Multistationarity and Hopf-bifurcations in families of ODE models of N-site phosphorylation

N 位磷酸化 ODE 模型族中的多平稳性和 Hopf 分岔

• 批准号: 517274113
• 负责人: Professor Dr.-Ing. Carsten Conradi
• 金额: --
• 依托单位: Studiengang Life Science Engineering
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/517274113?language=en
• 关键词: Multistationarity; Hopf; bifurcations; families; ODE

Phosphorylation is a biochemical mechanism underlying important biological processes. Biochemically a protein is altered by adding (or removing) a phosphate group at a designated binding site. Proteins often have more than one binding site. If spatial effects may be neglected, these models come in the form of systems of Ordinary Differential Equations with polynomial right-hand side. Mathematically one can distinguish three families of models parameterized by N, the number of binding sites. Measuring individual concentration variables is prohibitively expensive and technically challenging. Hence parameter values can only be given with large error bounds, if at all. In studying phosphorylation networks therefore, the following mathematical question arises naturally: are there (positive) parameter values, such that the ODEs admit a solution with property xyz - for some or all (positive) initial conditions. Here we will focus on the following three properties (i) uniqueness of steady states, (ii) existence of multiple steady states and (iii) existence of Hopf bifurcations. These are mathematically interesting and challenging and, at the same time, of interest in systems biology and medicine. In essence, we propose to establish (i), (ii) or (iii) in three families of ODEs with polynomial right-hand side and unknown but positive parameters and initial conditions.

磷酸化是一种重要的生物过程背后的生化机制。在生物化学上，蛋白质通过在指定的结合位点添加（或去除）磷酸基团而改变。蛋白质通常有不止一个结合位点。如果可以忽略空间效应，这些模型以右手边为多项式的常微分方程系统的形式出现。从数学上讲，我们可以区分三种以结合位点数目N为参数的模型族。测量单个浓度变量非常昂贵，而且在技术上具有挑战性。因此，参数值只能在较大的误差范围内给出，如果有的话。因此，在研究磷酸化网络时，自然会出现以下数学问题：是否存在（正）参数值，使得ode在某些或全部（正）初始条件下承认具有性质xyz -的解？这里我们将重点讨论以下三个性质(i)稳态的唯一性，（ii）多个稳态的存在性和（iii） Hopf分岔的存在性。这些在数学上是有趣的，具有挑战性的，同时，对系统生物学和医学也很感兴趣。本质上，我们提出在三类方程中建立(i)、（ii）或（iii），这些方程的右侧为多项式，参数和初始条件未知但为正。