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

生物化学网络的复杂性源于这样一个事实：它们是由远离平衡的非线性动力学控制的。近年来，系统生物学作为一门学科的出现，从整体系统的角度来研究生物复杂性，而不是从单独的组成部分(还原论者？的观点来看)。系统方法可能有助于解决生命科学中与系统级相互作用有关的许多基本问题，而不是与单个生物分子物种有关的问题。为了实现完整的系统级描述，除了新的实验进展之外，还需要开发许多新的分析技术和理论。一个主要的挑战是通过微分方程计算生物动力学模型。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。为了获得预期的预测能力，需要以足够的精度确定这些参数的值，但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划分为主子系统和次子系统。通过系统的信息收缩，人们可以关注主要系统，其中包含主要感兴趣的变量（特别是实验可解决的变量）。次要子系统没有被明确地处理，但它对主系统的影响在数学上得到了适当的解释。在这项提议中，研究人员将为非哈密顿系统开发一种通用的形式和数值算法，而不需要详细的平衡，重点是细胞调节网络。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一个更大的网络中；3)想要进行多尺度建模的地方。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，例如：金融交易、电力供应网络、流行病或生物恐怖袭击期间病毒的传播和进化，在处理信息不完全的复杂系统时，也会遇到类似的情况。

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.