Derivation of the Michaelis-Menten Equation. The Michaelis-Menten equation is an important equation in biochemistry and as

Transcription

1 Derivaion of he Michaelis-Menen Equaion The Michaelis-Menen equaion is an iporan equaion in biocheisry and as such i is iperaive ha you undersand he derivaion of his equaion. By undersanding he derivaion, you will have insigh ino he assupions ha wen ino his odel, and herefore you will have a beer appreciaion for he proper use of his equaion as well as he liiaions of his odel. In he following secions you will see wo differen derivaions of he Michaelis-Menen equaion. When one is learning a subjec for he firs ie, i ofen helps o have he sae or siilar inforaion presened fro alernaive perspecives. One way igh be clearer o you whereas he oher way igh be clearer o soeone else. Tha is o! You should failiarize yourself wih boh approaches, and hen sele on he one ha you prefer. The firs derivaion was adaped fro An Inroducion o Enzye ineics by Addison Aul (J. Che. Ed. 974, 5, 8 86). Firs Derivaion. We sar wih he ineic echanis shown in equaion (eq) : E + S ES E + P () In eq, E is enzye, S is subsrae, ES is he enzye-subsrae coplex, and P is produc. This equaion includes he assupion ha during he early sages of he reacion so lile produc is fored ha he reverse reacion (produc cobining wih enzye and re-foring subsrae) can be ignored (hence he unidirecional arrow under ). Anoher assupion is ha he concenraion of subsrae is uch greaer han ha of oal enzye ([S] >> [E ]), so i can essenially be reaed as a consan.

2 Fro General Cheisry we can equae he rae of his process ( ) o he change in produc concenraion as a funcion of ie (d[p]/d), or, equivalenly, we can designae he rae wih an ialicized v (v) as follows in eq : d[p] v () d Because he concenraion of he enzye subsrae coplex () canno be easured experienally, we need an alernaive expression for his er. Because he enzye ha we add o he reacion will eiher be unbound (E) or bound (ES) we can express he fracion of bound enzye as follows: () [E ] [E] In eq E is he concenraion of oal enzye, and he oher variables are as defined above. If we uliply boh sides of eq by E we arrive a eq 4: [E ] (4) [E] If we uliply he nueraor and denoinaor of he righ-hand side of eq 4 by /, we are, in effec, uliplying by one and we do no change he value of his expression. When we do his we obain eq 5: [E ] (5) [E] We have alos achieved our goal of isolaing. Nex, we need o coe up wih an alernaive expression for he raio [E]/. We do his by recalling ha a ajor assupion in enzye ineics is he seady-sae assupion. Basically, i says he rae of change of as a funcion of ie is zero: d/d = 0. Anoher way o express he

3 seady-sae assupion is ha he rae of foraion of ES equals he rae of breadown of ES. We can express his laer saeen aheaically as in eq 6: [ E ( ) (6) The lef-hand side of eq 6 expresses he rae of foraion of ES (according o eq ), and he righ-hand side expresses he wo ways ha ES can brea down (also according o eq ). We can rearrange eq 6 o isolae he raio [E]/. When we do we ge eq 7: [E] ( ) (7) [S] We now define a new consan, he Michaelis consan ( ), as follows in eq 8: ( (8) ) If we subsiue bac ino eq 7 we obain eq 9: E] (9) [S] [ We now subsiue he raio /[S] fro eq 9 in place of he raio [E]/ in eq 5 and we obain eq 0: [E ] (0) [S] If we uliply he nueraor and denoinaor of he righ-hand side of eq 0 by [S], we are, in effec, uliplying by one and we do no change he value of his expression. When we do his we obain eq : [E [S] [E [S] ()

4 Now we have achieved our goal of isolaing and we can subsiue his alernaive expression of ino eq and obain eq : [E v () [S] Nex, we iagine wha happens o eq when [S] > > as follows in eq : [E v [E ] ca[e ] () [S] The consan ca in he righ-hand os er of eq is used o signify ha is considered he caalyic consan. Under such condiions, when [S] is said o be sauraing, he enzye is funcioning as fas as i can and we define [E] (or ca [E ]) o be equal o V ax, he axiu velociy ha can be obained. Therefore, eq can be rewrien ino he failiar for of he Michaelis-Menen equaion (eq 4): Vax[S] v (4) [S] Nex, we iagine wha happens when > > [S] as follows in eq 5: Vax[S] v [S] (5) Since = V ax / in eq 5, we refer o V ax / as an apparen (or pseudo) firs order rae consan. Anoher way o loo a a siilar, relaed concep is o rewrie eq 4 as follows: ca[e v (6) [S] Since we are iagining he case where > > [S] we neglec [S] in he denoinaor and include he assupion ha [E ] [E] since a very low [S] relaively lile should for: 4

5 ca[e v [E (7) Once again, since = ca / in eq 7, we refer o ca / as an apparen second order rae consan. Because ca / is a easure of he rae of he reacion divided by he er ha reflecs he seady-sae affiniy of he enzye for he subsrae, i is considered an indicaor of he caalyic efficiency of he enzye and soeies is called he specificiy consan. I also is ore relevan o he physiological siuaion because in cells, [S] generally is equal o or less han. Is here an upper lii o he value ha ca / can approach? Yes, here is and he following shows how we can deerine his lii. To illusrae his lii we firs need o rewrie ca / as follows: ca (8) Nex, we iagine he case where >> : ca (9) So we see ha ca / can approach as a liiing value, and is he second-order rae consan for he producive collision of enzye and subsrae and as such i is liied by diffusion o abou M s. Thus, if we see an enzye ha has a ca / value in he neighborhood of M s we say ha he enzye has aained caalyic perfecion. You will see laer in he class ha a nuber of enzyes ha caalyze nearequilibriu reacions in eabolic pahways are caalyically perfec. Nex, we reurn o eq 6 and consider wha happens when v = ½ V ax : 5

6 V V ax ax [S] [S] (0) When we siplify eq 0 we find ha = [S] (under he above condiions; i.e., v = ½ V ax ). So, in oher words, is forally defined as a collecion of rae consans (eq. 8), bu i is also equal o he subsrae concenraion ha gives half-axial velociy of he enzye-caalyzed reacion. Before we discuss he second derivaion, we will consider wha happens when we ae he reciprocal of boh sides of eq 4. When we do his we obain eq : () v V [ S] V ax ax Eq is in he for of an equaion for a sraigh line (i.e., y = x + b, wih y = /v; = /V ax ; x = /[S]; and b = /[V ax ]). When experienal daa are ploed using his ransforaion he resuling plos are called double-reciprocal plos or Lineweaver-Bur plos in honor of he researchers who pioneered his ehod. The auhors of any exboos exol he virues of using Lineweaver-Bur plos o obain esiaes of V ax and. I disagree srongly wih his pracice because iniial velociy daa deerined a low subsrae concenraions (where here is inherenly ore uncerainy since [S] ) end up being he poins in a Lineweaver-Bur plo ha have oo uch sway in deerining he bes-fi line hrough he daa (see for exaple, Disadvanages of Double Reciprocal Plos by R. Bruce Marin, J. Che. Ed. 997, 74, 8 40). In fac, Lineweaver and Bur recognized his proble and in heir faous paper (J. A. Che. Soc. 94, 56, ) hey had consuled wih a saisician o deerine he proper weighing facors for he daa poins. To repea: eq is no a useful for of he equaion for obaining esiaes of V ax and. Insead, one should use a 6

7 odern sofware progra ha allows for ieraive fiing of he experienal daa o esiae hese paraeers. (This ype of fiing is called nonlinear leas-squares fiing.) Eq is useful for ploing daa ha have been obained in he presence of increasing concenraions of an inhibior. Such plos allow a researcher o diagnose he ype of inhibiion ha is occurring. We will discuss his laer on in class. Second Derivaion. For he second approach, we consider equaions,, and 6 (repeaed for your convenience): E + S ES E + P () d[p] v () d [ E ( ) (6) Once again, because we canno experienally easure, we see an alernaive expression for his paraeer so ha when we obain his alernaive expression we can uliply i by in eq and hus obain our desired rae equaion. To proceed, we will divide he righ-hand-os and lef-hand-os expressions in eq 6 by ( + ). Doing so will isolae. While we are a i, we will collec rae consans and define as we did above in eq 8: [E [E [E () ( ) ( ) I loos lie we succeeded in isolaing, bu noice ha on he righ-hand side of eq we have free enzye (i.e., [E]). Jus as we are unable o experienally easure, we also are unable o experienally easure [E]. All we can say definiively is 7

8 ha free enzye ([E]) is equal o oal enzye ([E ]) inus he for of he enzye ha is bound o subsrae (): [ E] [E ] () Subsiuing he righ-hand side of eq in place of [E] in eq leads o eq 4: ([E ] )[S] [ ES] (4) Muliplying boh sides of eq 4 by and uliplying [E ] and by [S] in he nueraor of he righ-hand side of his equaion leads o eq 5: [ ES] [E [S] (5) Adding [S] o boh sides of eq 5 and hen facoring ou leads o eq 6: [ ES]( [S]) [E (6) Finally, dividing boh sides of eq 6 by ( + [S]) effecively isolaes and yields eq 7: [E ES] ( [S]) [ (7) Eq 7 is he sae expression ha we obained in eq. Our wor is done. (Well, i s done once we subsiue he righ-hand side of eq 7 ino eq and hen subsiue V ax for [E ] as we did above when we obained eq 4; hese las few seps yield he sough afer Michaelis-Menen equaion.) 8