ON THE MICHAELIS-MENTEN ENZYME MECHANISM

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1 Romanan Reports n Physcs Vol. 57 No. 3 P ON THE MICHAELIS-MENTEN ENZME MECHANISM C. TIMOFTE epartment o Mathematcs Faclty o Physcs Unersty o Bcharest P.O. Box MG- Bcharest Magrele Romana E-mal: cladatmote@hotmal.com Receed Jne Abstract. The am o ths paper s to stdy the asymptotc behaor o the solton o a nonlnear problem arsng n the modelng o enzymatc reactons throgh poros meda. The doman s consdered to be a xed bonded open sbset R n n whch dentcal and perodcally dstrbted peroratons o sze are made. The asymptotc behaor o the solton o sch a problem s goerned by a new ellptc bondary-ale problem wth an extra zero-order term that captres the eect o the enzymatc reactons. Key words: homogenzaton enzymes Mchaels-Menten model.. INTROUCTION The am o ths paper s to stdy the homogenzaton o some nonlnear reacte lows throgh perodcally perorated meda. Sch problems are ery natral n the stdy o enzymatc reactons throgh poros meda and more precsely n the stdy o the so-called Mchaels-Menten model or a nce presentaton o the chemcal aspects noled n sch a model and or some hstorcal backgronds see [8]-[9] and the reerences theren. Enzymes are protens that speed p the rate o a chemcal reacton wthot beng sed p. Enzymes are sally specc to partclar sbstrates. The sbstrates n the reacton bnd to acte stes on the srace o the enzyme. The enzymesbstrate complex then ndergoes a reacton to orm a prodct along wth the orgnal enzyme. The rate o chemcal reactons ncreases wth the sbstrate concentraton. Howeer enzymes become satrated when the sbstrate concentraton s hgh. Addtonally the reacton rate depends on the propertes o the enzyme and the enzyme concentraton. We can descrbe the reacton rate wth a smple eqaton to nderstand how enzymes aect chemcal reactons. Mchaels- Menten eqaton remans the most generally applcable eqaton or descrbng enzymatc reactons. Let be an open bonded set n R n and let s perorate t by holes. As a reslt we obtan an open set whch wll be reerred to as beng the perorated doman "!#$% '&$ *& *-&'.-/!23$ peroratons. We shall deal wth the case n whch the peroratons are dentcal and perodcally dstrbted and ther sze s o the order o. The nonlnear problem stded n ths paper concerns the statonary low o a ld conned n o concentraton reactng nsde and on the

2 297 On the Mchaels-Menten enzyme mechansm bondary o the peroratons: ν β a g on where ν s the exteror nt normal to a > L 2 S s the bondary o the peroratons and s the external bondary o s a constant dson coecent characterzng the homogeneos and sotropc ld. We shall consder that the ncton β n s a contnosly derentable ncton monotonosly non-decreasng and sch that β. Also n the semlnear bondary condton on the srace o the peroratons n the ncton g s assmed to be gen and we shall address here the case n whch g s a snglealed maxmal monotone graph wth g.e. the case n whch g s the sbderental o a conex lower semcontnos ncton G. Ths general staton s well llstrated by the ollowng mportant practcal example arsng n the dson o enzymes and more precsely n the so-called Mchaels-Menten model: n on δ γ g δ γ >. < The exstence and nqeness o a weak solton o can be settled by sng the theory o semlnear monotone problems see [] and [7]. As a reslt we know that there exsts a nqe weak solton V H 2 where I wth then V { H on }. we assocate the nonempty conex sbset o K S V { V G L S } s also the nqe solton o the aratonal problem S

3 298 C. Tmote that sch Fnd µ β K G G a K 2 where µ s the lnear orm on W dened by ϕ σ ϕ ϕ µ S W d. We shall proe that the solton extended to the whole o conerges weakly n H to the nqe solton o the ollowng aratonal neqalty: H G G T a Q H β 3 Here j q Q s the homogenzed matrx whose entres are dened by: j j j dy y q δ 4 n terms o the nctons j soltons o the so-called cell problems perodc. on n T y ν 5 The approach we sed s the so-called energy method ntrodced by L. Tartar [] or stdyng homogenzaton problems. It conssts o constrctng stable test nctons that are sed n or aratonal problems.

4 % & 299 On the Mchaels-Menten enzyme mechansm The strctre o or paper s as ollows: rst let s menton that we shall jst ocs on the case n 3 whch wll be treated explctly. The case n 2 s mch smpler and we shall omt to treat t here. In Chapter 2 we ntrodce some sel notatons and assmptons and we ge the man reslt. In Chapter 3 we ge the proo o the man conergence reslt o ths paper. 2. SETTING OF THE PROBLEM AN THE MAIN RESULT 2.. Notaton and assmptons! " R n wth bondary # $ 2 class C. Let l [ [ l [... [ l [ be the representate cell n R n and T [ 2 n 2 an open sbset o wth bondary T o class C sch that T. We shall reer to T as beng the elementary hole. We shall denote by T k the translated mage o T by kl k Z n. Also we shall denote by T the set o all the holes contaned n and by \ T. Hence s a perodcally perorated doman wth holes o the same sze as the perod. We shall se the ollowng notatons: \ T S T θ. Also we shall denote by the characterstc ncton o the doman Settng o the problem As already mentoned we are nterested n stdyng the behaor o the solton n sch a perorated doman o the ollowng problem: β n ag on S ν 6 on Here ν s the exteror nt normal to a > S s the bondary o the holes and s the external bondary o s a constant dson coecent characterzng the homogeneos and sotropc ld. We shall consder that the ncton β n 6 s a contnosly derentable ncton monotonosly non-decreasng and sch that β. Also n the semlnear bondary condton on the srace o the peroratons n

5 3 C. Tmote 6 the ncton g s assmed to be gen and we shall address here the case n whch g s a sngle-aled maxmal monotone graph n R R wth g.e. the case n whch g s the sbderental o a proper conex lower semcontnos ncton G. Moreoer we shall sppose that the doman g R g s contnos and there exst a poste constant C and an exponent q wth q < n / n 2 sch that β C g C q q. 7 Let G g s ds. Ths general staton s well llstrated by the aboe mentoned mportant practcal example Mchaels-Menten model. The exstence and nqeness o a weak solton o can be settled by sng the classcal theory o semlnear monotone problems see [] and [7]. As a reslt we know that there exsts a nqe weak solton V H 2 where I wth V { H on }. we assocate the nonempty conex sbset o V K { V G L S } S then s also the nqe solton o the aratonal problem Fnd K sch that β a µ G G K 8 where µ s the lnear orm on W dened by µ S ϕ ϕ d σ ϕ W. 9 In order to descrbe the asymptotc behaor o the solton o problem 8 let s recall the ollowng well-known extenson reslts see [2]-[3]:

6 3 On the Mchaels-Menten enzyme mechansm Lemma 2.. There exsts a lnear contnos extenson operator 2 2 P L L ; L L V ; H and a poste constant C ndependent o sch that or any V P 2 C L L 2 and P C 2 2 L L. As a conseqence the ollowng Poncaré s neqalty holds tre n V : Lemma 2.2. There exsts a poste constant C ndependent o sch that or any V. 2 C L L The man reslt The man reslt o ths paper s the ollowng one: Theorem 2.3. Let be the nqe solton o the problem 8. Then there exsts an extenson P o nto all sch that P w weakly n H and s the nqe solton o the ollowng problem: H Q β T a G G H. Here Q q s the homogenzed matrx whose entres are dened by 4-5. j

7 32 C. Tmote 3. PROOF OF THE MAIN RESULT Proo o Theorem 2.3. Let be the solton o the aratonal problem 8 and let P be the extenson gen by Lemma 2.. Takng ϕ as a test ncton n 8 t s not dclt to see that P s bonded n H. So by extractng a sbseqence one can assme that there exsts H sch that P w weakly n H. It remans to denty the lmt eqaton satsed by. Let ϕ C. By classcal reglarty reslts L. Usng the bondedness o ϕ and there exsts M sch that ϕ < M. L x Let L ϕ x ϕ x. 2 x Then K. Moreoer ϕ strongly n 2 L. Let s compte : Usng ϕ x ϕ x x e x. x x as a test ncton n 8 we hae ~ P β. a µ G G 3 enote ρ Qe j. j e j dy 4 ϕ x Neglectng the term x whch tends strongly to zero we can x pass to the lmt n the let-hand sde o 3. Hence ~ P ρqϕ. 5

8 33 On the Mchaels-Menten enzyme mechansm For the second term n the let-hand sde o 3 let s notce that exactly lke n [6] one can easly proe that or any w ϕ C and or any z z weakly n H we get ϕβ z ϕβ z strongly n L q where Thereore we hae q 2n. q n 2 n β β ρ ϕ. 6 It s not dclt to pass to the lmt n the second term o the rght-hand sde o 3. We get P ρ.. 7 ϕ In order to pass to the lmt n the last term o 3 ollowng [3] and [6] let s ntrodce the lnear orm µ on W dened by µ S ϕ ϕ d σ ϕ W. From [3] we know that µ µ strongly n W 8 where T µ ϕ ϕ. On the other hand F s a contnosly derentable ncton monotonosly non-decreasng wth F and sch that there exst a poste constant and an exponent q wth q < n / n 2 sch that F q C t s not dclt to proe see [6] that or any ϕ C and or any z w z weakly n H one has

9 34 C. Tmote w ϕf z ϕf z weakly n W q 9 where q 2n. q n 2 n Now or the last term n the rght-hand sde o 3 assmng 7 or the monotone graph g and sng 9 wrtten or G and or z P we hae G P G weakly n q W. Combnng ths wth the conergence 8 we obtan T µ G P G. Hence we get T µ G G P G ϕ G. 2 It remans to pass to the lmt only n the last term o 3. For dong ths we can wrte the sbderental neqalty w w 2 w w 2 or any w H. Reasonng as beore and choosng w ϕ x ϕ x x where ϕ and M enjoy smlar propertes as the correspondng ϕ and M the rght-hand sde o the neqalty 2 passes to the lmt and one has lmn ρ Qϕϕ 2 ρ Qϕ ϕ or any ϕ C and by densty or any ϕ H. Hence or H we conclde

10 35 On the Mchaels-Menten enzyme mechansm lmn A ρ Q. 22 Pttng together 5-22 we get ρ Qϕ β ρ ϕ T ρ Q ρ ϕ a G ϕ G or any ϕ C and hence by densty or any H. So nally we obtan Q β T a G G whch s jst the lmt problem. Snce H.e. on and s nqely determned the whole seqence proed. P conerges and Theorem 2.3. s REFERENCES. H. Brézs Analyse Fonctonelle. Théore et Applcatons Masson Coranesc P. onato An Introdcton to Homogenzaton Oxord Lectre Seres n Mathematcs and ts Applcatons 7 The Clarendon Press Oxord Unersty Press New ork Coranesc P. onato H. Ene Homogenzaton o the Stokes problem wth non homogeneos slp bondary condtons Math. Meth. Appl. Sc Coranesc J. Sant Jean Paln Homogenzaton n open sets wth holes J. Math. Anal. Appl C. Conca J. I. az C. Tmote Eecte chemcal processes n poros meda Math. Models Methods Appl. Sc. M3AS C. Conca J. I. az A. Lñán C. Tmote Homogenzaton n chemcal reacte lows throgh poros meda Preprnt Unersdad Compltense de Madrd J. L. Lons G. Stampaccha Varatonal neqaltes Comm. Pre Appl. Math L. Mchaels M. L. Menten e knetk der nertnwrkng. Bochem. Z L. A. Segel Modelng dynamc phenomena n moleclar and celllar bology Cambrdge Unersty Press New ork L. Tartar Problèmes d homogénésaton dans les éqatons ax derées partelles n Cors Peccot Collège de France 977.