Understanding the complex interactions of biochemical processes underlying human disease represents the holy grail of systems biology. When processes are modelled in ordinary differential equation (ODE) fashion, the most common tool for their analysis is linear stability analysis where the long-term behaviour of the model is determined by linearizing the system around its steady states. However, this asymptotic behaviour is often insufficient for completely determining the structure of the underlying system. A complementary technique for analysing a system of ODEs is to consider the set of symmetries of its solutions. Symmetries provide a powerful concept for the development of mechanistic models by describing structures corresponding to the underlying dynamics of biological systems. To demonstrate their capability, we consider symmetries of the nonlinear Hill model describing enzymatic reaction kinetics and derive a class of symmetry transformations for each order of the model. We consider a minimal example consisting of the application of symmetry-based methods to a model selection problem, where we are able to demonstrate superior performance compared to ordinary residual-based model selection. Moreover, we demonstrate that symmetries reveal the intrinsic properties of a system of interest based on a single time series. Finally, we show and propose that symmetry-based methodology should be considered as the first step in a systematic model building and in the case when multiple time series are available it should complement the commonly used statistical methodologies.

中文翻译：

---

生化系统动态模型中的对称结构

了解人类疾病背后生化过程的复杂相互作用代表了系统生物学的圣杯。当以常微分方程 (ODE) 方式对过程建模时，最常用的分析工具是线性稳定性分析，其中模型的长期行为是通过围绕其稳态对系统进行线性化来确定的。然而，这种渐近行为通常不足以完全确定底层系统的结构。分析 ODE 系统的一种补充技术是考虑其解的对称性集。对称性通过描述与生物系统的潜在动力学相对应的结构，为机械模型的开发提供了一个强大的概念。为了展示他们的能力，我们考虑了描述酶反应动力学的非线性希尔模型的对称性，并为模型的每个阶推导出一类对称变换。我们考虑一个最小的例子，该例子包括将基于对称的方法应用于模型选择问题，与普通的基于残差的模型选择相比，我们能够展示出优越的性能。此外，我们证明对称性揭示了基于单个时间序列的感兴趣系统的内在属性。最后，我们展示并建议应将基于对称性的方法视为系统模型构建的第一步，并且在有多个时间序列可用的情况下，它应补充常用的统计方法。