Mathematical Modeling and Dynamics of Cooperative Bi‐Enzymatic Chemical Reaction in Response to pH and Thermal Factors

The structure of a cell is well organized whereby enzymes catalyze complicated metabolic reactions. Considering the inherent complexity of these reactions, this paper has proposed a mathematical model of a bi‐enzymatic cooperative chemical reaction. The key aim of this study is to examine how the pH and temperature control the model dynamics and stability through mathematical analysis. They are combined into a system of nonlinear coupled differential equations, whose invariant region, non‐negativity are proved. The chemical equilibrium and velocity of the reaction is calculated through which the approximate stationary concentration of substrate is derived. Lyapunov functions and eigenvalues are used to illustrate the model's stability and boundedness. By using the derived expression for the steady‐state concentration of , a biochemical application associated with the formation of cyclic adenosine monophosphate (cAMP) and mononucleotide (MN) is investigated within the proposed model. In particular, the stationary concentration of cAMP is analyzed. The impact of pH and temperature on the stability of the system is examined by incorporating the Henderson–Hasselbalch relation and the Arrhenius law. The analytical results demonstrate that the system exhibits stable behavior when the pH lies in the interval and the temperature range is . Additionally, the analysis of ionic strength, allosteric regulation of the model with cooperativity, and the variations in rate constants , are also investigated by varying the pH and temperature. For different pH and temperatures the velocity of reaction against the concentration of is also examined, which shows that by making a small change in the pH or temperature, the velocity changes rapidly. The system is solved numerically by using Adams Bash‐forth predictor corrector method and at the end all the discussed results are displayed in graphs, tables along with their respective eigenvalues, that show best matches with the derived results. This research help as an excellent foundation to the control of enzymes under different conditions. These results do not only improve our knowledge on the complex pH‐temperature relationship in enzyme reactions, but also provide close guidelines to the maximization of the reaction conditions under different commercial and biotechnological applications.