An Analytical Approximate Solution for the Quasi …downloads.hindawi.com/journals/ddns/2019/8901508.pdfDiscreteDynamicsinNatureandSociety eaboveequationdenotesaconservationlawfor

Research ArticleAn Analytical Approximate Solution for theQuasi-Steady State Michaelis-Menten Problem

U Filobello-Nino1 H Vazquez-Leal 1 R Castaneda-Sheissa1 V M Jimenez-Fernandez1

A L Herrera-May23 J Huerta-Chua 4 L Gil-Adalid1 and J E Pretelin-Canela1

1Facultad de Instrumentacion Electronica Universidad Veracruzana Cto Gonzalo Aguirre Beltran SN 91000 Xalapa VER Mexico2Micro andNanotechnology ResearchCenter Universidad Veracruzana Calzada Ruiz Cortines 455 94292 Boca del Rıo VERMexico3Maestrıa en Ingenierıa Aplicada Facultad de Ingenierıa de la Construccion y el Habitat Universidad VeracruzanaCalzada Ruiz Cortines 455 94292 Boca del Rıo VER Mexico

4Instituto Tecnologico Superior de Poza Rica Tecnologico Nacional de Mexico Luis Donaldo Colosio Murrieta SN Arroyo del Maız93230 Poza Rica VER Mexico

Correspondence should be addressed to H Vazquez-Leal hvazquezuvmx

Received 7 September 2018 Revised 8 February 2019 Accepted 11 February 2019 Published 27 March 2019

Academic Editor Seenith Sivasundaram

Copyright copy 2019 U Filobello-Nino et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

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 itis shown that the obtained solutions are accurate and therefore they explain the general behaviour of the Michaelis-Mentenmechanism

1 Introduction

TheMichaelis-Menten mechanism is employed to model thekinetics of enzyme-catalyzed reactions in cases where theconcentration of substrate is greater than the concentration ofenzyme The importance of such reactions relies on the factthat the majority of cell processes require enzymes to get asignificant rate [1 2]

Enzymes are protein molecules that act as catalystswith the purpose to accelerate chemical reactions in liv-ing organisms These enzymes work on specific moleculesdenominated substrates Without the existence of enzymesmost chemical reactions that keep living organisms alivewould be slow to sustain life [2]

Thus haemoglobin in red blood cells is an enzymewhile the oxygen is the substrate with which it combinesOther example would be the case of oxidation of glucosethis process provides water carbon dioxide and energynevertheless it is extremely slow when it is carried outexposed just to open air and there is no appreciable oxidationafter years of exposure [1 2]

Enzymes play a significant role in the regulation ofbiological processes they work as activators or inhibitors ina reaction To comprehend their role in the rate of reactionsthe study of enzyme kinetics is necessary its study consistsfundamentally in understanding the temporary behaviour ofthe several reactions and the conditions that influence them[1] This article proposes a study of the Michaelis-Mentenmechanism which describes one of the most basic enzymaticreactions starting from perturbation method technique

The classical perturbation method (PM) is a pioneer tech-nique to solve some cases of nonlinear differential equationsPM was proposed by S D Poisson and later extended byJ H Poincare Even though PM appeared in the early 19thcentury the application of this method to analyse nonlinearproblems was not performed until later on that century Themost important application of this method was in the subjectof fluid mechanics celestial mechanics and aerodynamics[3 4]

In general we will suppose that a differential equationcan be expressed as the sum of two parts one linear and one

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 8901508 9 pageshttpsdoiorg10115520198901508

2 Discrete Dynamics in Nature and Society

nonlinear The nonlinear part is considered as a small pertur-bation through a small perturbation parameter As a matterof fact this last assumption is considered as a disadvantageof PM As a consequence other methods have emergedto study the problem of finding approximate solutions forsome nonlinear problems such as variational approaches[5 6] tanh method [7] exp-function [8 9] Adomianrsquosdecomposition method [10 11] parameter expansion [12]homotopy perturbation method [13ndash17] homotopy analysismethod [18 19] Boubaker Polynomials Expansion Scheme(BPES) [20 21] among many others

Although PM provides generally better results for smallperturbation parameters 120576 ≪ 1 we will see that ourapproximate solution is handy and has good precision evenfor relatively large values of the perturbative parameter suchas what was reported in [22ndash24] This work will employPM with the purpose to get an approximate solution for theMichaelis-Menten mechanism previously mentioned

The rest of this work is presented as follows Section 2introduces the basic idea of PM method In Section 3 wewill present the basic enzyme reaction proposed byMichaelisand Menten Section 4 provides an analytical approximatesolution for a Quasi-Steady State Approximation Michaelis-Menten problem Section 5 discusses the results obtained Abrief conclusion is given in Section 6

2 Basic Idea of Perturbation Method

Let the differential equation of one dimensional nonlinearsystem be in the form

119871 (119909) + 120576119873 (119909) = 0 (1)

where we assume that 119909 is a function of one variable119909 = 119909(119905) 119871(119909) is a linear operator which in general containsderivativeswith respect to 119905119873(119909) is a nonlinear operator and120576 is a small parameter

Assuming that the nonlinear term in (1) is a smallperturbation and that the solution for (1) can be written asa power series in terms of the perturbative parameter 120576

119909 (119905) = 1199090 (119905) + 1205761199091 (119905) + 12057621199092 (119905) + sdot sdot sdot (2)

Then by substituting (2) into (1) and equating termshaving identical powers of 120576 we obtain a number of differen-tial equations that can be integrated recursively to find thefunctions 1199090(119905) 1199091(119905) 1199092(119905) 3 Michaelis-Menten Enzyme Kinetics

One of the most important enzyme reactions was proposedby Leonor Michaelis and Maud Leonora Menten [1 25] Itinvolves a substrate 119878 reacting with an enzyme 119864 to form anenzyme-substrate complex 119878119864 It is a reversible reaction withforward and backward rate constants 1198961 and 119896minus1 respectively(see below)

The complex SE undergoes unimolecular decompositionto form irreversibly a product P and the enzyme E Thisautocatalytic reaction is expressed as

119878 + 119864 1198701999445999468119870minus1119878119864 1198702997888997888rarr 119875 + 119864 (3)

and this equation says that one molecule of 119878 combineswith one molecule of 119864 to form one molecule of 119878119864 whichproduces one molecule of 119875 and one molecule of 119864 again

Next we will apply to (3) the law of mass action whichsays that the rate of reaction is proportional to the product ofthe concentrations of the reactants [1 2]

For the sake of simplicity we will use the same symbols todenote the concentrations thus119878 = [119878] 119864 = [119864] 119862 = [119878119864] 119875 = [119875]

(4)

where [ ] denotes concentrationThe law ofmass action applied to (3) leads to one equation

for each reactant thus the relevant chemical reaction equa-tions are 119878 = minus1198961119864119878 + 119896minus1119862 = minus1198961119864119878 + (119896minus1 + 1198962) 119862 = 1198961119864119878 minus (119896minus1 + 1198962) 119862 = 1198962119862

(5)

where the dot denotes differentiation with respect to 119905The initial values of concentrations are119878 (0) = 1198780119864 (0) = 1198640119862 (0) = 0119875 (0) = 0

(6)

The 119896119904 called rate constants are constants of proportion-ality in the application of the law of mass action

In order to show that system (5) can be reduced to twocoupled ordinary differential equations (ODES) for 119878(119905) and119862(119905) we add the second and third equations of (5) to obtain + = 0 (7)

and after integrating (7) we get119864 (119905) + 119862 (119905) = 1198640 (8)

or 119864 (119905) = minus119862 (119905) + 1198640 (9)

Discrete Dynamics in Nature and Society 3

The above equation denotes a conservation law for 119864(119905)It is expected that since 119864 is a catalyst and it only facilitatesthe reaction therefore its concentration has to be constant

By substituting (9) into the first and third equations of (5)we get the following system119878 = minus11989611198640119878 + (1198961119878 + 119896minus1) 119862 = 11989611198640119878 minus (1198961119878 + 119896minus1 + 1198962) 119862 (10)

We note that the last equation of (5) is uncoupled fromother three and can be expressed as follows

119875 (119905) = 1198962int1199050119862 (1199051015840) 1198891199051015840 (11)

Equation (11) determines 119875(119905) once 119862(119905) is knownNext we will obtain a dimensionless form of (10) by

introducing the quantities [1 2]120591 = 11989611198640119905119909 (120591) = 119878 (120591)1198780 119910 (120591) = 119862 (120591)1198640 120582 = 119896211989611198780 119870 = 119896minus1 + 119896211989611198780 equiv 1198701198721198780 120576 = 11986401198780

(12)

After substituting (12) into (10) we get the dimensionlesssystem (120591) = minus119909 + (119870 minus 120582) 119910 + 119909119910 119909 (0) = 1120576 119910 (120591) = 119909 minus 119870119910 minus 119909119910 119910 (0) = 0 (13)

where 119870119872 is called the Michaelis constant [1 2] and thedot denotes differentiation with respect to 120591 It is relevant tonote that (12) implies the inequality 119870 minus 120582 gt 0

Although (13) does not have an exact analytical solutionthere exists a useful approximation for this system notingthat in many biological process [1 2] a very small amountof enzyme is required in a reaction in comparison to theamount of substrate This implies that 120576 ≪ 1 (see (12)) sothat neglecting the term 120576 119910(120591) of (13) we get 119910 in terms of 119909as follows 119910 = 119909119870 + 119909 (14)

Substituting (14) into the first equation of (13) yields

= minus 120582119909119870 + 119909 (15)

Equations (14) and (15) define the Quasi-Steady StateApproximation (QSSA)

Noting that 119909(0) = 1 it is clear from (14) that 119910 does notsatisfy the initial condition 119910(0) = 0 for which it is expectedthat (14) results in a good approximation only for times notso close to initial 120591 = 0

Nevertheless it results that there exists a more gen-eral condition for which QSSA is valid This condition isexpressed in terms of Michaelis constant 119870119872 as follows [1]11986401198780 + 119870119872 11 + (119896minus11198962) + (119878011989611198962) ≪ 1 (16)

From this condition we see that even for a case such that120576 = 11986401198780 = 119874(1) (16) could be still satisfied if 119870119872 is largesuch as it occurs in many reactions [1]

An alternative way to express (16) useful for this study isin terms of some parameters defined by (12)1205761 + 119870 11 + (119896minus11198962) + (119878011989611198962) ≪ 1 (17)

Thus the condition that 119870119872 is large is also equivalent to119870 large values (119870 ≫ 1)Despite its simple appearance the nonlinear equation (15)

does not admit an explicit exact solution In order to obtaininformation numerical methods are used which require theintegration of (15) to obtain an implicit solution for 119909(120591) asfollows 119909 (120591) + 119870 ln 119909 (120591) = 1 minus 120582120591 (18)

Next information is obtained integrating (18) numericallyfor different values of time 120591

From the above it is clear that the mathematical descrip-tion of QSSA is just numerical until this point

4 An Analytical Approximate Solution for theQuasi-Steady State Approximation

Next we will show that PM provides an efficient tool in orderto get a good approximation for QSSA problem

We will see that despite the approximated origin of (15)its solutions will have the required accuracy to be consideredas one of the solutions for (13) As a matter of fact we will seethat it is also possible to employ these same solutions with thepurpose of obtaining an approximate solution 119910 = 119910(120591) fromthe solution of a linear equation

In order to obtain an approximate solution to (15) we willstart from the general hypothesis on the validity of QSSA (16)(or (17))

We begin rewriting (15) as

= minus 1198961199091 + 120572119909 = minus119896119909 (1 + 120572119909)minus1 (19)

where we have defined119896 = 120582119870120572 = 1119870 (20)


Although this work assumes small 120576 values QSSArequires preferentially large values of 119870 and in this limit 120572can be employed as an adequate perturbation parameter (see(20))

Employing the Newtonrsquos binomial formula (19) gets thesimpler form

+ 119896119909 minus 1205721198961199092 = 0 (21)

where for the sake of simplicity we kept only the firstorder power of 120572 since the rest of the terms contain higherpowers of the mentioned perturbation parameter Of coursein general to get more accurate solutions it is necessaryconsidering more terms of (19)

Identifying 120572 with the PM parameter we assume asolution for (21) in the form

119909 (120591) = 1199090 (120591) + 1205721199091 (120591) + 12057221199092 (120591) + 12057231199093 (120591)+ 12057241199094 (120591) + sdot sdot sdot (22)

(see (2))After substituting (22) into (21) and equating the

terms with identical powers of 120572 it can be solved for1199090(120591) 1199091(120591) 1199092(120591) 1199093(120591) and so on Later it will be seenthat a precise handy result is obtained by keeping up to fourthorder approximation

1205720)0 + 1198961199090 = 0 1199090 (0) = 1 (23)1205721)1 + 1198961199091 minus 11989611990920 = 0 1199091 (0) = 0 (24)1205722)2 + 1198961199092 minus 211989611990901199091 = 0 1199092 (0) = 0 (25)

1205723)3 + 1198961199093 minus 211989611990901199092 minus 11989611990921 = 0 1199093 (0) = 0 (26)1205724) 1199094 + 1198961199094 minus 211989611990901199093 minus 211989611990911199092 = 0 1199094 (0) = 0 (27) (28)

Thus after solving iteratively (23)ndash(27) we get

1199090 (120591) = 119890minus119896120591 (29)1199091 (120591) = 119890minus119896120591 minus 119890minus2119896120591 (30)1199092 (120591) = 119890minus119896120591 minus 2119890minus2119896120591 + 119890minus3119896120591 (31)1199093 (120591) = 119890minus119896120591 minus 3119890minus2119896120591 + 3119890minus3119896120591 minus 119890minus4119896120591 (32)1199094 (120591) = 3119890minus119896120591 minus 4119890minus2119896120591 + 6119890minus3119896120591 minus 4119890minus4119896120591 minus 119890minus5119896119905 (33) (34)

By substituting (29)ndash(33) into (22) we obtain a handyforth order approximation for the solution of (21) as it isshown119909 (120591)= 119890minus119896120591 + 120572 (119890minus119896120591 minus 119890minus2119896120591)

+ 1205722 (119890minus119896120591 minus 2119890minus2119896120591 + 119890minus3119896120591)+ 1205723 (119890minus119896120591 minus 3119890minus2119896120591 + 3119890minus3119896120591 minus 119890minus4119896120591)+ 1205724 (3119890minus119896120591 minus 4119890minus2119896120591 + 6119890minus3119896120591 minus 4119890minus4119896120591 minus 119890minus5119896119905)

(35)

The corresponding value of 119910(120591) is obtained by substitut-ing (35) into (14)

119910 (120591)= [ 119890minus119896120591 + 120572 (119890minus119896120591 minus 119890minus2119896120591) + 1205722 (119890minus119896120591 minus 2119890minus2119896120591 + 119890minus3119896120591) + 1205723 (119890minus119896120591 minus 3119890minus2119896120591 + 3119890minus3119896120591 minus 119890minus4119896120591) + 1205724 (3119890minus119896120591 minus 4119890minus2119896120591 + 6119890minus3119896120591 minus 4119890minus4119896120591)119870 + 119890minus119896120591 + 120572 (119890minus119896120591 minus 119890minus2119896120591) + 1205722 (119890minus119896120591 minus 2119890minus2119896120591 + 119890minus3119896120591) + 1205723 (119890minus119896120591 minus 3119890minus2119896120591 + 3119890minus3119896120591 minus 119890minus4119896120591) + 1205724 (3119890minus119896120591 minus 4119890minus2119896120591 + 6119890minus3119896120591 minus 4119890minus4119896120591)] (36)

Next we consider as a case study the value of theparameters 120582 = 01 119870 = 8 120576 = 01 (120572 = 0125 119896 = 00125)in such way that (35) and (36) get the form

119909 (120591) = minus0001953125119890minus005120591 + 0021484375119890minus003750120591 minus 0162109375119890minus0025120591 + 1142578125119890minus001250120591 (37)

119910 (120591) = [ minus0001953125119890minus0050120591 + 0021484375119890minus00375120591 minus 0162109375119890minus00250120591 + 1142578125119890minus0012501205918 minus 0001953125119890minus0050120591 + 0021484375119890minus003750120591 minus 0162109375119890minus00250120591 + 1142578125119890minus001250120591] (38)

From Figure 1 it is clear that the numerical solution andanalytical approximate solution (37) and (38) are in goodagreement but 119910(120591) given by (38) does not satisfy the initial

condition 119910(0) = 0 therefore we will provide a way toimprove it

After directly adding the two equations (13) we obtain


120576 119910 (120591) + 120582119910 = minus (120591) (39)

unlike (21) we note that (39) was obtained withoutemploying any kind of approximated process

Since the proposed solution (37) for 119909(120591) provided a goodaccuracy next we propose to substitute (35) into (39) in orderobtain to a linear differential equation for 119910(120591) (We wouldexpect good results considering adequate values of 120576 and 119870)

120576 119910 (120591) + 120582119910= 119896119890minus119896120591 + 120572119896 (119890minus119896120591 minus 2119890minus2119896120591)+ 1205722119896 (119890minus119896120591 minus 4119890minus2119896120591 + 3119890minus3119896120591)+ 1205723119896 (119890minus119896120591 minus 6119890minus2119896120591 + 9119890minus3119896120591 minus 4119890minus4119896120591)

(40)

Next we will get approximate solutions for 119909 and 119910 bysubstituting for instance the following sets of parameters

120582 = 1119870 = 9120576 = 0045 (120572 = 19 119896 = 19)

(41)

120582 = 13119870 = 5120576 = 0045 (120572 = 23 119896 = 1350)

(42)

(where we have taken into account the perturbativecharacter of 120576)

Theunknown function 119909 is determined for each case bysubstituting the above parameters into (35) while the corre-sponding 119910 is obtained by solving (40) through elementarymethods

The solutions for the first set of parameters (41) are givenby

119909 (120591) = minus014052755119890minus022222120591 + 00173754119890minus033333120591 minus 000198140119890minus044444120591 minus 000015241119890minus055555120591+ 112528577119890minus011111120591119910 (120591) = minus000005897(minus2130755636721198902211111120591 + 5348716512411990910minus711989022120591minus99704264441198902188888120591 + 15237058191198902177777120591+1472615091198902166666 + 167887857663 )119890minus2222222120591 (43)

and for the second set

119909 (120591) = minus03104119890minus052120591 + 00736119890minus078120591 minus 00144119890minus104120591 minus 00016119890minus13120591 + 12528119890minus026120591119910 (120591) = minus0000039528(minus6396296486641198902862888120591 + 3198605771691198902836888120591minus1148171400381198902810888120591 + 302319350321198902784888120591+42384504611198902758888120591 + 400115826039 )119890minus2888888120591 (44)

5 Discussion

The aim of this study was to explore the possibility offinding analytical approximate solutions for the Michaelis-Menten enzyme kinetics problem The study of this problemis relevant because it involves enzyme reactions and themost of the cell processes require enzymes to obtain asignificant rate Because the nonlinear system (13) has noexact analytical solution we proposed PM method in orderto get a perturbative solution for QSSA (see (14) and (15))

In order to test the potentiality of PM to describe QSSAwe considered as case study the values of the parameters120582 = 01 119870 = 8 and 120576 = 01 (120572 = 0125 119896 = 00125) (notethat the condition 119870 gt 120582 is satisfied and 120576 small) Although

we obtained a handy accurate approximate solution in almostthe entire domain of 120591 (see Figure 1) (38) does not satisfythe initial condition 119910(0) = 0 and its precision is subject inprinciple for times not too close to 120591 = 0

Although (38) failed which was assumed from the begin-ning forQSSA (sincewe adopted the approximate result (14))we employed the approximate solution (35) for 119909 as startingpoint to obtain a better analytical approximation for 119910 (infact since (21) was obtained after keeping only first orderpower of 120572 from Newtonrsquos binomial formula applied to (19)we considered as case studies values of 120572 less than one so thatgood approximations were expected for 119909 since (unlike 119910)the solution (35) for the ODE (21) satisfies the correct initialcondition 119909(0) = 1) With this purpose we substituted (35)


100908070605040302

0 20 40 60 80 100 120 140 160 180

x()

Exact x()Approximation x()

(a) For 119909(120591)

00 20 40 60 80 100 120 140 160 180

Approximation y()Exact y()

y()

010

008

006

004

002

(b) For 119910(120591)

Figure 1 Comparison between numerical solutions and approximations (37) and (38)

into (39) (obtained without employing approximations) togive rise the linear differential equation (40)

Figures 2 and 3 show the comparison between numericaland approximate solutions for parameter sets (41) and (42)respectively for which the condition 119870 gt 120582 was satisfied

Although it is clear from the figures mentioned that theproposed approximate solutions have good precision it isnecessary to provide a more analytical criterion to ensurethe accuracy of our solutions For this purpose Figures 4and 5 show the absolute errors that result from using theapproximations proposed in this paper Thus for the first setof parameters (41) we note that the absolute maximum errorscommitted by using 119909 and 119910 respectively (see (43)) are ofthe order of 00045 and 001 while for the second set (42)the absolute errors for 119909 and 119910 (see (44)) reach maximumvalues of 0012 and 00155 From all the mentioned above themost complicated region to model just right at the beginningseems natural especially for the case of (43) and (44) for thevariable y (see Figures 4 and 5) However it is worth notingthat our procedure provides flexibility to get more accuracyif necessary adding more terms to (21) and increasing higherorder terms in the approximate solution (35) which makesit useful for practical applications Thus from Figures 2ndash5we deduced that our approximations (43) and (44) havegood precision and they explain the general behaviour ofMichaelis-Menten mechanism in the limit studied

A relevant fact to mention is that although the firstcase (41) belongs to the domain of validity (120572 small) (42)corresponds to 120572 cong 067 which cannot be consideredsmall at all and any way we got an analytical approximationwith good precision which provides a certain additionalmargin of applicability to our proposed solutions Thereforethe coupled system of differential equations (21) and (39)preserves a lot of the chemical information content in thenonlinear system (13) (for QSSA approximation)

Equations (44) for instance correctly describe theasymptotic final state 119909 997888rarr 0 and 119910 997888rarr 0 as 120591 997888rarr infinthis implies that both substrate and the substrate-enzyme

complex concentrations tend to zero in this limit This isexpected because reaction (3) converts S into a product P

A brief qualitative description based on [1] is as followsfrom (21) it is clear that in the limit 120591 997888rarr 0 119889119909119889120591 lt 0(because 119909(0) = 1 and |120572| lt 1) thus 119909 begins to decreasefrom its initial value 119909(0) = 1 On the other hand sinceminus119889119909119889120591 gt 0 near 120591 cong 0 120576 gt 0 and 119910(0) = 0 then (39) impliesthat 119889119910119889120591 gt 0 What is more 119910 increases from its initialvalue 119910(0) = 0 until a maximum value which corresponds tothe condition 119889119910119889120591 = 0 from (39) 119910 = minus120582 is obtainedEven further in accordance with the second equation of (13)this value is given also by 119910 = 119909(119870 + 119909)

Besides from conservation law (9) and for instance of119910(120591) in (44) we can describe the general behaviour for 119864(119905)Enzyme concentration decreases from its initial value 1198640while the one of 119862 is increased to its maximum value Fromthis moment 119862 decreases and E asymptotically approachesagain to its initial value (see (9))

From all the above it is highlighted that the first orderordinary differential equations system composed by (21) and(39) besides providingmany information of the phenomenonunder study is simpler than (13) and very useful in the searchfor approximate solutions in the domain of QSAA

Finally we note that the loss of information assumed byusing (14) and (15) instead of system (13) is partly reducedfor the case of large values of 119870 because the incorrectinitial condition resulting from (14) 119910(0) = 1119870 + 1 (since119909(0) = 1) differs less from the real value 119910(0) = 0 as 119870increases These considerations and the fact that (14) has thecorrect asymptotic behaviour undoubtedly contributed to theaccuracy of the proposed results

6 Conclusions

The main contribution of this work was the proposal toemploy the pair of differential equations (21) and (39) whichprovided handy analytical approximate solutions for theQSSA Michaelis-Menten mechanism through the whole


0

y()

010

008

006

004

002

0 10 20 30 40 50 60 70

Exact y()Approximation y()

(a) For 119910(120591)

0 10 20 30 40 50 60 70

1009080706050403

0102

x()

Approximation x()Exact x()

(b) For 119909(120591)

Figure 2 Comparison between numerical solutions and approximations (43)

016

014

012

010

008

006

004

002

0

y()

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1

08

06

04

02

0

x()


(b) For 119909(120591)


0

0010

0008

0006

0004

0002

0 10 20 30 40 50 60 70(a) For 119910(120591)

0

0004

0003

0002

0001

0 10 20 30 40 50 60 70(b) For 119909(120591)

Figure 4 Absolute error for solutions (43)


0

0015

0010

0005

0 10 20 30 40 50 60 70(a) For 119910(120591)

0 10 20 30 40 50 60 700

0012

0010

0008

0006

0004

0002

(b) For 119909(120591)


domain of time 119905 ge 0 It is worth mentioning the notable rollof PM method whose application contributed to obtain anaccurate analytical approximate solution for (21) Althoughthis work assumed large values of 119870 this condition is notso restrictive because this case occurs in many reactions[1] Besides our case studies showed the potentiality of (21)and (39) even for values of 119870 gt 1 but not 119870 ≫ 1Finally given the flexibility to improve precision if necessaryit is expected that proposal of this article will be useful forpractical applications of the studied enzyme reactions

Data Availability

All data generated or analysed during this study is includedwithin this research article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

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

References

[1] J D Murray Mathematical Biology I An Introduction vol 1Springer New York NY USA 3rd edition 2002

[2] R H Enns Itrsquos a Nonlinear World Springer UndergraduateTexts in Mathematics and Technology Springer New York NYUSA 2011

[3] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[4] M H Holmes Introduction to Perturbation MethodsSpringerndashVerlag New York NY USA 1995

[5] L M B Assas ldquoApproximate solutions for the generalizedkdvburgers equation by hes variational iteration methodrdquoPhysica Scripta vol 76 no 2 p 161 2007

[6] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007

[7] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005

[8] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung A vol 62 no 12 pp 685ndash6882007

[9] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008

[10] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[11] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquoApplied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002

[12] L Zhang and L Xu ldquoDetermination of the limit cycle by hersquosparameter-expansion for oscillators in a u3 (1 + u2) potentialrdquoZeitschrift fur Naturforschung A vol 62 no 7-8 pp 396ndash3982007

[13] J H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[14] J H He ldquoHomotopy perturbation techniquerdquoComputer Meth-ods Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[15] J HHe ldquoHomotopy perturbationmethod for solving boundaryvalue problemsrdquo Physics Letters A vol 350 no 1-2 pp 87-882006

[16] M Y Adamu and P Ogenyi ldquoParameterized homotopy pertur-bation methodrdquo Nonlinear Science Letters A vol 8 no 2 pp240ndash243 2017

[17] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics vol34 no 1 pp 1ndash16 2015

[18] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by homotopy


analysis methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[19] HNHassan andMA El-Tawil ldquoAn efficient analytic approachfor solving two-point nonlinear boundary value problems byhomotopy analysis methodrdquo Mathematical Methods in theApplied Sciences vol 34 no 8 pp 977ndash989 2011

[20] M Agida and A S Kumar ldquoA boubaker polynomials expansionscheme solution to random loversquos equation in the case of arational kernelrdquo Journal of Theoretical Physics vol 7 no 24 pp319ndash326 2010

[21] A Yildirim S T Mohyud-Din and D H Zhang ldquoAnalyticalsolutions to the pulsed klein-gordon equation using modifiedvariational iterationmethod (MVIM) and boubaker polynomi-als expansion scheme (BPES)rdquo Computers amp Mathematics withApplications vol 59 no 8 pp 2473ndash2477 2010

[22] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoUsing per-turbation methods and Laplace-Pade approximation to solvenonlinear problemsrdquo Miskolc Mathematical Notesc vol 14 no1 pp 89ndash101 2013

[23] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of gelfands equationrdquo Asian Journal ofMathematics amp Statistics vol 6 no 2 pp 76ndash82 2013

[24] U Filobello-NinoH Vazquez-Leal A Sarmiento-Reyes and etal ldquoThe study of heat transfer phenomenausing pm for approx-imate solution with dirichlet and mixed boundary conditionsrdquoApplied and Computational Mathematics vol 2 no 6 pp 143ndash148 2013

[25] L Michaelis and M L Menten ldquoDie kinetik der invertin-wirkungrdquo Biochemistry Z vol 49 pp 333ndash369 1913

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nonlinear The nonlinear part is considered as a small pertur-bation through a small perturbation parameter As a matterof fact this last assumption is considered as a disadvantageof PM As a consequence other methods have emergedto study the problem of finding approximate solutions forsome nonlinear problems such as variational approaches[5 6] tanh method [7] exp-function [8 9] Adomianrsquosdecomposition method [10 11] parameter expansion [12]homotopy perturbation method [13ndash17] homotopy analysismethod [18 19] Boubaker Polynomials Expansion Scheme(BPES) [20 21] among many others

Although PM provides generally better results for smallperturbation parameters 120576 ≪ 1 we will see that ourapproximate solution is handy and has good precision evenfor relatively large values of the perturbative parameter suchas what was reported in [22ndash24] This work will employPM with the purpose to get an approximate solution for theMichaelis-Menten mechanism previously mentioned

The rest of this work is presented as follows Section 2introduces the basic idea of PM method In Section 3 wewill present the basic enzyme reaction proposed byMichaelisand Menten Section 4 provides an analytical approximatesolution for a Quasi-Steady State Approximation Michaelis-Menten problem Section 5 discusses the results obtained Abrief conclusion is given in Section 6

2 Basic Idea of Perturbation Method

Let the differential equation of one dimensional nonlinearsystem be in the form

119871 (119909) + 120576119873 (119909) = 0 (1)

where we assume that 119909 is a function of one variable119909 = 119909(119905) 119871(119909) is a linear operator which in general containsderivativeswith respect to 119905119873(119909) is a nonlinear operator and120576 is a small parameter

Assuming that the nonlinear term in (1) is a smallperturbation and that the solution for (1) can be written asa power series in terms of the perturbative parameter 120576

119909 (119905) = 1199090 (119905) + 1205761199091 (119905) + 12057621199092 (119905) + sdot sdot sdot (2)

Then by substituting (2) into (1) and equating termshaving identical powers of 120576 we obtain a number of differen-tial equations that can be integrated recursively to find thefunctions 1199090(119905) 1199091(119905) 1199092(119905) 3 Michaelis-Menten Enzyme Kinetics

One of the most important enzyme reactions was proposedby Leonor Michaelis and Maud Leonora Menten [1 25] Itinvolves a substrate 119878 reacting with an enzyme 119864 to form anenzyme-substrate complex 119878119864 It is a reversible reaction withforward and backward rate constants 1198961 and 119896minus1 respectively(see below)

The complex SE undergoes unimolecular decompositionto form irreversibly a product P and the enzyme E Thisautocatalytic reaction is expressed as

119878 + 119864 1198701999445999468119870minus1119878119864 1198702997888997888rarr 119875 + 119864 (3)

and this equation says that one molecule of 119878 combineswith one molecule of 119864 to form one molecule of 119878119864 whichproduces one molecule of 119875 and one molecule of 119864 again

Next we will apply to (3) the law of mass action whichsays that the rate of reaction is proportional to the product ofthe concentrations of the reactants [1 2]

For the sake of simplicity we will use the same symbols todenote the concentrations thus119878 = [119878] 119864 = [119864] 119862 = [119878119864] 119875 = [119875]

(4)

where [ ] denotes concentrationThe law ofmass action applied to (3) leads to one equation

for each reactant thus the relevant chemical reaction equa-tions are 119878 = minus1198961119864119878 + 119896minus1119862 = minus1198961119864119878 + (119896minus1 + 1198962) 119862 = 1198961119864119878 minus (119896minus1 + 1198962) 119862 = 1198962119862

(5)

where the dot denotes differentiation with respect to 119905The initial values of concentrations are119878 (0) = 1198780119864 (0) = 1198640119862 (0) = 0119875 (0) = 0

(6)

The 119896119904 called rate constants are constants of proportion-ality in the application of the law of mass action

In order to show that system (5) can be reduced to twocoupled ordinary differential equations (ODES) for 119878(119905) and119862(119905) we add the second and third equations of (5) to obtain + = 0 (7)

and after integrating (7) we get119864 (119905) + 119862 (119905) = 1198640 (8)

or 119864 (119905) = minus119862 (119905) + 1198640 (9)





119875 (119905) = 1198962int1199050119862 (1199051015840) 1198891199051015840 (11)



(12)





= minus 120582119909119870 + 119909 (15)




















+ 119896119909 minus 1205721198961199092 = 0 (21)



119909 (120591) = 1199090 (120591) + 1205721199091 (120591) + 12057221199092 (120591) + 12057231199093 (120591)+ 12057241199094 (120591) + sdot sdot sdot (22)



1205720)0 + 1198961199090 = 0 1199090 (0) = 1 (23)1205721)1 + 1198961199091 minus 11989611990920 = 0 1199091 (0) = 0 (24)1205722)2 + 1198961199092 minus 211989611990901199091 = 0 1199092 (0) = 0 (25)






(35)










120576 119910 (120591) + 120582119910 = minus (120591) (39)




(40)


120582 = 1119870 = 9120576 = 0045 (120572 = 19 119896 = 19)

(41)

120582 = 13119870 = 5120576 = 0045 (120572 = 23 119896 = 1350)

(42)







5 Discussion






100908070605040302

0 20 40 60 80 100 120 140 160 180

x()


(a) For 119909(120591)

00 20 40 60 80 100 120 140 160 180


y()

010

008

006

004

002

(b) For 119910(120591)












6 Conclusions



0

y()

010

008

006

004

002

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1009080706050403

0102

x()


(b) For 119909(120591)


016

014

012

010

008

006

004

002

0

y()

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1

08

06

04

02

0

x()


(b) For 119909(120591)


0

0010

0008

0006

0004

0002

0 10 20 30 40 50 60 70(a) For 119910(120591)

0

0004

0003

0002

0001

0 10 20 30 40 50 60 70(b) For 119909(120591)



0

0015

0010

0005

0 10 20 30 40 50 60 70(a) For 119910(120591)

0 10 20 30 40 50 60 700

0012

0010

0008

0006

0004

0002

(b) For 119909(120591)



Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018












119875 (119905) = 1198962int1199050119862 (1199051015840) 1198891199051015840 (11)



(12)





= minus 120582119909119870 + 119909 (15)




















+ 119896119909 minus 1205721198961199092 = 0 (21)



119909 (120591) = 1199090 (120591) + 1205721199091 (120591) + 12057221199092 (120591) + 12057231199093 (120591)+ 12057241199094 (120591) + sdot sdot sdot (22)



1205720)0 + 1198961199090 = 0 1199090 (0) = 1 (23)1205721)1 + 1198961199091 minus 11989611990920 = 0 1199091 (0) = 0 (24)1205722)2 + 1198961199092 minus 211989611990901199091 = 0 1199092 (0) = 0 (25)






(35)










120576 119910 (120591) + 120582119910 = minus (120591) (39)




(40)


120582 = 1119870 = 9120576 = 0045 (120572 = 19 119896 = 19)

(41)

120582 = 13119870 = 5120576 = 0045 (120572 = 23 119896 = 1350)

(42)







5 Discussion






100908070605040302

0 20 40 60 80 100 120 140 160 180

x()


(a) For 119909(120591)

00 20 40 60 80 100 120 140 160 180


y()

010

008

006

004

002

(b) For 119910(120591)












6 Conclusions



0

y()

010

008

006

004

002

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1009080706050403

0102

x()


(b) For 119909(120591)


016

014

012

010

008

006

004

002

0

y()

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1

08

06

04

02

0

x()


(b) For 119909(120591)


0

0010

0008

0006

0004

0002

0 10 20 30 40 50 60 70(a) For 119910(120591)

0

0004

0003

0002

0001

0 10 20 30 40 50 60 70(b) For 119909(120591)



0

0015

0010

0005

0 10 20 30 40 50 60 70(a) For 119910(120591)

0 10 20 30 40 50 60 700

0012

0010

0008

0006

0004

0002

(b) For 119909(120591)



Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018











+ 119896119909 minus 1205721198961199092 = 0 (21)



119909 (120591) = 1199090 (120591) + 1205721199091 (120591) + 12057221199092 (120591) + 12057231199093 (120591)+ 12057241199094 (120591) + sdot sdot sdot (22)



1205720)0 + 1198961199090 = 0 1199090 (0) = 1 (23)1205721)1 + 1198961199091 minus 11989611990920 = 0 1199091 (0) = 0 (24)1205722)2 + 1198961199092 minus 211989611990901199091 = 0 1199092 (0) = 0 (25)






(35)










120576 119910 (120591) + 120582119910 = minus (120591) (39)




(40)


120582 = 1119870 = 9120576 = 0045 (120572 = 19 119896 = 19)

(41)

120582 = 13119870 = 5120576 = 0045 (120572 = 23 119896 = 1350)

(42)







5 Discussion






100908070605040302

0 20 40 60 80 100 120 140 160 180

x()


(a) For 119909(120591)

00 20 40 60 80 100 120 140 160 180


y()

010

008

006

004

002

(b) For 119910(120591)












6 Conclusions



0

y()

010

008

006

004

002

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1009080706050403

0102

x()


(b) For 119909(120591)


016

014

012

010

008

006

004

002

0

y()

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1

08

06

04

02

0

x()


(b) For 119909(120591)


0

0010

0008

0006

0004

0002

0 10 20 30 40 50 60 70(a) For 119910(120591)

0

0004

0003

0002

0001

0 10 20 30 40 50 60 70(b) For 119909(120591)



0

0015

0010

0005

0 10 20 30 40 50 60 70(a) For 119910(120591)

0 10 20 30 40 50 60 700

0012

0010

0008

0006

0004

0002

(b) For 119909(120591)



Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









120576 119910 (120591) + 120582119910 = minus (120591) (39)




(40)


120582 = 1119870 = 9120576 = 0045 (120572 = 19 119896 = 19)

(41)

120582 = 13119870 = 5120576 = 0045 (120572 = 23 119896 = 1350)

(42)







5 Discussion






100908070605040302

0 20 40 60 80 100 120 140 160 180

x()


(a) For 119909(120591)

00 20 40 60 80 100 120 140 160 180


y()

010

008

006

004

002

(b) For 119910(120591)












6 Conclusions



0

y()

010

008

006

004

002

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1009080706050403

0102

x()


(b) For 119909(120591)


016

014

012

010

008

006

004

002

0

y()

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1

08

06

04

02

0

x()


(b) For 119909(120591)


0

0010

0008

0006

0004

0002

0 10 20 30 40 50 60 70(a) For 119910(120591)

0

0004

0003

0002

0001

0 10 20 30 40 50 60 70(b) For 119909(120591)



0

0015

0010

0005

0 10 20 30 40 50 60 70(a) For 119910(120591)

0 10 20 30 40 50 60 700

0012

0010

0008

0006

0004

0002

(b) For 119909(120591)



Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









100908070605040302

0 20 40 60 80 100 120 140 160 180

x()


(a) For 119909(120591)

00 20 40 60 80 100 120 140 160 180


y()

010

008

006

004

002

(b) For 119910(120591)












6 Conclusions



0

y()

010

008

006

004

002

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1009080706050403

0102

x()


(b) For 119909(120591)


016

014

012

010

008

006

004

002

0

y()

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1

08

06

04

02

0

x()


(b) For 119909(120591)


0

0010

0008

0006

0004

0002

0 10 20 30 40 50 60 70(a) For 119910(120591)

0

0004

0003

0002

0001

0 10 20 30 40 50 60 70(b) For 119909(120591)



0

0015

0010

0005

0 10 20 30 40 50 60 70(a) For 119910(120591)

0 10 20 30 40 50 60 700

0012

0010

0008

0006

0004

0002

(b) For 119909(120591)



Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









0

y()

010

008

006

004

002

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1009080706050403

0102

x()


(b) For 119909(120591)


016

014

012

010

008

006

004

002

0

y()

0 10 20 30 40 50 60 70


(a) For 119910(120591)

0 10 20 30 40 50 60 70

1

08

06

04

02

0

x()


(b) For 119909(120591)


0

0010

0008

0006

0004

0002

0 10 20 30 40 50 60 70(a) For 119910(120591)

0

0004

0003

0002

0001

0 10 20 30 40 50 60 70(b) For 119909(120591)



0

0015

0010

0005

0 10 20 30 40 50 60 70(a) For 119910(120591)

0 10 20 30 40 50 60 700

0012

0010

0008

0006

0004

0002

(b) For 119909(120591)



Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









0

0015

0010

0005

0 10 20 30 40 50 60 70(a) For 119910(120591)

0 10 20 30 40 50 60 700

0012

0010

0008

0006

0004

0002

(b) For 119909(120591)



Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018
























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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An Analytical Approximate Solution for the Quasi …downloads.hindawi.com/journals/ddns/2019/8901508.pdfDiscreteDynamicsinNatureandSociety eaboveequationdenotesaconservationlawfor