Enzyme reactions have numerous applications in diverse disciplines of science like chemistry, biology and biomechanics. In this study, we examine the role and act of enzymes in chemical reactions which is considered in the frame of fractional order model. The proposed model includes system of four equations which are studied via Caputo fractional operator. The systems of non-linear equations are evaluated by a semi-analytical approach called \(q\) -homotopy analysis transform method. The uniqueness and existence of the solutions has been investigated through fixed point theorem. The solutions of the proposed model are achieved through the considered method and the obtained outcomes are in the form of series which shows rapid convergence. The solutions are computed and graphs are plotted for the obtained results using mathematica software. The achieved results by the proposed method are unique and illustrate the significant dynamics of the considered model via 3D plots and graphs. The results of this study demonstrate the importance and effectiveness of projected derivative and technique in the analysis of time dependent fractional mathematical models. This study also gives an idea to extend the applications of enzymatic reactions in drug development, bio mechanics, and chemical reactions in various cellular metabolisms. Also, enzymatic reactions have a vital role in the fields of the food industry for processing food, in biotechnology for the manufacture of biofuels, and in metabolic engineering to design metabolic pathways.
