DynSyst_Special_Topics: Polynomial Dynamical s Systems Over Finite Fields: From Structure to Dynamics

我们对日益复杂的工程系统和自然系统的理解的进步，关键取决于使用数学模型作为计算分析和预测的基础。这种模型允许在计算机上模拟由几种不同的相互作用的力驱动的动态过程。对于生命科学中研究的系统来说尤其如此，生命科学是生物医学和生物工程进步的基础。人类细胞内处理外部信号并驱动细胞代谢的分子网络提供了这种过程的重要例子。目前对这种网络的设计原理知之甚少。获得更多理解的一种方法是研究数学模型的特性，这些特性捕获了它们的关键特征。了解模型的结构特征和这些特征对模型动力学的约束之间的关系，将允许根据观察到的动力学来制定关于生物网络设计特征的假设。这些假设可以在实验室中进行测试。该项目的目标是研究一种模型的结构和动力学之间的关系，该模型已被证明在捕获各种细胞内分子网络的关键特征方面非常有用。除了分子网络之外，这种模型类型的各个方面已经用于电气工程和计算机科学，因此这个项目可能会产生生命科学之外的影响。时间离散动力系统模型不仅在工程领域，而且在生命科学领域都是普遍存在的。特别是在过去的十年中，有限动力系统，即具有有限状态空间的时间离散动力系统，已越来越多地用于系统生物学中的各种生化网络，如基因调控网络和信号转导网络的建模。 在许多情况下，现有数据的数量和质量不足以建立详细的定量模型，如常微分方程组，这需要许多经常未知的参数。此外，离散模型往往更直观，更容易为生命科学家所用。这个项目的前提是有限域上的多项式动力系统形成了一个统一的，数学上丰富的动力系统，越来越多地用于系统生物学。该项目的目标是确定将其结构与其动态联系起来的结果。为了获得强有力的结果，有必要把重点放在这些系统的特定家庭。这个项目的选择是从所谓的嵌套canalyzing函数构造的布尔网络类，以及它们的多态推广。分子生物学中的许多调控机制都可以用这些函数来描述，并且所得到的网络具有良好的动力学性质。

Progress in our understanding of increasingly complex engineered and natural systems depends crucially on the use of mathematical models as the basis for computational analysis and prediction. Such models allow the simulation on a computer of dynamic processes that are driven by several different interacting forces. This is true in particular for systems studied in the life sciences, which form the basis for advances in biomedicine and bioengineering. Molecular networks inside human cells which process external signals and drive cellular metabolism provide important examples of such processes. Relatively little is currently known about the design principles of such networks. One approach to gaining increased understanding is to study properties of the mathematical models that capture their key features. An understanding of the relationship between structural features of the models and the constraints these features put on model dynamics will allow the formulation of hypotheses about design features of biological networks based on observed dynamics. These hypotheses can then be tested in the laboratory. The goal of this project is to study the relationship between structure and dynamics for a type of model that has proven to be very useful in capturing key features of a variety of intracellular molecular networks. Beyond molecular networks, aspects of this model type have been used in electrical engineering and computer science, so this project might have an impact beyond the life sciences. Time-discrete dynamical systems models are ubiquitous not only in engineering but also the life sciences. Especially during the last decade finite dynamical systems, that is, time-discrete dynamical systems with a finite state space, have been used increasingly in systems biology to model a variety of biochemical networks, such as gene regulatory networks and signal transduction networks. In many cases, the available data quantity and quality is not sufficient to build detailed quantitative models such as systems of ordinary differential equations, which require many parameters that are frequently unknown. In addition, discrete models tend to be more intuitive and more easily accessible to life scientists. The premise of this project is that polynomial dynamical systems over finite fields form a unified, mathematically rich class of dynamical systems that are increasingly used in systems biology. The goal of the project is to establish results that relate their structure to their dynamics. To obtain strong results it is necessary to focus on specific families of such systems. The choice for this project is the class of Boolean networks constructed from so-called nested canalyzing functions, and their multi-state generalizations. Many regulatory mechanisms in molecular biology can be described by such functions, and the resulting networks have good dynamic properties.