In most cases, the protein concentration will remain at its initial
value ([P 0 ]) throughout the reaction because the total concentration of binding sites on the sensor is vanishingly small compared
with the protein concentration in the well. Therefore:
d PN
½
dt
¼ k on P o
½ N o
½ À PN
½
ð
ÞÀk off PN
½
ð3Þ
If there is no nonspecific binding of the protein to the sensor,
then the instrument response must be directly proportional to
[PN], and this equation may therefore be rewritten as:
dR
dt
¼ k on P o
½ R max À R
ð
ÞÀk off R
ð4Þ
where R denotes the response at time t and R max is the maximal
response that would be obtained if all available binding sites on the
sensor were saturated (i.e., when [PN] ¼ [N 0 ]). Integration of this
equation gives:
R ¼
k on P o
½ R max 1 À e
Àt k on P o
½ þk off
ð
Þ
È
É
k on P o
½ þ k off
ð5Þ
Assuming that the maximum possible response (R max ) and the
equilibrium response at the end of the association phase (R eq ) must
be proportional to [N 0 ] and [PN], respectively, one may write:
k on
k off
¼
PN
½
P
½ N
½
¼
PN
½
P o
½ N o
½ À PN
½
ð
Þ
¼
R eq
P o
½ R max À R eq
À
Á
ð6Þ
and R eq ¼
k on P o
½ R max
k on P o
½ þ k off
ð7Þ
Substitution in Eq. 5 then gives:
R ¼ R eq 1 À e
Àt k on P o
½ þk off
ð
Þ
n
o
ð8Þ
and further substituting k obs ¼ k on [P o ] + k off gives:
R ¼ R eq 1 À e
Àtk obs
È
É
ð9Þ
The time dependence of the biosensor response in the association phase is then expressed in terms of a pseudo-first-order rate
constant k obs and R eq , the response at equilibrium. Values for k on
and k off can then, in favorable cases, be obtained as the slope and yaxis intercept of a plot of k obs versus [P 0 ], and the equilibrium
dissociation constant (K d ) can be calculated as k off /k on .
A value for the K d can also be obtained from the variation of the
R eq value with protein concentration. Dividing Eq. 7 by k on and
substituting K d ¼ k off /k on gives:
R eq ¼
P o
½ R max
P o
½ þ K d
ð10Þ
354
Stephen R. Martin et al.
value ([P 0 ]) throughout the reaction because the total concentration of binding sites on the sensor is vanishingly small compared
with the protein concentration in the well. Therefore:
d PN
½
dt
¼ k on P o
½ N o
½ À PN
½
ð
ÞÀk off PN
½
ð3Þ
If there is no nonspecific binding of the protein to the sensor,
then the instrument response must be directly proportional to
[PN], and this equation may therefore be rewritten as:
dR
dt
¼ k on P o
½ R max À R
ð
ÞÀk off R
ð4Þ
where R denotes the response at time t and R max is the maximal
response that would be obtained if all available binding sites on the
sensor were saturated (i.e., when [PN] ¼ [N 0 ]). Integration of this
equation gives:
R ¼
k on P o
½ R max 1 À e
Àt k on P o
½ þk off
ð
Þ
È
É
k on P o
½ þ k off
ð5Þ
Assuming that the maximum possible response (R max ) and the
equilibrium response at the end of the association phase (R eq ) must
be proportional to [N 0 ] and [PN], respectively, one may write:
k on
k off
¼
PN
½
P
½ N
½
¼
PN
½
P o
½ N o
½ À PN
½
ð
Þ
¼
R eq
P o
½ R max À R eq
À
Á
ð6Þ
and R eq ¼
k on P o
½ R max
k on P o
½ þ k off
ð7Þ
Substitution in Eq. 5 then gives:
R ¼ R eq 1 À e
Àt k on P o
½ þk off
ð
Þ
n
o
ð8Þ
and further substituting k obs ¼ k on [P o ] + k off gives:
R ¼ R eq 1 À e
Àtk obs
È
É
ð9Þ
The time dependence of the biosensor response in the association phase is then expressed in terms of a pseudo-first-order rate
constant k obs and R eq , the response at equilibrium. Values for k on
and k off can then, in favorable cases, be obtained as the slope and yaxis intercept of a plot of k obs versus [P 0 ], and the equilibrium
dissociation constant (K d ) can be calculated as k off /k on .
A value for the K d can also be obtained from the variation of the
R eq value with protein concentration. Dividing Eq. 7 by k on and
substituting K d ¼ k off /k on gives:
R eq ¼
P o
½ R max
P o
½ þ K d
ð10Þ
354
Stephen R. Martin et al.