Transient behaviour towards the stable limit cycle in the Selkov model of Glycolysis: A physiological disorder

TL;DR

This paper investigates how long it takes for the Selkov glycolysis model to return to its stable limit cycle after deviation, revealing saturation effects that may relate to physiological disorders.

Contribution

It provides a detailed analysis of convergence times in the Selkov model and proposes a mathematical modification to incorporate physiological disorder effects.

Findings

Convergence time saturates with distance from the limit cycle.

Deviations from the cycle can be linked to physiological disorders.

Proposed model modification accounts for saturation behavior.

Abstract

A simplified model for the complex glycolytic process was historically proposed by Selkov. It showed the existence of stable limit cycle as an example of Poincare-Bendixson theorem. This limit cycle is nothing but the time eliminated Lissajous plot of the concentrations of Adenosine-diphosphate (ADP) and Fructose-6-phosphate (F6P) of a normal/healthy human. Deviation from this limit cycle is equivalent to the deviation of normal physiological behaviour. It is very important to know how long a human body will take to reach the glycolytic stable limit cycle, if deviated from it. However, till now the convergence time, depending upon different initial parameter values, was not studied in detail. This may have great importance in understanding the recovery time for a diseased individual deviated from normal cycle. Here the convergence time for different initial conditions has been calculated in original Selkov model. It is observed that convergence time, as a function of the distance from the limit cycle, gets saturated away from the cycle. This result seems to be a physiological disorder. A possible mathematical way to incorporate this in the Selkov model, has been proposed.