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This article utilizes perturbation method (PM) to find an analytical approximate solution for the Quasi-Steady-State Michaelis-Menten problem. From the comparison of Figures and absolute error values, between approximate and numerical solutions, it is shown that the obtained solutions are accurate, and therefore, they explain the general behaviour of the Michaelis-Menten mechanism.

The Michaelis-Menten mechanism is employed to model the kinetics of enzyme-catalyzed reactions in cases where the concentration of substrate is greater than the concentration of enzyme. The importance of such reactions relies on the fact that the majority of cell processes require enzymes to get a significant rate [

Enzymes are protein molecules that act as catalysts with the purpose to accelerate chemical reactions in living organisms. These enzymes work on specific molecules denominated substrates. Without the existence of enzymes, most chemical reactions that keep living organisms alive would be slow to sustain life [

Thus, haemoglobin in red blood cells is an enzyme, while the oxygen is the substrate with which it combines. Other example would be the case of oxidation of glucose; this process provides water, carbon dioxide, and energy; nevertheless, it is extremely slow when it is carried out exposed just to open air and there is no appreciable oxidation after years of exposure [

Enzymes play a significant role in the regulation of biological processes; they work as activators or inhibitors in a reaction. To comprehend their role in the rate of reactions, the study of enzyme kinetics is necessary; its study consists fundamentally in understanding the temporary behaviour of the several reactions and the conditions that influence them [

The classical perturbation method (PM) is a pioneer technique to solve some cases of nonlinear differential equations. PM was proposed by S. D. Poisson and later extended by J. H. Poincare. Even though PM appeared in the early 19th century, the application of this method to analyse nonlinear problems was not performed until later on that century. The most important application of this method was in the subject of fluid mechanics, celestial mechanics, and aerodynamics [

In general, we will suppose that a differential equation can be expressed as the sum of two parts, one linear and one nonlinear. The nonlinear part is considered as a small perturbation through a small perturbation parameter. As a matter of fact, this last assumption is considered as a disadvantage of PM. As a consequence, other methods have emerged to study the problem of finding approximate solutions for some nonlinear problems such as variational approaches [

Although PM provides, generally, better results for small perturbation parameters

The rest of this work is presented as follows. Section

Let the differential equation of one dimensional nonlinear system be in the form

where we assume that

Assuming that the nonlinear term in (

Then, by substituting (

One of the most important enzyme reactions was proposed by Leonor Michaelis and Maud Leonora Menten [

The complex SE undergoes unimolecular decomposition to form irreversibly a product P and the enzyme E. This autocatalytic reaction is expressed as

and this equation says that one molecule of

Next, we will apply to (

For the sake of simplicity, we will use the same symbols to denote the concentrations; thus

where

The law of mass action applied to (

where the dot denotes differentiation with respect to

The initial values of concentrations are

The

In order to show that system (

and after integrating (

or

The above equation denotes a conservation law for

By substituting (

We note that the last equation of (

Equation (

Next, we will obtain a dimensionless form of (

After substituting (

where

Although (

Substituting (

Equations (

Noting that

Nevertheless, it results that there exists a more general condition for which QSSA is valid. This condition is expressed in terms of Michaelis constant

From this condition, we see that, even for a case, such that

An alternative way to express (

Thus, the condition that

Despite its simple appearance, the nonlinear equation (

From the above it is clear that the mathematical description of QSSA is just numerical until this point.

Next, we will show that PM provides an efficient tool in order to get a good approximation for QSSA problem.

We will see that despite the approximated origin of (

In order to obtain an approximate solution to (

We begin rewriting (

where we have defined

Although this work assumes small

Employing the Newton’s binomial formula, (

where, for the sake of simplicity, we kept only the first order power of

Identifying

After substituting (

Thus, after solving iteratively (

By substituting (

The corresponding value of

Next, we consider, as a case study, the value of the parameters

From Figure

Comparison between numerical solutions and approximations (

For

For

After directly adding the two equations (

unlike (

Since the proposed solution (

Next, we will get approximate solutions for

(where we have taken into account the perturbative character of

The unknown function,

The solutions for the first set of parameters (

and for the second set

The aim of this study was to explore the possibility of finding analytical approximate solutions for the Michaelis-Menten enzyme kinetics problem. The study of this problem is relevant, because it involves enzyme reactions, and the most of the cell processes require enzymes to obtain a significant rate. Because the nonlinear system (

In order to test the potentiality of PM to describe QSSA, we considered as case study the values of the parameters

Although (

Figures

Comparison between numerical solutions and approximations (

For

For

Comparison between numerical solutions and approximations (

For

For

Although it is clear from the figures mentioned that the proposed approximate solutions have good precision, it is necessary to provide a more analytical criterion to ensure the accuracy of our solutions. For this purpose, Figures

Absolute error for solutions (

For

For

Absolute error for solutions (

For

For

A relevant fact to mention is that although the first case (

Equations (

A brief qualitative description based on [

Besides, from conservation law (

From all the above, it is highlighted that the first order ordinary differential equations system, composed by (

Finally we note that the loss of information assumed, by using (

The main contribution of this work was the proposal to employ the pair of differential equations (

All data generated or analysed during this study is included within this research article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to thank Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution to this project.