-scale modeling. The proposed methods will provide a powerful tool for systems biologists to approach the ultimate goals of understanding complex biological processes and of improving human health. One encounters similar situations, of dealing with a complex system with incomplete information, in many other research areas, for example: financial transactions, the power supply network, and the spread and evolution of viruses during an epidemic or a bioterrorist attack.