MEDICAL SCHOOL Wit
`University of Wisconsin
`
`Bioclii
`
`itmnfiriiiiucolouv
`
`L CONTENTS
`
`ARNO ZIMMER, AXEL PROX, HELMUT PELZER and RALNER HANKWITZ: Simultaneous labelling
`with stable and radioactive isotopes in drug metabolism studies
`
`VOLKER DINNENDAHL, HANS D. PETERS and PETER S. ScHoNHoFER: Effects of sodium salicy-
`late and acetylsalicylic acid on cyclic 3’,5’—AMP-dependent protein kinase
`KARL-HEINZ KIESSLING and LARS P.I.LsTRoM: Cytochrome c stimulated oxidation of ethanol
`by liver mitochondria
`HEITAROH IWATA, ITARU YAMAMOTO, Encrn GOHDA, KYOJ1 MORITA, MITSUTAKA NAKAMURA
`and KEIKO SUM1: Potent competitive uricase inhibitors——2,8-diazahypoxanthine and related
`compounds
`YASUKO MURAKAMI and KA'lI‘ASI-II MAKINOI On the convulsive action of castrix
`
`Z. HUszTI, E. KASZTREINER, M. KURTI, M. FEKETE and J. BoRsY: 2-I-Iydroxy—5-carbometh-
`oxybenzyloxyamine: a new potent inhibitor of histidine decarboxylase
`«‘
`Z. Huszn, E. KASZTREINER, G. SzrLAGY1,J. KOSARY and J. BORSY: Decarboxylase inhibition
`and structure—activity relationship studies with some newly synthetized benzyloxyamine and
`pyridylmethoxyamine derivatives
`MICHAEL BRIGGS and MAXINE BRIGGS: Effects of some contraceptive steroids on serum
`proteins of women
`C.
`J’. PETER ERIKSSON: Ethanol and acetaldehyde metabolism in rat strains genetically
`selected for their ethanol preference
`
`7
`
`SUSANNE KELDING, PER BUCH ANDREASEN and LIS FAUERHOLDT: Effect of phenobarbital
`induction on galactose elimination capacity in the rat
`RUSSELL J, TAYLOR,
`;l1<., FRANZ J. LEINWEBER and GEORGE A. BRAUNZ 4-Imidazo1yl-3-
`amino-2-butanone (MCN-A-1293), a new specific inhibitor of histidine decarboxylase
`CHARLES L. HOPIPEL and BERNARD TANDLER: Biochemical effects of cuprizone on mouse
`liver and heart mitochondria
`
`KENNETH W. MILLER, EUGENE C. HEATH, KIRK H. EASTON and JAMES V. DINGELL: Effect of
`Triton X-100 on the conjugation of tetrahydrocortisone, in vitro
`JERALD M. McKENzxE and HELEN L. WHITE: Evidence for the methylation of apomorphine
`by catechol-0~methyl—transferase in vivo and in vitro
`
`2213
`
`2223
`
`2229
`
`2237
`2247
`
`2253
`
`2267
`
`2277
`
`2283
`
`2293
`
`2299
`
`23.11
`
`2319
`
`2329
`
`D Pergumon Press
`
`Continued on outside back cover
`
`BCPCA6 22(18) 2213-2356 (1973)
`
`AURO — EXHIBIT 1025
`
`
`
`

`
`ii,
`
`BIOCHEMICAL PHARMACOLOGY
`
`EDITORIAL BOARD
`
`Chairman: RUDOLPH A. PETERS
`
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`Prof. M. WELscH
`
`American Continent
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`Associate Editors
`Dr. DAVID G. JOHNS
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`Prof. P. A. SHORE
`
`Yale University School of Medicine, Department of Pharmacology, Sterling
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`HONORARY EDITORIAL ADVISORY BOARD
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`TAG E. MANSOUR-—-Department of Pharmacology, Stanford
`J. R. CO0PER——Yale University School of Medicine, New Haven,
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`University Medical Center, Stanford, Calif.
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`G. H. HITCHL\IGS4~Wel1c0me Research Laboratories, Research
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`ARNOLD D. WELCI-I
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`European Continent
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`J. MIcHELsoN—Seehenov
`Institute of Evolutionary
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`RUDOLPH A. PETERS [Chairman)—Institute of Biochemistry,
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`London
`D. F. HEArH«M.R.C. Research Labs., Carshalton, Surrey
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`L. L.
`IVER.sEN——Medical School, University of Cambridge,
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`
`Biocheniical Pharmacology, Vol. 22, pp. 3099-3108. Pergamon Press, 1973. Printed in Great Britain.
`
`RELATIONSHIP BETWEEN THE INHIBITION CONSTANT
`(IQ) AND THE CONCENTRATION OF INHIBITOR WHICH
`CAUSES 50 PER CENT INHIBITION (Isa) OF AN ENZYMATIC
`REACTION*
`
`YUNG-CHI CHENG and WILLIAM H. Pnusorr
`
`Department of Pharmacology, Yale University School of Medicine, New Haven, Conn. 06510, U.S.A.
`
`(Received 15 March 1973; accepted 27 April 1973)
`
`Abstract—-A theoretical analysis has been made of the relationship between the inhibi-
`tion constant (K,) of a substance and the (150) value which expresses the concentration
`of inhibitor required to produce 50 per cent inhibition of an enzymic reaction at a
`specific substrate concentration. A comparison has been made of the relationships
`between KI and 150 for monosubstrate reactions when noncompetitive or uncompetitive
`inhibition kinetics apply, as well as for bisubstrate reactions under conditions of coin-
`petitive, noncompetitive and uncompetitive inhibition kinetics. Precautions have been
`indicated against the indiscriminate use of I50 values in agreement with the admonitions
`previously described in the literature. The analysis described shows K, does not equal
`150 when competitive inhibition kinetics apply; however, K, is equal to 150 under condi-
`tions of either noncompetitive or uncompetitive kinetics.
`
`MANY DRUGS are believed to exert their biological effect as a consequence of enzyme
`inhibition. One approach to the understanding of the mechanism of action of such
`drugs has been to study the effect of drug concentration on the rate of reaction of an
`isolated enzyme. Several approaches have been used to describe the extent of in-
`hibition such as 150 (concentration of inhibitor producing 50 per cent inhibition),
`(I/S)50 (concentration of inhibitor relative to substrate concentration producing 50
`per cent inhibition), and KI (the dissociation constant of the enzyme—inhibitor complex,
`or the reciprocal of the binding aflinity of the inhibitor to the enzyme).
`Although the relationship between the inhibition constant (K,) and 150 of a com-
`petitive inhibitor of a monosubstrate reaction has been discussed,“ a detailed
`comparison of such a relationship for either bisubstrate reactions when competitive,
`noncompetitive or uncompetitive inhibition kinetics exist, or for monosubstrate
`reactions when the latter two types of inhibition kinetics apply has not been presented.
`An understanding of the relationship between [50 and K1 under these conditions and
`the theoretical basis for their determination is critical to appropriate interpretation of
`the experimental data, as well as for comparison of the literature values of 150 or
`U/S)5o. Blakley3 has indicated the limitations in the use of (I/S)50 relative to the K1.
`Although what is presented is no doubt readily apparent to the enzyme kineticist,
`those who are less familiar with enzyme kinetics and yet concerned with studying the
`effect of drugs on enzymes may find this communication useful.
`
`* This research was supported by United States Public Health Research Grant CA-05262.
`3099
`
`

`
`3100
`
`YUNG-CH1 CHENG and W. H. PRUSOI-‘F
`
`THEORETICAL ANALYSIS
`
`Several kinetic situations are described below that have the following limitations;
`(1) the reaction in the absence of the inhibitor follows a simple Michaelis—Menten
`equation; (2) the rate of the reaction depends on the amount of the enzyme—substrate
`complex; (3) a rapid equilibrium steady state method is used;4 and (4) only reversible
`inhibitors are discussed.
`
`Reactions iiwolving one substrate
`
`V-__I/maxs
`
`1
`
`Vmax =inaXi1num velocity; V0 : velocity in the absence of the inhibitor; Km =
`Michaelis constant of the substrate (S); S : substrate concentration.
`Case I. When a competitive in/iibitor (I) is present.
`
`E—S’—7ES~>E-|—P
`ll!
`E!
`
`V; =
`
`Km (1 + E) + s
`
`I
`
`(2)
`
`V, = velocity in the presence of inhibitor; I = inhibitor concentration; K, = dis-
`sociation constant of El.
`
`When I2: 150, V0 = 2 V1, then
`
`By rearrangement:
`
`2Vmax S
`
`m
`
`1 — S
`
`I50)
`
`__
`
`Vmax S
`
`W Km Ti‘ S ‘
`
`[50 -'-‘—".K1
`
`S
`
`Equation (3) is identical to that described by Webbzz
`
`I
`__
`
`(5)50
`
`K,
`K,
`= _ _.
`
`Km + S
`
`(4)
`
`The equation derived by Baker1 makes the assumption that V0 = Vmax in the
`beginning of the derivation:
`
`(1) _.£_£
`
`S5o—K,,,
`
`S
`
`(5)
`
`

`
`Relationship between K, and 150
`
`3101
`
`which is different from equation (4). However, since most investigators have S > Km
`in their assay condition, there would be no significant difference between equations
`(4) and (5), and hence these equations may be transformed into equation (6).
`
`Since 150 will depend on the substrate concentration used in the assay (equation 3),
`it is impossible to compare I50 values from one laboratory with those from another
`unless identical assay conditions are used. Bakerl has indeed considered the substrate
`concentration by his use of (I/S):-,0. Appropriate comparison of the effect of one
`compound relative to another may be made, provided that S > Km, and both com-
`pounds are competive inhibitors. Without prior determination of the type of
`inhibition such compounds exert, the relative values of (1/S)50 have questionable mean-
`ing. Thus, for example, one might have assumed that 3—N—methy1-5-iodo-2’-deoxy—
`uridine would be like 5-iodo-2’-deoxyuridine, a competitive inhibitor of thymidine
`kinase when thymidine is the variable substrate. However, uncompetitive inhibition is
`observed with N-methyl—S-iodo-2’-deoxyuridine when thymidine is the variable sub-
`strate, and competitive inhibition kinetics when ATP—Mg“ is the variable substrate.5
`Having determined that the two compounds being compared are indeed com-
`petitiveinhibitors, one can effectively use (I/S)50 values for the purpose of comparison.
`If the concentrations of inhibitors A and B required to produce 50 per cent inhibition
`at a particular substrate concentration are significantly different yet close, one may
`amplify the difference by augmenting the substrate concentration.
`Case II. When a noncompetitive inhibitor is present.
`
`EéEs_>E +P
`IJFKISIJYKH
`EI:—=ESI
`
`Vmax S
`V, = —-——-—:~—.
`I
`I
`
`"m(‘+k‘.;i+S(1+"i
`
`When I = I50, V0 = 2 V, and:
`
`Vmax S __
`
`ZVITIEIX S
`mi-w>+s<1+*-0)
`
`IS
`
`By rearrangement of the above equation:
`
`I = K
`
`so
`
`S
`K
`S —m — .
`
`(,,.+ )/(KIS+K”)
`
`(7)
`
`(8)
`
`When K,5 = K", that is when the affinity of inhibitor to the free enzyme (E) and the
`enzyme—substrate complex (ES) is the same, then equation (8) may be transformed into:
`
`150 = K“ or K15, or more simply, K1.
`
`(9)
`
`

`
`3102
`
`YUNG-CHI CHENG and W. H. PRUsoF1=
`
`Since S > Km in most assays performed, then equation (8) may be transformed into;
`
`1 +i)
`_1NK,..
`0*
`5 K15
`Kit
`
`and hence,
`
`[50 = K1s[(TS.‘ "i"
`
`K,,,
`
`K
`
`(10)
`
`Provided that Km/S < K15/Kn (since Km/S may be adjusted) apply, equation 10 may
`then be transformed into either:
`
`01'
`
`[50 = Kn
`
`(11)
`
`(§)w=1/(ii-:+%)~
`
`Thus it is quite apparent that there is no value in comparing the efi"ect of inhibitors on
`the basis of (I/S)5o, because I50 equals K, (equation 11).
`It should be emphasized that the dependence of 150 on S is different from that
`observed above with the competitive inhibitor.
`Case III. When an uncompetitive inhibitor is present.
`
`E#ES—->E—l—P
`KI1LI
`ESI
`
`VI :
`
`V
`
`Km
`
`+ i
`
`1
`
`I
`—— S
`
`‘L K)
`
`(13)
`
`When I = 150, V0 = 2 V,, then
`
`Vmax S
`
`Ii
`
`2Vmax S
`
`K,,,+<1+%)>SI
`
`Rearrangement of the above equation results in:
`
`K-n+S=S_‘9
`
`

`
`Relationship between K, and 150
`
`3103
`
`and
`
`I50:K]
`
`When S > Km, equation (14) may be simplified into
`
`150 2‘ KI.
`
`In this situation, 150 is independent of S provided that S > Km. Thus there is no
`value to express the data in terms of (I/S)50.
`
`Reactions involving two substrates
`
`When both substrates are added sequentially to the enzyme, the reaction follows
`either a rapid equilibrium random or ordered mechanism. The rate equation5‘3 is:
`
`Vmax
`
`V0 =
`
`(16)
`
`K” = dissociation constant of substrate A.
`Case IV. If the inhibitor competes for the free enzyme (E) with either substrate A
`or B in a random mechanism reaction, or with the first substrate in an ordered sequen-
`tial mechanism, then
`
`/\
`£1éEj
`X/
`
`EB
`
`EAB —> Products
`
`01'
`
`El é E é EA ; EAB » Products, and the rate equation will be:
`
`V1 2
`
`K.-,.K,, (1 + E) + K..B + KDA + A3
`
`I
`
`(17)
`
`By mathematical treatment similar to that performed in the previous cases, and
`When V0 = 2 V,, I = 150, then
`
`‘
`
`2V,,m AB
`
`Vmax AB
`
`K,.,,K,,
`
`(1 +1729) + K,,B + K,,A + AB
`
`I
`
`K“‘K” + K“B + K”A + AB
`
`

`
`3104
`
`YUNG-CHI CHENG and W. H. PRUSOFF
`
`This equation may be rearranged into:
`
`I =K 1
`
`5°
`
`’( +K...K. +19.
`
`A
`K
`“ B —+
`
`AB
`
`KiaKb)
`
`When K“ —= K,-.1, equation (18) may be simplified to:
`
`o=K1(l+%) <1-I-£7),
`
`.
`
`(18)
`
`(19)
`
`which basically is similar to equation (3); however, equation (19) takes into account an
`additional substrate.
`Equations (18) and (16) may alsolbe transformed into:
`
`I =
`so
`
`Vmax A B
`-——K.
`V0 KmKb
`1
`
`2
`(0)
`
`Since most assays are performed under optimal conditions in which either V0 =
`Vmax, or A > K, and B > K,,, then both equations (19 and 20) may be transformed into:
`
`[so = ““ " Kb
`
`(21)
`
`Thus the 150 value will depend on the concentration of both substrates A and B.
`Case V. When the reaction follows either an ordered sequential or a rapid equili-
`brium random mechanism, the inhibitor acts as a noncompetitive inhibitor and can
`bind to all of the enzyme species in the reaction with the same affinity:
`
`EA!
`/
`(LXI
`
`E1 ‘——‘-
`
`‘EAB —> Products.
`
`E E3/K Jfxr
`
`EABI
`
`JP Kr
`EBI
`
`Then, the velocity is described by:
`
`VI “‘
`
`Vmx AB
`
`(Ki(IKb + K..B + KbA + AB) (1 + E)I
`
`I .
`
`(22)
`
`When V0 = 2 V_,, I = I50, and upon substitution of equation 16 for V0 and equation
`(22) for V,, one obtains:
`
`I
`K,-“Kb + K,,B + K,,A + AB = 1%? (K,~aK,, + KaB —§— K,,A + AB)
`1
`
`

`
`Relationship between K, and 15,,
`
`and hence,
`
`150 = KI-
`
`3105
`
`(23)
`
`Thus the value of 150 does not depend on the substrate concentration and will be
`equal to K1. The 150 value obtained under these conditions can be compared between
`Iaboratories without concern with the substrate concentration.
`Case VI. When the reaction follows an ordered sequential mechanism, the inhibitor
`can compete with either the first substrate A for the free enzyme, or with the second
`substrate for the EA complex, or both:
`
`E :9: EA vi EAB —> Products.
`
`EI :EAI
`
`Then
`
`V, =
`
`I
`Kin-Kb (1-l-—-) + K..B + K:,/1 (1 +
`KIS
`
`I
`Kli
`
`.
`
`)+ A3
`
`(24)
`
`Vmax
`
`When V0 = 2 V,, I = 150, and by combining equations (23) and (16), one obtains:
`
`Ki,,K,, + KaB -i— K,,A + AB = I5o(
`
`K,.,,K,,
`
`K15
`
`+
`
`K,,/1)
`
`I.
`K1:
`
`.
`
`This equation may be transformed into:
`
`:
`
`15°
`
`)[(KiaKb
`Vmax
`( Va A .
`K15
`
`B
`
`-—j
`
`‘L K1:
`
`L)
`_ Vmax
`‘(VJ/B K1sA+K“'
`
`(25)
`
`Under the rare condition when K,,- = K” and V0 = Vmax, equation (25) can be
`simplified to:
`
`o = K1
`
`b
`
`(26)
`
`and
`
`I
`
`K,
`
`(gm,
`
`

`
`3106
`
`YUNG-CHI CHENG and W. H. PRUSOFF
`
`Case VII. When the reaction follows an ordered sequential or rapid random mech_
`anism, and the inhibitor can only bind to the EAB complex:
`
`/ K
`
`E < >EAB —> Products.
`B ‘EB K 4”’ Kl
`EA BI
`
`Then,
`
`Vmax
`VI = —¥————————:—~—.
`I
`KmK,, + KaB + K,,A + (1 —l— -1?) AB
`I
`
`(28)
`
`When V0 = 2 V,, I = 150, and by combining equations (28) and (16), one obtains:
`
`I50
`Ki,,K,, + KaB + K,,A + AB 2 — AB,
`1
`
`which may be transformed into:
`
`Vmax
`[so = V0 K1
`
`Thus, when
`
`V0 : Vmaxs [50 = Kb
`
`(29)
`
`When the reaction follows a “ping-pong” mechanism5‘3 and involves two substrates,
`the rate equation will be:
`
`Vo
`
`Vmax
`= ._j————.
`K,,A + K,,B + AB
`
`Case VIII. An inhibitor, 1, affects both forms of the enzyme, E and E ~ X I
`
`(30)
`
`(31)
`
`l
`
`Productl
`T
`
` A
`
`B
`l
`
`Product,
`T
`
`E
`
`lllm
`E1
`
`E ~ X
`
`TLK»-2
`E1 ~ X
`
`E
`
`The rate equation9 is:
`
`VI =
`
`V X AB
`ma
`Kb<1+I7<I—)A+Ka(1+Ki)B+AB
`
`:2
`
`1'1
`
`

`
`Relationship between K, and [50
`
`3107
`
`When I = 150, V0 = 2 V1, and by combining equations (30) and (31), one obtains:
`
`KA K,,B AB: —— —— I
`
`1, +
`
`+
`
`(B Ki2+AK”
`K, 1)
`1
`K,,
`
`so
`
`and
`
`I =
`
`5°
`
`Ka
`1
`Kb
`Vniax
`V0 /(B K” + A K“
`
`—— ——.
`
`(32)
`
`(33)
`
`Although K11 is generally not equal to K2, in the specific situation when K” does
`equal K,-2, equation (31) can be transformed into:
`
`150 = (1 +
`
`IQA + KaB
`
`Ki1-
`
`(34)
`
`Case IX. When the reaction follows a ping-pong mechanism, an inhibitor affects
`only one form of the enzyme——E or E ~ X. The rate equation will be:
`
`01'
`
`OK‘
`
`V, =
`
`Vmax
`
`I
`K,,A+K,,<1+—)B+AB
`Kil
`
`V, —
`
`Vmax
`
`I
`
`K,,(1+E-)A+KaB+AB
`
`.
`
`(35)
`
`(36)
`
`When I = I50, V0 = 2 V,, and by combining equations (35) and (30), one obtains:
`
`I
`K,,A + K,,B + AB = K,,A 5°
`Kil
`
`I ~K (1+K“B+B)
`50 — i1
`KbA
`Kb '
`
`Similarly, equation (36) can be transformed into:
`
`1 —K(1+K“B+A)
`50 — .-2
`KbA
`Kg .
`
`DISCUSSION
`
`(37
`
`)
`
`38
`
`(
`
`)
`
`The effect of an enzyme inhibitor in a variety of situations has been analyzed.
`Before the equations described above may be used, one must determine the type of
`
`

`
`3108
`
`YUNG-CH1 CHENG and W. H. PRUSOFF
`
`inhibition involved. This is readily established by applying the rules discussed by
`Cleland.5‘3 The relationship between K, and 150 varies. For instance, when a non-
`competitive or an uncompetitive.inhibitor is studied in a monosubstrate enzymatic
`reaction, [50 will be equal to K,, provided certain conditions are met (equations 11 and
`15). Howeveifif the inhibitor is a competitive inhibitor, I50 will be equal to K, (1 +
`S/Km) (equation 3). The equations (3, 8, 14, 18, 23, 25, 29, 33, 37) have described the
`relationship of [50 to K, when both the type of inhibitor and the reaction mechanism
`vary. It is readily apparent that the relationship of [50 to K, is dependent upon the
`type of inhibition and the mechanism of the reaction. It has been established that
`(I/S)50 may not be used in the absence of such knowledge without producing great
`uncertainties as to its meaning. K, does not equal [50 when competitive inhibition
`kinetics apply; however, K,
`is equal to 150 under the conditions of either non-
`competitive or uncompetitive kinetics.
`When a group of inhibitory compounds have an identical mechanism of action, a
`direct comparison of the I50 values among them will suflice to determine the relative
`eflicacy, provided the assays are performed under the same conditions. However, in
`certain cases, when the K, value of each compound is required, it may be impractical
`to perform the kinetic studies required to determine the K, for each. In this situation,
`it is still possible to calculate the K, values, provided one knows the K, of one coin-
`pound, by using the relationship:
`
`‘ (K01
`(I50)1
`(1so)2 —_ (Kz)2'
`
`This may be done without knowing the reaction mechanism or type of inhibitor in
`detail, except for Cases II, VI and IX, in which a certain assumption must be made
`before this generalrule applies. In Cases II and VI, the assumption is that K,.,- = K,-,-,
`and in Case IX, that K” = ,-,.
`When comparing the I50 values of compounds that inhibit a specific enzyme derived
`from the same source, but reported from different laboratories, a few important
`factors must be considered: (1) Are the assay conditions the same? (2) Do the com-
`pounds have the same reaction mechanism for their inhibitory eflect ?
`
`REFERENCES
`
`1. B. R. BAKER, Design of Active-sz'te—directed Irreversible Enzyme Inhibitor p. 202. Wiley, New Yofk
`(1967).
`2. J. L. WEBB, Enzyme and Metabolic Inhibitors p. 106. Academic Press, New York (1963).
`3. R. L. BLAKLEY, The Biochemistry of Folic Acid and Related Pteridines (Eds. A. Neuberger and
`E. L. Tatum), p. 162. North Holland, Amsterdam (1969).
`4. S. Cl-IA, J. biol. Chem. 243, 820 (1968).
`5. P. VOYTEK, P. K. CHANG and W. H. PRUSOFF, J. biol. Chem. 247, 367 (1972).
`6. W. W. CLELAND, Biochim. biophys. Acta 67, 104 (1963).
`7. W. W. CLELAND, Biochim. biophys. Acta 67, 173 (1963).
`8. W. W. CLELAND, Biochim. biophys. Acta 67, 188 (1963).
`9. N. MOURAD and R. E. PARKS, JR., J. biol. Chem. 24], 271 (1963).