THEORETICAL BASISFOR SYNERGISM
March, 1963
145
Sequential Blockage as a Theoretical Basis for Drug Synergism MARTINL. BLACK T h e Research Laboratories, Parke, Davis and Go., A n n Arbor, Ilichzgan Received September 1, 196d Kinetics analysis of a linear model enzyme system shows that, in theory, the combined effect of two inhibitors acting by sequential blockage is necessarily synergistic. The rate equations for the system describe a theoretical dose-effect surface for the drug pair which results in a formal definition of synergism that correlates well with experimental observation over the concentration range for which the effect is demonstrable. This definition provides a rationale for the isobole technique for demonstrating synergism experimentally and a means for defining and calculating the “amount” of synergism shown by two drugs.
The term synergism1 has been applied by pharmacologists2 and chemotherapist^^-^ to instances in which two or more inhibitors of a biological response are more effective when acting together than would be expected’ from additivity of their individual effects.8 Synergism is often encounteredg and is always of interest for the possibility that host toxicity may be only additive,1° or that the emergence of drug resistance may be delayed by the combined use of two or more inhibitors with different modes of action.” The effect is unmistakable12 when drugs A and B, which produce a certain sub-maximal effect in concentrations a and b alone, produce the same effect when combined in con(1) Observations of synergism and use of t h e isobole method for describing them are a t least 90 years old. l’. R. FFaser, Brit. Med. J.. 2, 457, 485 (1872), did not use the term but observed synergism, a s well as antagonism, between atropine and physostigmine, depending on relative doses. Since Fraser, t h e term has had a controversial career, a s emphasized b y a recent exchange of Letters t o t h e Editor. Various Authors, The A’ew Scienl i s t , 14, 481, 600, 662 (1962). T h e term even has a metaphysical connotation; i b i d . , 16, 45 (1962). (2) G . Chen, Arch. Intern. Pharmacodyn., 111, 322 (1957). (3) G. A. H. Buttle, Proc. Roy. SOC.)Wed., 49, 873 (1956). (4) E. Jawetz and J. B. Gunniaon, Pharmacol. Reus., 5, 175 (1953). (5) E. Jawetz, J. B. Gunnison and V. R. Coleman, J . Cen. Microbiol., 10, 191 (1954). (6) Anon., Chem. Eng. Xews. 32, 4473 (1954). (7) One person will “expect” more t h a n another, so this “definition” is operationally worthless. A. J. Zwart Voorspuij and C. A. G. Nass. Arch. Intern. Pharmacodyn., 109, 211 (19!7), and S. Loewe, Armeimittel-Forsch., 3, 285 (1953), have in fact shown how it can lead t o t h e absurd conclusion t h a t a drug can be synergistic with itself. As Loewe contends, t h e decisive test for synergism lies in treatment of t h e data b y the boloform method. This method bas been applied t o double inhibition by G. H. Hitchings, A m . J . Clin. iYulrilion, 3, 321 (1955). by S. B. Kendall, Proc. Roy. SOC. ’Wed., 49, 874 (19561, and by many others. Chen, ref. 2. and 4. J. Zwart Voorspuij and L. H. Bokma, Ann. inst. Pasteur, 96, 404 (1958), have used 4-dimensional boloforms t o describe the joint effects of 3 inhibitors. (8) Loewe (see ref. 7) emphasizes t h a t a definition of synergism based on non-additivity us. additivity of effects is meaningless since effects are never with certainty additive. He discusses i n detail t h e question of “what is properly additive t o what.” See also ref. 21. (9) T h e literature on synergism has been reviewed b y H. Veldstra, Pharmacol. Reus., 8, 339 (1956). A well documented case results from t h e action of sulfadiazine (11)and pyrimethamine (1%)on a variety of organisms, both in vitro and in v i v o . presumably by blockage of t h e essential sequence
p-Aminobenzoic acid
I1 -L
Folic acid
I?
Folinic acid
See L. G. Goodwin, Proc. R o y . Src. Med., 49, 871 (1956): S. B. Kendall, see ref. 7 : L. G . Goodwin and 1. hI. Rollo in ”The Biochemistry and Physiology of Protozoa,” S. H. Hutner a n d A. Lwoff. eds., Academic Press, New York, N. Y . , 1955, Vol. 2, pp. 24.5-246; L. P. Joyner and S. B. Kendall, Kature, 176, 975 (1955): G. H. Hitchings, see ref. 7. 110) As was shown by J. Greenberg, B. L. Boyd, and E. S. Josephson, J . Pharmacol. E x p t l . Therap. 94, 60 (1948), for experimental treatment of Plasmodium gdlinaceum infections with sulfadiazine-chloroguanide combinations. (11) D. A. hIitcbison. Bril. Med. Bull.. 18, 77 (1962). (12) See, for example, h l . W. Fisher and L. Doub, Baochem. Pharmacol., 3, 10 (1959). hl. W. Fisher, Anlihiot. Chemotherapy, 7 , 315 (1957), also has shown t h a t a concerted inhibitory effect, consisting of antimetabolic and immunogenic components. can achieve t h e same net result. B. A. Kaisbren, ibid., 7, 322 (1957), has reported t h e clinical effectiveness of this technique.
+
centrations m and yb, with ( T y) (>> i l and e? >>> i,. Since competition between inhibitors and substrates for the enzyme surfaces is required for inhibition, s1 il and s2 i:! are also realistic conditions. The only models, therefore, that will permit a proof of the desired type are: (1) 1st-order in SI, s2, i,, i2 and zero-order in el and e p ; i.e., el >>> $1 and i l ; e2 >>> s? and i?. A proof based on this niodel will be given in detail. el and s2 i2 (2) 2nd-order in which s1 i l e?. This procedure gives a more general but less useful result than the above. The differential equations that result caniiot be integrated, but they reduce to the 1storder set when 1st-order conditions are imposed. The 2nd-order case may be safely ignored for the present purpose, for reasoiis to be given later. The arguments to follow do not require the assumption of an intimately detailed mechanism, but the 1storder case may be depicted in highly schematic form as shown : (1) Inhibition by 11 alone
-
-
-
-
--
('2)
Irihibitioii by
12
- -
alone
147
alytic sense. I n the 2nd-order model the concentrations el and e2 of these enzymes must, however, be treated as variables. All reactions are assumed to be irreversible because of the mass-action effect of e >>> (i s). Steadystate conditions are not deliberately imposed, but the net result, as will be shown later, is equivalent to having done so. The condition for competitive inhibition (Assumption 3, see Introduction) will appear in the following development as negative signs for operators of the type -ds/di, and for the functions that they represent. This coiiditioii is justified by the following argument. Competition prevails as long as both S and I are present, so any change in s occurs in the face of competition from I. and conversely. Consider two hypothetical experiments that differ only in the infiiiitesimal extent to which competition has occurred after a given time interval; the relationship existing between the variables at the stated time is,
-
ds
=
kd(t' -
2)
Le., the increased extent, ds, to which S has not reacted with E is due to more efficient conipetition by I, as measured by the increased extent, d(io - i), to which I has reacted with E within the same time interval. This relationship carries a linear proportionality constant, k , since competition occurs on a one-for-one basis. Performing the indicated differentiation gives = k
-ds/dt
The precise form of the function corresponding to the operator -dsldi will appear later, but the above argument shows that it can be interpreted as a condition for competition and that, as such, it is always negative. The negative sign will hereafter be affixed to dsldi as a reminder that the function that it represents must likewise be negative. This argument acknowledges oiily those changes in s and i that are due to competition proper; other contributions to the net changes in these quantities mill be incorporated, as needed, in the form of the appropriate operators. I n this sense, -ds/di is actually a partial differential coefficient in which all influences upon s and i that do not result from competition are considered constant. As will be shown later, this mathematical interpretation of a metabolic block leads to variants of Huxley'sZ5 well-known allometric equation, dy/ dz = liy/x, which has been applied to competitive biological interactions of widely variable types.Z6 Derivation for First-order Model with Inhibition by 11 Alone.-From definition (e)
(3) Inhibition by
I1
and I?simultaneously
(1)
Kt
From definition (i), with (2)
f(sz)
=
sIo,
s2 =
sz0, sA0 and s3constant
f(s1)
1;roiii tlie coiiditioii for coiiipctition hctweeii S1 a i d Il for E1 (3,
From definition 'The hackcts ciiclosc tlie spccics that react competitivcly. The symbols (E1) and (E2) have no mathematical meaning but are included as reminders that enzyme-1 and enzyme-2 are involved in the usual cat-
(1)
SI
= f(21,
( a ) ,when t is constant i,
= AZlO)
(26) J. S Huxley, "Problems of Relatixe Grontli." Aletliuen dind Co , Ltd , London, 1932, p 7 (26) H G. Bray and K W h i t e ref 17. p p 3 3 2 - 3 3 3
AI. L. BLACK
145
1'01. G
So, froiii theorem I3
which the negative term is the condition for conipetitioii between I, and SI for El. l l i e operations indicated in cq. ( 5 ) are performed 011 definitions (a), (b), ( e ) , (e) and (g). Differentiation of definition (e) gives iii
dlitldsz
(6)
2
(7)
A?.?: -
dsl
Iiitcgratioii of mi. ( 2 0 ) pi\.(>;-
- In 1 2 ~ =
(21)
I,Z
Definitioii (c) divided by definition (a) gires dSJ
duhstituting definition (11) for i, ant1 definition ( e ) for k ? s : , cmicclling idciit ical tcrnis aiid rearranging gi\ (>-
Dcftiiitioii (a) divided by dehnition (b) gircs
+ 111 c
In
+ In C
or, by dehnition (f) 111 E , =
(22)
kisl
In 7211
' 2
k?
k?
x.,
22"
Derivation for First-order Model Inhibited by I1 and both inhibitors present
Iz Combined.-With
I?, = f ( i 1 0 ,
(23)
Differentiation of definition (g) n ith t constant gives ('3)
dZl/dZ,Q =
C'ombining eq. (G-'3) (5) givw
bo,
2.0)
by tlieoreni
p-kxl
in tlir manner indicated by cq. I3ut tlic partial differential coefficients on the right side of eq. (24) are giveii by eq. (12) and (20) so eq. ('11) heroines
Substituting definition (g) for i, in eq. (10) and caiicelling identical terms gives
(25)
Rearranging aiid integrating cq. (25) gives Ilisappeara~iccof the t term still leaves time as a hidden variable in all subsequent equations; thus Rt will always refer to response, aiid E , to effect, after a constant timc interval dating from introduction of the inhibitor(s) to the system. By definition (e), L2s2 = lit,so eq. (11) beconics, after rrnrranging dRt
(12)
I, -2
=
k,
x
/l!t
-d
21 0
210
Iiitcgratioii of eq. (12) givrz (13) 01
~
-
111
Rt
=
I,
In
7,"
+ ln c
hy definition (f)
(11)
I n Et
=
i, 2 kx
111 zl"
+ In
('
Derivation for First-order Model with Inhibition by derivation follon s from definitions ( e ) , (d), (e), and (h), theorem B, and the condition for rompetition by a method similar to the above. Thus
I? Alone.-The
(15)
Rt
= f(s2)
(l(J)
82
=
-
111 l f t =
' 2 In kx
+ knk, In + ln c
or, hy definition (f)
'lIi(>locus of CY!. ( 2 7 ) is tlic desired theoretical doseeffect surface for the drug pair. Although the addition required by eq. (24) would seem to imply additive effects for I, and I*,the net result is multiplicative -and therefore synergistic--because of the presence of the logarithmic factors, diln/il" = d (In ilo) and dizO/i10 = d (111i?"), a i seen in eq. ('15). It is this feature that leads to cq. ('27) and its associated (lose-effect surface as the first recorded indication of the obligatory nature of synergism. This point will be PXplored later in more detail. Derivations for the Second-order Model.--The procedure (See Appendix) is analogous to that for thc 1st-order case except that c1 and e? must be treated a;variables and freer u5e of partial differentials must therefore be made in handling the appropriate definitions, (f), (i), (k)- i n ) . The equations that result are, for single inliibitioii
f(4 (28)
aid, at collstallt t t 17)
(26)
22 = j ( 7 2 " )
SC)
and ( 39)
and, for double inhibition
A s hfore, the negative term is the condition for compet i t i o l i between Szand I? for Typc found cxpcrimeiit>ally i t i cases of syiicrgisrn. Wlicii X.2 > k , and li, > /iy tliis siirfarc. lias t,he gciicral shape shoivn in Fig. 1, adapted from I,ot.\ve.' \Then 1 ~ 2< Ii, or li, < the individual , or BB', have curvatures opposite doscvffect e i i r v ~ sAIAI' t o tliosc slioivii, hiit--as will appear later--the direction of rurvature of isoboles A41Baiict A'13'3 n-hich is the crit,crioii for synergism, is the samc for all combinations of tlic vnliws of lc2, and I;,. Esperimeiital application of eq. ( 2 7 ) is most coiiveiii r i i t whcn oiic of thr coneeiitratioa variables is held roiistaiit ; t'Iic rcsultiiig graph is tlicii a vertical profile of' tlw holoform surface. &-Iprofilra of this t'ype is also liiic.ar i i i h i E , rs. I i i i" siiiw t,Iie constaticy of the J i w d inhibitor coiictliitratioii ticcomes a part of 111 f'. 'rli(> usf1 of ccl. ( 2 7 ) io tliis maiiiicr is particularly coiivciiieiit i i i tlic iri r ' i t , ~ study ~ of in1iibitr.d bacterial growth fw(,:~iisilof thr simple proportioiialitics rclatiiig E,, 72%a i d .1,. as cxprcwcd 1)y clcfiiiitioiis (1) a i i c l (f).40 I o r t11is piirpo~~ ~ ( ,1 . ( 2 7 ) 1)ecomc.s -
111 - t t =
lir - Ir1
I,.
y
ir"
-+ I,.., I, >
~* I l l
iz'l
+
Ill
L'
i i i \vhicli the proportionality coiist,aiits in definitions (j) and (f) are part of new intercept. In C. Applicability of eq. ( 2 7 ) is iiot. of course, limited t,o bacterial growth. I:cliiatioii ( 2 7 ) lias a number of properties of interest (31) I f . I.:.\Vatsun, i b t d . , J . I . 1:. A l a c k u o r t l ~a n (
nl(,r.iteTn. .I., 4 2 , !+I( 1 ~ 1 8 ) . c'c, ibid.. 22, 689 (1928). .J. 11. Q i i a s t r l a n d &' . ('amp. P h p i o l . . 16, I (JWO), ) K.f ' . Fislirr ilmi 11. ) K. C . Fisher a n d .I. R . S t e a m . 7b?d., 19, 109 (18.12). ) .I h. l . ( % n e . i h i d . , 19, 173 (14.18). ('lark, " G m p r a l Pliartiiir(.i,lr)ey," i n "Heffter'a IIandbiirh 111.r p n I'Iiariii;tk,~losir~,''17. IIriihnrr a n d .J. Srlriillpr, r d s . . Vriliic irinerr. Rc,rliti, I ! l : { i , Rri. IT'. )
alsu tlir aiisor1,tion fiincticiri of .t. .J. ( ' l a r k , ref. 3 7 . 1'. Is) AI, I i u r o k a w a , AI. 1Iat:tnii. N. Kashiwaci, 7 Isliii1;r n i i l l 13. 1 I i r r 1 i i r 1 a . .I. L ~ w , , 63, 14 1962). 1'
(3.5)
expected. el and c 2 appear explicitly, aiid tlicl 15torder result. ( ~ 1 .(K4), is obtained when tlie 1st-ordcr condition, 02 >>> il" ~ 1 ' )i.b imposed on ('(1. (:35) E(1uation (34) describes a family of curves, some segments of wliicli are indicated in I'ig. 2 . h given (wrvc is rompletely determined for particular values of C aiid u' and tlic result can he considered a theoretical isobole for iyiic.rgism hetwecn a pair of drugs, I, and Iz,that react iii a 1st-order maiiiier (rate coiistaiits k , and k,) with E, mid EL in c*ompetitioii with SI and S,,respectively. Earl1 such isobole i q a constant-effect profile of tlic theoretical boloform surface for tlie drugs in questioii within the concentration raiige for which synergism i. demonstrable. The relationship of a theoretical isobolc t o a generalized t)oloform surface and to the individual clov-effect r u r ~ is~ >howl i in Fig. 1. F h c h thcorctical isobolt. of the type cxprc (:i 4) approac~hr~s I lie aws asymptotivally w tli:Lt i:, c i \ ~ i J yI P ~~ P Iipgativ(3 the point of iiltcrsertioii of tho l i i l o , itn= iLn, w i t l i itii i s o h k ~ Ah
i.{O) 1)III. ill
(.i7)
-
-
dztf'/(li2"- s l n p gl~lrc~I~:d
-
-w
THEORETICAL BASISFOR SITERGISM
March. 1963
151
eq. ( 3 3 ) and (3-1) for iIo= 0 or i 2 0 = 0 thus result from the assumption of a 1st-order mechanism. The 2nd-order result, eq. ( 3 5 ) , would presumably also lead to an indeterminate form, but this cannot be established formally since eq. ( 3 5 ) cannot be integrated under 2nd-order conditions. The 1st-order theoretical isobole, eq. (34), can be given meaningful intercepts, however, by translation to a new coordinate system with a new origin a t @,a). This is a valid, indeed, a realistic procedure since drugs must often exceed certain threshold concentrations ( b or a) before aiiy effect is o b s e r ~ e d . ~Letting ~ ? ~ ~ a and b represent the threshold concentrations for I1 and Iz,respectively, eq. ( 3 3 ) can be translated to the new coordinate system
c
I
-0
'2
Fig. 2.-Segments of the theoretical isobole, iIo = C/(izo)", for various values of C and w: ( a ) same C ( # 1 ) , different w's; (b) same C ( = l),different w'5; (c) different C's, same w ; (d) different C's, different w's.
by rearrangement of eq. (32). The symmetry of a theoretical isobole relative to the line i 1 O = izo is thus determined by the magnitude of w ; for the special case, w = I , the isobole is a rectangular hyperbola.41a Differei~tiation~~ of eq. ( 3 2 ) gives
For all meaningful values of (39)
it,izo, and
10
= z
izO
=
+a
Y+b
where x and y are the concentrations of I, and 1, in excess of threshold concentrations, a and b. b and a are both constants, so the new origin is (b,a) and eq. ( 3 3 ) becomes In (z
( 40)
+ a)
=
- w In ( y
+ b ) + In C
+
+
Intercept, hi C, is now the value of In (x a) for (y b ) = 1. The result of this translation of coordinates is, in isobole form (z
(41 I
+ a)
= CAY
+ b)"
with a general shape shown in Fig. 3 . The new isobole,
in eq. (38)
d2ii0/(diz0)2 >0
so every isobole is concave as shown, as required for synergism and as found experimentally. A similar test for the direction-of-curvature of the 2nd-order isobole, obtained by partial differentiation of eq. ( 3 5 ) , gives a result identical with eq. (39); this identity is another indication of the qualitative equivalence of the 1st- and 2nd-order treatments. The ratio, w,can be found from eq. (37) by visually fitting a tangent, di10jdi20, to any point, (Go, izo), on an isobole. This is more easily done, however, from a linear plot of In i1O us. In izo, from which (- w) = slope directly, as required by eq. ( 3 3 ) . C, in a logarithmic plot of eq. ( : 3 3 ) , is the extrapolated value of il0 for i z o = 1. The asymptotic approach of each elid of a theoretical isobole, eq. (34), to the axes implies that, contrary to experience, an infinite concentration of either drug alone would be required to produce the standard response; this feature, though inconvenient, is in itself only a formal detriment to the theory,43for the same limitation is inherent in the kinetics analysis of aiiy 1st-order reaction. Thus, for k
A + B
the rate equation, -dA/dt = kA, is indeterminate in t for A = 0, but first-order reactions do, in practice, go to completion in finite time. The indeterminate forms of (4la) R. A . Edgren. .4?1n. S.Y . Acad. Sei., BS, l i 0 (196Y), has referred t o the hyperbolic nature of experimentally established isoboler. (42) By logarithmic differentiation as described by G. J . Kynoh. ref. 24, p. 48.
210
ha)
y(=ii-b)
6
+
+
Fig. 3.-Graph of the function (z a ) = C/(y b)" relative to an origin a t ( b , a ) ; a theoretical isobole in translated coordinates.
nom plotted on an x vs. y scale, is of course identical with the previous one, but relative to the new origin @,a), and has finite intercepts on the x and y axes. C and (- w)are best determined from ey. (-10) by a linear plot of In ( a 2) 'us. In ( b y). The intercepts, A and B, in Fig. 3 are now finite, for if, in eq. (41), y = 0, then x = A = (C/b") - a, and if x = 0, then
+
+
y = B = (C/a)ku/kz
-b
Intercept -1is the coiicentjratioii of drug 11 in excess of its minimum effective concentration (a) that, when act(43) A . .J. Clark, ref. 37,p . 7 .
iiig i i i the pres1 ce of a tliresliold coiicc.iitratioii ( h ) of IS, causes the standard respoilbe. The meaning of intercept B is clear by analogy. S o t e that 1'ig. 1 is plotted relative t o the same traiislat(xd codsdiiiatc system, and that the Et-coordinate of tlic origiii, (Et)i, is therefore iiot zero; it must be foiiiid by translation of pq. (27) and casting the result iiito exponential form, giving Et
(12) \\
=
L (U
1ie11cc, by settnig .L
+
=
( 1 31
Xj'L / A = ( i , t
0 2nd =
i/ =
Cd
0
=
/2t
f(c2,s.j
(.12j
7 2 = .f(Sl)
i .I;;)
.xi =
j(il)
and ( .I1)
41 hl-iku
I n theory, a tliresliold-rffect isobo'e should connect poiuts b aiid u in the Kt = 0 plane of a boloform surface. This predicts the existence of synergism when z10 = gu mid i20 = hb, y slid h he!iig fractions such that g h < 1 . This isobole is iiot seen, houe~-er,in rig. 1 since tlic El = 0 planc 15 iiot slion ii therc. TIIPcrobs-liatchrd arra I~etwcenthe additivity and .yticrgism isoboles iii Fig 3 l i a i heeii defined as tlic "amount" of synergism.4' '1'111s quantity, 2 , can t x dctermincd directly from tlie graph hy mechanical iiitegratioii or by integration of eq. (41). Letting 0 repreirnt tlie orrgiii, tlieii
+
=
l'roiii these definitions ill)
y)L+L,
l'lic individual doqe-effect curves (BB' and AX') in l'ig. 1 are dcfiiied by c y . (43) 11lien .c = 0 or y = 0.
X r w AOU
Appendix Derivation of Equation (28j.-The procedure is analogous ( ( I that follon-ed for the M-order model except that el and el niiist be treated as variables and freer iise of partial differentiation mimt be mnde in handling the relevant definitions, ( f ) , ( i ) , (kj-(ii\.
il
=
,f(Lll')
So, I Jtheorem ~ B (A3)
I n the strictest sense, eq. (A2)-(A4j and the operators in eq. (A5) should be written as partial dependencies; i t will be seen latrr, however, that certain convenient cancellation properties require that only R! he treated explicitly as a function of two variables, as specified in eq. ( A I . ) Applying thcoreni A to eq. ( A l \ gives
niid these operations, applied to definition (k), give
tlRt
(-47)
=
(k2ez)dsz
+ (k2s2)del
nr
(AB)/?
a1Nl
B u t de2/tls:
5 3
- AOB =
hOCB
=
1, so cq. (Asj becollies
1) c h i t i o n (k)divided by definition (1 j is
and definition (1) divided by definition (in) gives
Integration of definition ( m ) gives
I,'or thci spccial cas:', w
z
=
(AB)/?
=
1 , tlie integration gives
- C In ( B +-
0)
+ C In b + aB
Y is t,hus dct,ermined Pnt,irc.!y by t)he graphical constants for a given syst'cm. Acknowledgments.-I have Ixnefited from many discussions of this problem n-ith many of my associates; :imoiig them. I on-e pnrticdar thanks to RIr. Leonard 1)ouk) who first, aroused my interest in the subject and offered many va1ual;lc rriticisms of the manuscript t1i:ring its prcp::rution. I am also grat,eful to the reof t his p ; : p c ~for citing three reference^^^^^^^^^^^ \\.hirh cwntribi;tt~ t o thc record and hare, therefore, I ) c Y ~ I:iddcd. ~ 'I'iiestx rrfi~rewes,though relevant t o the prcscnt, n-ork, do riot in any n-sy:inticipat,e its methods 0r
111
il ==
J
-/;%
cldt
+ 111 ilo
or
(X13j
i , = 2'p- k , f
i,l,il
and partial t1ifferenti:ition of q. (.II :1) gives
Ikluations (rig), (\.I!()), ( . i l l ~ land ( X l 4 j are the terms in eq. (A5), so eq. (A5j beconies, after these substitutions
Sulmtituting definition ili I i'or (b2e.sz), eq. (A13) for i,, cmcelling identical terms, factoring, : m I rearranging gives
c 0 1 IC 111sio 1 is.
fig)
( 1 . 13. I:lion, ?,. ? i n w r zinc1 (;. 11. Ilirrliings, J . B i d . Citem., 208, 477
(1054). ' l i ) ,J. 13, I . i , i i t s
iiiiiJ
(19:fi).
III
eci. ( A l G ) gives
( 3 ; l
I'haf?nnro7. I : e c s . , 9 , 211 (19371. ( & 7 )(:I) k. 1;. dr .l,)nql-i i r i , "Qiisntit,!tire RIetliods i n Pharrnacolo~y," 11. di. ,lc>ngr.,m i , , N~,vtii-€Id!:~rid I'uld, ] > . 328-339. (18) .I. 11.
Substitution of definitic)ii( t I
I ? . 1 ). 1 5 1 i ~ ~ l i .I. c ~ .I7iiiwmnr.ol. E r p e r . T l i P m p . , 116, 480
(;zdd1~10.
Derivation of Equation (29j.--As hibition by 1 2 alone
in the 1st-ordcr case for in-
~ I O N O A MOXIDASE I N E ISHIBITORS.IT'
March, 1963
153
Integration of definition ( n ) gives
(Ali)
(A21) But
In i2= - k k , J e 2 d t
+ In
i20
or
Rt = f(st,ez)
(AIS)
(A25)
so, by theorem A
iz =
i,0e-kyJr2dt
and partial differentiation of eq. (A25) gives
or Substitution of eq. (.422), (A2R) and ( h 2 6 ) in eq. ( A l i ) gives The dependence of R, on iz is omitted in eq. (A18)-(A20) since i t has already been acknowledged in the first two terms of the righthand side of eq. (A17). Since de*/dsz = 1, eq. (A20) may be simplified, giving
SubFtitliting definition ( I d for X - Z W ~ and eq. ( ~ 1 2 5for ) i?, cancelling identical terms, factorin%and rearranging g i w s
Applied to definition (k), eq. (A21) becomes (422)
dS2
=
hlez
Substitution of definiti-Jn i f ) in e?. (:4'23) givw
+ k?s?
Definition ( k ) divided by definition ( n ) gives
Derivation of Equetion (3O'.---ry the reasoning given i n t b ~ 1st-order cape, eq. (3)and (29) may be combined d'rrctly t o give eq. (30).
Monoamine Oxidase Inhibitors. JV. Some Dialliylaminophenylalkylhydrazines and Related Compounds JACOB FINKELSTEIN, JOHN A. ROMAKO, ELLIOT CHIASG,asu JOHSLEE Research Laboratories, Hoffmann-La Roche, Ine., A-utlqj, -Y.J . Received October 5, 1962 Two sequences of chemical reactions leading to several new basically substituted phenylalkylhydrazines are described. The compounds were less active and/or more toxic than the unsubstituted parent compound in limited animal tests.
Certain aralkylhydrazines and selected acyl derivatives of them are potent, long-acting monoamine oxidase inhibitors.' Substitution of the phenyl ring with amino or dialkylamino residues converted the length of activity from periods of the order of 25 days to less than 1 day. It was noted2 that with increasing length of the alkylene bridge, compounds of intermediate length of activity (4-5 days) were obtained. This paper describes the preparation of these latter materials. (4-Dimethylamino-a-methylphenethy1)hydrazine (V) was synthesized as follows p-OzNCsHaCH2CCCH3
I
-
treating 4-nitrophenylacetyl chloride with diethvl ethoxymagnesium m a l ~ n a t e . ~It was reduced readily in the presence of formalin to produce 1-(4-dimethylaminophenyl)-2-propanone (11) in 91% yield as a yellow distillable oil, which was treated with acetyl hydrazine to form the corresponding acetylhydrazone (111) which was reduced to IV in acetic acid by H2/Pt02, stopping the absorption of one equivalent of hydrogen. If the reduction was permitted to continue, the cyclohexyl compound VI was formed which, on deacetylation, gave VII. Deacetylation of IV gave V.
p-(CH,)2NC,H,CHzCOCH,
I1
p-(CH3)2NC6HaCH&H(CH3)NH?;"3 IV
p-(CHs)zNC6H&HzCH(CH3)NHNH2 V
1-(4-Nitrophenyl)-2-propanone (I) was prepared by
-+
~D-(CH,),NC,€~,CH,C(GH ) = P \ ' S H C Q C H ~---* I11 p-(CH3)2NC,H "CRlCH(CH3)?;HSIICOCIT, VI
I
p - ( CH,)zNCeH,&H2CH(CH,)NH?r",z VI1
The synthesis of the higher homolog, 1-(4-dimethyl-
aminophenyl)-3-hydrazinobutaneand related products, (I) T. S. Gardner. E. Wenis, and J. Lee, J . Med. Pharm. Chem., 2, 133 (1960). (2) T. S. Gardner, E. Wenis, and J. Lee, ihid.. 3, 241 (1961).
is shomn in the scheme a t the top
Of
the next page.
(3) C. G. Overberaer and II. Biletch, J . A m . Chem. Soc., 73, 4881 (1951).
