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(lily) #1
wherekþ 1 is the association rate constant andk 1 is the dissociation rate constant.
Therefore:
k 1
kþ 1

¼Kd¼

1

Ka

¼

ð½RtŠ½RLŠÞ½LŠ
½RLŠ

ð 17 : 1 Þ

whereKdis the dissociation constant for RL andKais the association or affinity
constant. Rearranging gives:

½RLŠ¼

½LŠ½RtŠ
Kdþ½LŠ

ð 17 : 2 Þ

Determination ofKd
Equation 17.2 is of the form of a rectangular hyperbola which predicts that ligand
binding will reach a limiting value as the ligand concentration is increased and
therefore that receptor binding is a saturable process. The equation is of precisely
the same form as equations 15.1 and 15.2 that define the binding of the substrate to its
enzyme in terms ofKmandVmax. For the experimental determination ofKd, equation
17.2 can be used directly by analysing the experimental data by non-linear regression
curve-fitting programs (see Section 15.2.1). However, several linear transformations
of equation 17.2 have been developed. One such is theScatchardequation (17.3):

½RLŠ
½LŠ

¼½RtŠ
Kd

½RLŠ

Kd

ð 17 : 3 Þ

This equation predicts that a plot of [RL]/[L] against [RL] will be a straight-line
slope1/KdallowingKdto be calculated. However, in many studies the relative
molecular mass of the receptor protein is unknown so that the concentration term
[RL] cannot be calculated in molar terms. In such cases it is acceptable to express the
extent of ligand binding in any convenient unit (B), e.g. pmoles 10^6 cells, pmoles
mg^1 protein or more simply as an observed change, for example in fluorescence
(▵F), under the defined experimental conditions. Since maximum binding (Bmax) will
occur when all the receptor sites are occupied, i.e. when [Rt]¼Bmax, equation 17.3 can
be written in the form:
B
½LŠ¼

Bmax
Kd 

B

Kd ð^17 :^4 Þ

Hence a plot ofB/[L] againstBwill be a straight line, slope1/Kdand intercept on the
y-axis ofBmax/Kd(Fig. 17.4). In cases where the relative molecular mass of the
receptor protein is known the Scatchard equation can be expressed in the form:
B
½LŠBmax¼

n
Kd

B

BmaxKd ð^17 :^5 Þ

wherenis the number of independent ligand-binding sites on the receptor. The
expressionB/Bmaxis the number of moles of ligand bound to one mole of receptor.
If this expression is defined asr, then:
r
½LŠ

¼

n
Kd



r
Kd

ð 17 : 6 Þ

672 Cell membrane receptors and cell signalling

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