
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39256517
71937
10.1038/s41598-024-71937-8
Article
High performance computational approach to study model describing reversible two-step enzymatic reaction with time fractional derivative
Chethan H. B. 1
Turki Nasser Bin 2
Prakasha D. G. prakashadg@gmail.com
prakashadg@davangereuniversity.ac.in

1
1 https://ror.org/05w9k9t67 grid.449028.3 0000 0004 1773 8378 Department of Mathematics, Davangere University, Shivagangotri, Davangere, 577 007 India
2 https://ror.org/02f81g417 grid.56302.32 0000 0004 1773 5396 Department of Mathematics, College of Science, King Saud University, P.O.Box-2455, 11451 Riyadh, Saudi Arabia
10 9 2024
10 9 2024
2024
14 2111416 7 2024
2 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
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.

Keywords

Fractional derivative and integrals
Enzymatic reaction
q-homotopy analysis transform method
Fixed point theory
Subject terms

Chemical biology
Chemistry
Mathematics and computing
http://dx.doi.org/10.13039/501100002383 King Saud University RSP2024R413 Turki Nasser Bin issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Chemical kinetics is a discipline of chemistry that focuses on studying chemical reactions. Numerous chemical reactions inside the cells occur due to proteins called enzymes. Enzymes are the proteins which are obtainable in nature and act as catalysts in various chemical reactions. They shrink the activation energy by transforming substrates into products and speed up the reaction rate and also toughen the chemical bonds between the molecules. Enzymes are extremely specific as they catalyze only small amount of substrate reaction and accelerate the reactions, biological processes by 10 million times. Hence the enzyme-catalyzed reactions, enzyme kinetics are studied. The study of enzyme –catalyzed reaction rate is called enzyme kinetics which involves statistical formulation, coefficients in the rate of reaction and quantification1,2. Enzyme kinetics has significance in determining the rate of reaction and examines the consequence of change in reaction. This type of kinetics research provides insight into how an enzyme works as a catalyst and how it contributes to metabolism. About 100 years ago Michaelis and Menten proposed the basic fundamental equation of enzyme kinetics3 as it indicates substantial development in the depiction of enzymes. Further, Cha4 examined the true velocities and errors of the total substrate and total enzyme concentration. Urban et al.5 reported a systematic coverage of examples on enzymatic micro reactors. Gan et al.6 conducted an experimental study and mathematical analysis to determine how the changing dynamics and nature of interactions between enzymes, cellulose substrate, products (reducing sugars), and non-hydrolytic materials in a heterogeneous solid–liquid reaction system influence reaction kinetics of enzymatic cellulose hydrolysis. Kinetic and thermodynamic investigations7 show that the trait of greatly increased catalytic activity with colloidal stability is due to an effective method of modifying enzyme–substrate association by altering rate constants in the presence of functionalized gold nanoparticles. Numerous mathematical methods have been used to study the dynamics of the enzyme kinetics model; these methods are cited in8–10 and the references therein.

Fractional calculus (FC) is a tool to study derivatives and integrals of rational order. As we know, classical calculus has been developed as a vast subject, and many researchers have been working on it till now. Due to the ideas of German mathematician Leibniz and L-Hospital, the theory of fractional calculus came into existence about 300 years ago. Fractional calculus can be assumed as a well-developed and established subject. The main advantage of fractional order is it gives the solution in between the intervals which helps us to analyse the results more understandably. Nowadays their exist some modified fractional operators like Riemann–Liouville (R.L) integral, Caputo fractional derivative, Caputo-Fabrizio, Hilfer derivative, Atangana-Baleanu derivative and other derivative. In this study we utilize Caputo derivative which has the significance like it is non-local in behaviour, well-suited for initial value problems, it is bounded and it also gives smoother behaviour compared to other fractional operators. These behaviours of Caputo operator makes easier to model systems having initial conditions11–14. The differential equations with rational order are called fractional differential equations (FDEs). The benefits of FDEs are they give memory effects, hereditary property and also possess uncertainty property. Hence FDEs are more utilized in modelling the natural and complex phenomena in real world like electrodynamics15, signal processing16, fluid dynamics17, chaos behaviour18, financial models19, optics20, a noisy environment21, human diseases22, image processing23 and many others24–27. Finding the solution to non-linear fractional differential equations is not easy and needs some computational work, hence many researchers have applied diverse methods to solve FDEs like variational iteration method28,29, differential transform method30,31, residual power series method32, homotopy perturbation method33,34 and many analytical, numerical methods35–47.

In this article we apply a novel semi-analytical technique called q-homotopy analysis transform method (q-HATM). This method is a mixture of HAM and the Laplace transform (LT), which provides the solution to the fractional model. The q-HATM is a unique technique which is highly effective in resolving nonlinear partial differential equations. Unlike Liao's homotopy analysis approach, which was introduced in 199248, q-HATM incorporates the Laplace transform into the homotopy analysis approach (HAM)49,50 boosting its capacity to handle fractional-order model’s non-locality and nonlinearity. Moreover, q-HATM differs from q-HAM51 by proposing a more general structure that contains an auxiliary parameter, allowing for a wider convergence region and faster series solution. This creates q-HATM a dominant tool for researchers dealing with complex differential equations in many areas of science and engineering. The advantage of proposed method is it requires less computational work which gives series solution and also the solutions converge rapidly. It is well suited to solve non-linear differential equations because it discretize the non-linear terms in equations. The applications of considered method are cited in52–54. The novelty of this article is we applied Caputo fractional derivative to the model which is time-dependent and the solutions are obtained in form of series using q-HATM and graphs are plotted through the solutions demonstrates the dynamics of the model. The q-HATM has a significant effect in the development of enzymatic reactions. The methods importance is it gives rapid convergence and flexibility, which allows for the effective investigation of non-linear differential equations similar to such biological systems. This novel technique improves modeling precision and offers deeper understandings into enzyme kinetic characteristics, which is critical for the development of new medications and therapies. Compared to previous studies in literature our investigation demonstrated more precise dynamics of the model and the solutions are very effective in understanding the behavior of the model which will be useful in further studies on this considered model. This article also describes the uniqueness, existence and convergence of the systems solution.

Preliminaries

The basic definitions of fractional-order operator and Laplace transform are presented here11–13.

Definition.2.1.

For a function f(t)∈C-1n, the fractional integral of order α≥0 in terms of Riemann–Liouville is specified as.1 0Jtαft=1Γα∫0tt-ϑα-1fϑdϑ,

J0ft=ft.

where Γ represents the Gamma function.

Definition.2.2.

The fractional derivative for a function f∈C-1n is defined as follows in terms of Caputo:2 Dtαft=dnftdtn,α=n∈N,1Γn-α∫0tt-ϑn-α-1fnϑdϑ,α∈n-1,n,n∈N.

Linear property of Caputo fractional derivative isDcα(λx(t)+μy(t))=λDcαx(t)+μDcαy(t),

where λ and μ are some constants.

Definition.2.3.

A Caputo fractional derivative's Laplace transform (LT) for the function Dtαf(t) is stated as follow.3 LDtαft=sαFs-∑r=0n-1sα-r-1fr0+,n-1<α≤n,

where F(s) symbolizes the LT of the function f(t).

Formulation of the reversible two-step enzymatic reaction model of fractional order

We investigate the two-step procedure10 in which an input S is transformed into an output P by an enzyme F. Due to the amalgamation of F and S with constant positive rate β a complex B is produced. Then with a positive rate ω, F is manufactured due to the reduction of B to P and S is also degraded to F with positive rate λ. At a rate of reverse reaction ψ, B is produced by the elements formed by the breakdown of F and P.

Figure 1A provides a schematic representation of the reaction plan. Yet the opposite reaction is avoided when the product P is continuously removed during time t (minutes). Thus, it is common to believe that the rate of reversal of a reaction is zero. As a result, Fig. 1B displays the reaction’s characteristic shape.Fig. 1 Two-step reversible enzymatic reaction. (A) With reverse response. (B) Without reverse response.

The system of equations of chemical reactions10 with initial conditions expressed as:dSdt=λB-βSF,

4 dFdt=λ+ωB-βSF,

dBdt=βSF-λ+ωB,

dPdt=ωB,

with S0=S0, F0=F0, B0=B0, P0=P0.

The proposed model is considered in fractional order via Caputo fractional derivative:0cDtαSt=λB-βSF,

5 0cDtαFt=λ+ωB-βSF,

0cDtαBt=βSF-λ+ωB,

0cDtαPt=ωB,

where S0=S0≥0, F0=F0≥0, B0=B0≥0, P0=P0≥0.

Khan et al.10 defined that the speed of reaction of the considered process as:6 V=Se0ωβSβ+λ.

When the substrate complex and enzyme has formed with reaction speed vmax=e0, the above equation modified as7 V=SvmaxωβSβ+λ

Fundamental procedure for suggested method

In this part of the paper, we illustrate the solution algorithm of the q-HATM39–41. Consider a general non-linear and non-homogeneous fractional-order differential equation which is in the form given below8 DtαUx,t+RUx,t+NUx,t=fx,t,n-1<α≤n

where DtαUx,t indicates the Caputo fractional derivative of the function Ux,t,fx,t is the source terms, the nonlinear differential operator is denoted by N and R denotes the linear differential operator which is bounded in x and t.

Now, by implementing the LT on Eq. (8), we obtain9 sαLUx,t-∑k=0n-1sα-k-1Ukx,0+LRUx,t+LNUx,t=Lfx,t

On simplifying Eq. (9), we get10 LUx,t-1sα∑k=0n-1sα-k-1Ukx,0+1sαLRUx,t+LNUx,t-Lfx,t=0

In accordance with the HAM, the non-linear operator N is demonstrated as11 Nφx,t;q=Lφx,t;q-1sα∑k=0n-1sα-k-1φkx,t;q0++

1sαLRφx,t;q+LNφx,t;q-Lfx,t,

where q∈0,1nn≥1, and φ(x,t;q) specifies the real function with respect to x,t and q.

The deformation equation of zero’th-order involving H(x,t) is characterized as follows:12 1-nqLφx,t;q-U0x,t=ħqHx,tNφx,t;q,

where φx,t;q is an anonymous function and U0x,t is an initial guess of Ux,t, q∈0,1n refers to the embedding parameter, L represents the LT,ℏ≠0 is an auxiliary parameter. The following outcomes are respectively satisfied for q=0 and q=1n,13 φx,t;0=U0x,t,φx,t;1n=Ux,t.

Hence, the solution φ(x,t;q) converge from U0x,t to the solution Ux,t by varying q between 0 and 1n. Subsequently, the function φζ,t;q in series form is extended as follows by employing Taylor theorem throughout q

where14 φx,t;q=U0x,t+∑m=1∞Umx,tqm

15 Umx,t=1m!∂mφ(x,t;q)∂qm|q=0.

By a specific choice of auxiliary parameter ℏ and n, auxiliary linear operator, the initial estimation U0x,t and Hx,tappropriately, the series (14) converges at q=1n, which provides the solution to the original non-linear Eq. (8) of the type16 Ux,t=U0x,t+∑m=1∞Umx,t1nm

The next step is to differentiate the deformation Eq. (12) up to m-times with regard to q, then divide by m! and ultimately taking q=0, we acquire the deformation equation of order m as17 L[Umx,t-KmUm-1x,t]=ℏH(x,t)RmU→m-1

where18 Km=0,m≤1,n,m>1,

and19 RmU→m-1=1m-1!∂m-1N[φx,t;q]∂qm-1|q=0,

and the vectors are considered as20 U→m=U0x,t,U1x,t,⋯,Umx,t.

By performing inverse Laplace transform to the deformation Eq. (17), which yields the recursive equation and can be expressed as follows21 Umx,t=KmUm-1x,t+ℏL-1H(x,t)RmU→m-1.

Finally, resolving Eq. (21), we attain the Umx,t iterative terms and Hx,t=1. This is how the q-HATM series solution is described:22 Ux,t=∑m=0∞Umx,t

Analysis of the solution of the model

In this section we analyse the solution of the fractional order model and examined its uniqueness, convergence and existence47.

Existence of the solution

This section provides existence of the solution via fixed point theory.

Definition 5.1.114

Let k,d be a non-empty couchy space and λ,0≤λ<1 then consider mapping S:X→X∋ for every x,x¯∈X then then dSx,Sx¯≤λx,x¯ holds, the x∗∈X is the unique fixed point of S. If Skk∈N the sequence defined bySk=SSk-1L∈N1,S1=S;

Thus for any x0∈X,{Sx0k}k=1k=∞ reaches to the fixed x∗.

Definition 5.1.214

Let m∈N,H⊂Rm,p,q⊂Randh;p,q×H→R be a function.

For any x1,x2,⋯xmx1∗,x2∗,⋯xm∗∈H, it satisfies the generalized Lipschitzian condition.hS,x1,x2,⋯xm-hS,x1∗,x2∗,⋯xm∗≤A1x1-x1∗+A2x2-x2∗+⋯+Amxm-xm∗,

Aj≥0,j=0,1,2,3⋯m

Specifically, h satisfies the Lipschitzian condition.

If ∀ζϵp,qand for anyx,x∗ϵG,hζ,x-hζ,x∗≤Ax-x∗,A>0,

Let us consider the system of equation0cDtαSt=ϕ1x,t,S,

0cDtαFt=ϕ2x,t,F,

0cDtαBt=ϕ3x,t,B,

0cDtαPt=ϕ4x,t,P.

Now using above equation we haveSx,t-Sx,0=0ItαλB-βSF,

Fx,t-Fx,0=0Itαλ+ωB-βSF,

Bx,t-Bx,0=0ItαβSF-λ+ωB,

Px,t-Px,0=0ItαωB.

Then by the definition of Riemann–Liouville fractional integral, we getSx,t-Sx,0=1Γα∫t0t-vα-1ϕ1x,v,Sdv,

Fx,t-Fx,0=1Γα∫t0t-vα-1ϕ2x,v,Fdv,

Bx,t-Bx,0=1Γα∫t0t-vα-1ϕ3x,v,Bdv,

Px,t-Px,0=1Γα∫t0t-vα-1ϕ4x,v,Pdv.

In this section we have provided the existence of solution. By the aid of fixed point theory, Cauchy space and with the help of Lipschitzian condition, Riemann–Liouville fractional integral we have shown the existence of solution for the considered model. Hence we can conclude that their exist unique solution for the proposed time fractional mathematical model.

Convergence of the solution

Consider a mapping H:G→G is non-linear with Banach space G. Let us take‖Hu-H(v)‖≤μi‖u-v‖,∀u,v∈G

Hence their exist fixed point converge to a singular point is H and‖vm-vp‖≤μip1-μi‖v1-v0‖,i=1,2,3,4⋯

Proof

Let cj,‖∙‖ be a Banach space with norm specified as ‖gt‖=maxt∈jg(t) function on J.

Now we verify Sp,Fp,Bp,Pp is a Cauchy sequence in cj,‖·‖

For S consider,‖Sm-Sp‖=maxtϵJSSm-Sp,

=maxtϵJ∫Kp+hSm-1-Sp-1-hL-11SαλBm-1-λBp-1-βSm-1Fm-1-βSp-1Fp-1

≤maxtεJKp+hSm-1-Sp-1-h∫0tλBm-1-λBp-1-βSm-1Fm-1-βSp-1Fp-1t-vαΓ1+αdvBy convolution theorem

≤Kp+hSm-1-Sp-1-h∫t0λδ1+βδ2+δ3t-vαΓ1+αsm-1-Sp-1dv

The above inequality is reduced to‖Sm-Sp‖≤μ1‖Sm-1-Sp-1‖,

where δ1=Bm-1-Bp-1,δ2=Sm-1-Sp-1,δ3=Fm-1-Fp-1

Then takes m=p+1 it yields‖Sp+1-Sp‖≤μ1‖Sp-Sp-1‖≤μ12‖Sp-1-Sp-2‖⋯μ1p‖S1-S0‖.

Using triangular inequality‖Sp-Sp-1‖≤‖Sp+1-Sp‖+‖Sp+2-Sp+1‖+⋯‖Sp-Sp-1‖,

≤μ1p+μ1p-1+μ1p-2+⋯+μ1m-1‖S1-S0‖,

≤μ1p1-μ1m-p-11-μ1‖S1-S0‖.

As 0<μ<1,so1-μ1m-p-1<1, then we have‖Sp+1-Sp‖≤μ1p1-μ1‖S1-S0‖

But ‖S1-S0‖<∞ consequently as m→∞ then ‖Sp+1-Sp‖→0 proves Sp is a Cauchy sequence.

This proves theorem.

Simultaneously we have,‖Fp+1-Fp‖≤μ2p1-μ1‖F1-F0‖,

‖Bp+1-Bp‖≤μ3p1-μ3‖B1-B0‖,

‖Pp+1-Pp‖≤μ4p1-μ4‖P1-P0‖.

Uniqueness of the solution

The solution of considered fractional differential equation via q-HATM is unique,

Whenever 0<μi<1,i=1,2,3,4

Proof

The solution of equation is illustrated as follows

In general,vx,t=∑p=0∞vpx,t.

For i = 1, suppose S,S∗ be two different values ∋S-S∗≤maxt∈JS-S∗,≤Kp+hS-S∗-hL-11SαλB-βSF,

≤Kp+hS-S∗-h∫t0λB-βSFt-vαΓ1+αdv

(By convolution theorem)≤Kp+hS-S∗-h∫t0λS1+S2+S3βt-vαΓ1+αS-S∗dv

The above inequality related toS-S∗≤μSS-S∗

μS=Kp+hS-S∗-h∫t0λδ1+δ2+δ3βt-vαΓ1+αdv,

where δ1=Bm-1-Bp-1,δ2=Sm-1-Sp-1,δ3=Fm-1-Fp-1,

We get1-μ5S-S∗≤0,

S-S∗=0,0<μ<1,

S=S1∗

Similarly,F=F1∗

B=B1∗

P=P1∗

Solution of the considered model using q-HATM

Consider the time fractional enzymatic reaction model10:23 0cDtαSx,t=λB-βSF,0cDtαFx,t=λ+ωB-βSF,0cDtαBx,t=βSF-λ+ωB,0cDtαPx,t=ωB.

With initial conditions,24 Sx,0=S0,Fx,0=F0,Bx,0=B0,Px,0=P0.

Applying the LT on considered model (23) and utilizing above initial conditions (24).25 LSx,t-1sS0-1sαLλB-βSF=0,LFx,t-1sF0-1sαLλ+ωB-βSF=0,LBx,t-1sB0-1sαLβSF-λ+ωB=0,LPx,t-1s(P0)-1sαLωB=0.

Defining non-linear operator as26 N1φ1x,t;q,φ2x,t;q,φ3x,t;q,φ4x,t;q=Lφ1x,t;q-1sS0-1sαLλφ3x,t;q-βφ1x,t;qφ2x,t;q,N2φ1x,t;q,φ2x,t;q,φ3x,t;q,φ4x,t;q=Lφ2x,t;q-1sF0-1sαLλ+ωφ3x,t;q-βφ1x,t;qφ2x,t;q,N3φ1x,t;q,φ2x,t;q,φ3x,t;q,φ4x,t;q=Lφ3x,t;q-1sB0-1sαLβφ1x,t;qφ2x,t;q-λ+ωφ3x,t;q,N4φ1x,t;q,φ2x,t;q,φ3x,t;q,φ4x,t;q=Lφ4x,y,t;q-1sP0-1sαL{ωφ3x,t;q}.

By using the proposed algorithm the deformation equation at Hx,t=1, can be written as 27 LSmx,t-kmSm-1x,t=ℏL-1R1,mS→m-1,F→m-1,B→m-1,P→m-1,LFmx,t-kmFm-1x,t=ℏL-1R2,mS→m-1,F→m-1,B→m-1,P→m-1,LBmx,t-kmBm-1x,t=ℏL-1R3,mS→m-1,F→m-1,B→m-1,P→m-1,LPmx,t-kmPm-1x,t=ℏL-1R4,mS→m-1,F→m-1,B→m-1,P→m-1

where28 R1,mS→m-1,F→m-1,B→m-1,P→m-1=LSm-1x,t-1-kmn1sS0-1sαLλBm-1-β∑i=0m-1SiFm-1-i,R2,mS→m-1,F→m-1,B→m-1,P→m-1=LFm-1x,t-1-kmn1sF0-1sαLλ+ωBm-1-β∑i=0m-1SiFm-1-i,R3,mS→m-1,F→m-1,B→m-1,P→m-1=LBm-1x,t-1-kmn1sB0-1sαLβ∑i=0m-1SiFm-1-i-λ+ωBm-1,R4,mS→m-1,F→m-1,B→m-1,P→m-1=LPm-1x,t-1-kmn1sB0-1sαLωBm-1.

Employing Laplace inverse transform on Eq. (27) we obtain29 Smx,t=kmSm-1x,t+ℏL-1R1,mS→m-1,F→m-1,B→m-1,P→m-1,Fmx,t=kmFm-1x,t+ℏL-1R2,mS→m-1,F→m-1,B→m-1,P→m-1,Bmx,t=kmBm-1x,t+ℏL-1R3,mS→m-1,F→m-1,B→m-1,P→m-1,Pmx,t=kmPm-1x,t+ℏL-1R4,mS→m-1,F→m-1,B→m-1,P→m-1.

Computing the above system of equations, we have30 S0=S0,F0=F0,B0=B0,P0=P0.

S1t=-ℏtαB0λ-βF0S0Γα+1,

F1t=-ℏtαB0λ+ω-βF0S0Γα+1,

B1t=-ℏtαβF0S0-B0λ+ωΓα+1,

P1t=-B0ωℏtαΓα+1.

In the same manner the series can be attained. Then, the q-HATM solution for the proposed model (23) is given by31 Sx,t=S0x,t+∑m=1∞Smx,t1nm,Fx,t=F0x,t+∑m=1∞Fmx,t1nm,Bx,t=B0x,t+∑m=1∞Bmx,t1nm,Px,t=P0x,t+∑m=1∞Pmx,t1nm.

Numerical results and discussion

In this segment, we demonstrate the numerical simulation for the considered fractional model with graphical results to investigate reversible enzymatic reactions. The time-fractional derivative is investigated via Caputo operator and the results are obtained through a semi analytical approach called q-homotopy analysis transform method. In Sect.  “Analysis of the solution of the model” we have demonstrated the existence, uniqueness and convergence of the system’s solution via fixed point theory, Banach fixed point theorem and Laplace transform. In the next part we calculated the series solution of the proposed model using the considered semi analytical approach. To solve the system of equations we have considered values of the initial conditions as Sx=0.7, Fx=0.7, Bx=0, Px=0 and the values of parameters as ω=0.1,β=0.1,λ=0.2. Figure 2 shows the 3D plots of the obtained solutions, here we can observe the decline of the substrate S and increase in the enzyme activity F. Due to the action of enzyme activity the substrate S is converted into the product P which is increasing with time t and the complex B is formed by these reactions. If we take larger fractional values there will be increase in concentration of F, P and decrease in S, B. The rise of concentration in P, F is due to deformation of B as it does not convert fully into substrate S. Figure 3 defines the nature of alpha curves which are obtained through solutions for various alpha values. Figure 4 shows the behavior of ℏ curves for diverse fractional values and for suitable value of ℏ and the obtained series solutions quickly converges. The ℏ-curves provide a graphical representation that aids in determining the optimal range of ℏ for different problems, which is essential for the stability and convergence of the solution. Based on the practical observations provided, we have seen that the use of fractional derivative helps us to more effectively model the structure of reversible enzymatic reactions.Fig. 2 Nature of the q-HATM solution for the proposed model via 3D plots at α=0.75.

Fig. 3 Behavior of the α curves for the obtained solution at diverse values.

Fig. 4 Nature of ℏ curves for the solution of system of equations at n=1 and α=0.5,0.75,1.

Conclusion

The expected analytical solutions for the fractional reversible enzymatic reactions model are studied using q-homotopy analysis transform method in the present work. Here we considered the fractional Caputo derivative to study the considered problem. This method has the potential to be applied to various fractional, classical models arising in the fields of physics, chemistry and other branches of science. The current investigation helps us to study applications of enzyme reaction in bio mechanics which includes bone remodelling, cellular metabolism and wound healing. The study on enzyme reactions helps us to carry out research on drug development, to model disorders caused by genetics and plays crucial role in modelling neurodegenerative diseases. The concept of enzyme kinetics can be extended to study evolutionary biology, biochemical oscillations and to model gene regulation networks. The following conclusions can be observed by this study.The existence and uniqueness of the model’s system of solutions with fixed point theorems are explained.

The solutions we got using q-HATM are in the form of series and converge rapidly.

The plots indicate a clear influence of both the arbitrary order, applied parameters on the model and these parameters have the ability to influence stability of the model.

The model’s behaviour is also dependent on both time instant and time history, which can be easily analysed by the fractional calculus concept.

The current research can be expanded to include chemical processes in many plant and animal cells, as well as industrial applications.

Analysis confirms that the proposed method is exceptionally effective and successful in describing reversible two-step enzymatic reaction.

Acknowledgements

This project was supported by the Researchers Supporting Project number (RSP2024R413), King Saud University, Riyadh, Saudi Arabia.

Author contributions

H. B. Chethan developed the theoretical formalism and writing—original draft preparation and a formal analysis. D. G. Prakasha and Nasser Bin Turki performed investigation, software coding, validation and Writing—review & editing. H. B. Chethan and D, G. Prakasha did a substantial contribution to the conception, design of the work. All authors discussed the results and contributed to the final manuscript. D. G. Prakasha supervised the findings of this work. All authors reviewed the manuscript.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Code availability

Codes that support the findings of this study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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