- Email: [email protected]
Effects of contamination on radioligand binding parameters
LETTERS
VIP and PACAP in pain and inflammation
University of New York, HSC T17 040, Stony Brook, NY 11794-8172, USA. E-mail: [email protected]
Selected references
In a recent TiPS article1, Dickinson and Fleetwood-Walker make a solid case for an important role for vasoactive intestinal peptide (VIP) and pituitary adenylate cyclase-activating polypeptide (PACAP) in the perception of pain. This view is based on the effects of selective agonists and antagonists of receptors for the two neuropeptides on electrophysiological activity of dorsal horn neurones in normal and neuropathic rats2, and on the upregulation of VIP and PACAP and some of their receptors following peripheral nerve injury. The authors’ conclusion is justified by the data presented and reviewed. However, because pain is often linked with inflammation in the minds of many, it might be assumed that VIP and PACAP generally promote inflammation. In reality, a large body of evidence from different experimental models suggests that VIP, and in many instances also PACAP, can prevent or suppress tissue injury and inflammation. Such evidence (summarized in Ref. 3) includes: (1) prevention of acute high-permeability
pulmonary oedema caused by oxidant stress or glutamate; (2) attenuation of bronchoconstriction and airway inflammation induced by capsaicin, which releases the proinflammatory sensory neuropeptide tachykinins; (3) protection of CNS neurones against death induced by a variety of insults4,5; and (4) inhibition of the production of tumour necrosis factor a and other inflammatory cytokines by lipopolysaccharidestimulated macrophages6. The upregulation of VIP or PACAP, and of their receptors, elicited by acute injury or axotomy7, which is an example of messenger plasticity8, might represent adaptive responses designed to modulate the inflammatory reaction, rather than to help bring it about9. Therefore, the role of VIP or PACAP in the genesis and alleviation of pain and inflammation appears to be complex and its full clarification awaits further exploration. Sami I. Said Professor of Medicine, VA Medical Center, Northport NY & State
1 Dickinson, T. and Fleetwood-Walker, S.M. (1999) VIP and PACAP: very important in pain? Trends Pharmacol. Sci. 20, 324–329 2 Dickinson, T. et al. (1999) The role of VIP/ PACAP receptor subtypes in spinal somatosensory processing in rats with an experimental peripheral mononeuropathy. Neuropharmacology 38, 167–180 3 Said, S.I. (1998) Antiinflammatory actions of VIP in the lungs and airways. In ProInflammatory and Anti-Inflammatory Peptides (Said, S.I., ed.), pp. 345–361, Marcel Dekker 4 Brenneman, D.E. et al. (1998) Activitydependent neurotrophic factor: structure– activity relationships of femtomolar-acting peptides. J. Pharmacol. Exp. Ther. 285, 619–627 5 Said, S.I. et al. (1998) Glutamate toxicity in the lung and neuronal cells: prevention or attenuation by VIP and PACAP. Ann. New York Acad. Sci. 865, 226–237 6 Delgado, M. et al. (1999) Vasoactive intestinal peptide (VIP) and pituitary adenylate cyclase-activating polypeptide (PACAP) protect mice from lethal endotoxemia through the inhibition of TNF-a and IL-6. J. Immunol. 162, 1200–1205 7 Mohney, R.P. et al. (1994) Galanin and vasoactive intestinal peptide messenger RNAs increase following axotomy of adult sympathetic neruons. J. Neurobiol. 25, 108–118 8 Hökfelt, T. et al. (1994) Messenger plasticity in primary sensory neurons following axotomy and its functional implications. Trends Neurosci. 17, 22–30 9 Said, S.I. (1994) VIP and messenger plasticity. Trends Neurosci. 17, 339
PRINCIPLES
Effects of contamination on radioligand binding parameters Sebastian Lazareno and Nigel J.M. Birdsall Radioligand binding studies are used to provide quantitative estimates of parameters such as the receptor density of a tissue and the affinity values of labelled and unlabelled ligands. The presence of an unlabelled competing contaminant, which might be present because of actual contamination, inadequate radioligand purification or the breakdown of the radioligand to an active species, has surprising effects on these estimates: the apparent affinity of the radioligand is increased but the Ki values of unlabelled ligands are unaffected. The most striking and sensitive effects are on radioligand association kinetics, which become independent of radioligand concentration at high radioligand concentrations. The manner in which a radioligand behaves if it is contaminated with an unlabelled competitor is rather unexpected. In this article, the simplest case in which two ligands compete for a single receptor that does not undergo isomerization or
interact with another membrane component is considered. In this scenario, it is assumed that the concentrations of radioligand and contaminant are not affected by their binding to membrane components.
0165-6147/00/$ – see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S0165-6147(99)01412-1
TiPS – February 2000 (Vol. 21)
S. Lazareno, Group Leader, MRC Collaborative Centre, 1–3 Burtonhole Lane, Mill Hill, London, UK NW7 1AD. E-mail: slazare@ nimr.mrc.ac.uk and N.J.M. Birdsall, Group Leader, Division of Physical Biochemistry, National Institute for Medical Research, Mill Hill, London, UK NW7 1AA. E-mail: nbirdsa@ nimr.mrc.ac.uk
57
PRINCIPLES (a)
(b)
0.6 0.4
1
0 0
Fraction of Bmax
0.8
0.5
Scatchard plots Bound / nM free
Fraction of Bmax
1.0
1
Bound
(c)
0.2
Fractional Radioligand contamination observed Kd log M
0.4 0.3
0 0.25 0.5 1 2 4
0.2 0.1 0.0
0.0 −12 −11 −10 −9 −8 −7 [Radioligand] log M
−6
−9
Fitted parameters
−8
−7 −6 −5 −4 [Inhibitor] log M
−9.00 −9.10 −9.17 −9.28 −9.48 −9.69
Inhibitor IC50 1/Ki log M −5.70 −5.65 −5.60 −5.52 −5.40 −5.22
−6 −6 −6 −6 −6 −6
−3 trends in Pharmacological Sciences
Fig. 1. Theoretical saturation (a) and inhibition (b) curves with a contaminated radioligand. The radioligand (L) and contaminant (C) had Kd values of 1 nM and the inhibitor (I) had a Kd (51/KI) value of 1 mM. The contaminant concentration was c.[L], where c is the fractional contamination (Box 1). (a) Each point was calculated using the formula [L].KL/(11[L].KL1c.[L].KL), where KL is the affinity (1/Kd) of both the radioligand and contaminant. (b) Each point was calculated using the formula: [L].KL/(11[L].KL1c.[L].KL1[I].KI), where KI is the affinity (1/Kd) of the inhibitor and [L] is 1 nM. (c) The observed radioligand Kd and inhibitor IC50 were estimated from the data in a and b, respectively, using nonlinear regression. The Ki of the inhibitor was calculated using the Cheng–Prusoff equation.
Figure 1 shows simulations of saturation curves of a radioligand with different degrees of contamination. As might be expected, the apparent Bmax was reduced with increasing contamination. More surprisingly, however, the apparent affinity (1/Kd, where Kd is the equilibrium dissociation constant) of the radioligand increased with increasing contamination. This decrease in the apparent Kd occurs because the presence of contamination causes the true concentration of receptor-binding ligands to be underestimated. Figure 1 also shows the binding of a fixed concentration of radioligand in competition with an unlabelled inhibitor at
Fraction of true Bmax bound
(a) 1.2
Contaminant koff = 10 min−1
0.25
(c) 0.25
1.0
0.20
0.20
0.8
0.15
0.15
0.10
0.10
0.2
0.05
0.05
0.0
0.00
0.00
Contaminant koff = 0.2 min−1
0.6 0.4
−0.2 0
1
(d) Fraction of true Bmax bound
(b)
Uncontaminated radioligand koff = 0.06 min−1
different levels of radioligand contamination. The presence of contamination reduces the potency of the inhibitor but because it also increases the apparent affinity of the radioligand the corrected (Ki) values are, perhaps at first sight surprisingly, unaffected by radioligand contamination. It can be shown that this holds true for any radioligand concentration (Box 1). Binding of the pure radioligand (Fig. 2a) is a monoexponential function of time, and the observed association rate constant (kobs) increases linearly with increasing radioligand concentrations (Fig. 3). The effect of a competitor on
2 3 4 5 Time (min)
6
0 10 20 30 40 50 60 70 Time (min)
(e)
Contaminant koff = 0.06 min−1
0.25
7
Contaminant koff = 0.01 min−1
0.7 0.6
0.20
0.4 0.3
0.10
0.2
0.05
0.1 0.00
0.0 0
1 2 Time (min)
3
[Radioligand] as a fraction of its true Kd 0.1 0.3 1 3 10 30 100 Exponential fit Smooth curve
0.5 0.15
0 10 20 30 40 50 60 70 Time (min)
0
5
10 15 20 25 30 Time (min)
trends in Pharmacological Sciences
Fig. 2. Theoretical time-dependent association of a contaminated radioligand. The fractional contamination was 0 (a) or 4 (b–e). The data were calculated using Eqn 1 of Ref. 1. Note the different scales on the x and y axes.
58
TiPS – February 2000 (Vol. 21)
PRINCIPLES
Box 1. Description of the binding of a contaminated radioligand
[]
Effect of contamination of a radioligand with a competitive antagonist
L and C are the radioligand and contaminant, with affinities of KL and KC, respectively. The equilibrium dissociation constant of the radioligand is Kd (51/KL ). The fractional contamination is c (i.e. at the Kd concentration of L, the concentration of C is c/KC ). The concentration of C is therefore:
[ ] []
c C = L . KL . KC
[1]
If the total concentration of binding sites is Bmax, bound L in the presence of C is:
[]
Bmax . L . K L
[]
[ ]
1 + L . K L + C . KC
[2]
Substituting Eqn 1, the equilibrium binding of contaminated radioligand is given by: BLC =
[]
Bmax . L . K L
[]
1 + L . K L . (1 + c)
[3]
BLC =
[]
BmaxLCobs . L . K LCobs
[]
1 + L . K LCobs
K L = kon / koff
[4]
[12]
it follows that:
[
]
[13]
The effect of a contaminant on radioligand association kinetics depends on the relative dissociation rates of the two ligands1. If the contaminant has very fast kinetics compared with the radioligand, radioligand association is still exponential and the observed association rate constant (kobs) can be derived from the fixed radioligand concentration [L], its apparent affinity KLC and its dissociation rate constant (koff, which is unaffected by the contaminant)3. A particular radioligand concentration [L] has a contaminant concentration of [C]. From Eqn 2, in the presence of this fixed concentration of C, the radioligand has an apparent affinity from saturation assays, KLC, given by: K LC =
With rearrangement, this equation has the form:
[11]
Because the affinity of L, KL, is given by:
kobs = koff 1 + L . K L
Observed Kd and Bmax values
BLC =
kobs = kon . L + koff
KL 1 + C . KC
[14]
[ ]
and, by analogy with Eqn 13, an apparent association rate constant:
( []
kobsLC = koff . 1 + L . K LC
where BmaxLCobs, the observed Bmax, is:
)
[15]
Substituting Eqns 1 and 14 into 15: B = max 1+ c
BmaxLCobs
[5]
and KLCobs, the observed affinity from saturation curves of the contaminated radioligand with constant fractional contamination, is: K LCobs = K L . (1 + c)
[6]
from which it follows that the observed Kd of the contaminated radioligand is: K dLCobs
K = d 1+ c
[7]
which is less than the true Kd. Estimation of the affinity, KI, of an unlabelled inhibitor I
The apparent affinity (KILC, 5 1/IC50) of an inhibitor I in the presence of radioligand concentration [L] and contaminant concentration [C] is: K ILC
KI = 1 + L . K L + C . KC
[]
[ ]
KI 1 + L . K LCobs
[]
[]
1 + L . K LCobs IC50
1 kobsLCmax = koff . 1 + c
[17]
the basal association rate constant at very low radioligand concentrations is koff, and [L]0.5, the radioligand concentration causing 50% of its own maximal effect (change in rate constant), is:
[L ]0.5 = K 1 . c
[18]
L
which corresponds, in terms of the measured affinity (KLCobs, from Eqn 6), to:
[9]
[L ]0.5 = K 1 + c . c
[19]
LCobs
Hence [L]0.5 decreases as c increases.
Thus, the correct value for the affinity of the inhibitor, KI, is obtained by application of the usual Cheng–Prusoff 2 correction using the IC50 (1/KILC) and the observed radioligand affinity, KLCobs: KI =
[16]
If c50 then there is no contamination and kobs is a linear function of [L]. If c.0 then the asymptotic association rate constant at high radioligand concentrations is:
[8]
Substituting Eqns 1 and 6: K ILC =
[] []
L . KL kobsLC = koff . 1 + 1+ L . KL . c
[10]
Association kinetics
The association of pure radioligand over time is exponential: the observed association rate constant kobs is a function of the radioligand concentration [L], its association rate constant kon, and its dissociation rate constant koff, and is given by:
If the contaminant has the same dissociation rate as the radioligand, the contaminant behaves as though it were unlabelled radioligand and the specific activity of the radioligand appears to be overestimated. In this case, the kobs of the contaminated radioligand, like that of the pure radioligand, is a linear function of radioligand, concentration and, from Eqn 6, is given by:
{ []
}
kobsLC = koff . 1 + L . K L . (1 + c)
[20]
Thus, the observed association constant is faster than expected. If the contaminant does not have very fast kinetics, or the same dissociation rate as the radioligand, the association of contaminated radioligand is not a simple exponential function of time but it can be simulated with Eqn 1 of Ref. 1.
TiPS – February 2000 (Vol. 21)
59
PRINCIPLES Fractional contamination 0 0.2 0.4 0.6 0.8 1.0
k obsLC (fraction of k off)
4
3
2
1 0
1 2 3 [Radioligand] (fraction of observed K d) trends in Pharmacological Sciences
Fig. 3. Theoretical dependency of observed association rate constant (kobsLC) (as a fraction of koff) and radioligand concentration (as a fraction of observed Kd) with various degrees of contamination by a contaminant with fast dissociation kinetics.
radioligand association depends on the relative dissociation rates of each ligand1. In the presence of a contaminant with much more rapid dissociation kinetics than the radioligand (Fig. 2b), radioligand binding is still monoexponential but kobs is no longer linearly dependent on radioligand concentration and becomes independent of it at high concentrations (Fig. 3). This is because the concentration of contaminant is a constant fraction of the radioligand concentration and at high ligand concentrations a change in radioligand concentration has an effect on kobs that is equal to, and opposite that of, the corresponding change in the contaminant concentration. As the dissociation rate of the contaminant approaches that of the radioligand (Fig. 2c) the simple kobs measure is still relatively independent of radioligand concentration but the curves are no longer monoexponential. When the contaminant and radioligand have the same dissociation rates (or the specific activity of the radioligand is overestimated) (Fig. 2d) the patterns of radioligand association are the same as those of the uncontaminated material, although the association rate
appears to be faster. When the contaminant has slower kinetics than the radioligand (Fig. 2e) the association curves are dependent on radioligand concentration but ‘overshoot’ at shorter times. If the contaminant has fast kinetics compared with the radioligand, the measurement of kobs over a range of radioligand concentrations provides a sensitive assay for the detection and quantitation of the contamination. Figure 3 shows that contamination ratios as low as 0.2, which would have little discernible effect on apparent Kd or Bmax, cause a clear deviation from linearity in the plot of kobs vs [radioligand], using realistic concentrations of radioligand (i.e. the data could be generated from saturation experiments using different incubation times). It is worth noting that such a nonlinear relationship between kobs and [radioligand] caused by radioligand contamination might be misinterpreted as indicating a second, rate-limiting, process in the binding reaction, such as receptor isomerization or interaction with another molecule. In conclusion, radioligand contamination with an active, unlabelled ligand might arise from accidental (or deliberate) addition, inadequate radioligand purification or breakdown of the radioligand to an active, unlabelled species. If a radioligand has Kd and Bmax values that are lower than expected, and if its observed rate of association becomes independent of radioligand concentration at high radioligand concentrations, the possibility should be considered that the radioligand is contaminated with an unlabelled competitor. If the Kd and Bmax values are lower than expected but the association rate is linearly related to radioligand concentration the specific activity of the radioligand might have been overestimated. Fortunately, even contaminated radioligands might be acceptable for the task of estimating the affinity values of unlabelled competitors. Selected references 1 Motulsky, H.J. and Mahan, L.C. (1984) The kinetics of competitive radioligand binding predicted by the law of mass action. Mol. Pharmacol. 25, 1–9 2 Cheng, Y. and Prusoff, W.H. (1973) Relationship between the inhibition constant (Ki) and the concentration of an inhibitor which causes 50 per cent inhibition (I50) of an enzymatic reaction. Biochem. Pharmacol. 22, 3099–3108 3 Lazareno, S. and Birdsall, N.J.M. (1995) Detection, quantitation, and verification of allosteric interactions of agents with labeled and unlabeled ligands at G protein-coupled receptors: interactions of strychnine and acetylcholine at muscarinic receptors. Mol. Pharmacol. 48, 362–378
Editorial policy Most articles published in TiPS are commissioned by the Editor. However, authors who wish to contribute to any section of the journal should contact the Editor with the names of all authors and a point-by-point outline of the proposed article including 10–12 key references. Outlines might also be sent to members of the Advisory Editorial Board. Completed manuscripts submitted without liaison with the TiPS Editorial Office cannot be considered. TiPS readers who wish to suggest that a particular topic be covered are also invited to send their suggestions to the Editor.
60
TiPS – February 2000 (Vol. 21)
VIP and PACAP in pain and inflammation
University of New York, HSC T17 040, Stony Brook, NY 11794-8172, USA. E-mail: [email protected]
Selected references
In a recent TiPS article1, Dickinson and Fleetwood-Walker make a solid case for an important role for vasoactive intestinal peptide (VIP) and pituitary adenylate cyclase-activating polypeptide (PACAP) in the perception of pain. This view is based on the effects of selective agonists and antagonists of receptors for the two neuropeptides on electrophysiological activity of dorsal horn neurones in normal and neuropathic rats2, and on the upregulation of VIP and PACAP and some of their receptors following peripheral nerve injury. The authors’ conclusion is justified by the data presented and reviewed. However, because pain is often linked with inflammation in the minds of many, it might be assumed that VIP and PACAP generally promote inflammation. In reality, a large body of evidence from different experimental models suggests that VIP, and in many instances also PACAP, can prevent or suppress tissue injury and inflammation. Such evidence (summarized in Ref. 3) includes: (1) prevention of acute high-permeability
pulmonary oedema caused by oxidant stress or glutamate; (2) attenuation of bronchoconstriction and airway inflammation induced by capsaicin, which releases the proinflammatory sensory neuropeptide tachykinins; (3) protection of CNS neurones against death induced by a variety of insults4,5; and (4) inhibition of the production of tumour necrosis factor a and other inflammatory cytokines by lipopolysaccharidestimulated macrophages6. The upregulation of VIP or PACAP, and of their receptors, elicited by acute injury or axotomy7, which is an example of messenger plasticity8, might represent adaptive responses designed to modulate the inflammatory reaction, rather than to help bring it about9. Therefore, the role of VIP or PACAP in the genesis and alleviation of pain and inflammation appears to be complex and its full clarification awaits further exploration. Sami I. Said Professor of Medicine, VA Medical Center, Northport NY & State
1 Dickinson, T. and Fleetwood-Walker, S.M. (1999) VIP and PACAP: very important in pain? Trends Pharmacol. Sci. 20, 324–329 2 Dickinson, T. et al. (1999) The role of VIP/ PACAP receptor subtypes in spinal somatosensory processing in rats with an experimental peripheral mononeuropathy. Neuropharmacology 38, 167–180 3 Said, S.I. (1998) Antiinflammatory actions of VIP in the lungs and airways. In ProInflammatory and Anti-Inflammatory Peptides (Said, S.I., ed.), pp. 345–361, Marcel Dekker 4 Brenneman, D.E. et al. (1998) Activitydependent neurotrophic factor: structure– activity relationships of femtomolar-acting peptides. J. Pharmacol. Exp. Ther. 285, 619–627 5 Said, S.I. et al. (1998) Glutamate toxicity in the lung and neuronal cells: prevention or attenuation by VIP and PACAP. Ann. New York Acad. Sci. 865, 226–237 6 Delgado, M. et al. (1999) Vasoactive intestinal peptide (VIP) and pituitary adenylate cyclase-activating polypeptide (PACAP) protect mice from lethal endotoxemia through the inhibition of TNF-a and IL-6. J. Immunol. 162, 1200–1205 7 Mohney, R.P. et al. (1994) Galanin and vasoactive intestinal peptide messenger RNAs increase following axotomy of adult sympathetic neruons. J. Neurobiol. 25, 108–118 8 Hökfelt, T. et al. (1994) Messenger plasticity in primary sensory neurons following axotomy and its functional implications. Trends Neurosci. 17, 22–30 9 Said, S.I. (1994) VIP and messenger plasticity. Trends Neurosci. 17, 339
PRINCIPLES
Effects of contamination on radioligand binding parameters Sebastian Lazareno and Nigel J.M. Birdsall Radioligand binding studies are used to provide quantitative estimates of parameters such as the receptor density of a tissue and the affinity values of labelled and unlabelled ligands. The presence of an unlabelled competing contaminant, which might be present because of actual contamination, inadequate radioligand purification or the breakdown of the radioligand to an active species, has surprising effects on these estimates: the apparent affinity of the radioligand is increased but the Ki values of unlabelled ligands are unaffected. The most striking and sensitive effects are on radioligand association kinetics, which become independent of radioligand concentration at high radioligand concentrations. The manner in which a radioligand behaves if it is contaminated with an unlabelled competitor is rather unexpected. In this article, the simplest case in which two ligands compete for a single receptor that does not undergo isomerization or
interact with another membrane component is considered. In this scenario, it is assumed that the concentrations of radioligand and contaminant are not affected by their binding to membrane components.
0165-6147/00/$ – see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S0165-6147(99)01412-1
TiPS – February 2000 (Vol. 21)
S. Lazareno, Group Leader, MRC Collaborative Centre, 1–3 Burtonhole Lane, Mill Hill, London, UK NW7 1AD. E-mail: slazare@ nimr.mrc.ac.uk and N.J.M. Birdsall, Group Leader, Division of Physical Biochemistry, National Institute for Medical Research, Mill Hill, London, UK NW7 1AA. E-mail: nbirdsa@ nimr.mrc.ac.uk
57
PRINCIPLES (a)
(b)
0.6 0.4
1
0 0
Fraction of Bmax
0.8
0.5
Scatchard plots Bound / nM free
Fraction of Bmax
1.0
1
Bound
(c)
0.2
Fractional Radioligand contamination observed Kd log M
0.4 0.3
0 0.25 0.5 1 2 4
0.2 0.1 0.0
0.0 −12 −11 −10 −9 −8 −7 [Radioligand] log M
−6
−9
Fitted parameters
−8
−7 −6 −5 −4 [Inhibitor] log M
−9.00 −9.10 −9.17 −9.28 −9.48 −9.69
Inhibitor IC50 1/Ki log M −5.70 −5.65 −5.60 −5.52 −5.40 −5.22
−6 −6 −6 −6 −6 −6
−3 trends in Pharmacological Sciences
Fig. 1. Theoretical saturation (a) and inhibition (b) curves with a contaminated radioligand. The radioligand (L) and contaminant (C) had Kd values of 1 nM and the inhibitor (I) had a Kd (51/KI) value of 1 mM. The contaminant concentration was c.[L], where c is the fractional contamination (Box 1). (a) Each point was calculated using the formula [L].KL/(11[L].KL1c.[L].KL), where KL is the affinity (1/Kd) of both the radioligand and contaminant. (b) Each point was calculated using the formula: [L].KL/(11[L].KL1c.[L].KL1[I].KI), where KI is the affinity (1/Kd) of the inhibitor and [L] is 1 nM. (c) The observed radioligand Kd and inhibitor IC50 were estimated from the data in a and b, respectively, using nonlinear regression. The Ki of the inhibitor was calculated using the Cheng–Prusoff equation.
Figure 1 shows simulations of saturation curves of a radioligand with different degrees of contamination. As might be expected, the apparent Bmax was reduced with increasing contamination. More surprisingly, however, the apparent affinity (1/Kd, where Kd is the equilibrium dissociation constant) of the radioligand increased with increasing contamination. This decrease in the apparent Kd occurs because the presence of contamination causes the true concentration of receptor-binding ligands to be underestimated. Figure 1 also shows the binding of a fixed concentration of radioligand in competition with an unlabelled inhibitor at
Fraction of true Bmax bound
(a) 1.2
Contaminant koff = 10 min−1
0.25
(c) 0.25
1.0
0.20
0.20
0.8
0.15
0.15
0.10
0.10
0.2
0.05
0.05
0.0
0.00
0.00
Contaminant koff = 0.2 min−1
0.6 0.4
−0.2 0
1
(d) Fraction of true Bmax bound
(b)
Uncontaminated radioligand koff = 0.06 min−1
different levels of radioligand contamination. The presence of contamination reduces the potency of the inhibitor but because it also increases the apparent affinity of the radioligand the corrected (Ki) values are, perhaps at first sight surprisingly, unaffected by radioligand contamination. It can be shown that this holds true for any radioligand concentration (Box 1). Binding of the pure radioligand (Fig. 2a) is a monoexponential function of time, and the observed association rate constant (kobs) increases linearly with increasing radioligand concentrations (Fig. 3). The effect of a competitor on
2 3 4 5 Time (min)
6
0 10 20 30 40 50 60 70 Time (min)
(e)
Contaminant koff = 0.06 min−1
0.25
7
Contaminant koff = 0.01 min−1
0.7 0.6
0.20
0.4 0.3
0.10
0.2
0.05
0.1 0.00
0.0 0
1 2 Time (min)
3
[Radioligand] as a fraction of its true Kd 0.1 0.3 1 3 10 30 100 Exponential fit Smooth curve
0.5 0.15
0 10 20 30 40 50 60 70 Time (min)
0
5
10 15 20 25 30 Time (min)
trends in Pharmacological Sciences
Fig. 2. Theoretical time-dependent association of a contaminated radioligand. The fractional contamination was 0 (a) or 4 (b–e). The data were calculated using Eqn 1 of Ref. 1. Note the different scales on the x and y axes.
58
TiPS – February 2000 (Vol. 21)
PRINCIPLES
Box 1. Description of the binding of a contaminated radioligand
[]
Effect of contamination of a radioligand with a competitive antagonist
L and C are the radioligand and contaminant, with affinities of KL and KC, respectively. The equilibrium dissociation constant of the radioligand is Kd (51/KL ). The fractional contamination is c (i.e. at the Kd concentration of L, the concentration of C is c/KC ). The concentration of C is therefore:
[ ] []
c C = L . KL . KC
[1]
If the total concentration of binding sites is Bmax, bound L in the presence of C is:
[]
Bmax . L . K L
[]
[ ]
1 + L . K L + C . KC
[2]
Substituting Eqn 1, the equilibrium binding of contaminated radioligand is given by: BLC =
[]
Bmax . L . K L
[]
1 + L . K L . (1 + c)
[3]
BLC =
[]
BmaxLCobs . L . K LCobs
[]
1 + L . K LCobs
K L = kon / koff
[4]
[12]
it follows that:
[
]
[13]
The effect of a contaminant on radioligand association kinetics depends on the relative dissociation rates of the two ligands1. If the contaminant has very fast kinetics compared with the radioligand, radioligand association is still exponential and the observed association rate constant (kobs) can be derived from the fixed radioligand concentration [L], its apparent affinity KLC and its dissociation rate constant (koff, which is unaffected by the contaminant)3. A particular radioligand concentration [L] has a contaminant concentration of [C]. From Eqn 2, in the presence of this fixed concentration of C, the radioligand has an apparent affinity from saturation assays, KLC, given by: K LC =
With rearrangement, this equation has the form:
[11]
Because the affinity of L, KL, is given by:
kobs = koff 1 + L . K L
Observed Kd and Bmax values
BLC =
kobs = kon . L + koff
KL 1 + C . KC
[14]
[ ]
and, by analogy with Eqn 13, an apparent association rate constant:
( []
kobsLC = koff . 1 + L . K LC
where BmaxLCobs, the observed Bmax, is:
)
[15]
Substituting Eqns 1 and 14 into 15: B = max 1+ c
BmaxLCobs
[5]
and KLCobs, the observed affinity from saturation curves of the contaminated radioligand with constant fractional contamination, is: K LCobs = K L . (1 + c)
[6]
from which it follows that the observed Kd of the contaminated radioligand is: K dLCobs
K = d 1+ c
[7]
which is less than the true Kd. Estimation of the affinity, KI, of an unlabelled inhibitor I
The apparent affinity (KILC, 5 1/IC50) of an inhibitor I in the presence of radioligand concentration [L] and contaminant concentration [C] is: K ILC
KI = 1 + L . K L + C . KC
[]
[ ]
KI 1 + L . K LCobs
[]
[]
1 + L . K LCobs IC50
1 kobsLCmax = koff . 1 + c
[17]
the basal association rate constant at very low radioligand concentrations is koff, and [L]0.5, the radioligand concentration causing 50% of its own maximal effect (change in rate constant), is:
[L ]0.5 = K 1 . c
[18]
L
which corresponds, in terms of the measured affinity (KLCobs, from Eqn 6), to:
[9]
[L ]0.5 = K 1 + c . c
[19]
LCobs
Hence [L]0.5 decreases as c increases.
Thus, the correct value for the affinity of the inhibitor, KI, is obtained by application of the usual Cheng–Prusoff 2 correction using the IC50 (1/KILC) and the observed radioligand affinity, KLCobs: KI =
[16]
If c50 then there is no contamination and kobs is a linear function of [L]. If c.0 then the asymptotic association rate constant at high radioligand concentrations is:
[8]
Substituting Eqns 1 and 6: K ILC =
[] []
L . KL kobsLC = koff . 1 + 1+ L . KL . c
[10]
Association kinetics
The association of pure radioligand over time is exponential: the observed association rate constant kobs is a function of the radioligand concentration [L], its association rate constant kon, and its dissociation rate constant koff, and is given by:
If the contaminant has the same dissociation rate as the radioligand, the contaminant behaves as though it were unlabelled radioligand and the specific activity of the radioligand appears to be overestimated. In this case, the kobs of the contaminated radioligand, like that of the pure radioligand, is a linear function of radioligand, concentration and, from Eqn 6, is given by:
{ []
}
kobsLC = koff . 1 + L . K L . (1 + c)
[20]
Thus, the observed association constant is faster than expected. If the contaminant does not have very fast kinetics, or the same dissociation rate as the radioligand, the association of contaminated radioligand is not a simple exponential function of time but it can be simulated with Eqn 1 of Ref. 1.
TiPS – February 2000 (Vol. 21)
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PRINCIPLES Fractional contamination 0 0.2 0.4 0.6 0.8 1.0
k obsLC (fraction of k off)
4
3
2
1 0
1 2 3 [Radioligand] (fraction of observed K d) trends in Pharmacological Sciences
Fig. 3. Theoretical dependency of observed association rate constant (kobsLC) (as a fraction of koff) and radioligand concentration (as a fraction of observed Kd) with various degrees of contamination by a contaminant with fast dissociation kinetics.
radioligand association depends on the relative dissociation rates of each ligand1. In the presence of a contaminant with much more rapid dissociation kinetics than the radioligand (Fig. 2b), radioligand binding is still monoexponential but kobs is no longer linearly dependent on radioligand concentration and becomes independent of it at high concentrations (Fig. 3). This is because the concentration of contaminant is a constant fraction of the radioligand concentration and at high ligand concentrations a change in radioligand concentration has an effect on kobs that is equal to, and opposite that of, the corresponding change in the contaminant concentration. As the dissociation rate of the contaminant approaches that of the radioligand (Fig. 2c) the simple kobs measure is still relatively independent of radioligand concentration but the curves are no longer monoexponential. When the contaminant and radioligand have the same dissociation rates (or the specific activity of the radioligand is overestimated) (Fig. 2d) the patterns of radioligand association are the same as those of the uncontaminated material, although the association rate
appears to be faster. When the contaminant has slower kinetics than the radioligand (Fig. 2e) the association curves are dependent on radioligand concentration but ‘overshoot’ at shorter times. If the contaminant has fast kinetics compared with the radioligand, the measurement of kobs over a range of radioligand concentrations provides a sensitive assay for the detection and quantitation of the contamination. Figure 3 shows that contamination ratios as low as 0.2, which would have little discernible effect on apparent Kd or Bmax, cause a clear deviation from linearity in the plot of kobs vs [radioligand], using realistic concentrations of radioligand (i.e. the data could be generated from saturation experiments using different incubation times). It is worth noting that such a nonlinear relationship between kobs and [radioligand] caused by radioligand contamination might be misinterpreted as indicating a second, rate-limiting, process in the binding reaction, such as receptor isomerization or interaction with another molecule. In conclusion, radioligand contamination with an active, unlabelled ligand might arise from accidental (or deliberate) addition, inadequate radioligand purification or breakdown of the radioligand to an active, unlabelled species. If a radioligand has Kd and Bmax values that are lower than expected, and if its observed rate of association becomes independent of radioligand concentration at high radioligand concentrations, the possibility should be considered that the radioligand is contaminated with an unlabelled competitor. If the Kd and Bmax values are lower than expected but the association rate is linearly related to radioligand concentration the specific activity of the radioligand might have been overestimated. Fortunately, even contaminated radioligands might be acceptable for the task of estimating the affinity values of unlabelled competitors. Selected references 1 Motulsky, H.J. and Mahan, L.C. (1984) The kinetics of competitive radioligand binding predicted by the law of mass action. Mol. Pharmacol. 25, 1–9 2 Cheng, Y. and Prusoff, W.H. (1973) Relationship between the inhibition constant (Ki) and the concentration of an inhibitor which causes 50 per cent inhibition (I50) of an enzymatic reaction. Biochem. Pharmacol. 22, 3099–3108 3 Lazareno, S. and Birdsall, N.J.M. (1995) Detection, quantitation, and verification of allosteric interactions of agents with labeled and unlabeled ligands at G protein-coupled receptors: interactions of strychnine and acetylcholine at muscarinic receptors. Mol. Pharmacol. 48, 362–378
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TiPS – February 2000 (Vol. 21)