Parallelism of log dose-response curves in competitive antagonism

Pharmacological Research Communications, VoL 1, No. 1, 1969

PARALLELISM C

I

LOG D O S E - R E S P O N S E C U R V E S

IN C O M P E T I T I V E A N T A G O N I S M H.O.

Schild,

D e p a r t m e n t of Pharmacology~ Street,

Received

U n i v e r s i t y C o l l e g e London, London,

Gower

W.C.I.

22 N o v e m b e r 1968

It is generally c o n s i d e r e d that in c o m p e t i t i v e

drug

a n t a g o n i s m the log d o s e - r e s p o n s e

curves are parallel

example A r u n l a k s h a n a

1959).

novel m a t h e m a t i c a l

and Schild,

In the f o l l o w i n g a

proof of this p r o p o s i t i o n

ment is based on simple

receptor theory,

(see for

is given.

The argu-

and it is a s s u m e d that

the activation of an equal f r a c t i o n of receptors leads to equal responses of the preparation. By d e f i n i t i o n parallel

curves must have the same slope at

any given value of y. Hence if it can be shown that the differential c o e f f i c i e n t

of the equation for c o m p e t i t i v e

anbagonism

p l o t t e d on a log dose axis is itself a single v a l u e d function of y, all curves o b e y i n g this equation must be parallel. can f u r t h e r be shown that the functions

If it

g i v i n g receptor o c c u p a t i o n

by agonist in the presence and absence of a n t a g o n i s t have the same differential

coefficient,

their plots on a log dose axis must be

parallel. The equation for simple c o m p e t i t i v e

a n t a g o n i s m is (Gaddum,

i936) Y 1 where

y is

the

concentrations

-

=

KIA

y

fraction

K2B of

+ 1

activated

receptorsz

of agonist and a n t a g o n i s t

A and

respectively,

B are

and K 1 and

2

Pharmacological Research Communications, Vol. 1, No. 1, 1969

K 2 are

affinity

constants.

The

derivative

of y w i t h

respect

to l o g e A is

dy

dy

dA

dy

KIA (K2B + i) . . . . . . = y(1 (KIA + KzB + i) 2

A= d log e A

dA

Similarly

d log e A

in t h e

dA

absence

of

- y).

antagonist

Y KIA

=

1 - y and

dy

y(1

=

-

y).

d log e A It as

follows

a function

of v a r i o u s agonist given axis

that

curves

of l o g d o s e

concentrations

is p l o t t e d

b y the

slope

The

arguunent - y)

is

2.3 that

also

representing

of d g o n i s t

term

y(l the

applies

in the

of a n t a g o n i s t

on a l o g e a x i s

"logistic"

the

A is y(l

the

y(l

the

are

slope

- y);

receptor absence

the

1 - y curves

in

K2B the

presence dose

of

is

on a l O g l o

- y). derivative to the

case

of y w i t h

respect

Y

KIA

derivative

becomes

to l o g e

K Bn + 1

KlAn Y

If

point

if p l o t t e d

1 - y When

and

parallel. at a n y

occupation

y(l

2

- y)n but

the

+ 1

absence

and

presence

of

antagonist

remain

parallel.

References Arunlakshana, O. and Schild, H.O. ; Brit. J. Pharmac. 14., 48-58, 1 9 5 9 Gaddum, J.H.; J. Physiol. (Lond)., 89, 7P, 1936

Chemother.