Iterative approximate solutions of kinetic equations for reversible enzyme reactions

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1 Vol.5, No.6, (0) doi:0.46/ns Natural Scinc Itrativ approximat solutions of kintic quations for rvrsibl nzym ractions Sarbaz H. A. Khoshnaw Dpartmnt of Mathmatics, Univrsity of Licstr, Licstr, UK Rcivd 7 March 0 rvisd 9 April 0 6 May 0 Copyright 0 Sarbaz H. A. Khoshnaw. This is an opn accss articl distributd undr th Crativ Commons Attribution Licns, which prmits unrstrictd us, distribution, and rproduction in any mdium, providd th original work is proprly citd. ABSTRACT W study kintic modls of rvrsibl nzym ractions and compar two tchniqus for analytic approximat solutions of th modl. Analytic approximat solutions of non-linar raction quations for rvrsibl nzym ractions ar calculatd using th Homotopy Prturbation Mthod (HPM) and th Simpl Itration Mthod (SIM). Th rsults of th approximations ar similar. Th Matlab programs ar includd in appndics. Kywords: Enzym Kintics Homotopy Prturbation Mthod Itration Mthod Michalis-Mntn Kintics Quasi-Stady Stat Approximation. INTRODUCTION Th varity of chmical ractions in a living organism is carrid out by nzyms. It appars that th rat of chmical ractions (both forward and backward) is acclratd by nzyms. Thy ar ssntial bcaus many chmical ractions occur without th activity of nzyms. Such ractions ar linkd withan nzym s activ sit, and thy bcom a product aftr a sris of stags. Ths stags ar known as th nzymatic mchanism. Thr ar two typs of mchanisms, singl substrat and multipl substrat mchanisms [-4]. An important branch of nzymology is nzym kintics which is usd to study th rat of chmical ractions. Diffrntial quations ar usd to charactriz th nzym kintics basd on som principls of chmical kintics [5-8]. Th singl nzym raction is on of th most powrful kinds of kintic raction. Simply put, this nzym raction is dfind as follows: ES ES EP k k k () whr th concntrations of nzym, substrat, nzymsubstrat complx and product ar dfind by [E], [S], [ES] and [P], rspctivly. Also, k, k and k rprsnt th raction rat constants. By using th ida of mass action, w can dscrib th raction Eq. in trms of a systm of non-linar ordinary diffrntial quations []. Thr ar varitis of possibl simplifications for th systm (Eq.) to dscrib analytic approximat solutions of th systm. On of th most common approachs to simplify this systm is th us of quasi-stady stat approxmation (QSSA). Th quasi-stady stat assumptions occur as fundamntal assumptions for nzym kintics, and th history of this subjct bgan 80 yars ago. It plays a ky rol with rgard to th analysis of th nzym kintic quations [5]. Anothr simplification is th Michalis-Mntn quation cratd in 9which pointd that th nzym raction (Eq.) should b k k, thrfor ES k. It mans that thr is quilibrium btwn ES k [E], [S] and [ES] to produc [P] and [E]. In 95, Briggs and Haldan proposd that th Michalis-Mntn assumption is not always applid. Thy said that it should b rplacd by th assumption that [ES] is prsnt, not ncssarily at quilibrium, but in a stady stat undr condition S0 E 0.This mans that th concntrations of [ES] occur as a stady stat. This is known as th stady stat assumption (SSA) or is somtims calld th quasi-stady stat approximation (QSSA), or psudostady sat approximation [9]. Th first dscription of QSS was givn by Briggs and Haldan in 95 [0]. Thy dscribd th simplst nzym raction in Eq., and pointd out th total concntration of nzym [E], whr E E ES is a tiny valu in comparison with th tot concntration of substrat [S]. Also, thy hav shown th trm of d ES is ngligibl compard to d and dt ds t dp. As a rsult, thy found th Michalis-Mntn dt Copyright 0 SciRs.

2 S. H. A. Khoshnaw / Natural Scinc 5 (0) quation, which is a diffrntial quation usd to dscrib th rat of nzymatic ractions. Th classical Michalis-Mntn quation is dfind as, k ES k kes, or ES dp k ES ES, k ES k S dt () k S whr k M k k M M is th Michalis-Mntn constant k (for mor dtails s []). Th purpos of this work is to driv asymptotic approximat xprssions for th substrat, product, nzym and nzym-substrat concntrations for Eq. by using (HPM) and (SIM), and to point out th similaritis and diffrncs btwn th mthods of (HPM) and (SIM) for all valus of dimnsionlss raction diffusion paramtrs,, and k. Anothr aim of this projct is to find out th appropriat itration in (SIM) compard to (HPM).. MATHEMATICAL FORMULATION Th Michalis-Mntn Eq. was applid by Kuhn in 94 [] to svral cass of nzym kntics. Th modl of biochmical raction was dvlopd by Briggs and Haldan in 95 []. Th modl of an nzym action considrs a raction that includs a substrat [S] which binds an nzym [E] rvrsibly to asubstrat-nzym [ES]. Th substrat-nzym lads rvrsibly to product [P] and nzym [E]. This mchanism is oftn writtn as follows: k k k k4 E S ES P E () Th mchanism shows th binding of substrat [S] and th rlas of product [P] whr th fr nzym is [E] and th nzym-substrat complx is [ES]. In addition, k, k and, k k4 dnot th rats of raction. It is clar from Eq. that substrat binding and product ar rvrsibl. Th concntration of th ractants in Eq. is dnotd by lowr cas lttrs:,,, E s S c ES p P (4) Th tim of volution of Eqs. and 4 ar found by th law of mass action to obtain th st of systm of th following non-linar raction quations: ds ks kc (5) dt d ks k kc k4p (6) dt dc d dp kc k4p dt (8) ks k k c k4p t (7) whn th initial conditions at t = 0 ar givn by 0, 0, 0, 0 0 s s0 c c0 p p0 (9) Adding Eqs.6 and 7, and using initial conditions Eq.9, w obtain c 0 (0) Also, adding Eqs.5, 7 and 8, and using initial conditions Eq.9, w gt s c p s 0 () By using Eqs.0 and, th systm of ordinary diffrntial quations (Eqs.5-8) rduc to only two variabls, s and c, as follows: ds ks 0 ks k c () dt dc ks 0 ks kkck40cs0s c () dt with initial conditions s 0 s0, c0 c0. By introducing th following paramtrs: 0,,, 0 0 kt s t c t p t u v w, s 0 t k k k 0 k4 E,, k,,, 0 ks0 ks0 s0 k m. (4) W us th dimnsionlss tchniqu to rduc th numbr of paramtrs for th systm of Eqs. and and th initial conditions Eq.9. This can b rprsntd in dimnsionlss form as follows: du uuk v (5) d dv u u kv v u v d (6) dw uuvv d (7) u 0, v 0 0, w 0 0. (8) In this papr, w stimat th analytic approximat solution for a systm of non-linar ODE (Eqs.5-8), by using th mthods of (HPM) and (SIM).. ANALYTICAL APPROXIMATE SOLUTION USING THE HOMOTOPY PERTURBATION METHOD Th basic ida of th Homotopy-Prturbation Mthod (HPM) is dfind in this sction. It is thn applid to find th approximat solution of th problm in Eqs.5-8. It is considrd from th following function: A x f r 0, r (9) Copyright 0 SciRs.

3 74 S. H. A. Khoshnaw / Natural Scinc 5 (0) with th boundary conditions B x, x 0, r (0) n whr AB,, f r and ar gnral diffrntial oprators, boundary oprators, a known analytic function, and th boundary of th domain, rspctivly []. Th function A consists of linar part L and nonlinar part N. So, th Eq.9 can b writtn as: LxNx f x 0 () Th Homotopy function is dfind z r, q : 0, R, which satisfis by Hz, q qlz Lx0 qaz f r q 0, () r or, 0 0 0,,, H zq L z L x ql x q N x f r whr 0, () q is an mbdding paramtr. At th sam tim, x 0 is an initial approximation of Eq.9, which satisfis Eq.0. Basically, from Eqs. and w can obtain: H z,0 L z L x 0, (4) 0 H z, A z f r 0, (5) Changing, z r q from x 0 to x r dpnds on th valus of q from zro to unity. It is calld dformation in th fild of topology. At th sam tim, L z Lx 0 and Az f r ar calld Homotopy. W us q as a small paramtr initially, and w dfind th Eqs. and as a powr sris in q : Lt q = z z qz q z (6) 0 to gt th approximat solution of Eq.9 x lim z z z z (7) q 0 Thus, HPM includs a combination of th prturbation mthod and th Homotopy mthod. Eqs.5-7 can b solvd analytically in a simpl and closd form by using th Homotopy Prturbation Mthod (HPM) (Rf Appndix A). So, th approximat solutions of th systm of non-linar diffrntial quations (Eqs.5 and 6) bcom: (s Eqs.8 and 9). Th analytic xprssions of th substrat u and nzym substrat v concntrations can b rprsntd in Eqs.8 and 9. Th dimnsionlss concntration of nzym E can b obtaind from Eqs.0 and 4 as follows: t E v (0) 0 Th dimnsionlss concntration of th product obtaind ithr by Eq.7 as follows: w u t u t v t v t mv t dt 0 () or w can us Eqs. and 4 to find th concntration of th product w as follows: u v w. () Th simpl analytic approximat solution form of th concntrations of nzym E and product w for all valus of paramtrs,, and k, ar rprsntd in Eqs SIMPLE ITERATION METHOD In this sction, w us a simpl tchniqu to find th analytic approximat solution for th systm of Eqs.5 and 6. W introduc this mthod by rwriting Eqs.5 and 6 as follows: du uk v uv () d dv u k v uv v d (4) w is u ab abc a a c c cb c a b c c c c c c c c c a abc ac a b bc c c c c c c c (8) v b bc c c b b b b cb bc c c c c cc cc c c c b b b cb bc b c cc c c c c c c (9) Copyright 0 SciRs.

4 S. H. A. Khoshnaw / Natural Scinc 5 (0) Lt a k, b and c k, thn th Eqs. and 4 can b writtn as: u n un A Gun, vn, for n 0,,, (5) v v n n uv n n whr Gun, vn is a non-linar bunvn un a part of th systm (Eq.5), and A is a b c matrix of th linar part of th systm (Eq.5). To valuatan approximat solution of Eq.5 with th initial conditions implid by Eq.8, w introduc th following stps to approach th approximat solution. Stp. For n 0, u0, v0 0 and, if possibl suppos that 0 (just in this stp). It mans w assum th non-linar part of Eq.5 approachs zro. Consquntly, w obtain th following systm: u u A v v (6) W can solv th systm of ordinary diffrntial Eq.6 analytically [4]. So, th solution of Eq.6 with initial conditions (Eq.8) is p p u d d p p v (7) d d whr p and p ar ignvalus of matrix A, and p p d, d p and a p p a p p d p p p. W substitut and v in Eq.0 and u Eq., thn obtain E and w, rspctivly. Th bhaviour of th componnts in Eq.7 ar dscribd in Figurs -5 (s Appndix C). Stp. For n, and substituting Eqs.7 in 5, w obtain th following systm of non-linar ODE: u u A Gu, v v v (8) It is clar that th systm of non-linar diffrntial quations (Eq.8) is solvd analytically [4]. Th solution of th systm with initial conditions (Eq.8) is obtaind as follows: u ac ac d p p p 4 p p p d4 5 d d v c p c p d p p p 4 6 p p p d d d (9) (40) whr d and,, d9 c, c4 ar constants. W substitut u and in Eqs.0 and, and obtain and v E w, rspctivly. Th bhaviour of concntrations in this stp is dscribd in Figurs 6-0 (S Appndix D). Stp. For n, and substituting Eqs.9 and 40 in Eq.5, w gt th following systm of non-linar ODE: u u A Gu, v v v (4) Th systm of non-linar diffrntial quations (Eq.4) is solvd analytically. Th solution of th systm with initial conditions (Eq.8) is obtaind as follows (s Eqs.4 and 4). Whr d and 90,, d c5, c6 ar constants. W substitut u and in Eq.0 and Eq., and obtain v E and w, rspctivly. Th bhaviour of concntrations in this stp is dscribd in Figurs -5 (S Appndix E). On th othr hand, w can asily raliz that th bhaviour of concntrations u, v, E and w of (HPM) ar dscribd in Figurs ASYMPTOTIC ANALYSIS An important dvlopmnt of asymptotic analysis was suggstd by Kruskal (96) for diffrntial quations [5]. H dfind asymptotology as th art of dscribing th bhaviour of spcifid solution (or family of solutions) of a systm in limiting cas. Th following thr diffrnt conditions can b idntifid basd on th initial E 0 ratio [6]. S 0 ) If th initial concntration of nzym E 0 is much gratr than th initial concntration of substrat p p p pp p pp pp p u ac ac d d d d d d p p p p p 4p pp pp p p 4 p d d d d d d d d d d d p p p pp p pp pp p v hc h c d d d d d d p p p p p 4p pp pp p p 4 p d d d d d d d d d d d (4) (4) Copyright 0 SciRs.

5 744 S. H. A. Khoshnaw / Natural Scinc 5 (0) Figur.,, 0.4 and k.. Figur 4.., 0., 0.9 and k.. Figur..6,., 0.9 and k.7. Figur ,.,. and k.7. Figur. 0.8, 0., 0.9 and k.. Figur 6.,, 0.4 and k.. Copyright 0 SciRs.

6 S. H. A. Khoshnaw / Natural Scinc 5 (0) Figur 7..6,., 0.9 and k.7. Figur 9.., 0., 0.9 and k.. Figur , 0., 0.6 and k.. S 0 E 0. This mans that S 0. Also, Schnlland Maini in [] mphasizd that th initial concntration of nzym gratly xcds th concntration of substrat, that ise0 S 0. So, from Eq.4, w gt. In this cas, th part of th nzym concntration which binds to th concntration of th substrat is small. This mans that thr is a fr rat of nzym. This rat is basd on th availability of th substrat, and is incrasd whnvr th concntrations of substrat ar incrasd, or by adding additional substrat to th chmical raction. ) If th initial concntration of substrat S 0 is much gratr than th initial concntration of nzym E 0 E 0. This mans that. So, from Eq.4, w S 0 Figur ,.,. and k.7. obtain. In this cas, thr is a small part of substrat that links to th nzym, whil a part of it is fr. In this cas, nzym molculs usually bind to substrat molculs which man that a small amount of nzym is fr. Th availability of nzym in this cas dpnds on this rat, and incrass whn th rat of nzym is incrasd, or by adding som xtra nzym to th chmical raction. ) If th initial concntration of nzym and substrat E 0 ar qual. This mans, so from Eq.4, w gt S0. In this cas, thr ar no any fr molculs of nzym or substrat. In othr words, all substrat molculs ar occupid by th nzym molculs, and all n- Copyright 0 SciRs.

7 746 S. H. A. Khoshnaw / Natural Scinc 5 (0) zym molculs ar also limitd by th numbr molculs of th substrat. Furthrmor, if w look th constant rat of ractions k4 and k from Eq.4, w can dfin th following conditions: 4) If k4 k, thn. 5) If k4 k, thn. 6) If k4 k, thn. In addition, according to th dfinition of and k from Eq.4, w obtain k, bcaus k always has a positiv valu. As rsult, w can asily combin th Conditions -6. W thn gt th following fiv basic cass in this papr: Cas. Th valu of and, Cas. Th valu of and, Cas. Th valu of and, Cas 4. Th valu of and, Cas 5. Th valu of and. Figur. 0.8, 0., 0.6 and k.. Figur..00,.00, 0.4 and k.. Figur 4.., 0., 0.8 and k.. Figur ,.,. and k.7. Figur..6,., 0.9 and k.7. W apply th abov cass sparatly in th analytic approximat solution for both mthods (HPM) and (SIM). Copyright 0 SciRs.

8 S. H. A. Khoshnaw / Natural Scinc 5 (0) Figur 6.,, 0.4 and k.. Figur 9.., 0.9, 0.8 and k.. Figur 7..6,., 0.9 and k.7. Figur ,.,. and k.7. Figur , 0.6, and k.. 6. RESULTS AND DISCUSSIONS Th figurs in this sction ar dividd in to four groups. Th first thr groups ar rlatd to thr itrations of SIM and th last group rfrs to th HPM. Figurs -0 show th analytic approximat solution of substrat u, nzym E, nzym-substrat complx v and product w. Each figur in this work corrsponds to on cas in th prvious sction. Th figurs chang in trms of th valus of th dimnsionlss paramtrs,, and k. W hav applid two diffrnt mthods which ar SIM and HPM to find th analytical approximat solutions for Eqs.5 and 6. Th HPM has bn usd by many rsarchrs for th systm (Eq.) [,,4]. Th main purpos of this discussion is to find th similaritis and diffrncs btwn th mthods which ar Copyright 0 SciRs.

9 748 S. H. A. Khoshnaw / Natural Scinc 5 (0) usd in this study. Anothr purpos is to rcogniz th bst itration of th SIM compard to th HPM. Thr ar a varity of data rsults that tll us th scond itration in our approach (SIM) is similar to HPM. First of all, th scond itration has many significant similaritis compard to (HPM), and som of thm provid xcllnt rsults in trms of our work. For instanc, Figurs 6-0 show that th valu of th concntration of substrat u slightly dcrass from its initial valu u 0 and thr ar a fw changs in th valu of th concntration of th nzym-substrat complx v. Gnrally, thy rach som constant valus aftr 4. Also, in Figurs 6-0, it appars that th concntration of th componnts ar somwhat similar to thos of corrsponding Figurs 6-0. Anothr xampl is that th valu of th concntration of nzym E in both sts of figurs is mor or lss is th sam, spcially in Cass, and 5. Anothr crucial point is that th valu of concntration v in Figur rachs a maximum whn 0. Also, in th sam intrval of tim, th valu of th concntration v rachs a maximum in Figur 8 as wll. W can also raliz that th valu of th nzym in both figurs nds up at a minimum valu whn 0. In addition, Figurs -5 and Figurs 6-0 show that thr is a gradual dcras in th rat of substrat u btwn 0 which thn lvls off aftr 4. On th othr hand, th concntration of th product w slightly incrass btwn 0 in both st of figurs, and is likly to rmain stabl aftr 4. Howvr, thr ar som diffrncs btwn our simpl tchniqu (SIM) and th classical tchniqu (HPM). For xampl, Figurs -5 show that th valu of th concntration of substrat u slightly dcrass from its initial valu u 0, and thr ar a fw changs in th valu of th concntration of th nzym-substrat complx v. Gnrally, thy bcom zro aftr 5. Manwhil, in Figurs 6-0, it appars that th concntration of th componnts do not fall to zro, but instad rach som constant valus. Basically, it could b pointd out that th diffrncs btwn thm ar small and can b thrfor b ignord. Ovrall, it can b said that th scond and third itrations of SIM ar appropriat for obtaining a good approximat solution for our cas study. In particular, th rsults of th scond itration ar mor fittd to an approximat solution in comparison with th classical tchniqu (HPM). Howvr, although thr ar som diffrnt valus in trms of rsults btwn HPM and th scond itration mthod, thy ar tiny. Figurs -5. In ths profils of th normalizd concntrations of th substrat u, nzym-substrat complx v, nzym E and product w corrspond to Cass -5, rspctivly. Th quations of Stp ar applid to plot th figurs (s Appndix C). Figurs 6-0. In ths profils of th normalizd concntrations of th substrat u, nzym-substrat complx v, nzym E and product w corrspond to Cass -5, rspctivly. Th quations of Stp ar applid to plot th Figurs (s Appndix D). Figurs -5. In ths profils of th normalizd concntrations of th substrat u, nzym-substrat complx v, nzym E and product w corrspond to Cass -5, rspctivly. Th quations of Stp ar applid to plot th figurs (s Appndix E). Figurs 6-0. In ths profils of th normalizd concntrations of th substrat u, nzym-substrat complx v, nzym E and product w corrspond to Cass -5, rspctivly. Th Eqs.8 and ar applid to plot th figurs (s Appndix A). 7. FINDINGS W hav usd th man of th scond norm (Eq.44) to find th total diffrncs btwn th HPM and ach itration of th SIM (s Tabls -). Th rat of convrgnc btwn th SIM and th HPM is shown in Figur. Thus, w us th following quation to find this rat of convrgnc: N i fi g f g i (44) N N whr f and g ar th valu of th concntrations of substancs u, v, E and w for th SIM and th HPM rspctivly, and N is th numbr of timscal itrations. Th avrag norm btwn th scond itration and HPM is small in valu. For instanc, th avrag valu of th norm concntration of E is small (qual to 0.0) (s H-S in Figur ). This mans that th scond itration mthod is th most appropriat itration in this cas study in trms of approaching th approximat solution. Although th rat of th scond norm for th third itration is also small (s H-S), but th scond itration mthod of our work is th bst itration in ordr to obtain th convrgnc in trms of th solution in comparison with th classical mthod (HPM). Figur. Th avrag valu of th scond norms convrgnc btwn th HPM and th itrations of th SIM. Copyright 0 SciRs.

10 S. H. A. Khoshnaw / Natural Scinc 5 (0) Tabl. Th avrag numbr of scond norms btwn th first itration (SIM) and HPM, th rsults ar calculatd by using Matlab program (s Appndics B and C). Nor.Con. Cas Cas Cas Cas 4 Cas 5 Av. UU VV EE WW Tabl. Th avrag numbr of scond norms btwn th scond itration (SIM) and HPM, th rsults ar calculatd by using Matlab program (s Appndics B and D). Nor.Con. Cas Cas Cas Cas 4 Cas 5 Av. UU VV EE WW Tabl. Th avrag numbr of scond norms btwn th third itration (SIM) and HPM, th rsults ar calculatd by using Matlab program (s Appndics B and E). Nor.Con. Cas Cas Cas Cas 4 Cas 5 Av. UU VV EE WW On th othr hand, th diffrncs btwn our approach (SIM) and th HPM occurrd mor frquntly in th first itration than in othr itrations (s H-S valus). It could b said that this itration is not quit appropriat in this cas study. This may b causd by giving th non-linar part in this itration a zro valu (s Stp ). Th blu column (H-S), rd column (H-S) and grn column (H-S) dscrib th scond norm diffrncs btwn (HPM) and th itrations of (SIM), rspctivly. Th figur is plottd by using th rsults of Tabls -. Th scond norm diffrncs ar rprsntd by u, v, E and w. 8. CONCLUSION Th simpl itration mthod (SIM) and th Homotopy Prturbation Mthod (HPM) ar usd to find approximat analytic solutions to non-linar diffrntial Eqs.5 and 6. Straightforward mthods ar drivd for stimating th concntrations of substrat u, product w, n- zym-substrat complx v and nzym E. Th dimnsionlss tchniqu applis to rduc th non-linar systm of ODE. Th HPM was usd for a simpl nzym raction (Eq.) [,]. W hav usd this mthod for our cas study, and hav obtaind an analytical approximat solution. Furthrmor, a simpl approach tchniqu (SIM) was applid. This consistd of thr itrations (stps). Th approximat solution of th scond stp is similar to th classical mthod (HPM) (s Figurs 6-0 and Figurs 6-0). W hav also usd th ida of th scond norm to dtrmin th bst itration for th problm. So, it is clar that th scond itration mthod is quit similar to th HPM. Consquntly, Figur shows that th scond itration is th appropriat on (s Figur for th H-S valus). Thus, th SIM tchniqu could b applid to som othr complx chmical ractions to find appropriat solutions, and to dscrib th bhaviour of thir paramtrs. For xampl, it could b applid to many opn path ways in trms of biochmical ractions [7]. In addition, w highly rcommnd applying th simpl approach (SIM) to dscrib th approximat solutions of complx nzym ractions [8], th raction mchanism of comptitiv inhibitions, and th raction schm of allostric inhibitions [9]. REFERENCES [] Mahswari, M.U. and Rajndran, L. (0) Analytical solution of non-linar nzym raction quations arising in mathmatical chmistry. Journal of Mathmatical Chmistry, 49, doi:0.007/s [] Schnll, S. and Maini, P.K. (000) Enzym kintics at high nzym concntration. Bulltin of Mathmatical Biology, 6, doi:0.006/bulm [] Varadharajan, G. and Rajndran, L. (0) Analytical solutions of systm of non-linar diffrntial quations in th singl-nzym, singl-substrat raction with non mchanism-basd nzym inactivation. Applid Mathmatics,, doi:0.46/am [4] Varadharajan, G. and Rajndran, L. (0) Analytical solution of coupld non-linar scond ordr raction diffrntial quations in nzym kintics. Natural Scinc,, doi:0.46/ns [5] Li, B., Shn, Y. and Li, B. (008) Quasi-stady stat laws in nzym kintics. Th Journal of Physical Chmistry A,, -. doi:0.0/jp077597q [6] Murray, J.D. (989) Mathmatical biology. Springr, Brlin. doi:0.007/ _5 [7] Rubinow, S.I. (975) Introduction to mathmatical boilogy. Wily, Nw York. [8] Sgl, L.A. (980) Mathmatical modls in molcular and cllular biology. Cambridg Univrsity Prss, Cambridg. [9] Hanson, S.M. and Schnll, S. (008) Ractant stationary Copyright 0 SciRs.

11 750 S. H. A. Khoshnaw / Natural Scinc 5 (0) approximation in nzym kintics. Th Journal of Physical Chmistry A,, doi:0.0/jp8066 [0] Briggs, G.E. and Haldan, J.B.S. (95) A not on th kintics of nzym action. Biochmical Journal, 9, 8-9. [] Gorban, A.N. and Shahzad, M. (0) Th michalis-mntn-stucklbrg thorm. Entropy,, [] Mna, A., Eswari, A. and Rajndran, L. (00) Mathmatical modling of nzym kintics raction mchanisms and analytical solutions of non-linar raction quations. Journal of Mathmatical Chmistry, 48, doi:0.007/s [] H, J.H. (999) Homotopy prturbation tchniqu. Computr Mthods in Applid Mchanics and Enginring, 78, doi:0.06/s (99)0008- [4] Dawkins, P. (007) Diffrntial quations. [5] Gorban, A.N., Radulscu, O. and Zinvyv, A.Y. (00) Asymptotology of chmical raction ntworks. Chmical Enginring Scinc, 65, 0-4. doi:0.06/j.cs [6] Kargi, F. (009) Gnralizd rat quation for singlsubstrat nzym catalyzd ractions. Biochmical and Biophysical Rsarch Communications, 8, [7] Flach, E.H. and Schnll, S. (00) Stability of opn pathways. Mathmatical Bioscincs, 8, doi:0.06/j.mbs [8] Pdrsn, M.G., Brsani, A.M., Brsani, E. and Corts, G. (008) Th total quasi stady-stat approximation for complx nzym ractions. Mathmatics and Computrs in Simulation, 79, doi:0.06/j.matcom [9] Gok, Ch.S., Walchr, S. and Zrz, E. (0) Computing quasi-stady stat rductions. Journal of Mathmatical Chmistry, 50, doi:0.007/s x Copyright 0 SciRs.

12 S. H. A. Khoshnaw / Natural Scinc 5 (0) Appndix A. This appndix consists of th solution of Eqs.5 and 6 by using th HPM. Furthrmor, this mthod is usd to driv Eqs.8 and 9 from Eqs.5 and 6, lt a k, b and c k, q u uq u uavv 0, (45) d d d d dv dv 0, d d (46) q cv q bu cv buv v with initial conditions, u 0, v 0 0. (47) Thus, by using th HPM [,,4], th approximat solution of Eqs.45 and 46 ar: u u qu q u (48) 0 v v qv q v (49) 0 Substituting th Eqs.48 and 49 in Eqs.45 and 46, and comparing th cofficints of th lik powr q, w gt th following systm of ordinary diffrntial quations: 0 du0 q : u d 0 0, (50) du : 0 d, (5) q u av0 u0v0 du : 0, (5) d q u av u0v uv0 and. 0 dv0 q : cv d 0 0, (5) dv q : cv bu0 bu0v0 v0 0 d, (54) dv 0, (55) d q : cv bu bu0v buv0 v0v Th analytical solutions of Eqs with initial conditions Eq.47 ar: v u 0 (56) u (57) and, v0 0, (59) b v bc c c, c c c b b b c cc cc b cb cb c b b c c c c c b cb cb b c c c c c c (60) According to th HPM, w can asily find that lim 0 q (6) u u u u u (6) and, v lim v v0 v v (6) q By putting Eqs in Eq.6 and Eqs.59-6 in Eq.6, w obtain th approximat solution for th systm of non-linar ODE quations (Eqs.5 and 6) which is dscribd in Eqs.8 and 9. Appndix B. Lt k, k, k and t, and w us th following Matlab programming to plot th functions in Eqs.8-. t 0 for i :0 k k. k 0.9 k. a k k k b k c k k k k u ab abc a ac c cb c a b c c c c c c c c c a abc ac a b bc c c c c c c c u xp k t ab ck k k c txp k t abcak k ak c c k c ^ c t b k c k t k k c b k k c k c k c xp xp ^ ^ k c t a k c k a k k a b c a k c c k c xp ^ ak ck b ck abk k k ^k ck c^ k c xp k t (58) Copyright 0 SciRs.

13 75 S. H. A. Khoshnaw / Natural Scinc 5 (0) xp xp xp ^ xp cb kkbkcb ckk ckb ck c xp ct v b ck k t c bk k k c c k c c t k c b ck k b c ck k t b ck c k k t k k bc b^k c b c k k c xp ck t b k c b^ k c ck E vw kuk v Ai ubi vci EDi wt i tt t0. nd plot (T,A, r,t,b, k.,t,c, b.,t,d, g ) y labl ( Concntration of u, v, E and w ) x labl ( Dimnsionlss Tim (t) ) axis squar. Appndix C. Lt k, k, k and t, and w us th following Matlab programming to plot th functions of Stp. t 0 for i :0 k k. k 0 k. a k k k b k c k k k k p k csqrt k c ^4 ck a b p k csqrt k c ^4 ck a b d p k p p d p k p p d p k p k a p p u d xp p t d xp p t v d xp p t d xp p t v w k u k v A i u B i v C i D i w T i t t t 0.nd plot (T,A, r,t,b, k.,t,c, b.,t,d, g ) y labl ( Concntration of u, v, E and w ) x labl ( Dimnsionlss Tim (t) ) axis squar. Appndix D. Lt k, k, k and t, and w us th following Matlab programming to plot th functions in Stp. t 0 for i :0 k k. k 0. k. a k kk b k ckk k k p k csqrt k c ^4 ck a b p k csqrt k c ^4 ck a b d p k p p d p k p p d p k p k a p p d k d d d k d d k d d 4 5 d k d d d bd d ( k k d ^ 6 7 d ( bd d ) bd d k k d ^ 8 d bd d k k d^ 9 d p k a p p d p p 0 d p k a p p d p p d d d d d p d d d d d p d d d d d p p d k d p d d d d d p p d d d d d p d d d d d p d k d p d ad ad d ad a d d ad ad d ad a d d p k d p k d d p k d p k d d p k d p k d d p k d p k d 9 7 c d d d d a a d d d d p k d d d d a p p c a d d d d p k d d d d a p p u c axp p t c axp p t d xp p t d xp p p t d xp p t d xp xp xp p p t d8 p t d9 v c p k p t c p k p t d p t d xp xp Copyright 0 SciRs.

14 S. H. A. Khoshnaw / Natural Scinc 5 (0) v w k u k v Ai u Bi v Ci Di w T i tt t 0. nd plot (T,A, r,t,b, k.,t,c, b.,t,d, g ) y labl ( Concntration of u, v, E and w ) x labl ( Dimnsionlss Tim (t) ) axis squar. Appndix E. Lt k, k, k and t, and w us th following Matlab programming to plot th functions in Stp. t 0 for i :0 k k. k 0. k.4 a k k k b k c k k k k p k csqrt k c ^4 ck a b p k csqrt k c ^4 ck a b d p k p p d p k p p d p k p k a p p d k d d d k d d k d d 4 5 d k d d d bd d ( k k d ^ ) 6 7 d bd d bd d (k k d ^ ) 8 d bd d ( k k d ) d p k a p p 9 0 d p p d p k a p p d p p d d d d d p d d d d d p d d d d d p p d k d p d d d d d p p d d d d d p d d d d d p d k d p d ad a d 4 8 d ad ad d ad a d d ad ad d p k d p k d d p k d p k d d p k d p k d d p k d p k d 9 7 c d d d d a a d d d d p k d d d d a p p c a d d d d p k d d d d a p p d k c a p k k d d k d d d k c k d d c4 a p k k c4 c a p k k d d d k c a d k c p k d 6 d k c ad7 k c4 ad6 k c p k d k c ( p k ) d 4 d k c a d k c a d k c p k d k c p k d ( ) 4 d k c ad k c 5 9 p k d 5 d k c ^ap k k d d k d d d k c a d k c p k d ( d k c a d k c p k d d k d d d k d d k d d d k d d k d d k d d d k d d k d d d k d d d k d d d bd k k k c ^ p k ^k k 45 0 d d k k d d d bd k k k c c p k p k 46 4 k k c c p k p k k k d d k k d d 7 9 d bd k k k c d p k 47 6 k k c d p k 6 d bd k k k c d p k 48 7 k k c d p k 4 6 p k k k c d p k d bd k k k c d p k k k c d p k k k c d Copyright 0 SciRs.

15 754 S. H. A. Khoshnaw / Natural Scinc 5 (0) d bd k k k c d p k k k c d p k 9 d bd k k k c ^ p k k k d d k k d d d bd k k k c d p k k k c d p k 4 8 d bd k k k c d p k k k c d p k 4 9 d b d k k k d ^ d bd k k k d d k k d d d bd k k k d d k k d ^ k k d d d bd k k k d d k k d d d b d k k k d ^ ^ d bd k d k d d d d d p d d d d d p d d d d d p d d d d d p p d d d d d p d ( d d d d ) d d d d d p p d d d d d p p d d d d d p p d d d d d p d d d d d p p d d d d d p p d d d d d p d d d d d p 4 p d d d d d p d d d d d p p d d d d d p d d d d d p p d d d d d p d d d d d p p d d d d d / p p d d d d d p d d d d d p d d d d d d d d d d 4 p p d d d d d p d d d d d p p d d d d d p p d d d d d p d d d d d p d ad ad d ad a d d ad ad d ad a d d ad ad d a d d ad d ad a d d ad ad d ad d a d d ad ad d ad a d d ad ad d ad a d d ad ad d ad a d h p k h p k d h d h d d h d h d d h d h d d h d h d d h d h d d h d d h d d h d h d d h d h d d h d d h d d h d h d d h d h d d h d h d d h d h d d h d d d d d d d d0 d0 d 0 d04 d05 d06 N d d d d d d d d d d d d d d d c5 M h an a h h c N h h h c 5 6 Copyright 0 SciRs.

16 S. H. A. Khoshnaw / Natural Scinc 5 (0) xp xp xp xp d9xp ptd9xp p ptd94xpp ptd95txppt d96 xpp td97 xp p td98 xp p td99 xp p t d00 txp p td0 xp4 0 xp 0 xp d04 xpp p td05 xp4 p td06 v c5 h xp p t c6 h xp p t d07 xp p t d08 xp p p t d09 xp p td0 xp p p td xpp p td txpp t d xpp td4 xp p td5 xp p td6 xp p t d7 txp p td8 xp4 p td9 xp p p td0 xp p p t d xpp p td xp4 p td u c a p t c a p t d p t d p p t p t d p p t d p p t v w ku kv A i u B i v nd plot (T,A, r,t,b, k.,t,c, b.,t,d, g ) y labl ( Con- cntration of u, v, E and w ) x labl ( Dimnsionlss C i D i w T i t t t0. Tim (t) ) axis squar. Copyright 0 SciRs.