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658 HANSLINEWEAVERN D D E A N BURK Vol. 56[CONTRIBUTIONRO M THE FERTILIZERNVESTIGATIONSNIT,BUREAUOF CHEMISTRYND SOILS, NITEDSTATESDEPARTMENTF AGRICULTURE]

The Determination of Enzyme Dissociation ConstantsBY HANS INEWEAVERND DEANBURK

IntroductionKinetic studies of enzyme reactions have led

to the theory of equilibrium intermediate com-pound formation between enzyme and substrate(general reference, Haldane,' p. 38). The sim-plest case of equilibrium may be represented bythe dissociation constantOn the basis of the assumed theory the ra te of theobserved reaction is directly proportional to theconcentration of the enzyme-substrate compound,(ES), a t all values of the concentration of thesubstrate, (S). It is proportional to (S) only atlow values of (S). The numerical value of thedissociation constant is given by the substrateconcentration a t half-maximum velocity, where(E:l = (ES).

The equilibrium in equation 1 may be hetero-geneous or homogeneous. Hitchcock'" has pointed0111.he formal identity of the simplest Langmuiradsorption isotherm with an equation he derivedexpressing the application of t he law of massaction to a reversible homogeneous reaction be-tween two substances in solution, the total con-centration of one, (E) + (ES), being kept con-stant and the other, (S), varied. In the lattercase the observed initial reaction velocity ZI isgiven by the well-known equation (cf. Michaelisand Menten2)

21 = VUl&X(S)/(& + (SI)where VI is a numerical constant representingthe maximum velocity obtained when the enzymeE exists completely in the form ES (V,,, =k(EtotaJ). In the simpiest Langmuir surfacereaction, the velocity is proportional to theamount q of gas adsorbed a t an equilibrium pres-sure fl, where

(3)a being the maximum amount of gas adsorbablewhen the adsorbing surface is saturated. b is adissociation constant corresponding to K, andrepresents the rat io of the velocity constants ofevaporation and condensation of gas from and

Ka = (E) (S)/(ES). (1)

(2)

II= W/(b + P)

(1) J. B. S. Haldane, "Enzymes," 1.ongmans. Green arid Co ,( la) D. I. Hitchcock, THISOURNAL, 48,2870 (1926).( 2 ) L Michaelis and M. L. Menten Biochem. Z . , 49 , 1333 (1913)

London, 1930.

onto the surface, providing the observed reactionvelocity is negligible compared to the velocity ofevaporation. Agreement of experimental datawith either equation, therefore, does not neces-sarily decide whether an enzyme-substrate equi-librium is heterogeneous or homogeneous sinceboth represent rectangular hyperbolas. Data ofan independent nature in addition to the kineticstudies a re required.

Enzyme properties are determined chiefly bymeans of kinetic studies. I n many cases investi-gated heretofore, the mechanism and equations ofthe simplest case just outlined have been assumedto hold without regard to other possibilities.The dissociation constant K,, hether true orapparent, has been evaluated arbitrarily byplotting activity (initial velocity) against (S) orlog (S) and taking the value of (S) at half-maxi-mum activity. In certain cases detailed ana-lytical methods have also been employed. Gener-ally considered, however, there have been no con-venient and direct methods available for ascer-taining which of several mechanisms may, andwhich m ay mt, be involved. Although (S)a t half-maximum activity is in any case a characterizingproperty of an enzyme, it does not necessarilyrepresent a thermodynamic dissociation constant.In many cases the kinetics involved do not corre-spond to equation 2 (Case I), but to Cases I1 toV II , briefly outlined as followsCase I E + S ES (active)Case I1 E + nSe S, (active)Case I11 E + S ES (active), ES f 12 - 1)seES, inactive)Case IV General inhibition: non-competitive; and com-petitive (2E + S + Ie S (active) +E1 (inactive))Case V E + In + n')S e S, (active) f ES,?(active)Case VI E + S+ S (active), (steady-state con-centration of ES)Case VI1 S ---f S', steady-state (S')), S' + Ee S(active)In all these cases the velocity is assumed to be adirect function of the concentration of an activeintermediate or active intermediates.

I t is proposed in this paper to present graphicalmethods of testing velocity equations and evalu-ating constants involved in the various funda-


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March, 1934 THEDETERMINATIONF ENZYME ISSOCIATIONONSTANTS 659mental type mechanisms postulated above, byputting a given equation in a form that is linearand employing stra ight line extrapolations. Therelative weighting of the experimental observa-tions alters iii a definite manner when the form ofan equation is altered, and if not taken intoaccount may alter slightly the parameter con-stants obtaiiied, whether graphical or analyticalmethods art! employed. This possible disad-vantage will rarely outweigh the convenience ofthe graphical method, where proper weighting isless easily applied. In this connection i t is ofteninstructive to plot a given set of da ta in severalways. This is true whether straight lines areinvolved or not. Complementary analytical re-finements leading to the determination of theactual probabilities of the constants and functionsby means of Pearsons Chi test, involving properweighting, are considered e l s e ~ h e r e ~ ~ , ~ ~n con-nection with experimental studies of the writerson the nitrogen fixing enzyme nitrogenase inAzotobacter.

Each case will be illustrated by experimentaldata specifically consistent with, but not neces-sarily proving, th at case only., Needless to say,caution is necessary in concluding that a reactionfollows a certain mechanism because a set of datafits a certain equation. Lack of extensive datamight also be responsible for a misleading agree-ment of da ta with an assigned mechanism. Inany case, the direction of purposeful experimenta-tion should be more evident. It is desired toencourage detailed analyses of kinetic da ta relat-ing to enzymes in order to determine what mecha-nisms m a y , but not necessarily do, hold, andparticularly to eliminate certain mechanismswhich definitely do not hold. Non-conformity of aset of da ta with a given mechanism eliminatesthat mechanism unless closer analysis indicatesthat some consistent experimental error might beinvolved. The various experimental data dis-cussed are given chiefly as illustrations of varioustypes of kinetic data ; their intrinsic interest issecondary hue.

Various combinations of Cases I to VI1 will notbe considered except by implication, nor casesinvolving reaction courses such as irreversibleconsecutive reactions, reaction product inhibition,irreversible inactivation. No consideration will

(3) (a ) H. I i n e w e u v c r , I) .Burk and W. E. Deming, TIIIS O U X X A L ,56, 225 (1932); it,) Elran Burk, Azotase and Nitrogenase inAzotobacter, a rsview chapter in Ergebnisse der Enzymforschung,by F. F. Nord ami R. Weidenhagen, Vol. 111. Leipzig, 1934.

be given to postulated enzyme mechanisms otherthan those involving formation of intermediatecompounds or complexes.

Experimental(Invertase, Raffinase, Amylase.)-

Equation 2 (likewise equation 3) becomes linear inform upon taking the reciprocal of both sides

l / v = KS/Vmax(S) l /vmas (4)When l / v is plotted against l/(S), the ordinateintercept is l /Vmax and the slope of t he straightline is K,/ max, thus evaluating K,. Multipliedthrough by (S) equation 4 becomes

( S ) / U= (S)/Vme.x + K J V m s xand when (S) /v is plotted against (S), he ordinateintercept is K,/Vma, and the constant slope1/ Vmax. Hitchcock4 suggested this la ttermethod in testing the Langmuir equation cited.Hanes6 determined K , and Vmax or the amylasesystem by the method of least squares based onequation 5 , rather than graphically. Kuhn6also used an analytical method with invertase.It has been our experience that usually themethod of plotting based on equation 4 ather than5 gives a better placement of the experimentalvalues at low substrate concentrations. Figure 1and Table I illustrate the use of this method.

Case I.

( 5 )

TABLEK, ND VmaxVALUES N CASEI

---Figure 1- -Original paper--Michaelis an d

Enzyme K , Vmax K , Vmax ObserverCurve I 0.0166 M 3.94 0 .0166 M M enten2Curve I1 ,017 M 23 . 4 , 017 M KuhnsCurve111 . 23 M 100 . 2 4 M KuhnBCurve I V . 076% 0 . 478 . 0 7 i % 0 . 478 HanessCurve V .079% . 552 .0790/0 ,554 Hanesl

Invertase,Invertase,Raffinase,Amylase,Amylase,

Case 11. (Citric Dehydrogenase.)-The ve-locity equation for Case I1 corresponding to equa-tion 2 for Case I may easily be shown to be

Y = Vmsx(S)/(K#+ (S))A plot of l / v against l/(S)yields a straight line,the ordinate intercept and slope having the samesignificance as before, but K , is now the nth powerof the substrate concentration a t half-limitingvelocity. The data of D am 7 on citric dehydro-genase appear to represent a case where n is 2 , as

(6)

(4 ) D I. Hitchcock, Physical Chemistry, Thomas, Baltimore,

(5) C .S. Hanes, Biochem. J . , 26, 1406 (1932).(6 ) R . Kuhn, 2. hysiol . Ch em . , 126, 28 (1923).( 7 ) W. J . D a nn , B i o c h e m . J . , 26 , 178 (1931).

1931, p. 93.


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660 HANS INEWEAVERN D DEANBURK Vol. 56indicated in Curve I, Fig. 2. Any straight lineextrapolation of Curve I1 ( a = unity) wouldalmost certainly intersect the abscissa at a finite

l2r r

I I5 10 15[l/(S)l X b.

Fig. ].-Evaluation of K, and Vmaxin Case I.Curve I, invertase, data of Michaelis and Menten,8Table I, Fig. l a, scale: ( l / v ) X 10; (l/(S)) X 0.1.Curve IC, invertase, data of Kuhn.6 Table 8, scale:( l / z i ) X 100; (l/(S)) X 0.1. Curve 111, raffinase,data of Kuhn,6 Table 14, scale: (l/v ) X 2 5 ; (1/(S)) X 0.1. Curve IV, amylase, data of H a ~ i e s , ~Table 111, scalc: (l/u) X 1; (l/(S)) X 0.5. CurveV , amylase, data of Hanes,' Table V, scale: ( l / v ) X1 , ( l / ( S ) ) X 0.5. v scale values in original in-vestigators' relative units. (S) values: Curves IIF, 111, molal; Curves IV and V, %.

00

substrate concentration, implying attainable in-finite velocity. The data in this case are probablynot as trustworthy as those of the much studiedhydrolytic enzymes. We do not wish to dis-courage possible reinterpretations, leading toeither simpler or more complex mechanisms, byindicating tha t the velocity-substrate measure-ments, a3 reported by the original investigator,are consistent with a mechanism in which n = 2 .In this connection, for instance, one might wish toinvestigate the effect of concomitant substance?in the enzyme preparation which reduce methyl-ene blue and which might compete with citrate forcitric dehydrogenase and affect thereby the calcu-lation of the desired true velocity, and thuspm5ibIy the value of n.

In cases where each component of the ratio of(ES,)/(E) can be evaluated experimentally, theequation

may be thrown into linear form by taking thelog of both sides, and the slope obtained on plot-ting log (ES,)/(E) against log (S) evaluates n,equal to unity in the simplest case, and the inter-cept at S = 1 evaluates log l, 'Ks. Redfield andIngallss used this method in studying the oxygendissociation of hemocyanin. For the enzymiccase, if the maximum velocity observed experi-mentally (Vmax-obs)corresponds accurately tokE,,,,, (E saturated) , then Case 11 may be solvedin the same manner, using log v , I / m a x o h s - Y)instead of log (ES,)/(E) .

The case 2E + S E& superficially theconverse of Case 11, would seem incapable ofaccounting for the observed rates of enzymicreactions. This mechanism kinetically requiresthe entrance into the reaction of two individualparticles or colloidal carriers, each with one (ormore) active E groups. If one active group, E,pcr single colloidal carrier of large molecular-,-- ~-

(ESJ/(E) = ( S ) " / K s ( 7 )

- 1,/p--2O I-

I I I1 2[l/(S)"IX b.

Fig. 2.-Evaluation of K , and Vmsr inCase 11. Citric dehydrogenase, data ofDann,' 25", average of 5 experiments.Curve I, scale: ( l h ) 1 ; (l/(S)z)X 0.001;K, = 0.007 M; ubstrate concentration athalf-maximum velocity = 0.084 M, TTmax =1.7. CurveI1,scale: ( l / v ) X 1 ; (l/(S)l) X0.04. Dann's values: "K," = substrateconcentration at half-maximum velocity =0.172 M, I,,, = 1.0. 'ii scale values in rela-tive velocity units of Dann. (S)values:molal.

weight is assumed, the molecular concentration,rate of diffusion, and probability of proper spa-tial orientation of the active group at the momentof collision would presumably be too low. The2 5 3 (1932).( R ) A C. Redfield a n d 15. N. I nEalis, J. Cel l . C o m p . I'hysiol .. 1.


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March, 198 Z THEDETERMIXATIONr EXZYMEISSOCIATIOXONSTAXTS f i(iassumption of several E groups per single col-loidal carrier acting independently or together isequivalent merely to one reactant from the stand-point of kinetic theory and represents cases of thetypes already considered.

-0.6 -0.4 -0.2 0 (S)+ ( S ) 4 / K 2 .Log (S).

Fig. S.-Evaluation of K 8 , Vmma,, P , nd n in Case111. Curve I, A . B , C, D, Azotobacter chroococcumrespiralion as a function of oxygen pressure, data ofBurk.9 Curve 11. A , B, C, D , catalase activity as afunction of Hz02 concentration, data of Stern.lo vvalues: relxtive. (S)values: CurvesI.4, IB, I 'B , (02)>< L !114, 0. (14, .1 YG-atm., respectively; Curves IIA ,I I H , I I ' B , (HpO,) Y 8.6, 8.6, 14.2 M , respectively.I.og (S)va!utis: Curve IC , ( log % 02) - 2; CurveITC. [log , ' I L O ~ )+ 0.3.Case 111. (Oxygenase and Catalase.)-The

wlo.;ity equation involved in substrate inhibitionis (Ref. 1, !.S f )whereKz s the dissociation constant (ES) S ) c n - l ) /(ES,) of the inactive enzyme-substrate com-pound. Equation 8 can be written in the form

plot of ( S ) / v against (S) yields a curve whichbecomes cclncave upward a t high values of (S),where (S)"l'Kzs no longer negligible. The linearlimiting slope a t low (S) values is I /Vma X.(V,,,, is not 2::. experimentally observed maxi-mum velocity, see Vopt later.) The intercept isKS!Vrnax,hus evaluating K,. Figure 3B illus-trates this method. Curves I and I1 are plotsof data for oxygenase activity (Azotobacter

v = 1 z7 t n ax ( S ) / ( ( S ) + K,+ ( S ) " / K s ) (8 )( S ) / v = KJV,,x + ( l / V d ( ( S ) + ( S ) n l K 2 ) (9)

0 5 10 16 20Concentration of glucose, Yo.

Fig. 4.-Distribution of E as a functionof glucose concentration (Azotobacter re -spiration8). Curve I, calculated concentra-tion of ESa (inactive). Curve 11, calculatedconcentration of ES (active) without ES3formation. Curve 111, calculated concentra-tion of ES with ES3formation; (O ) , bservvecivalues. Curve IV, calculated concentrationof E (Curve V - I - 111). Curve V, totalconcentration of enzyme, free and combined,E + ES + ESs (Curve I + I11 + I\')

It follows from equation 9 that, having deter-mined Kz and n, the curves of Fig. 3A may bethrown into linear form by plotting ( S ) / v against((S) + (S)",'Kz). Providing the mechanism in

(9) D. Burk, J . P h y s . Chem., 34 , 1207 (1930).(10) K. G . Stern, 2. h y s i o l . C h c m . ,109, 176 (1932).


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662 HANSLINEWEAVERN D DEANBURK VOI. 56Case I11 holds, a straight line will be obtainedwhose slope and intercept (as in Fig. 3B) are1/ Vma, and K,/ V,,,, respectively. Figure 3Dillustrates this method of testing the applicabilityof this mechanism over the entire substrate con-centration range.

Figure 4 hows the calculated relative distribu-tion between E, ES (active) and ES, (inactive)for Azotobacter respiration as a function of glucosecon~entration,~he curves being constructed onthe basis of constants K,, Vma,, K2 and n deter-mined by the method illustrated in Fig. 3. TableI1 presents the various constants determined inconnection with Figs. 3 and 4.

'The optimum velocity (Vopt) is obtained at afinite value of (S) (see Table 11) and hence isalways less than V,,,, which is theoreticallyobtained, even in the absence of inhibition, onlya t infinite (S). This is illustrated in Fig. 4,Curves 111and V, and Table 11,Columns 3 and 4.

Mechanisms are conceivable in which n need notbe a whole number. A #riori, also, n might bequi te large, instead of 3 to 4 as found here, or 2assumed by Stern. Haldane (Ref. 1, p. 85)assumed a value of 2 in the case of the data ofBamannl' on sheep's liver lipase and Murray12confirmed this later for ethyl butyrate with morecomplete experimental data.

The case of Azotobacter respiration is especiallyinteresting as a bimolecular reaction involvingtwo substrates where high substrate concentrationinhibition is given by both reactants, oxygen andglucose, and to fairly high powers of each. Theoxygen function reported in Fig. 3,4 was obtaineda t a glucose concentration yielding approximatelyoptimum rate (80% a t 1% glucose instead of100% at 32y0 as in Table 11). The glucosefunction of Fig. 4 was obtained a t an oxygenconcentration yielding an approximately optimumrate (95 to 100% a t 21 % oxygen instead of 100%

TABLE1K,, max, KZA N D n VALUESN CASE111

Column No. 1 2 3 4 5 6 7 8 9 10Low S) Hi & (S) . . (S), - . , . Ia t a t (Shigh) (n ) a tFunction Substrate Ka Vmax Vopt n K? l / tVopt l /zVapt at I/zVOpt VoptaType figure . . . . . . 3 B 3 B 3 A 3 C C 3 3 A 3 A 3 c 3 AAzotobacter respira-

Azotobacter respira-Catalase (Fig. 3) HzOz . 0 5 7 M 141 100 4.0 0 ,043M . 0 3 3 M 0 . 4 M 0 . 0 6 4A4 0 . 17MCatalase (Sternlo) H& . 033M . . . 100 2.0 . 4 M . . . . . . . . . . . ( . I 1 M )

tionQ 0 2 0.75% 106 100 4. 0 2 X lo5% 0.70% 76% 4. 4 X l o s % 15%tionQ Glucose .S l% 100 78 3 . 0 10% .45% 12.4 % 154% 3 .2 %

Calculated from (S)vopt= (K&/(n - l))*'" Ref . 1, p. 85).The subsirate concentrations a t which v = l/zV0are not given by K , (as in Case I) and KZ'/(~- '~.Use of (S) a t '/ZVopt as an approximation to (S)a t I/zVma, will lead to low values of K , and highvalties of K2 for catalase activity, as shown bycomparing Stern's values with those derived fromFig. 3 (Table 11, Rows 6 and 7) , and for all threecases in another manner comparing Columns 1and7, and Columns 6 and 9. Stern's value of K Z =( S ) ( n - l ) := 0.4 M was based on a value of S at'/2 ITopt and an assumed value of n = 2 (as appearsto be the case in certain lipases). Using n = 4,as found in Fig. 3 , Kz would be 0.064, still higherthan the value KZ 0.043, from Fig. 3, becauseV,,, is less than V,,,.

'The experimental values of n in Table I1 arewhole numbers to within the first decimal pointand the values of K2 reported are based upon inter-cepts at log (S) = 0 in Fig. 3C when the slopesare drawn with the exact whole number assumed.

a t 15%). As pointed out in a previous paper,geither chain reaction kinetics or contact catalysismight be involved in the conjugate maxima. Com-petitive inhibition between oxygen and glucosewas held possible. Th e constants K, and K2 ofTable 11, however, were derived upon a basis ofnon-competitive inhibition between oxygen andglucose, and this postulated mechanism is appar-ently confirmed by the experimental data, CurveID, Fig. 3, and Curve 111, Fig. 4. Additional ex-periments of a direct nature substantiate this view.The percentage inhibition of respiration a t differ-ent glucose concentrations was found to be practi-cally independent of the oxygen pressure over a 22-fold range, 4.5 to 100 X atm., and a t differentoxygen pressures independent of the glucose con-centration over a 25-fold range, 0.2 and 5a/,, show-ing no competitive inhibition between substrates.

(11) E. Bamann, Bcr., 69, 1538 (1929).(12) D. R. . Murray, Biochrm. J . , 94, 1890 (1932).


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March, 1931 THEDETERMINATIONF ENZYMEISSOCIATION CONST4NTS 663Case I11 might be written, E + Se S

(active), E + nS =$ ES, (inactive); K3 =(E)(S)=/(ES,). This introduces no change inthe treatment except for employing the relationK3 = K 2 K s in Fig. 3C and 3D and Table 11.There is also the more general case where theactive form might be ES,f where n is a numberlarger than 1 but less than n in the inactive formES,. This would involve some of th e method ofCase 11, plotting l / v against I/(S), or (s)/vagainst (S) to obtain K , and Vma,, etc.

Case IV. (Invertase.)-Competitive inhibi-tion may be represented in the simplest case (cf.Case I) by the equationl / v = (l/Vm*.x)(K*-t *(I)/&)(l/(S)) + / V m x t (11)where (I) is the concentration of inhibitor and K ,the enzyme-inhibitor dissociation constant. Theterm Ks(I )/KI represents an increase in slope inthe 1/v X l/(S) plot, when Vmax is constant.Competitive inhibition occurs when the termslope X I fmax is increased by inhibitor as inCurves 44 and 45, compared with the control 42,Fig. 5 and Table I11 (data of Kuhn and Miinch13).Non-competitive inhibition is indicated by a lowerV,,, (higher ordinate intercept) as in Curve 43(and possibly slightly in 44 and 45) compared toCurve 42, Fig. %i.he case of two or morecompeting substrates (Ref. 1, p. 85) is a specialcase of competitive inhibition where both enzyme-substrate complexes decompose irreversibly.

TABLE11K , , V,,, AND K I VALUESN CASE V

ApparentK8Ks 1 (Kuhn andCurve Inhibitor K a + X I Vmax Miinch)

42 . . . . . . 0.035 .. 26.3 0.04043 0.175 M a-glucose ,034 . . 21.3 .04044 .183 Ad 8-glucose ,076 0.156 25.0 .07045 .196 M ructose .076 ,167 25.0 ,070Case V. (Esterase.)-A given substrate forms

more than one active enzyme complex. Thevelocity equation. is21 = (V*mx-n(s)fl/K3-t~ ~ ~ ~ - ~ , ( s ) n / ~ ~ - ~ , ) / ( lVma,-, and Vm,,-,t correspond to maximumvelocities theoretically obtained when E existscompletely in the active forms ES, and ES,,,respectively, and K , and Ks-,l are correspondingdissociation constants. In this case a plot of vagainst (S) may pass through an optimum as inCase 111, but approaches a constant, not zero,

( S ) /K , + W/K*-M) (12)

(13) R.Kuho and H.MClnch, 2. 9 h ~ ~ i O l .hcm.. 168, 1 (1927).

velocity, corresponding to Vmx-,,!. The valuesof the limiting slopes and intercepts of both l / v X1/(S) and ( S ) / v X (S) plots may be used inevaluating the various constants occurring inequation 10. Schwab, Bamann and Laeverenz14worked out this case in connection with theaction of human liver esterase on (-)-ethylmandelic ester, the active forms being ES1 andES2 (n = 1, n = 2), K , and Ks-,j being 4.3 X

and 3.8 X M . The ratio of l,,,, :Vop, : Vmax-ng is 1 : 0.81 : 0.584, Vopt occurringa t about 3.9 X M .

2or --iI-----T----i---T -I / Ii

161 / 4 , ry I

L10 20 30 40 50

I/@).Fig. 5.-Competitive and non-competitive

inhibition (Case IV) . Curves 42,4 3,44, 45,invertase, da ta of Kuhn an d Munch,13Tables42, 43, 44,45, respectively.

Case VI. (Urease.)-In velocity equations in-volving (ES) = steady state where k3 = k2, K,is replaced by an apparent dissociation constantKS = (kz + kZ)/k1 = K, + k3/k1 arrived at byBriggs and Haldane15 for the mechanism

ki ksE + S e S+ E + Pkz (13)P is the final reaction product. A similar con-stant K: = k3/kl was obtained by Van Slyke andCullenl*assuming k2 = 0. The velocity equationcorresponding to equation 4 is (cf. Ref. 1, p. 40)A plot of l / v against l/(S) yields a straight line asin Case I but Kj varies as a direct function ofC hc m . , 215, 121 (1933).

1,~ R : / V m s x ( S ) + / V m a x (14)(14) G. M. Schwab, E. Bamann and P. Laeverem, 2. ihysiol.(15) G . E. Briggs and J. B S. Haldane, Biochcm. J . , 19 , 338(16) D. D. Van Slrke and G. E. Cullen, J . Biol. C h c m . , 19 , 141(1925).

(1914).


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dB4 HANS INEWEAVERN D DEANBmrE 1'01. 56V,,, (i. e., of k3), approaching a limiting value,K,, a t the ordinate intercept, which will be zeroonly if kz = 0. The substrate concentration athalf -maximum velocity, S vm,/2, varies withVmax also. Urease activity appears to involveeither K : or K,! as a function of PH Ref. 1, p. 41),and competitive inhibition (Case IV ) by hydrogenion may also be involved. Instances of Case VIare probably uncommon. Obviously, K , valuesobtained in Cases I -V are true dissociation con-stants only if they remain constant while thecorresponl-ling TT is varied considerably.

1r : II16c

1 2 3[ l/(S)]X 4.1 X lo-' (volume %)-I .Fig. 6.-Evaluation of K aand K: , in Case

VI1 (photosynthesis). Curves 11, 111, IV,V and VI correspond to relative light in-tensities I of 31.5, 80.4,127, 175 and 191,respectively, da ta of Hoover, Johnston andBrackett,17 22".

Case VII. (Diffusion.)-In this case the con-centration of active intermediate is not directlyproportional to (E)(S) ut to (E)@'),where S' isformed from S a t a velocity comparable withES --f E + P. This case is very common andcovers the activities of many intracellular en-zymes, especially where rates of diffusion, as intissues, often become limiting. Actually, thereaction S + ' may be either chemical orphysical, but is more commonly the latter. Thevelocity equation can be shown to be

where k: is the velocity constant in S+ '. Anexplicit rather than the implicit solution of v inw = va&{(S)/(Vmax+ K K , + KI'(S) - V ) (15)

equation 15 involres a complex quadratic equa-tion.K,, the true dissociation constant of ES (T-+ S), and k i , may be evaluated by three methods(A , B , C) . Placing v = 1 , ' z b 7 , n a x in equation 15yields Sr7,J2 = K, + 17 , nax (L'k: , so thatplotting S ' , , , , , /2 against V,,,, yields K , asintercept and 1,'2ki as slope (Method A, rf. Curve8 , inset, Fig. 0) . As in Case VI, S~rm,,izariesdirectly with V,,,, but in Case iTIIa plot of l / ' z iagainst 1/(S) no longer yields a straight line but acurve whose slope is ( 1 ' k ; + K,,'Vmdx- a /Vmaxki)/(l - u2//(S)V,,,ki). This is true since(ES) is no longer a rectangular hyperbolic func-tion of (S). The limiting slope at low values ofl,'(S) is K,/VmaX nd the ordinate intercept isl!Vmax (Method B). As 1/(S) increases, theslope increases, approaching another limit, K JV,,, + 1/ki = K,t/Vmax,as v becomes negligi-ble. l / k i may thus be evaluated by taking thedifference between the two limiting slopes. Aplot of K,. (= slope X V m a x = Vmax/k; + KJagainst Vm,, yields a line whose slope is lik; andwhose intercept is K , (Method C, rf. Curve B,inset, Fig. 6 ) it is also possible to plot the limitingslope K,t/ Vm,, against 1! V,,, (slope =K,, inter-cept = l/ki).

I I1

0 1 2 0 2 4 6 8 1 0(S). l/(S).

Fig. 'i..--Elimination of diffusion steady state in CaseVI1 (photosynthesis). Curves I and 1', I1 and 11', dataof Van der P a a ~ i w , ~ ~U . 20 ', normal and narcotized Hor-midium; Curve I11 and 111', data of Warburg,20 TableIV, Row 4, 5" . Curves I, I1 and 111, (S) X lo%, 40%and 5 X lo 6 M . Curves I' and 11', ( l /a) X 100, 1/(S)) X 0.1; Curve 111', (l/v + 0.02) X 50 , (l /(S)) X5 X 108.

Figure 6 and Fig. 7B Curve 11' illustrate thenature of the curves one might obtain in a diffu-sion-limited enzymic process (data of Hoover,Johnston and Brackett"). Van den Honert'*showed by independent direct means that the ra te

l / n values in investigators' units.

(17) W H Hoover, E S Johnston an d F S Brackett , S m t f k -(18)T.H. Van den Honert, Rec . trau. bot . nlcri., 27, 149 (1930).

s o n i ~ nMisc C a l l . , 87, No. 18 (1933).


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March, 1934 TH EDETERMINATIONF ENZYME DISSOCIATIONONSTANTS 665of diffusion of dissolved carbon dioxide wasnormally limiting a t high light intensities and hightemperatures in the case of the filamentous algaHormidium. Van der PaauwIg later confirmedthis indirectly by lowering Vma, sufficiently withnarcotic (presumably without effect on the rate ofcarbon dioxide diffusion in the stomata-freeHormidium) to obtain a linear 1 , v against 1/(S)plot (Curve 1, compared to Curve 11, Fig. 7B).Curve 11, Fig. 7A , is a close approximation to t ha trequired bj, strict application of the well-knownBlackman Principle of Limiting Factors (cf. Ref.17, p. 17; Ref. 18, p. 149; Ref. 19, p. 500),whereas Curve I is the usual rectangular hyper-bola. This general principle, which limits theextent of the transition range where photosynthe-sis is limited simultaneously by two factors such ascarbon dioxide pressure and light intensity,would appear to receive analytical interpretationin the treatment of Case VI1 given here. Theunicellular green alga chlorella studied by War-burg20 and Emerson and Arnoldz1 (continuouslight) yielded a linear l / v against l/(S) plot, inagreement with Warburgs view that diffusion isnot here limiting (cf. Curve IT1 and 111,Fig. 7Aand 7B).

We do not propose that all of the known perti-nent data on the influence of carbon dioxidepressure in the photosynthetic mechanism can berepresented by equation 15, since, a priori, th enature of the light reaction, only partially under-stood, has not been included. The problem ofphotosynthesis, of widespread interest, involvesmuch data and numerous postulated mechanismswhich are generally of limited applicability andnot easily correlated. For these reasons numeri-cal values of the constant K , for wheat andHormidium, obtainable by Methods A , B, or C,are not for the time being given. Until moreuniform postulates, based to some extent on dataother than kinetic, can be found in the literature,this procedure seems desirable. The generalmethods presented, however, may be used in anydetailed consideration of proposed or to-be-pro-posed photosynthetic mechanisms. The qualita-tive analysis given for wheat indicates that, asactually found for Hormidium, the rate of diffu-sion of the substrate carbon dioxide is, at low andintermediate pressures, normally limiting, evena t low light intensities where the rate of photo-

(19) F Van der Paruw, Rcc troo bot n l e d , 29, 497 (1932)(20) 0 Warburg, Btochem Z , 100, 230 (1919)(21) R Emerson and W Arnold, J Gcn Physsol , 16. 409 (1932).

synthesis at high carbon dioxide pressures isdirectly proportional to the light intensity.Fur ther experimentation on the effect of tempera-ture at high light intensities and low carbondioxide pressures could assist in deciding thispoint further. Although Case VI1 is common,unfortunately no data on any case, particularly ofa simpler nature than photosynthesis, have cometo our attention which are susceptible of strictquantitative analysis. In this connection, interest-ing model experiments might be devised, usingeither enzymes or catalysts in general.

The treatment of Case VI1 does not resolve thedata into a linear plot over the entire substrateconcentration range. However, it possesses theadvantage of offering three methods of obtainingK , in the simple case, and is capable of evaluatingadditional constants as well in more complexcases. In applying Case VII , i t is necessary to beable to vary V,,,, as for example, by non-competi-t i re inhibitors, PH, ossibly temperature, etc.

DiscussionIf mathematical representation be given a set of

kinetic data one should be in a superior positionto test an assigned mechanism, first with respect toother known facts, and second with respect toexperimentation suggested by such representa-tions. The methods for correlating a givenmechanism and set of da ta based on the lineargraphical methods suggested in this paper placeemphasis upon testing the qualitative nature of amechanism before obtaining the numerical valuesof t he related constants involved.

Mathematical representations of kinetic da taare not necessarily limited to demonstrated casesof enzymic catalysis, but may cover complexphysiological processes like respiration, photo-synthesis, chemical autotropism, or nitrogenfixation, where the formal or arbi trary assignmentof enzymic nomenclature, based on methods ofphysical rather than organic chemistry, may, ormay not, be conventional. In vitro demonstra-tions need not be prerequisite to mathematicalanalysis of data obtained from in v i v o systems.

The writers are indebted to Dr. A. K. Balls, Dr.P. H. Emmett and Dr. W. E. Deming of theBureau of Chemistry and Soils for valuable criti-cism and suggestions.

Summary1. Graphical methods involving constantslopes and straight line extrapolations have been


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666 ROBERT . BURTNER Vol. 56developed for testing and interpreting kineticdata, and for determining dissociation constantsof enzyme-substrate and enzyme-inhibitor com-pounds and other related constants when the dataare found to be consistent with a n assigned mecha-nism.2. Representative analyses are given forinvertase, raffinase, amylase, citric dehydrogen-ase, catalase, oxygenase, esterase and lipase,involving substrate activation, substrate inhibi-tion, general competitive and non-competitiveinhibition, steady sta tes and reactions of variousorders.

3. A plot of the reciprocal of the observedvelcicity Y against the reciprocal of the substrateConcentration (S) yields in the simplest case (e. g.,invertase, amylase) a straight line whose slopeand ordinate intercept yieldK, (Michaelis dissocia-tion constant) and V,,, (theoretical maximumvelocity), In the presence of competitive inhibi-tors the slope is increased but the intercept isunchanged. With non-competitive inhibitors theintercept also is raised. When the active inter-mediate contains n molecules of S a straight line isobtained upon plotting the reciprocal of (S)",

instead of the reciprocal of (S),which would yielda curve concave upward. A steady-state occur-ring before the formation of the active inter-mediate, to whose concentration v is proportional,will yield a curve with two limiting slopes.Curves of v plotted directly against (S), passingthrough an optimum and approaching a zerovalue, indicate the existence of an additional in-active intermediate (catalase), whereas approach-ing a constant value indicates two or more activeintermediates (I-ethyl mandelic esterase). Theconstants involved in the three latter cases aredetermined from limiting slopes and intercepts ofvarious plots.

4. The various methods described are ap-plicable to general chemical catalysis, homogene-ous or heterogeneous, and possess many advan-tages of usefulness and convenience over lessextensively developed methods employed hereto-fore. They are capable of eliminating certainpostulated mechanisms] and indicating whatmechanism or mechanisms may be involved,though not necessarily proving what mechanismis involved, in a given case,WASHINGTON,. C. RECEIVEDUGUST18, 1933

[CONTRIBUTION F R O M THE CHEMICAL LABORATORYF IOWA STATE COLLEGE]Orientation in the Furan Nucleus. VIII. 8-Acylaminofurans

BY ROBERT . BURTNERThis report is concerned with the synthesis of

one of the more promising precursors of simple8-aminofurans, namely, the 8-acylaminofurans.The increased availability of 3-furoic acid' opensan attractive avenue of approach to 3-acylamino-furan. Th e following sequence of reactions illus-trates the synthesis of 3-benzoylaminofuran

NHzNHz3-CaHaOCOOH+ -CdHaOCOOCzHs-NOz~-C&OCONHNHS 3-C&OCONs +

CsHsMgBrIn like manner, 2-methyl-3-benzoylaminofuranwas prepared from 2-methyl-3-furoic acid, and thecorresponding 2-methyl-3-acetaminofuranwas ob-tained by the use of methylmagnesium iodide in-stead of phenylmagnesium bromide.

3-CdHsONCO- - C ~ H ~ O N H C O C B H ~(1) Oilman and Burtner, Tasa Jown?i&, 66,2906 (3983).

Experimental Part3-Furoyl Hydrazide.-This hydrazide was prepared from

ethyl 3-furoate and hydrazine hydrate in a customarymanner. The yield was 75% and the compound melteda t 124124.5 ' after crystallization from a benzene-rhethanol mixture.Anal . Calcd. for CsHeOzN2: C, 47.61; H, 4.76.Found: C, 47.37; H, 5.01.

3-Fury1 Isocyanate.-A solution of 12 g. (0.1 mole) of 3-furoyl hydrazide in 300 cc. of water was mixed with75 cc. of 2 N sulfuric acid and the resulting solutionchilled to 0'. To this was added dropwise and withshaking a solution of 7 g. (0.11 mole) of sodium nitrite in30 cc. of water. The azide was removed by extractionwith ether, and this extract was washed with water andthen dried over sodium sulfate. Approximately three-fourths of the ether was removed by distillation; then90 cc. of dry benzene was added; and , finally, the re-maining ether was removed by distillation. The benzenesolution of 3-furoyl azide was then carefully heated untilthe oalculated volume of nitrogen was evolved for mu-

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