Univalent positive polynomial maps and the equilibrium state of chemical networks of reversible binding reactions
Univalent positive polynomial maps and the equilibrium state of chemical networks of reversible binding reactions
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DOI：
10.1016/j.aam.2009.05.001
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发表时间：
2009-10-01
影响因子：
1.1
通讯作者：
Gnacadja, Gilles
中科院分区：
文献类型：
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作者：
Gnacadja, Gilles
我们考虑一个映射f =（f（1），.，f（n））：R->= 0（n）-> R->= 0（n）对于x =（x（1），.，x（n））by f（i）（x）= x（i）psi（i）（x（i））+ Sigma（alpha是I的元素）alpha（i）a（alpha）x（alpha）其中I是Z（>= 0）（n）的有限子集，a（alpha）是R中的常数，对于每个alpha是I的元素，并且psi（1），.，psi（n）是可微保序函数R->= 0 -> R->0。我们证明了f是一个双射。满射性是布劳威尔不动点定理的结果。对于内射性，我们证明了f的雅可比矩阵处处是P-矩阵，然后我们应用Gale-Nikaido全局单叶性定理。psi（1）= ... = psi（n）= 1，f是可逆结合反应的化学网络研究中感兴趣的正多项式映射。对于这些，我们提出了基本和复合物种和正常和完整的网络的概念。药理学和其他领域的许多网络都属于这些类别。我们证明了它们的平衡态和详细平衡态是一致的，是唯一的关于基本物种的总浓度。映射f产生了一个方程，该方程具有唯一的解，该解给出了平衡态。我们还证明了浓度总是收敛到平衡状态，从而解决了完整的网络的全局吸引子猜想，这肯定了这一性质的更大类的复杂平衡网络。(C)2009 Elsevier Inc. All rights reserved.
We consider a map f = (f(1),...,f(n)) : R->= 0(n) -> R->= 0(n) given for x = (x(1),...,x(n)) byf(i)(x) = x(i) psi(i) (x(i)) + Sigma(alpha is an element of I)alpha(i)a(alpha)x(alpha)where I is a finite subset of Z(>= 0)(n), a(alpha) is a constant in R->= 0 for each alpha is an element of I, and psi(1),...,psi(n) are differentiable order-preserving functions R->= 0 -> R->0. We prove that f is a bijection. Surjectivity arises as a consequence of the Brouwer Fixed-Point Theorem. For injectivity, we show that the Jacobian matrix of f is everywhere a P-matrix and we then apply the Gale-Nikaido Global Univalence Theorem. With psi(1) = ... = psi(n) = 1, f is a positive polynomial map of interest in the study of chemical networks of reversible binding reactions. For these, we propose notions of elementary and composite species and of normal and complete networks. Many networks in pharmacology and other fields fall in these classes. We prove that their equilibrium states and detailed-balanced states coincide and are unique with respect to total concentrations of elementary species. The map f gives rise to an equation that has a unique solution which gives the equilibrium state. We also prove that concentrations always converge to the equilibrium state, thereby settling for complete networks the Global Attractor Conjecture, which affirms this property for the larger class of complex-balancing networks. (C) 2009 Elsevier Inc. All rights reserved.