Re-Examination of the Sel’kov Model of Glycolysis and Its Symmetry-Breaking Instability Due to the Impact of Diffusion with Implications for Cancer Imitation Caused by the Warburg Effect

We revisit the seminal model of glycolysis first proposed by Sel’kov more than fifty years ago. We investigate the onset of instabilities in biological systems described by the Sel’kov model in order to determine the conditions of the model parameters that lead to bifurcations. We analyze the glycolysis reaction under the circumstances when the diffusivity of both ATP and ADP reactants are taken into account. We estimate the critical value of the model’s single compact dimensionless parameter, which is responsible for the onset of reaction instability and the system’s symmetry breaking. It appears that it leads to spatial inhomogeneities of reactants, leading to the formation of standing waves instead of a homogeneous distribution of ATP molecules. The consequences of this model and its results are discussed in the context of the Warburg effect, which signifies a transition from oxidative phosphorylation to glycolysis that is correlated with the initiation and progression of cancer. Our analysis may lead to the selection of therapeutic interventions in order to prevent the symmetry-breaking phenomenon described in our work.