89.

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		2.4.5— Example of Bifurcation Diagrams: Theory and Experiments of Glycolysis

		Markus et al. used a bifurcation diagram to compare their theoretical predictions with their experimental results. Each value on the horizontal axis corresponds to measurements at one value of the frequency of the input sugar flow. Along the vertical line above each point is plotted the ATP concentration measured at the same time during each cycle of the input sugarflow.

		When there is only one point vertically over one frequency on the horizontal axis, it means that there was the same value of the ATP concentration at the same time in the input sugar cycle, and thus the variation of ATP concentration was periodic. When there are two points at a given frequency, it means that at one input sugar cycle the ATP concentration had one value, and that at the next input sugar cycle it had a different value. These same values then repeated on alternating input sugar flow cycles. When this was the case, then the ATP concentration took twice as long to repeat as the period of the input sugar flow cycle. This change in behavior is called a period doubling bifurcation. When the ATP concentration is spread along a vertical line, it means that there was a different value of the ATP concentration at each input sugar cycle, and thus the variation was chaotic.

		The bifurcation plot predicted by the equations that model glycolysis is complex. At low frequencies, the period of the ATP concentration matches that of the input sugar flow. As the frequency of the input sugar flow increases, there are period doubling bifurcations, then a region of chaos, then a period 5 times as long as that of the input sugar flow cycle, then a second region of chaos, and finally a period 3 times as long as that of the input sugar cycle.

		The experiments found the same sequence of bifurcations predicted by the equations. The locations of the bifurcations are somewhat shifted in frequency. Nonetheless, even the fine details of the order of the bifurcations in the experimental results match the theoretical predictions. This match tells us that this system is deterministic. The fluctuations present at certain frequencies are chaotic and are not generated by a random mechanism.

 

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