Department of Biomedical Engineering, Boston University, Boston, MA 02215, USA.
J R Soc Interface. 2013 Apr 3;10(83):20130087. doi: 10.1098/rsif.2013.0087. Print 2013 Jun 6.
Efforts to catalogue the structure of metabolic networks have generated highly detailed, genome-scale atlases of biochemical reactions in the cell. Unfortunately, these atlases fall short of capturing the kinetic details of metabolic reactions, instead offering only topological information from which to make predictions. As a result, studies frequently consider the extent to which the topological structure of a metabolic network determines its dynamic behaviour, irrespective of kinetic details. Here, we study a class of metabolic networks known as non-autocatalytic metabolic cycles, and analytically prove an open conjecture regarding the stability of their steady states. Importantly, our results are invariant to the choice of kinetic parameters, rate laws, equilibrium fluxes and metabolite concentrations. Unexpectedly, our proof exposes an elementary but apparently open problem of locating the roots of a sum of two polynomials S = P + Q, when the roots of the summand polynomials P and Q are known. We derive two new results named the Stubborn Roots Theorems, which provide sufficient conditions under which the roots of S remain qualitatively identical to the roots of P. Our study illustrates how complementary feedback, from classical fields such as dynamical systems to biology and vice versa, can expose fundamental and potentially overlooked questions.
为了对代谢网络的结构进行编目，人们已经生成了高度详细的、基于基因组规模的细胞内生化反应图谱。然而，这些图谱未能捕捉到代谢反应的动力学细节，而只是提供了拓扑信息，以供进行预测。因此，研究人员经常考虑代谢网络的拓扑结构在多大程度上决定其动态行为，而不考虑动力学细节。在这里，我们研究了一类被称为非自催化代谢循环的代谢网络，并对其稳定态的稳定性这一开放性猜想进行了分析证明。重要的是，我们的结果不受动力学参数、速率定律、平衡通量和代谢物浓度选择的影响。出乎意料的是，我们的证明揭示了一个基本但显然尚未解决的问题，即在已知加项多项式 P 和 Q 的根的情况下，如何找到它们和的根 S = P + Q 的根。我们推导出了两个名为 Stubborn Roots Theorems 的新结果，它们提供了 S 的根与 P 的根在定性上保持相同的充分条件。我们的研究说明了来自诸如动力系统等经典领域与生物学之间的互补反馈如何能够揭示基本且潜在被忽视的问题。
