Hochschule für Technik und Wirtschaft, Berlin, Germany.
Joint Research Center for Computational Biomedicine, Aachen, Germany.
Bull Math Biol. 2019 Oct;81(10):4174-4209. doi: 10.1007/s11538-019-00639-4. Epub 2019 Jul 22.
We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. A particular focus is on the case of reaction networks whose steady states admit a monomial parametrization. For such systems, we show that in the space of total concentrations multistationarity is scale invariant: If there is multistationarity for some value of the total concentrations, then there is multistationarity on the entire ray containing this value (possibly for different rate constants)-and vice versa. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semi-algebraic conditions that involve only concentration variables. These conditions can easily be extended to include total concentrations. Hence, quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the total concentration of the kinase or the total concentration of the phosphatase. This result is enabled by the chamber decomposition of the space of total concentrations from polyhedral geometry. Together with the corresponding sufficiency result of Bihan et al., this yields a characterization of multistationarity up to lower-dimensional regions.
我们将实代数几何的工具应用于化学反应网络的多稳定性问题。特别关注的是那些稳态可以进行单项式参数化的反应网络。对于这样的系统，我们证明了在总浓度空间中多稳定性是尺度不变的：如果在某个总浓度值存在多稳定性，那么在包含该值的整个射线（可能具有不同的速率常数）上存在多稳定性-反之亦然。此外，对于这些网络，可以通过制定仅涉及浓度变量的半代数条件，在不依赖于速率常数的情况下决定多稳定性。这些条件可以很容易地扩展到包含总浓度。因此，量词消去法可能会为总浓度空间中的多稳定性区域提供新的见解。为了证明这一点，我们表明，对于在两个结合位点上的蛋白质的分布式磷酸化，如果底物的总浓度大于激酶或磷酸酶的总浓度，那么多稳定性才是可能的。这个结果是通过来自多面体几何的总浓度空间的腔分解实现的。与 Bihan 等人的相应充分性结果一起，这给出了多稳定性的特征，直到更低维的区域。
