Conradi Carsten, Feliu Elisenda, Mincheva Maya, Wiuf Carsten
Life Science Engineering, HTW Berlin, Berlin, Germany.
Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark.
PLoS Comput Biol. 2017 Oct 3;13(10):e1005751. doi: 10.1371/journal.pcbi.1005751. eCollection 2017 Oct.
Mathematical modelling has become an established tool for studying the dynamics of biological systems. Current applications range from building models that reproduce quantitative data to identifying systems with predefined qualitative features, such as switching behaviour, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce a procedure to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The procedure is based on the computation of the Brouwer degree, and it creates a multivariate polynomial with parameter depending coefficients. The signs of the coefficients determine parameter regions with and without multistationarity. A particular strength of the procedure is the avoidance of numerical analysis and parameter sampling. The procedure consists of a number of steps. Each of these steps might be addressed algorithmically using various computer programs and available software, or manually. We demonstrate our procedure on several models of gene transcription and cell signalling, and show that in many cases we obtain a complete partitioning of the parameter space with respect to multistationarity.
数学建模已成为研究生物系统动态的既定工具。当前的应用范围从构建能重现定量数据的模型到识别具有预定义定性特征的系统，如切换行为、双稳态或振荡。从数学角度看，后一个问题相当于识别与给定定性特征相关的参数值。我们引入一种程序，将常微分方程参数化系统的参数空间划分为系统具有唯一或多个平衡点的区域。该程序基于布劳威尔度的计算，并创建一个具有参数依赖系数的多元多项式。系数的符号决定了具有和不具有多稳态的参数区域。该程序的一个特别优势是避免了数值分析和参数采样。该程序由多个步骤组成。这些步骤中的每一个都可以使用各种计算机程序和可用软件通过算法解决，也可以手动解决。我们在几个基因转录和细胞信号模型上展示了我们的程序，并表明在许多情况下，我们获得了关于多稳态的参数空间的完整划分。
