论文标题
化学反应网络中特征值零的结构性障碍
Structural obstruction to the simplicity of the eigenvalue zero in chemical reaction networks
论文作者
论文摘要
多立场性是在原本相同条件下表现出两个不同平衡（稳态）的系统的特性，并且它对生化系统具有公认的重要性。由于鞍形节点分叉的结果，多立场性可能会出现在参数空间中，这必然需要在分叉平衡处进行简单的雅各比元素零特征值零。具有简单特征值零的矩阵在一组单数矩阵中是通用的：任何其雅各比式具有代数多特征值零的系统，都可以扰动其雅各布具有简单特征值零的系统。因此，人们期望在应用中，奇异的雅各布人总是以简单的特征值为零。但是，化学反应网络通常考虑固定网络结构，而自由却取决于动力学的各种和不同的选择。在这里，我们提出了一个化学反应网络的示例，该反应网络的Jacobian要么是非词性的，要么具有代数多重特征值零。特征值零的结构性障碍仅基于网络，并且与浓度的价值和动力学的选择无关。特别是这构成了标准鞍节分叉的阻塞。
Multistationarity is the property of a system to exhibit two distinct equilibria (steady-states) under otherwise identical conditions, and it is a phenomenon of recognized importance for biochemical systems. Multistationarity may appear in the parameter space as a consequence of saddle-node bifurcations, which necessarily require a simple eigenvalue zero of the Jacobian, at the bifurcating equilibrium. Matrices with a simple eigenvalue zero are generic in the set of singular matrices: any system whose Jacobian has an algebraically multiple eigenvalue zero can be perturbed to a system whose Jacobian has a simple eigenvalue zero. Thus, one would expect that in applications singular Jacobians are always with a simple eigenvalue zero. However, chemical reaction networks typically consider a fixed network structure, while the freedom rests with the various and different choices of kinetics. Here we present an example of a chemical reaction network, whose Jacobian is either nonsingular or has an algebraically multiple eigenvalue zero. The structural obstruction to the simplicity of the eigenvalue zero is based on the network alone, and it is independent of the value of concentrations and the choice of kinetics. This in particular constitutes an obstruction to standard saddle-node bifurcations.
