266
LLOYDINGRAHAM AND BENJAMIN MAKOWER
Vol. 68
VARIATION OF THE MICHAELIS CONSTANT WITH THE CONCENTRATIONS OF THE REACTANTS IN AN ENZYME-CATALYZED SYSTEM’ BY LLOYDL. INGRAHAM AND BENJAMTN MAKOWER Western Regional Research Laboratory,2 Albany, Calif.’ Received October 17, 1065
An analysis has been made of the kinetics of an enzymatic system of two reactants to determine the dependence of the Michaelis constant on the concentrations of the reactants. TWO previously suggested types of mechanism were examined. One type involves the familiar binary complex of the enzyme with one of the reactants; the other involves a ternary complex with both reactants. For the ternary complex type of mechanism consideration was given to four cases: (a) two mechanisms of formation of the ternary complex wherein the reactants combine with the enzyme in an ordered or in a random sequence; and to (b) two reaction conditions, equilibrium or steady-state. It was shown that for each of the above four cases there are four distinct experimental conditions under which the reaction kinetics can be studied accordin to the Michaelis-Menten analysis for maximal rates and that these four conditions give rise to three different expressions for maximal velocities and four different expressions for Michaelis constants. The four experimental conditions arise from the variation of the concentration of either reactant, while the other reactant’s concentration is ke t constant and is either at, or below, the concentration required for maximal rate. From the derived rate laws for the di8erent experimental conditions, it was found that in only two out of the four conditions (one of the reactants present in large excess) is the expression for the Michaelis constant analogous in form and meaning to that for the binary enzyme-substrate complex, frequently described. Under the other two conditions (both reactants a t limited concentrations), both the maximal velocit and the Michaelis constant for one reactant, vary with the concentration of the other reactant. This kind of variation (with the other reactant) is also expected for reactant concentrations required to give any fraction of the maximal velocity. This means that, in practice, such a variation may also be found in the experimental concentration of a given reactant required to roduce a rate approximately independent of that reactant. Recently published data on the kinetics of lipoxidase and gkcose oxidase are offered as examples of some of these predicted variations. Furthermore, the previously proposed binary-complex ty e of mechanism for glucose oxidase is ruled out and a ternary-complex type is offered as a reasonable possibility. It is ago shown that the concept of a ternary complex is not essential for the explanation of the kinetics of two reactants. A more generalized formulation involving a modified form of the enzyme might be applicable instead.
Our recently publisheda and continuing investigations on the enzymatic browning of fruits have led us to a consideration of the significance of the experimentally determined Michaelis constants in enzyme systems which involve two reactants (substrates) a t limited concentrations. As already pointed out by LuValle and Goddard4most enl;yme systems involve a t least two reactants, but one of them frequently is present in large excess (as might be the case when water is one of the reactants in a hydrolytic reaction) and the systems can then be analyzed by the familiar Briggs and Haldane6 treatment in terms of a binary enzyme-substrate complex. When both reactants are present a t relatively low concentration, the above treatment has been reported t o give anomalous variations, as large as seven to tenfold, in the measured Michaelis constant, with the concentration of one of the reactants.s No satisfactory explanation has yet been offered to account for these variations in terms of the kinetics that might hold for such systems. It is the purpose of this paper to present such an explanation. The variations noted above will be accounted for in terms of a reaction mechanism which involves the formation of a ternary complex of the enzyme with the two reactants. It will also be shown that the meaning of the measured Mi(1) Presented at A.C.S. meeting, Chicago, Sept. 6-11, 1953. (2) Bureau of Agricultural and Industrial Chemistry, Agricultural Research Administration, U. 8. Department of Agriculture. (3) L. L. Ingraham, J. Corse and B. Makower, J . Am. Chem. Soc., 74, 2623 (1952). (4) J. E. LuValle and D. R. Goddard, Quart. Rev. Bid., 28, 197 (1948). (5) G. E. Brigge and J. B. S. Haldane, Biochem. J., 19,338 (1925). (6) (a) H. Laser, Proc. Roy. SOC. (London), 14OB, 230 (1952); (b) A. L. Tappel, P. D. Boyer and W. 0. Lundberg, J . B i d . Chem., 199, 267 (1952).
chaelis constant is analogous t o that for a binary enzyme-substrate complex only under certain limited experimental conditions. Rate Laws and Their Significance, for a Two Reactant System Involving a Ternary Complex.Several investigators previously have postulated the existence of ternary complexes of enzymes with two r e a c t a n t ~ , ~with ~ ~ J one reactant and an activator or coenzyme,g~10.11and with one reactant and an inhibitor.12 The kinetics for two reactants have been deduced for the equilibrium condition by Laidler and Socquet,le and were extended to the steady-state conditions by Segal, Kachmar and Boyer.14 Some of the rate laws derived by these authors will serve as a basis for the discussions which are t o follow. These laws were therefore collected, together with some new derivations, in Table I. The previously derived expressions, have been appropriately revised so as to present all the results in terms of uniform symbols and definitions. Two mechanisms of formation of a ternary complex of reactants A and B with enzyme E have been considered. Mechanism I requires that reactant A combines with the enzyme before reactant B. Mechanism I1 is a generalized form of I t o include (7) L. Michaelis, “Currents in Biochemical Research,” (D. E. Green ed.), Interscience Publishera, Inc., New York, N. Y., 1946, p. 225. (8) R. A. Alberty, J . A m . Chem. Soc., 75, 1928 (1953). (9) J. F. Kachmar and P. D. Boyer, J . Bid. Chem., 200, 669 (1953). (10) G. W. Schwert and M. T. Hakala. Arch. Biochem. Biophya., 88, 55 (1952). (11) J. Botts and M. Moralea, Trans. Faraday Soc., 49,696 (1953). (12) H. T. Huang and C. Niemann, J . Am. Chem. Soc., 7 6 , 1395 (1953). (13) K. J. Laidler and I. M. Socquet, TEIS JOURNAL,64, 530 (1950). (14) H. L. Segal, J. F. Kachmar and P. D. Boyer, Enaymologia, 15,
187 (1952).
.
Mar., 1954
VARIATION OF
THE~MICHAELIS CONSTANT
267
TABLE I CALCULATED RATELAWSFOR SEVERAL MECHANISMS c
Meohanism
-
Rate of formation of product Pa Steady-atate conditions Equilibrium oonditions
I. Two substrates, ternary complex, ordered sequence E1
+A
kl
Eo
V
k s k fx(A,B)'J
V
= k&
frI(A,B)'J
+
ki Ea +P EI K1 = kl/kn, Ke ks/kr, Ks = ka/ki K4 W k s
Vmaxm
Vmm l sxAB ksEo KiKz = KsKd
= k&'o
111. Two substrates, binary complex
E1
+A
kk
kl
ksEo kiAB kiA ks ksB
+ +
Vka
V
ABK KA
+
= ksEo------
1
+
E2 +P EI Vmax ~ S E O Vmax ~SEO K I kllkr a VmaxA and VmalBare the maximum rates reached as a limit upon inoreasing the concentration of substrates A and B, respectively. VmaxAB= &EOis the maximum limiting rate for both substrates, A and B. Eo is the total enzyme concentration, complexed or uncomplexed. klkaAB f l ( A J = Ea'Eo kiksAB (kd ks)klA f kaksB kz(L ks)' KiKeAB This w a the ~ expression previously derived by K. J. Laidler and I. M. fII(A, B ) &/EO P 1 + K I A + KIK-B* Boo uet.1* Ea Eo AB(k1kak.r kikah-4 keksks krk&B)/AZ(kikaks kikika) A*B(k~ksks) Bz(kaksks 4'IIII(A, B ) kskake) ABs(krkskd A B b k a h kikakp f kikski 4- krkska kskska) A(kek4ks kok& +klkrkr klkbk? klkiko) B(k8ksk.r kakike kekrks kak.&e kzkske f h k 4 h k n M 7 kikiko. KI nKdB fxv(AJB, K, j-K I K A KlKJ3 KIKzR~AB' I These rate laws were prevlously denved by H. J. Segal, J. F. Kachmar and P. D. Boyer.14
'
+ +
++
+ + + p
++ + 2 .+
+
+
+
++ + +
+
++
+
+ ++
+
+
cases where enzyme E combines with substrates A and B in a stepwise manner but in a random
sequence to yield the- complex Es. Mechanism 111 involves the two reactants A and B again but they do not form a ternary complex." Mechanism IV, the familar one6 for a single substrate system, has been included for comparison and to aid in the later discussion. For each mechanism, the results are given for both steady-state and equilibrium treatments. The rate V , of the formation of product P, was (16) The transition state in the seoond step could be called the ternary complex but it differs from the ternary complexes in mechanisms I and I1 in that the potential energy of the system in I11 is a maximum whereas in the former ome8 (I and 11) it is a minimum.
calculated in a conventional manner by setting up steady-state or equilibrium equations for all the enzyme forms (El to Ed),calculating the concentration of El in terms of the concentrations of the substrates A and B and of total enzyme, EO,and finally, evaluating the function V = ~ E s . The calculated expressions for V are shown in Table I as the first equations for each of the four mechanisms. It is seen that a very complicated expression, containing 24 terms in the denominator, is obtained for the general steady-state case of two substrates. The other three general equations for V , for mechanisms I and 11, are essentially those of Segal, Kachmar and Boyer.I4
2 6%
LLOYDINGRAHAM AND BENJAMIN MAKOWER
Vol. 58
VrnaxAB
A
Vmax
>
t0
s
Vmax A
W
>
K,
K,
K,
K,
EXPANDED SCALE
CONCENTRATION O F A ( O R B)Fig. 1.-Curves of the velocity versus the concentration of the reactant A (or B), showing the three maximum velocities VmaxA,Vmaligand VmsxABand the four Michaelis constants, Kr, KII, KIIIand KIV corresponding to the four experimental conditions, V + VlnsxAlV --c VmmSxB, VmsxA+ VmaxAB and VmsxB+ VmaxAB, respectively. The left-hand half of the graph has been expanded greatly to show the initial part of the concentration range in greater detail.
The equations for steady state conditions are ample of such a study would be the dependence expressed in terms of rate constants; equations for on oxygen pressure of the rate of enzymatic oxidaequilibrium conditions, in terms of equilibrium tion of a reducing substrate. In that case it is constants. necessary to assure oneself that the maximum When these equations are analyzed according to rate with respect to the reducing substrate is the Michaelis-Mentenl6 method for the evaluation actually being maintained over the whole range of of maximum rates it becomes apparent a t once oxygen pressures under investigation. Obviously, that the two substrate systems have three possible concentration of the reducing substrate satismaximum rates as compared with one for the factory for the lowest oxygen pressure studied, single substrate. Thus, by increasing the concen- will also be satisfactory for the highest pressures trations of A, or of B, or of both A and B, (but not vice versa). different limiting maximum rates will be obIt is of interest to note that the equations for tained which will be designated here by the sym- VmaxA and VmalB are, in general, analogous in bols VmaxA, V m a x B and VmaxAB, respectively. The form to the general equation for V in the single equations for each of these, for the different substrate system (mechanism IV). This is to be mechanisms, are also shown in Table I. They were expected inasmuch as VmaxA(or B) rate is only a calculated from the general equations (for V) by function of the other substrate B (or A). The dropping out those terms which approach zero with dependence on the other substrate leads in turn to increasing concentration of the substrate involved. an important observation that VmaxA(or B , is not a Some of these terms were of the type, const./AB, unique value but represents a series of values, SO that a t low concentrations of A (or B) it may require a high concentration of B (or A) t o allow depending on the concentration of substrate B (or A). For example, a plot of V us. A could be this term to be neglected. The consequence of this relationship is that the shown as a family of curves, each curve representing maximum rate, VmaxA,for example, will be at- a different concentration of B, and each curve tained only a t very high concentrations of reactant levelling off a t a different value of VmaxA. The last equations for each mechanism in Table I A, when the concentration of reactant B is held very represent a condition where all of the enzyme exists low (and vice versa). A further important consequence relates to essentially in the form of the complex E,. The experiments where it is desired to study the rate corresponding rate, VmaxAR, is therefore a unique, as a function of the concentration of one reactant single valued quantity. while the concentration of the other reactant is Derivation of the Michaelis Constant.-The high enough to yield a maximum rate. An ex- Michaelis constant is obtained experimentally from a study of the variation of the rate (toward a (16) L. Michaelis and RI. L. Menten, Biochem. Z . , 49, 333 (1913).
VARIATION OF THE MICHAELIS CONSTANT
Mar., 1954 MICHAELISCONSTANTS‘
FOR
TABLE I1 VARIOUS MECHANISXS AND
FOR
DIFFERENT EXPERiMENT’AL CONDITIONSb
Experimental conditione
’-
269
--.
VmaxB
I. Two substrates, ternary complex, ordered sequence
+ + k,k& + + ksB)
Steadv state
hkr kzks ki(h ks
Equilibrium 1
KI
+ KI&B Steady state
11. Two substrates, ternary
f(A)d
complex, random sequence
+
1 KsB KI KiKzB
+
Equilibrium 1 KIA Ka KIKA Steady state kiA kz
+ +
+
111. Two substrates, binary complex
ks Equilibrium m
IV. One substrate, binary complex
kz
Steady state
+ k5
...
ki
Equilibrium
...
The first constant listed is that calculated from steady state conditions and the second is calculated from equilibrium conditions. b The experimental conditiolis designated as V + VmaxA,means that in these experiments concentration of B is constant and that of A is varied so that the velocity V approaches VmsxAas a limit; VmaXh+ VmaxAB means that concentration of A is constant and high enough to yield maximum rate (VmsxA)and concentration of B is varied to VmaxAB. Mechanisms are the same as in Table I. The other two conditions are similar to above with A and B interchanged. flB) and f ( A ) are functions of B and A , respectively, which could not be reduced to simple expressions and are therefore not presented here. In these cases, the Linewea~er-Burk~~ method does not give the same constants as those obtained from the substrate concentrations at half maximal velocity. a
maximum) with increasing concentration of the of the varying substrate when the rate is one-half of the maximum rate, the same results would have substrate. With a single-substrate enzyme system only one been obtained in most cases. An exception is set of such experiments is possible (all other factors found for the two instances V +- VmaxA (or B ) , being constant), whereas with the two-substrate for the general case (mechanism 11) of the two systems, four different sets of experimental condi- substrate systems. In these two instances linear tions can exist. Thus, the concentration of each sub- Lineweaver-Burk plots are not obtained (as noted strate, A (or B), can be varied while the concentra- previously8s14)and the expressions evaluated from tion of the other substrate B (or A) is held constant, half maximal rates are too complicated and are not to obtain VmaxA (or Vmaxg). Once these maxima cited. have been reached, the other corresponding subSignificance of the Derived Expressions for strates can be varied in each case to determine Michaelis Constant.-It is to be noted from Table I1 that only under conditions VmaxA (or B) -+ VmaxAB (see Fig. 1). For each type of experiment it is possible to VmaxABare the constants generally analogous to calculate a Michaelis “constant.” Table I1 shows those usually evaluated for a one-reactant system theresultsof such calculationsfor the four sets of con- (mechanism IV). Under all other conditions therc ditions. In analogy with the single substrate system, is no such similarity. the Michaelis “constant” calculated here is the value A corollary is that a linear Lineweaver-Burk plot that would be found by treating the data by the is no assurance that a one-substrate system is being Lineweaver-Burkl7 method, that is, from a plot of dealt with or that the evaluated Michaelis constant 1/V US. 1/S, of the expression V / V m a x = S / ( S analogous to the usual Michaelis constant for a Km) where S is the varying reactant, and K , is is the Michaelis constant. The results shown in one-substrate system. It is to be noted also that in all three of the two Table I1 have been calculated from the ratios of reactant cases (mechanisms I, I1 and 111) under V / V m a x A and VmaxB/VmaxAB, etC., Using the equations from Table I. If the “constants” in Table experimental conditions where V+ VmaxA ( o , B) (see I1 were calculated as equal to the concentration Table 11) the experimental Michaelis constant is not a single valued quantity but its magtiitude is a (17) H. Lineweaver and D. Burk. J . Am. Chem. Soc., 66, G58 function of the concentration of that sulxtratc (1934).
+
270
WILLIAME. HAINES,R. VERNON HELM,C. W. BAILEYAND JOHNS. BALL
VOl. 58
which is being held constant in the given series of conditions V + VmaxA (or B) instead of the condiexperiments. tions VmaxA (or B) 4 VmaxAB. Perhaps the term The results found for mechanism I11 are of spe- Michaelis constant should be reserved for the latter cial interest, since this mechanism has been postu- conditions, which yield a constant analogous to lated in order to explain the anomalous variation that derived for the classical one-reactant system.6 of Michaelis constant with substrate in the glucose There are certainly many other mechanisms or oxidase-glucose-oxygen system.Ba Although this variations of these mechanisms that could be dismechanism accounts for the observed 7-fold varia- cussed. We have pointed out one commonly postion in the Michaelis constant with 20-fold varia- tulated mechanism I11 that can usually be elimition in oxygen pressure, it also predicts that nated by the above analysis and have discussed two VmaxA (or B) is linear with the substrate concentramechanisms (I and 11) that demonstrate the pittion B (or A), which is contrary to experimental falls involved in measuring Michaelis constants. facts. Experiments shOw6a that both Vmarglucose It is important to note at this time that mechandependence on oxygen pressure and VmaxOxygen de- isms I and I1 are special and simpler cases of a more pendence on glucose concentration are not linear general formulation which yields essentially the relationships but appear to obey the usual Michaelis- same kinetics. This formulation, previously mentioned by Laidler and Socquet,13is Menten law. An explanation for, both, the variations of the E1 A Ez W Michaelis constant and of Vmax.4 (or B), can be ofEe B =Ea X fered in terms of mechanism I or I1 involving terE s + P + E nary complexes. Under the conditions, V VmaxA (or B), which prevailed during the experiwhere Ezand E3are modified forms of the enzyme, ments, the Michaelis constants would be ex- and W, X and P are products of the reactions. If pected to vary with concentrations of the react- W and X are non-existent, the above mechanism ants as shown in Table I1 and the maximum reduces to mechanism I or I1 and Ea becomes a velocities, VmaxA(or B), would also vary as shown in ternary enzyme-substrate complex. Thus, the concept of a ternary complex is not essential for the Table I. A similar explanation may be offered for the re- conclusions drawn concerning mechanisms I and sults of Tappel, Boyer and Lundberg,6bwho found 11. However, the more general formulation above a 10-fold increase in the Michaelis constant for oxy- would require that the rate of the reaction would gen as the reducing substrate (linoleate) was in- also be a function of the concentrations of the prodcreased 20-fold in a study of the reaction kinetics ucts W and X. Acknowledgment.-We greatly appreciate the of soybean lipoxidase. We see now that a variation in the Michaelis helpful discussions with Dr. Hans Lineweaver durconstant is to be expected if it is measured under ing the preparation of this paper.
+ +
-
+ +
PURIFICATION AND PROPERTIES OF T E N ORGANIC SULFUR COMPOUNDS1 BY WILLIAME. HAINES,R. VERNONHELM,C. W. BAILEYAND JOHN S. BALL Contributionfrom the Petroleum and Oil-Shale Experiment Station, Bureau of Mines, Laramie, Wyming Received October $8, 1866
Ten organic sulfur compounds, ethanethiol, 2-methyl-2-propanethiol 1-pentanethiol, 2-thiabutane, 3-thiapentane, thi& cyclobutane, thiacyclopentane, &%dithiabutane, 3,4-dlthiahexane and thiophene, were purified to a t least 99.9 mole p r ity. The stability of the compounds under storage and distillation conditions wa! investigated. I n addition to the oilin point and freeaing point, the following roperties were measured at 20, 25 and 30 : density, viscosity, surface tension an! refractive index for the r, C, D, e, v, $and g lines. Derived functions including reffactivity intercept, specific dispersion, molecular refraction, parachor and molar volume were calculated and are discussed briefly. Mass, infrared and ultraviolet spectra were determined permitting some correlation of the effect of sulfur groupings on spectra.
Literature surveys2 have shown that data on the physical properties of thiols, sulfides, disulfides and thiophenes are meager. Many of the data available are unreliable because they were determined on compounds of doubtful purity. Because these types of compounds occur in petroleum products, (1) This investigation w&8performed ea part of the work of American Petroleum Institute Research Project 48.4 on “The Production, Iaolation, and Purification of Sulfur Compounds and Measurement of their Properties,” which the Bureau of Mines aonducts at Bartlesville, Okla., and Laramie, Wyo. (2) W. E. Hainea, W. J. Wenger, R. V. Helm and J. S. Ball, U. S. Bur. Mines, Rept. Inueat., 4060 (1946); R. V. Helm, W. E. Hainea and J. S. Ball, ibid., 4666 (1949).
the Bureau of Mines and the American Petroleum Institute have cooperatively sponsored Research Project 48. This Project has among its purposes the production of sulfur compounds in high purity and the accurate determination of their physical properties. The attainment and measurement of the high purities are facilitated by the purification of a large quantity of each compound from which samples are available for sale by the American Petroleum Institute as calibration standards. Because of this latter activity, the stability of the compounds has been investigated.
L
