Analysis of Ligand-Receptor Interactions from the Molecular Level to the Whole-Body Level

Abstract

A system for the quantitative analysis of ligand-receptor interactions is presented, based on models of different levels of complexity. For two pools of receptors, binding of a radioactive ligand is described by b = [(Bml x A(nl))/(K(nl)dl + A(nl))] + [(Bm2 x A(n2))/(Kn2(d2) + A(n2))], (1) where b is the number of bound receptors at a ligand concentration [A], Bml and Bm2 are the receptor concentrations. Kdl and Kd2 are dissociation constants for the ligand-receptor complex, and n1 and n2 are Hill coefficients. The magnitude of the physiological response for a system consisting of two discrete pools of receptors with different affinities is given by p = [(Pm x A(nl))/(EC50(nl) + A(nl))] + [(Pm2 x A(n2)/(EC50(n2)2 + A(n2))], (2) where p is the magnitude of the response to an agonist (or antagonist) at concentration [A], Pml and Pm2 are the maximal magnitudes of the responses for the individual pools of receptors, EC50(1) and EC50(2) are the agonist concentrations giving responses of magnitudes Pm1/2 and Pm2/2, and n1 and n2 are Hill coefficients. The parameters of these equations show: the number of pools of receptors with different affinities for the ligand (Kd or EC50), the number of active receptors (Bmax) or the magnitudes of the maximal response (Pmax), and the numbers of ligand molecules binding with the receptor (n, the Hill coefficient). E is the efficiency (E = Bmax/2Kd, or E = Pmax/2EC50) and gives the overall characteristics of the activity of the effector system. This method of analysis can be applied to any biological reactions whose results can be presented quantitatively.

*u*n*s*t*u*r*t*u*r*e**t*e*x*t*

The common first stage in the action of a signalmolecule on an effector system is its binding with a specific receptor. At the molecular level, this is attachment ofa ligand molecule to a receptor, which can be assessed using radioactive ligands (the only direct stage in the lig-and-receptor interaction). This yields the maximum number of receptors ( Bmax) and their affinity ( Kd). In morecomplicated effector systems, the action of a signal molecule is manifest as a biochemical or electrophysio-logical reaction, reactions of an isolated organ, or reactions of a whole body, which can be measured in unitsappropriate to the system - changes in enzyme activity, magnitudes of contractile responses, changes in arterial

pressure, etc. These magnitudes - the maximal responsesof the effector system (

Pmax) and EC50 (the ligand con-centration inducing responses of

Pmax/2) are analogousbut not identical to Bmax and Kd, as they sum up theparameters of sequences of the biochemical reactions

involved in realizing the effect of the ligand-receptorinteraction. Each of these reactions has its own parameters, which are reflected in the overall parameters of thephysiological reaction (

Pmax, EC50).The first stage in the action of a signal molecule (neurotransmitter, hormone, peptide, or other active molecule)on an effector system is its attachment to a specific receptor. This process has repeatedly been analyzed theoretical-ly and experimentally using enzymatic [4], radioligand [26, 28, 30, 32], and physiological reactions [8, 14, 15, 25].Since all these reactions follow similar rules, the successes achieved in the theoretical analysis of each of these proNeuroscience and Behavioral Physiology, Vol. 32, No. 3, 2002 Analysis of Ligand-Receptor Interactions from the MolecularLevel to the Whole-Body Level B. N. Manukhin

0097-0549/02/3203-0283$27.00 (C)2002 Plenum Publishing Corporation 283

Translated from Rossiiskii Fiziologicheskii Zhurnal imeni I. M. Sechenova, Vol. 86, No. 9, pp. 1220-1232, September, 2000. Original article submitted March 28, 2000.

A system for the quantitative analysis of ligand-receptor interactions is presented, based on models of different levels of complexity. For two pools of receptors, binding of a radioactive ligand is described by

b = [(Bm1*An1)/(K n1d1 + An1)] + [(Bm2*An2)/(Kn2d2 + An2)], (1) where b is the number of bound receptors at a ligand concentration [A], Bm1 and Bm2 are the receptor concentrations, Kd1 and Kd2 are dissociation constants for the ligand-receptor complex, and n1 and n2 are Hill coefficients. The magnitude of the physiological response for a system consisting of two discrete pools of receptors with different affinities is given by

p = [(Pm1*An1)/(EC501n1 + An1)] + [(Pm2*An2)/(EC502n2 + An2)], (2) where p is the magnitude of the response to an agonist (or antagonist) at concentration [A], Pm1 and Pm2 are the maximal magnitudes of the responses for the individual pools of receptors, EC501 and EC502 are the agonist concentrations giving responses of magnitudes Pm1/2 and Pm2/2, and n1 and n2 are Hill coefficients. The parameters of these equations show: the number of pools of receptors with different affinities for the ligand (Kd or EC50), the number of active receptors (Bmax) or the magnitudes of the maximal response (Pmax), and the numbers of ligand molecules binding with the receptor (n, the Hill coefficient).

E is the efficiency (E = Bmax/2Kd, or E = Pmax/2EC50) and gives the overall characteristics of the activity of the effector system. This method of analysis can be applied to any biological reactions whose results

can be presented quantitatively.

KEY WORDS: Neurotransmitter receptors, ligand-receptor interactions, physiological and radioligand reactions.

N. K. Kol'tsov Institute of Developmental Biology, RussianAcademy of Sciences, 26 Vavilov Street, 117334 Moscow, Russia.

cesses can, with some limitations, be used in studies of theactions of biologically active compounds at levels from the subcellular to the whole-body. Each new report in this areamakes one more step to deciphering the features of ligandreceptor interactions - the first, basic, reaction in the func-tioning of the body's regulatory systems [18, 35, 39].

Application of standard methods of analysis of exper-imental data obtained from radioligand and especially physiological experiments does not always give satisfactoryresults. This suggests that not all of the characteristics of the binding of receptor to ligand and the realization of its actionas a physiological response have been considered.

Previous theor etical and e xperimental studies ha vebeen based on methods in volving the quantita tive mea surement of the major par ameters in r adioligand [11, 32]and ph ysiological r esponses [9]. We pr esent her e the results of fur ther moder nization of biolo gical models ofligand-receptor interactions and systems for their analysis, allowing the quantita tive characteristics of the functionalactivity of receptors to be obtained, along with the characteristics of the binding of lig ands to these r eceptors. Ourown results, as well as published data, are used to provide examples of the analysis of the actions of signal moleculeson models a t different levels, ranging from the molecular to the whole-body.

METHODS

Previous graphical and mathematical analyses of theequilibrium binding of radioactive ligands [11, 32] have demonstrated the existence of two discrete pools of specif-ic adrenoreceptors and muscarinic receptors, whose

Kd val-ues in rat erythrocytes and cerebral cortical membranes

show significant differences in ligand affinity. Attachmentof ligands to specific receptors is described by

b = [(Bm1*An1)/(Kn1d1 + An1)] +

+ [(Bm2*An2)/(Kn2d2 + An2)], (1) where b is the number of receptors occupied at a ligand con-centration [

A], Bm1 and Bm2 are the concentrations of bind-ing sites in the individual pools of the effector system,

Kd1and Kd2 are the dissociation constants of the ligand-recep-tor complexes, and n1 and n2 are Hill coefficients (coefficients of cooperativity or measures of the numbers of ligandmolecules attached to the receptor).

In general [9], the actions of signal molecules in phys-iological responses involving attachment of different numbers of ligand molecules to one receptor in a pool of recep-tors with different affinities are described by

p = [(Pm1*An1)/(EC501n1 + An1)] +

+ [(Pm2*An2)/(EC502n2 + An2)], (2)

where p is the magnitude of the response to an agonist atconcentration [

A], Pm1 and Pm2 are the maximal magni-tudes of the responses for the individual pools of receptors,

EC501 and EC502 are the agonist concentrations givingresponses of magnitudes

Pm1/2 and Pm2/2, and n1 and n2are Hill coefficients (coefficients of cooperativity or measures of the numbers of ligand molecules attached to recep-tors [4]). The absence of comment indicates that the receptors in the pool of receptors with a given affinity areassumed to have the same Hill coefficient. Analysis of ligand-receptor interactions in enzymatic, radioligand, andphysiological experiments shows that calculated values of n are not always a multiple of a whole number. It wouldappear that in these cases, some of the receptors bind one ligand molecule, while others bind two, i.e., the pool ofreceptors consists of two sub-pools in terms of this criterion, even though they have the same ligand affinity. In thesecases, values for each sub-pool of receptors are defined, for radioligand experiments, by

b = [(B1*A + B2*A2)/(Kd1 + A) + (K2d1 + A2)] +

+ [(B3*A + B4*A2)/(Kd2 + A) + (K2d2 + A2)], (3)

where (B1 + B2) = Bm1 and (B3 + B4) = Bm2, while other variables are as in Eq. (1). For physiological experiments, the relationship is

p = [(P1*A + P2*A2)/(EC501 + A) + (EC5012 + A2)] +

+ [(P3*A + P4*A2)/(EC502 + A) + (EC5022 + A2)], (4)

where (P1 + P2) = Pm1, (P3 + P2) = Pm2, while other vari-ables are as in Eq. (2).

Experimental data were plotted in Cartesian coordi-nates as well as in coordinates defined by the transformation of Eqs. (1) and (2):

(b/An1 + b/An2) = [(Bm1 + Bm2)/(Kn1d1 + Kn2d2)] -

- b/(Kn1d1 + Kn2d2)], (5)

(p/An1 + p/An2) = [(Pm1 + Pm2)/(EC501n1 +

+ EC502n2)] - [p/(EC501n1 + EC502n2)]. (6) Experiments on isolated organs were performed bystandard methods [2], taking account of the features of the

experimental system [1, 3, 6-8]. The radioligand methodwas standard (with minor changes) [10]. Agonists and antagonists of cholinoreceptors and adrenoreceptors wereobtained from RBI and Sigma (USA); radioactive ligands were from Amersham (UK).The main parameters of ligand-receptor interactions were estimated, i.e., Kd1, Kd2, Bm1, Bm2, EC501, EC502, Pm1, Pm2, and n1 and n2, using the computer programSigmaPlot 5.0. The effectivenesses (

E) of the actions of sigManukhin284 nal molecules on individual pools of receptors and overall -for the effector system - were determined from

E = Bm/2Kdn (7)or E = Pm/2EC50n. (8) Effectiveness (E) is an integral measure which quanti-tatively characterizes the activity of a ligand at a concentration of Kd or EC50 [8, 9, 11].The significances of differences were assessed using Student's test (p < 0.05). All values are presented as arith-metic means and standard errors (

M +- m).

RESULTS

Studies of the characteristics of the equilibrium bind-ing with ligands of m-cholinoreceptors were based on use of the specific non-selective agonist [3H]quinuclidinyl ben-zylate. Specific binding of this ligand by m-cholinoreceptors in rat brain membranes was saturable (Fig. 1, A). TheHill coefficient, calculated from all experimental points, was 1.43. Hill coefficients of 1 < n < 2 indicate the exis-tence of at least two discrete pools of receptors and the binding of more than one ligand molecule per receptor. Theoptimum model of the ligand-receptor interaction included

Analysis of Ligand-Receptor Interactions from the Molecular Level to the Whole-Body Level 285

Fig. 1. Experimental points and theoretical curves for the specific binding of [3H]quinuclidinyl benzylate with m-cholinoreceptors in rat

cerebral cortex membranes. The abscissa in (A) shows the [3H]quinuclidinyl benzylate concentration (A); the abscissa in (B) shows the concentration of m-cholinoreceptors (b), pmol/mg protein; the ordinate in (A) shows the concentration of m-cholinoreceptors (b) pmol/mg protein; in (B), the ordinate on the right shows b/A and the ordinate on the left shows b/A2 (pmol/mg/nM and pmol/mg/nM2 respectively). Curves: 1) b/A; 2) b/A2, 3, 4) asymptotes (which show the specific binding of ligand by high-affinity and low-affinity pools of receptors). Theoretical curves were constructed using Equations (1) and (3) with parameters Kd1 = 0.43 nM, Kd2 = 2.83 nM, Bm1 = 712 fmol/kg, Bm2 = 677 fmol/mg, and n = 2, n = 7.

Fig. 2. Experimental points and theoretical curves for the specific binding of [3H]propranolol by rat erythrocyte b-adrenoreceptors. The abscissa in (A) shows the concentration of [3H]propranolol (A), nM; the abscissa in (B) shows the number of b-adrenoreceptors (b), U/cell; the ordinate in (A) shows the number of b-adrenoreceptors (b), U/kg; in (B), the ordinate on the right shows b/A and the ordinate on the left shows b/A2 (U/cell/nM and U/cell/nM2 respectively). Curves: 1) b/A; 2) b/A2; 3, 4) asymptotes. The theoretical curves were constructed using Eqs. (1) and (3) with parameters Kd1 = 0.74 nM, Kd2 = 14.40 nM, Bm1 = 24 U/cell, Bm2 = 283 U/cell, n = 2, n = 7.

two pools of receptors heterogeneous in terms of affinity forthe ligand, and binding two molecules of ligand per receptor, i.e., the model described by Equation (1). Figure 1, Aand

B, 2 shows theoretical curves constructed using Equa-tions (1) and (3) with parameters of

Kd1 = 0.43 +- 0.02 nM, Kd2 = 2.67 +- 0.13 nM, Bm1 = 712 +- 22 fmol/mg protein, Bm2 = 677 +- 19 fmol/mg protein, and n = 2. The experi-mental points are superimposed on the theoretical curves.

The good agreement between the experimental points andthe theoretical curves shows that the calculations are well founded. The asymptotes (Fig. 1, B, 3, 4) show the separatecharacteristics of ligand binding to the high- and low-affinity pools of receptors.Thus, graphical and mathematical analysis of the equilibrium binding of [3H]quinuclidinyl benzylate, over a widerange of concentrations, demonstrates binding to rat brain membrane muscarinic cholinoreceptors of two molecules ofspecific antagonist and the existence of two discrete pools of muscarinic receptors with significant differences in lig-and affinity.

Analysis of the equilibr ium binding of lig ands withnative r at er ythrocyte b-adrenoreceptors sho wed tha t the magnitude of specif ic [ 3H]propranolol binding incr easedover a r adioligand concentr ation r ange of 0.2 to 22.2 nM (Fig. 2, A). The non-linearity of the binding plot for [3H]pro-pranolol and r at er ythrocyte b-adrenoreceptors in standar d Scatchard coordinates (Fig. 2, B, 1) suggests a ligand-recep-tor interaction more complex than [L] + [R] = [LR], with the existence of several pools of ligand binding sites and/orattachment of several ligand molecules to the receptor. Thus, the main parameters of these curves were calculatedusing Equation (1). The best results were obtained using a model including two heterogeneous pools of receptors, withattachment of two ligand molecules to one receptor with

parameters: Kd1 = 0.74 +- 0.07 nM, Kd2 = 14.4 +- 0.41 nM, Mm1 = 24 +- 2 U/cells, Bm1 = 263 +- 5 U/cell, and n = 2. Plot-ted on coordinates of

b against b/A2 (Fig. 2, B, 2), the plotof specific binding of [3H]propranolol using mean values

from se ven e xperiments w as a conca ve cur ve, whichdemonstrates the e xistence of tw o pools of r eceptors [30, 33]. A theor etical cur ves w as plotted using Equa -tion (1) and these par ameters (F ig. 2,

A). Exper imentalpoints (mean v alues of se ven experiments) w ere in g ood

agreement with the theor etical curve, which provides evi-dence that the method used f or calculating the parameters was correct. A theoretical plot w as constructed, based onEquation (3), in coor dinates of

b against b/A2 and usingthe same par ameters; e xperimental points w ere super imposed on this plot (F ig. 2, B, 2). The coincidence of theexperimental points with the theoretical curve supports the agreement betw een the theor etical calcula tions and theexperimental data.

The r esults obtained her e lead to the conc lusion tha tthere ar e two pools of b-adrenoreceptors in na tive r at er ythrocytes, these significantly differing in terms of the disso-ciation constant and their n umbers per cell. The affinity for [3H]propranolol of the high-af finity r eceptor pool w as 20times higher, while their numbers were 10 times lower than those of low-affinity receptors (Table 1). The different ratiosof the main par ameters in the tw o pools has the ef fect that the effectiveness of lig and binding to the high-af finity poolof receptors (

E1) was twice that of the low-affinity pool.Thus, graphical and mathematical analysis of the equilibrium binding of [3H]propranolol, over a wide range ofconcentrations, demonstrated a ttachment to er ythrocyte b-adrenoreceptors of two molecules of specific antagonistand the existence of two discrete pools of receptors with significantly different ligand affinities.

Manukhin286 TABLE 1. Parameters of Radioligand and Biochemical Reactions

System/receptors Ligand Kd1 Kd2 n1 n2 Bm1 Bm2 Reference Erythrocyte/b-adreno- [3H]Propranolol 0.74 +- 0.07 14.4 +- 0.41 2 2 24 +- 2 263 +- 5 [11]

receptors nM nM U/cell U/cell

Erythrocyte/b-adreno- [3H]Dihydroalprenolol 0.22 +- 0.01 3.75 +- 0.44 2 2 71.2 +- 0.3 61 +- 4 [11]

receptors nM nM U/cel U/celll

Erythrocyte "ghosts"/ [3H]Propranolol 0.70 +- 0.17 19.59 +- 2.59 2 2 2 6 [11]b

-adrenoreceptors nM nM U/cell U/cell

Rat cerebral cortex mem- [3H]Dihydroalprenolol 0.74 +- 0.09 7.63 +- 0.70 2 2 25 +- 2 48 +- 2 [11]

branes/b-adrenoreceptors nM nM fmol/mg fmol/mg

Rat cerebral cortex mem- [3H]Quinuclidinyl benzylate 0.43 +- 0.02 2.67 +- 0.13 2 2 712 +- 22 677 +- 19 [11] branes/m-cholinoreceptors nM nM fmol/mg fmol/mg

Rat vas deferens mem- [3H]WB4101 1.19 +- 0.03 - 1 129 +- 9 - [1] branes/a-adrenoreceptors nM fmol/mg

Human lymphocytes/b- [125I]Iodocyanopindolol 7.58 +- 2.76 224.3 +- 51.2 2 2 1956 +- 30 5317 +- 579 [24]

adrenoreceptors pM pM U/cell U/cell

Rat cerebral cortex mem- Mianserin IC501 IC502 2 2 22 +- 5% 110 +- 11% [5]

branes/Na,K-ATPase 0.09 +- 0.02 mM 0.81 +- 0.15 mM

Similar analysis of the actions of ligands (neurotrans-mitters, agonists) were performed for isolated organs. Binding of ligand to a receptor is only the first stage in the mul-tistep process underlying the realization of physiological responses. There is evidently some quantitative relationshipbetween the f irst and last ste ps of the ph ysiological response. Therefore in texts, the link between the quantita-tive characteristics of measures of physiological reactions (EC50 and Pm) and the term "receptor" is arbitrary butnonetheless convenient and accepted in the literature.

The plot of the relationship between the contractileresponse of sea cucumber (

Holodeima edulis) smooth mus-cle (n-cholinoreceptors) and the acetylcholine concentration is hyperbolic (Fig. 3, A). The non-linearity of the plotof the m uscle r esponse to acetylc holine in standar d Scatchard coordinates (Fig. 3, B, 1) suggests the existenceof several pools of receptors and/or attachment of several ligand molecules to the receptor. Mathematical analysis of

the experimental data using Equation (2) showed that thebest results were obtained with the ligand-receptor interaction model including two heterogeneous pools of receptorsand attachment of two ligand molecules to one receptor, with the following parameters: EC501 = 3.04 +- 0.07 uM,EC50

2 = 12.70 +- 0.40 uM, Pm1 = 71 +- 2 U, Pm2 = 63 +- 2 U,and n = 2. On the basis of these calculations, the experimental data were plotted in coordinates of p against p/A2(Fig. 3,

B, 2). The plot of this response was a concave curve,demonstrating the existence of two pools of receptors, i.e.,

high- and low-affinity receptors [11]. The interactions ofligand with each pool of receptors separately are shown on the plot as straight lines, i.e., asymptotes (Fig. 3, B, 3, 4).For assessment of the correspondence between the calculated parameters and the experimental data, Equation (2) andthese parameters (Fig. 3, A) were used to construct the theoretical curve. The coincidence of the curve and the exper-imental points demonstrates that the method used for calcuAnalysis of Ligand-Receptor Interactions from the Molecular Level to the Whole-Body Level 287

Fig. 3. Relationship between sea cucumber m uscle contractile responses and the acetylc holine concentration. The abscissa in (A) shows the concentration of acetylcholine (A), uM; the abscissa in (B) shows response magnitude (p), U; the ordinate in (A) shows response magnitude, U; in (B), the ordinate on the right shows p/A and the ordinate on the left shows p/A2 (U/uM and U/uM2 respectively). Curves: 1) p/A; 2) p/A2; 3, 4) asymptotes (w hich show the d ynamics of acetylcholine responses of high- and lo w-affinity pools of r eceptors). The theoretical curves were constructed using Eqs. (2) and (4) with par ameters EC501 = 3.04 uM, EC502 = 12.70 uM, Pm1 = 71 U, Pm2 = 63 U, n = 2.

Fig. 4. Relationship between contractile responses of slices of rat small intestine and acetylcholine concentrations. In (B), the ordinate on the left is in (U/uM2) * 1000. For further details see caption to Fig. 3.

lation of EC50, Pm, and n was correct. Equation (4), plottedin coordinates of

p against p/A2, was used with the sameparameters to construct the theoretical curve. The coincidence of the experimental points with the theoretical curvein these coordinates again supports the relationship between the theor etical calcula tions and the e xperimental da ta(Fig. 3,

B, 2).

Thus, the relationship between the magnitude of thecontractile response of the sea cucumber muscle to acetylcholine was best described by a model including two dis-crete pools of receptors and attachment of two transmitter molecules to one receptor. The acetylcholine affinity of thereceptors of the high-affinity pool was four times greater and their numbers were slightly greater as compared with

Manukhin288 TABLE 2. Parameters of Physiological Reactions

System/receptors Ligand EC501 EC502 n1 n2 Pm1, % Pm2, % Reference Sea cucumber muscle/ Acetylcholine 3.04 +- 0.07 12.70 +- 0.40 2 - 71 +- 2 29 +- 2 [9]

n-cholinoreceptors uM uM

Chick embryo amnion/ Isopropylnoradrenaline 0.033 +- 0.001 0.499 +- 0.033 2.21 +- 0.08 - 60 +- 1 40 +- 30 [9]b

-adrenoreceptors uM uM

Rat vas deferens/ Noradrenaline 4.62 +- 0.56 17.22 +- 0.11 2 - 45 +- 5 55 +- 9 [9]a

1-adrenoreceptors uM uM

Rat portal vein/ Isopropylnoradrenaline 0.062 +- 0.010 1.13 +- 0.0 1.09 +- 0.00 - 52 +- 0 48 +- 0 [9]b

-adrenoreceptors uM uM

Rat portal vein/ Acetylcholine 0.044 +- 0.010 0.374 +- 0.0 3.07 +- 0.01 - 61 +- 0 39 +- 0 [9] m-cholinoreceptors uM uM

Rat portal vein/ Noradrenaline 0.016 +- 0.001 0.119 +- 0.0 2.25 +- 0.00 - 58 +- 0 42 +- 0 [9]a

-adrenoreceptors uM uM

Rabbit hindlimb vessels/ Phenylephrine 20.11 +- 1.31 - 1 - 192 +- 1 [31]a

-adrenoreceptors nmol/kg mmHg

Rabbit hindlimb vessels/ Isopropylnoradrenaline 1.68 +- 0.03 - 2 - 39 +- 1 [31]b

-adrenoreceptors nmol/kg mmHg

Spinal rabbit pressor res- Adrenaline 0.23 +- 0.01 - 1.31 +- 0.08 - 195 +- 6 [8]

ponse/a-adrenoreceptors uM mmHg

Rabbit pressor response/ Same 0.32 +- 0.01 - 2 - 83 +- 1 [8]a

-adrenoreceptors uM mmHg

Rabbit depressor res- Same 1.27 +- 0.01 - 2 - 52 +- 1 [8] ponse/b-adrenoreceptors uM mmHg

Fig. 5. Relationship between arterial pressure responses in a spinal rabbit and the adrenaline concentration. The abscissa in (A) shows the adrenaline concentration (A), uM; the abscissa in (B) shows the response magnitude (p), mmHg; the ordinate in (A) shows response magnitude, mmHg; in (B), the ordinate on the right shows p/A and the ordinate on the left shows p/A2, mmHg/uM and (mmHg/uM2) * 1000. Curves: 1) p/A; 2) p/A2; 3) p/A1.3.

figures for low-affinity receptors. The effectiveness (E) ofthe action of acetylcholine on the high-affinity pool was 11.8 U/ uM, while tha t of the lo w-affinity pool w as2.5 U/uM, i.e., cholinoreceptors of the high-affinity pool responded to the EC50 concentration of acetylcholine 4.7times more strongly than those of the low-affinity pool

Responses to acetylcholine (0.01-1.0 uM) were stud-ied in isolated rat small intestine strips. Figure 4,

A showsa plot from which it is evident that the experimental points

were loca ted along a h yperbolic cur ve. Calcula tionsshowed tha t the best r esults w ere obtained f or a single pool of m-c holinoreceptors and a ttachment of tw omolecules of acetylcholine to one receptor, i.e., the results corresponded to Equation (2) at EC50 = 0.079 +- 0.002 uM, Pmax = 78.9 +- 0.1 U, and n = 1.9 +- 0.1. In Fig. 4, B, theseexperimental da ta ar e shown as plots in coor dinates of

pagainst p/A (Fig. 4, B, 1) and p against p/A2 (Fig. 4, B, 2).The only coordinates in which the points were distributed

along a single line were p against p/A2, supporting the cor-rectness of the calculations of all parameters.

The possibility of analyzing ligand-receptor interactionsat higher levels, i.e., the whole-body level, was studied using the ar terial pr essure r esponse to adr enaline in r abbits. Theplot showing the relationship of the arterial pressure responses of spinal rabbits to adrenaline in Cartesian coordinates wasa h yperbola (F ig. 5,

A). The distr ibution of e xperimentalpoints corresponded to Equa tion (2) f or one pool of r eceptors, with EC50 = 0.23 +- 0.01 uM, Pmax = 196 +- 6 mmHg,and n = 1.3. Figure 5,

B shows three plots of the experimen-tal points f or n = 1, 2, and 1.3 (F ig. 5,

B, 1, 2, and 3). Theonly plot putting all the experimental points on the same line

was that with n = 1.3, demonstrating that the ma thematicaland graphical methods of analyzing these data gave concordant results. The Hill coef ficient suggests that in this case ,some of the receptors in a pool of receptors all with the same affinity (EC50) attach one ligand molecule each, while oth-ers attach two. Calculations based on Equa tion (6) sho wed that of the Pmax of 196 +- 6 mmHg, 125 +- 12 mmHg is real-ized via receptors with n = 1 and 72 +- 11 mmHg via receptors with n = 2.

DISCUSSION

Five biological models of different levels of complex-ity were used as examples which showed that the relationships between response magnitudes and ligand concentra-tions plotted in Cartesian coordinates were all hyperbolic. Only ad ditional ma thematical and g raphical anal ysisrevealed the characteristics of the actions of the ligands on the receptors mediating different biological responses.Studies using biological models with different receptors (m- and n-cholinoreceptors, a- and b-adrenoreceptors),ligands (a- and b-antagonists), and levels of complexity (cell membr anes, cells, biochemical r eactions, isolated

organs, arterial pr essure) demonstr ated tha t the mainparameters of lig and-receptor inter actions could be assessed quantitatively (Tables 1 and 2). As examples, someof these data are shown in Figs. 1-5.

Analysis of the radioligand data demonstrated the pos-sibility that one effector system can contain one or two pools of receptors - high- and low-affinity - and that twoligand molecules can attach to one receptor. The two pools of receptors in this case were not subtypes of b-adrenore-ceptors and m-cholinoreceptors, because quinuclidinyl benzylate and propranolol are not selective antagonists andhave identical affinities for all subtypes of these receptors [19]. A series of published data also provide evidence thatreceptors of a given pharmacological subtype can be heterogeneous in terms of affinity for agonists and antagonists.Thus, high- and low-affinity binding of [3H]quinuclidinyl benzylate with muscarinic receptors has been demonstratedin rat myocardial membranes [20]. In the case of rat cortex b2-adrenergic receptors [33], the existence of two pools ofbinding sites w as demonstr ated using [ 3H]dihydroalprenolol. Human l ymphocyte b2-adrenoreceptors w erefound to have two saturable and one unsaturable pools of binding sites for [125I]iodocyanopindolol binding [24]. As arule, receptor heterogeneity is only seen when blockers are used over a wide range of concentrations [16, 37].The literature contains significantly less direct data on the attachment of two ligand molecules to one receptor.Indirect evidence for this possibility is provided by experimental data yielding Hill coefficients of greater than unityfor myocardial membrane muscarinic cholinoreceptors [41] and membranes from the electric skate Torpedo californica[21]. Data have been obtained on the cooperative binding of [3H]quinuclidinyl benzyla te and other lig ands withm-cholinoreceptors [20, 27, 29, 41] and isoproterenol with b-adrenoreceptors [23].Two pools of r eceptors cannot be distinguished or ar e not significantly distinguished when their Kd (EC50) valuesare close or when the ligand concentrations activating mainly the low-affinity pool or the high-af finity pool are similar.Use only of graphical methods (Hofstee, Lineweaver-Burke, Hill, Schild, and other coordinates) for analyzing results doesnot always allow the non-linearity of the experimental curves to be esta blished. Two pools of r eceptors can c learly bedetected using the computer program "Ligand" [34]. However, this is not designed to assess the binding of mor e thanone ligand molecule to the receptor. Only the use of mathematical computer programs able to solve Equations (1, 2)make it possible to distinguish the number of discrete pools of receptors and the main par ameters of the lig and-recep-tor binding reaction (

Kd, Bm, EC50, Pm, and n).The Tables and F igures sho w e xamples pr ovided b y

analyses of the contractile responses of organ smooth mus-cles to neur otransmitters. In g eneral ter ms, the r elationships between the response magnitudes and the ligand con-centrations w ere descr ibed b y Equa tion (2). The main

Analysis of Ligand-Receptor Interactions from the Molecular Level to the Whole-Body Level 289 parameters of the r esponses, EC50, Pm, and n g ive thequantitative characteristics of the ph ysiological responses. The fit of the model to the experimental data were evaluat-ed in ter ms of the siz es of the er rors of the estima ted parameters and the concor dance of the calcula ted with theexperimental points. Theoretical curves constructed using calculated par ameters coincided with the e xperimentalcurves. Thus, these parameters quantitatively characterize the physiological response and its de pendence on the con -centration of the signal molecules. The EC50 par ameter corresponds to the lig and concentr ation e voking thehalf-maximal r esponse (

Pm/2). P arameter n is less w ellunderstood. In enzymolo gy, it is ter med the coef ficient of

cooperativity, or the Hill coef ficient. Ho wever, thebest-founded m ultistep model of cooper ativity descr ibed by Monod , Wymen, and Chang eux [4] does not al waysexplain the lig and-receptor inter action [36]. According to Hill's h ypothesis, the lig and molecule a ttaches to theenzyme in a single ste p. It can thus be sug gested tha t a t n > 1, more than one lig and molecule a ttaches to a r ecep-tor. It is unlik ely tha t fr actional n umbers of lig and molecules can attach to one receptor. This leads to the sug-gestion that the pool of r eceptors of a g iven affinity consists of tw o components, binding either one or tw o ligandmolecules. The possibility tha t tw o acetylc holine molecules a ttach to a single acetylc holine r eceptor in thefrog heart was demonstrated by Turpaev [14]. Ma thematical methods have started to be used in recent studies of thecontractile responses of smooth m uscles for calculation of EC50 and n. As a result, a single effector system was foundto contain high- and low-affinity pools of receptors [38], with 1 < n < 2 [40] and n > 2 [17].Studies of the arterial pressure pressor responses to adrenaline in normal and spinal rabbits have also demon-strated the effectiveness of this analytical system for experimental results. Along with the usual parameters for thepressor response for spinal animals, a group of receptors with the same affinity (EC50) was found but which can bindeither one or two adrenaline molecules.

In physiological responses, the process going from thereceptor binding of tr ansmitters to the ef fects in volves a series of intracellular stages. We cannot therefore definitive-ly tak e n as a measur e of the n umber of tr ansmitter molecules bound to receptors. At the same time , n is one ofthe main parameters characterizing the action of transmitters in physiological responses. It follows from Equation (2) thatwhen n > 1, the magnitude of a r esponse at a concentration [A] < EC50 is smaller , while at [A] > EC50 the r esponse isgreater than at n = 1. The converse obtains at n < 1. Thus, the parameter n is a measure of the modulation of the magnitudeof a ph ysiological r esponse w hich de pends on the lig and concentration, or it is a measur e of the heter ogeneity of thepool of receptors in terms of their ability to bind one or tw o ligand molecules. Tables 1 and 2 sho w the main par ametersof radioligand and ph ysiological responses calculated from

our own results and published data. Experimental treatmentsproduce dif ferent alter ations in the par ameters of these responses [1, 6, 31]. The ma gnitude and na ture of thesechanges identify the str ength and dir ection of the tr eatment on the process of interest, i.e., they provide quantitative char-acterization of the activity of a ph ysical or pharmacological treatment on the effector system.Thus, a system for quantitative analysis of ligandreceptor interactions has been presented, using models ofdifferent levels of complexity. The main parameters determined by this method characterize the properties of thestudy system: the number of pools of receptors with different ligand sensitivities, their affinities for the ligand (Kd orEC50), the number of active receptors (

Bmax), or the mag-nitude of the maximal response ( Pmax), and the number ofligand molecules bound per receptor (n, the Hill coefficient). The derivative parameter, effectiveness E = Bmax/2Kd(for radioligand experiments) or

E = Pmax/EC50 (for phys-iological experiments), provide a general quantitative characteristic for the activity of an effector system. This analyt-ical system for ligand-receptor interactions can be used in studies of any biological response whose results can bemeasured quantitatively.

This study was supported by the Russian Fund forBasic Research (Grant No. 99-04-49141).

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