Advantages of external periodic events to the evolution of biochemical oscillatory reactions

Abstract

We compare, by calculations on a simple model of glycolysis, the evolutionary development of oscillatory reaction mechanisms in the presence and absence of external periodic events, such as an oscillatory or constant influx of glucose in an open reaction system. The chosen model has autonomous oscillations for given choices of the parameters of the feedback loops responsible for the oscillations, and for a given range of the total adenylate pool concentration. We change first one, then two of the parameters, so that there are no autonomous oscillations, and then vary these parameters with a genetic algorithm method in which the parameters are represented by binary strings that evolve by selection, crossover, and mutations; the optimization goal is the attainment of a high ATP/ADP concentration ratio in the system. This goal is taken to provide evolutionary advantages and is shown to be achieved more quickly in the presence of external periodic events, rather than constant influx of glucose. The results suggest the possibility of the enhanced evolutionary development of oscillatory biological reactions at shores where waves impinge on rocks and bring nutrients periodically. Measurements have shown that animals and plants grow more rapidly in the presence of such wave action than in its absence.

Fig. 1.

A simple model of the reaction mechanism of glycolysis (see more details in ref. 5). In the EBR, there are four dissociation constants for ligands bound to the R and T conformations () for PFKase (step 2) and PKase (step 4), respectively.

Fig. 2.

Bifurcation diagram for the adenylate pool at 41.3 with α₂ = 0.075. The real part of the temporal eigenvalues of the linearized kinetic equations is plotted vs. the parameter α₁. The bifurcation occurs after α₁ = 0.334.

Fig. 3.

Plot of the ATP/ADP ratio (see definition under One Variable in Results of Computations) vs. the number of generations in the GA (a), α₁ for the oscillatory influx of glucose with the amplitude 0.5 mM/min (deterministic case) (b), and the number of generations for the constant influx of glucose (c). In a and c, the highest value of the ATP/ADP ratio among 24 individuals at each generation by the GA is plotted and shows that systems with the constant influx of glucose take about double the generations necessary to reach the autonomous oscillation than for systems with the oscillatory influx of glucose.

Fig. 4.

The frequency density vs. number of generations to reach the bifurcation. Shown are the oscillatory influx (average 30) (a) and the constant influx (average 59) (b).

Fig. 5.

ATP/ADP ratio vs. the number of generation in the GA necessary to reach the bifurcation for the two-variable case. (a) Bifurcation at the 178th generation for the constant influx. (b) Bifurcation occurs at the 79th generation for the oscillatory influx. The highest value of the ATP/ADP ratio among 24 individuals at each generation is plotted. At the bifurcation, the ATP/ADP ratio is 1.1 for the oscillatory influx, much higher than the value, 1.0004, for the constant flux.

Fig. 6.

The frequency density vs. number of generations for the two-variable case to reach the bifurcation. Shown are the constant influx (average, 167) (a) and the oscillatory influx (average generation, 76) (b).