Turing Instabilities and Rotating Spiral Waves in Glycolytic Processes

Abstract

We study single-frequency oscillations and pattern formation in the glycolytic process modeled by a reduction in the well-known Sel’kov’s equations (Sel’kov in Eur J Biochem 4:79, 1968), which describe, in the whole cell, the phosphofructokinase enzyme reaction. By using averaging theory, we establish the existence conditions for limit cycles and their limiting average radius in the kinetic reaction equations. We analytically establish conditions on the model parameters for the appearance of unstable nonlinear modes seeding the formation of two-dimensional patterns in the form of classical spots and stripes. We also establish the existence of a Hopf bifurcation, which characterizes the reaction dynamics, producing glycolytic rotating spiral waves. We numerically establish parameter regions for the existence of these spiral waves and address their linear stability. We show that as the model tends toward a suppression of the relative source rate, the spiral wave solution loses stability. All our findings are validated by full numerical simulations of the model equations. Finally, we discuss in vitro evidence of spatiotemporal activity patterns found in glycolytic experiments, and propose plausible biological implications of our model results.