Parameter estimation reliability in enzyme kinetics depends upon the substrate range concentrations under assay. An inappropriate concentration set may lead to spurious values of km and Vmax in the Michaelis-Menten approach. In this paper, the theoretical arguments for a practical criterium concerning the best work range of substrate concentration are discussed on a velocity ratio basis (V1/Vn) as response to the pertinent substrate concentration ratio (S1/Sn)

BRUNE, W.

FABRIS, J. D.

Repository Open Access to Scientific Information from Embrapa

Arq.Biol.Tecnol.31(4):527-538,nov.,1988FABRIS,J.D.1988PARAMETER EVALUATION IN MICHAELIS-MENTEN KINETICS(~ )Walter Brune and José "Domingos FabrisDepartamento de Qurmlca - Universidade Federal de Viçosa36570 - Viçosa - MG - BraZI I(~ )CNPMS/EMBRAPACa Ixa Posta I 15135700 - Sete Lagoas - MG - Brazl 1Recebido para publicação em 12 de agosto de 1988ABSTRACTParameter estlmation rellablllty In e nz vme kinetlcs a ee e na supon the substrate range concentratlons under assay. AnInapproprlate concentratlon set may lead to spurlous values of Kmand Vmax in the Michael is-Menten approach. In thls paper, thetheoretlcal arguments for a practlcal crlterlum concernlng thebest work range of substrate concentratlons are dlscussed 00 a~~1~~11y ratio basis (V1/Vo) as response to the pertioentsubstrate ~Qnk~nlL~11~n ratio (S1/50).INTRODUCTIONThe Mlchaells-Menten eqUatlon (I) has been sUbJected to avarlety of comments on what may be referred to as te~tbookdlstorttons, such as goodness-of-flt for estlmates of error (1)and the comparlson of reciprocal m"1crocomputerplota wlth527".Arq.Biol.Tecnol.JI(4):527-538,nov.,1988softwares (2).Vmall . S( 1 )vKm + SIn fact, ínaccurate graphs of velocíty vs concentratlon areparticularly common In biochemlcal telltbooks as has been polntedout recently by NAQUI (3), who a t t uces to the fact that In e t o t sof V vs 5 the e xp rees t on (11) applles(dV/d5)5=0Vmall/Km ( I I )As a suggestl&n we should Ilke to recai I a further helpfuldrawlng aid: the slope (I li)(dV/d5) = Vmax/~Km.5=Km( 11 I )At thls polnt V vmax/2 as evldenced by the Mlchaells-Menten e qu a t Ion (I) for 5 = Km.The estlmatlves of parameters (Km and Vmax) by Ilnearlzatlon(suCh as In Llneweaver-Burk representatlons, FIGURE 1) are atpresent somewhat obsolete havlng been superseded by computer•techn Iques (~).The following symbols apply:V: Inicial veloclty of an enzyme-catalyzed reactlon, obeylngHlchaells-Menten klnetlcs; Vmax: Maxlmum conventlonal veloclty ofsucn a r eec t lcn : Km: Mlchaells-Henten constant; 51 (I = , ••• n):Substrate concentratlon, 51 > ... > 5n; GC: GeOmetrlc Center. GC1/2= (51 x sn )528-.,c •Arq.Biol.Tecnol.31(4):527-538,nov.,1988-12 V_li-1V.os , .•--,4.16 V.n -------------------- __~~------_r----~--------------~~S-I3.16 = GC-l o0:1 S~i------~------------------~i--S~3..•6 0.3110.'316- GCFIGURE 1. Lineweaver-Burk plot. Km lIes wlthln the experimentalr a n 9e 5 n 151 = 10; Km = 0.31 fi •This, however, does not exempt from searchlng for the mostrellable range of 5'5 r n o rc e r to mt n t m lze t n e consequences ofexperimental error. As a premlse, t n t e rpo t e t r o ns cause smallerdlstortlons than extra~olatlons. In other words, a range ofexperimental 5's encompasslng Km Is preferable to those notcomprising.lt.In enzyme klnetics the array of 5's represents theIndependent variable and that of V's the dependent varlable.By geometrlc argumenta It 15 wel 1 establlshed that slopesnear unlt yield more dependable readlngs than those devlatingconslderably from it (5)'In Llneweaver-B rk plots the slope rep~esentlng Mlchaells-Menten klnetics Is Km/Vmax. The Intersectlon of the stralght Ilne-1 -1wlth the ordinate reglsters Vmax 50, 2 Vmax of the ordlnate-1yields Km on the absclssa (fiGURE 1). The value of Km In thisfigure may oe t a oe r t ee as "c e nt e re d " t s n < Km < 51> IdentlfylngItself wlth GC.If the range of experimental 5'5 does not encompass thehltherto unknown Km-value, toere fOi lows one of two posslblealternatives:529Arq.Biol.Tecnol.31(4):527-538,nov.,1988a) Km may ee t e s s than the smallest applled 5 of theexperimental range, Km < 5n. In llneaweaver-Burk representatlons-1sueh a Km -value takes a more dlstant posltlon than theexperimental range itself. The slope turns out to bepreearlously low, 50 that the aeeuraey of Km-measurements areendangered. In consequence, experimental error affects thécomputation of Vmax and Km more than In centered posltlons(FIGURE 2).J2::l--------:~=:-,I ,C:-t Ic} r- -io-.----,"'T"'c-----------r,-II I I S0.1 0.D5....,,~-~"----------""T,-II I SII(.20 2FIGURE 2. lineweaver-Burk Plot. Km 15 smaller t n a n s n , thesmallest experimental value. 51/Sn = 10; Km = 0.05b ) Km may turn out to be greater than the largest appl t e d 5,-1Km > 51. As a eonsequenee in lIneweever-Burk p t o t s Km 15located eloser to the ordlnate (FIGURE 3) whleh leads tounconvenlently steep slopes, ageln i mp e r I I I n s a c c u r e c yordlnate Intersectlon readlngs (5, 7).METHODOLOGYAn set of data dea IIng wl th the beta-experimentalglueosidase of a fungus (tl~ml~21~sp) hydrolyslng p-nitrophenylbeta-glueoside (8) may help make our polnt. The unabrldged set ofthese observations runs from 5n 0.05 to 51 = 2, expressedemp lr J ca l unlts. FIGURE 4 re co rds the observed Inltlal velocltyas a function of the appl led substrate eoneentratlon by way of530OfIn..Arq.Biol.Tecnol.31(4):527-538,nov.,1988v-tIIIIt----""*, :-5-110CilCt----..",O""'!I:-- 5t-----r, - S/K.o.OS0.5fiGURE 3. Llneweaver-Burk plot. Km is larger than 51, the largestexperimental value Sl/Sn = 10; Km =" 2the Llneweaver-Burk representation. Vmax and Km have beeneva I uated i n t n e usua I tiay, wh II e t n e geometr Ic center between-1 -1 -1 -1/2Sn and 51 Is located at GC = (51 x s n )DISCUSSION-1 -1In a range, e. 9., from 51 1 to Sn = 10, the se ome t r r e-1center I ies at 5 3.16 (see also FIGURE 1). The goodness-of-centerlng can be expressed by the ratlo Vl/Vn where thestibscripts correspond to those of their pertinent S's. ThlS ratlo'IdependS not only on the centering but also upon the width of thesubstrate range, but not on the empirical scale. This can betaken as a rei lable criterium for the centering of Km, stretchlngfrom unlt (Km « s n , fiGURE 2) to SliSn (for Km » Si, r r cuar 3),ten in the present case.The original equation of Michaelis-Menten (I) In Itswreduced" form, where yi = Vi/Vmax and xl = SI/Km, takes theshape-1Y i-11 + xi ( I V )531Arq.Biol.Tecnol.JI(4):527-538,nov.,198825í20 (a)(b)((c )•FIGURE 4. Llneweaver-Burk plot of a set of data deal Ing wlth thebeta-glucosldase of a fungus (tlYml~~l~sp) hydrOlysingp-nltrophenyl beta-glucoside (8). GG: Geometrlc center.(a) The whole set ranging from 51 = 2 5n = 0.05: (b) Asubset from 51 = 2 to 5n =0.2; (c) A subset from 51 =0.5 to 50 = 0.05. Arbltrary uolts.532-1 -1R a (1 + xn )/(1 + x1 ), (V)".0. Arq.Biol.Tecnol.Jl(4):527-538,nov.,1988"the veloclty ratlo R = V1/Vn may be expreaaed bylince It has the lame meanlng as Y1/yn, the pertlnent velocltyratlo (fiGURE 5).,-I2;11'----i-;--i--:---------.....;....---- s-'K.' $2-1----,i~_'rt------------------T~------- x-IXI Xi-'-I'.•fiGURE 5. Mlcheells-Menten eqUetlon In Its reduced form,-1expressed through the llneweaver-Burk plot. x = Km/s;-1y = Vmax/V.for goodness-of-flt Judgements arbltrary tolerance IImita at0.5 GC and 2 GC can be establlshed, belng equldlstant from the GCIn the geometrle sense by 100~.The llneweaver-Burk data treatment to the above quotedI:Blml~JüJl set y Ie Ids Vmax = 0.300 and Km :: 0.68'9 (TABlE I>.2The eorrelatlon eoeflelent (r = 99.6 ~) dlscloses a hlghlyrellable set: On the other hand the Km value 0.689 Iles fer apartfrom the geometrle center (GC = 0.316) of the whole set; Itsposltlon Iles above the proposed toleranee IImlts 0.158 (= 0.316x 0.5) and 0.632 (= 0.316 x 2). So the assayed range Df S's turnsout to be rather low for Km evaluatlon notwlthstandlng the faetthat Km st r t t t les wlthln the applled s r s . To emp nas í ze the533,.'Arq.Biol.Tecnol.31(4):527-538,nov.,1988TABLE I - Parameter an a t ys t s an d rellabillty test of a se t ofdata dealing with the beta-glucosidase of a fungus(~Yml~~l~ sp) hydrolysing p-nltrophenyl beta-glucoslde< B) •Range of S's (Sn to S1)Whole Set Bisected Sets0.05 to 2 0.2 to 2 0.05 to 0.5llneweaver-Burk Parameter SearchlngSlope (Km/Vmax) 2.29 1.94 2.33-1Intercept (Vmax ) 3.333 4.255 2.786Vmax 0.300 0.235 0.359..Km 0.689 0.456 0.8382r 'J,-Correlatloncoefflcient 99.6 98.8 99.7Rellabll ity TestVelocity Ratlo, R = V1/VnExpected (*) 1~.98 2.67 6.63Observed (**) 9.40 2.69 5.91R-Tolerance Iimita (***)lower 3.86 2.23 2.234.49, 4.492.193 1.1931.58 6.323.467 7.5471.243 2.0211.439 2.420Upper 10.37-1Km 1.451-1GC 3.16-1(Km.GC) 4.590u = in [1/(Km.GC)] 1.524-1s ' a t u 1.702(*) From the fltted llneweaver-Burk stralght Ilne.(**) From experimental data.<***) Replaclng Km elther by 0.5 x GC (Iower Ilmit) or by 2 x GC(upper Ilmlt) on equatlon (V).Importance Df the proper cholce of S's, we bl8ected the wholerange Into one portlon of low and another one Df hlgh S'8, e8chfragment comprlslng a ratlo of Sl/Sn = 10. Detal 18 of fractlónlngand results can be followed In TABlE I as wel I as In fiGURES 4and 6.534".. Arq.B101.Tecnol.31(4):527-538,nov.,1988v'105L---------------,-~~~~-------T~~----__S·1fiGURE 6. Set of data used In fiGURE q. demonstratlng thelnterference of range-wldth and posltlon on Km and Vmaxestlmatlons. (A) The whole set ranging from S1 • 2 Sn •0.05; (b) A s u ns et from S1 2 to Sn =0.2; (c) A subsetfrom 51 = 0.5 to Sn = 0.05.In search for the best posltlon of a range of S's it seemsadvlsable to introduce a logarlthmic scale (5, 9) whereby unlfledS1'Sn ratlos acquire equal lengths. 50, e. g., a range choaen asS1'Sn = e, i .e .•In 51 - r n Sn = 1. (V I )"has only one deflnlte length, wherever the range may lle.uIntroducing x = e • the equatlon (IV) assumes the form-1Y-u, + e (V 11 )A plot of y va u produces the graph In fiGURE 7. In thlsfigure one may verlfy the assumptlon that a range In theneighbourhood of u o exhlblts the hlghest slope. As aconsequence t n e experimental error n as t n e smallest Influence on535Arq.Biol.Tecnol.31(4):527-538,nov.,1988the results In thls reglon. Thus. a range In such a posltlonylelds the best flt.The response of two u-values to thelr respectlve V's varIesaccordlng to the pertlnent slope of the graph In FIGURE 7.r ne s t oee dy/du. a r eo c e i r ee v'. may ne u nc e rs t ood In t n t scontext as an e xe rese r o n of r e r r e e r r r rv of experImental values.-1Its reclprocal value v' on the contrary, tells about theeffect experimental error mav cause on calculated parameter-1estimates. The graphlcal express10n y' V8 U comes out to be anexponentlal curve (FIGURE 8) wlth remarkable Influences ofexperimental error even at dlstances apparently as amal I as u = ±2. In the above quoted set the factor Introduced bV a somewhatimproper range wlth u = 1.524 amounts to 1.702. The correct Km-value of the considered tl~ml~~l~experlment thus may"'""1through the expre~slon In Km ± In (V' ).be foundInto account "the whole set,In consequence, taklng-1labelled by Km = 0.689 and v ' ::1.702 (TABlE I), the most plauslble value for Km Iles between1.173 (= 0.689 x 1.702) and 0.405 (= 0.689/1.702). Thls span forthe slte of Km refers to what may be called a WsystematlcW error.It .contr Ibutes to the error obtalned through Inevltable,U-I.X-4 -z -I o .1 +Z +4I , i Iotxi i i i i i i-1.5 -I -0,1 -0.5 O .eu "'0,5 .0.1+1,0 .'.5XI I i i i i i I i01 o.z O,33Z 0.8 I 2 132 110-1 -uFIGURE 7. Reduced Hlchaells-Henten equatlon y :: 1 + euSubstrate concentratlons x are recorded as e .536Arq.Biol.Tecnol.Jl(4):527-538,nov.,1988measurement shortcomlngs expressed by Its standard devlatlon.In concluslon we euggeet the followlng p~ocedure for aIII,jI,~~iI-I",-,TABLE 1 aleo makee clear that the range from 0.2 to 2 offere a.more dependable readlng for Km than the Improperly centeredthough broader range from 0.05 to 2. An analogoue reeult can bedrawn of couree for Vmax-evaluatlone.re8eonably dependable eetlmate of Km and Vmax: In a prellmlnarytrlal, one may chooee two 5'e wlth the ratlo 51'Sn = 10. If theobserved V'e turn out to reeult In a veloclty ratlo R < 2.2,meaeurements wlth lower 5's should be trled. On the other hand,If R > Q.5, hlgher S's should be tested. Theee two arbltrary R-limite are then a practlcal prior crlterlum of rellablllty Inenzyme klnetlce etudles.Such recommendatlons enould not lead one to undereetlmatethe otherwlee us6ful Informatlon on low and hlgh S's. 50,cooperatlvlty, posltlve as wel I as negatlve, shows a peculiar•e re t t r e at low S/Km, whlle eubs t re t e+e xc eae Inhlbltlon Iteel·f Ismore etrongly characterlzed the hlgher S/Km le .•-1,-----r--.--t--.---i---- u-2 -I O +1 +2FIGURE 8. Reduced and derlved Hlchaells-Honten equatlon. Effect-1 u -uof experimental errore y , = e + 2 + e a8 functlon-1of U. The v: -valuee were etlll normallzed by dlvldlng-1them by 41, yleldlng y' •. 1 when u .. D. See text forfurther detalle.537Arq.Biol.Tecnol.31(4):527-538,nov.,1988ACKNOWlEOGEHENTWe are very grateful to Hrs. Cassla C. Harger, who klndlyprovlded U5 wlth her recent results on the beta-glucosldase of afungus (~Uml~~lAsp) hydrolyslng p-nltrophenyl beta-glucoslde.REFERENCES1. SAGNEllA, G. A. (1985). Model flttlng, parameter estlmatlon,linear and non-Ilnear regresslon. Trends In BlochemlcalSclenC8s, 10:100-103.2. GUTFREUNO, H. (1986). Where are the asymptotes of Hlchael Is-Menten? Trends In Blochemlcal Sclences, 11:203.3. NAQUI, A. (1986). Where are the asymptotes of Mlchaells-Menten? Trends In Blochemlcal SClences, 1':6~-65.1. BRUNE, W. Cálculo cln'tlco em reaçOes enzlmátlcae. Viçosa,Imprensa Universitária, UFV, 1978. 83p •5. GANNON, M. .N. (1986). Where are the asymptotes of Mlchael Is-Menten? A slmple method to vlsuallze and determine themaxlmum responee. Trende In Blochemlcal Sclencee, 11:509-510.8. RATKOWSKY, O. A. (1986). A sultable parameterlzatlon of theHlchaells-Menten enzyme reactlon. Blochem. J., 2~0:357-360.7. ENGEl, P. Enzyme klnetlce, the steady-state approach. In:Outllne Studlee In Blology (seconu Edltlon>. London,Chapman and Hall, 1981. 96p.8. HARGER, C. C. (1987>. Personal Communlcatlon.9.ASCENZI, P. and Amlconl, G. (1987>. logarlthmlc plote Inenzymology. Representatlon of the Hlchaelle-Mentenequatlon. Blochem. Educatlon, 15:83-84.538

Parameter evaluation in Michaelis-menten kinetics.

Abstract

Similar works

Full text

Available Versions

Repository Open Access to Scientific Information from Embrapa