  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 \textit{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 behavior, 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 work
illustrates how complementary feedback, from classical fields like dynamical
systems to biology and vice versa, can expose fundamental and potentially
overlooked questions.
