SIMPLE MODEL Direct Binding Analysis

Neurochemistry, 56:120:575 Dr. Patrick J. McIlroy Supplementary Notes SIMPLE MODEL Direct Binding Analysis The interaction of a (radio)ligand, L, with its receptor, R, to form a non-covalent complex, RL, is described by k +1 R + L RL (1). k -1 The concentration of the receptor-ligand complex, [RL], is frequently referred to as the amount bound [B]. According to the law of mass action, at equilibrium [R][L] K d = ))))) (2) [B] where K d is the equilibrium dissociation constant. The total number of receptors (B max or [R t ]) is the sum of the occupied, [B], and unoccupied, [R], receptors. Therefore [R] = B max - [B] (3) Substitution of this relationship for [R] in equation (2) yields (B max - [B])[L] K d = ))))))))))))))) (4) [B] which can be rearranged to B max [L] [B] = )))))))) (5) [L] + K d an equation describing a saturation function and analogous to the Michaelis-Menten equation describing the rate of an enzymatically catalyzed reaction. K d, then, is analogous to K m and represents the concentration of ligand necessary to half-saturate the receptor. Strictly speaking, one should also correct the concentration of ligand, [L], to reflect the effect of removing the concentration bound from the total ligand pool. Thus, the [L] term in equation (5) should really be free ligand, [F] [F] = [L t ] - [B] (6). However, in the case of neurotransmitter-receptor interactions, receptor concentrations are typically several orders of magnitude lower than those of ligand (i.e. pm to nm for receptor and approximately :M for ligand) and the removal of bound ligand from the BindingAnalysis2005.wpd - Page 1

free pool is insignificant. In a typical saturation experiment, increasing concentrations of (radio)ligand are incubated with a constant concentration of receptor, and the amount bound to the receptor (B) is determined. One can then plot [B] (y-axis) as a function of [L] (x-axis, Fig 1). Nonlinear regression analysis can now be used to fit the results to equation (5) and estimates of K d and B max determined. In practice, however, this analysis is complicated by the fact that the (radio)ligand non-specifically binds to various components of the assay system such as test tube walls, filters, etc.. This nonspecific association (non-specific binding; NSB) is usually nonsaturable and is a function of the ligand concentration [L] (fig. 1). NSB = n[l] (7) Thus, the true equation describing the total binding in an assay tube has to incorporate specific binding and non-specific binding. B max [L] [B] = )))))))) + n[l] (8) [L] + K d NSB can be determined experimentally by adding a large excess of unlabeled ligand to assay incubations and determining the amount of radio-ligand bound. Since the large excess of unlabeled ligand will displace the radio-ligand from the receptor (due to competition), the only radio-ligand bound at the end of the incubation will be that nonspecifically associated with the assay components. The NSB is then subtracted from the total binding to give specific binding. Instead of using non-linear regression analysis to obtain estimates of B max and K d, one can use a variety of linearizing transformations to obtain estimates. The most common of these transformations are Scatchard analysis and the double reciprocal plot. )))))))))))))))))))))))))))) TABLE I and Figure 1. Analysis of saturation data (arbitrary units). Ligand Bound Ligand Conc Specific Non-specific Total 0.50 0.42 0.04 0.46 1.00 0.71 0.08 0.79 2.00 1.11 0.16 1.27 4.00 1.54 0.32 1.86 8.00 1.90 0.64 2.54 16.00 2.16 1.28 3.44 32.00 2.32 2.56 4.88 BindingAnalysis2005.wpd - Page 2

Scatchard Analysis Expansion of equation (4) yields K d B = B max [L] - [B][L] (9) which can be rearranged to [B][L] B max [L] B = - )))))) + ))))) (10). K d K d Dividing through by [L] yields the Scatchard equation [B] 1 B max ))) = - ))[B] + ))) (11) [L] K d K d which says that a plot of the bound ligand, [B], versus the [B]/[L] ratio will yield a straight line with a slope that is the negative inverse of the K d and an x-intercept of B max (Figure 2A). Double Reciprocal Plot. This linearizing transformation is analogous to the Lineweaver- Burk plot. Take the inverse of both sides of equation (5), 1 [L] + K d ))) = )))))))) (12) [B] B max [L] then rearrange and cancel to get 1 K d 1 1 ))) = ))) ))) + ))) (13). [B] B max [L] B max An equation that says that a plot of the inverse of the ligand concentration, [L], versus the inverse of the bound ligand, [B], will yield a straight line with an y-intercept of the inverse of B max and an x-intercept of the negative inverse of K d (Figure 2B). The advantages of using a transforming analysis is that one can obtain estimates of B max without using saturating concentrations of (radio)ligand. The analysis, however, is subject to significant error if the highest concentration of (radio)ligand used does not exceed K d, particularly when high levels of NSB are encountered. A second advantage of linearized plots is that visual inspection of the plot provides insight into whether or not the reaction in question is a simple bimolecular reaction (i.e. see reaction (1)). A curve plot implies that the reaction is complex or that errors have been made in the experimental procedure. Discrimination between the various possibilities can be accomplished by a kinetic analysis. TABLE II. Scatchard and double reciprocal data transformations BindingAnalysis2005.wpd - Page 3

Ligand Specific Conc. Bound [B] / [L] 1 / [B] 1 / [L] 0.50 0.42 0.83 2.40 2.000 1.00 0.71 0.71 1.40 1.000 2.00 1.11 0.56 0.90 0.500 4.00 1.54 0.38 0.65 0.250 8.00 1.90 0.24 0.53 0.125 16.00 2.16 0.14 0.46 0.063 32.00 2.32 0.07 0.43 0.031 Figure 2. Scatchard (A) and double reciprocal (B) plots. )))))))))))))))))))))))))))) Kinetic Analysis In the mechanism under discussion, reaction (1), the forward reaction is described by a second order rate equation d[b] )))) = k +1 [R][L] (14) dt and the reverse reaction by a first order rate equation d[b] - )))) = k -1 [RL] (15) dt At equilibrium, the forward reaction rate will equal the reverse reaction rate. Therefore k +1 [R][L] = k -1 [RL] (16). Rearranging gives BindingAnalysis2005.wpd - Page 4

[R][L] k -1 )))))) = ))) = K d (17). [RL] k +1 Thus determination of the two rate constants can provide an alternative method of estimating the K d. On Kinetics Classically, the on rate (k +1 ) is determined by measuring the amount of (radio)ligand bound as a function of time during the initial interaction of ligand and receptor, and applying the data to the integrated form of the rate equation. From (13) above, written to account for the effects of the binding of ligand on the ligand and receptor concentrations, one gets d[b] )))) = k +1 ([R t ] - [B])([L t ] - [B]) (18) dt which can be rearranged, with like terms on the same side of the equation, to d[b] ))))))))))))))))))))))) = k +1 dt (19) ([R t ] - [B])([L t ] - [B]) Integration of both sides (with the incorporation of the constant term into the equation) gives 1 [L t ]([R t ] - [B]) k +1 t = ))))))))))))) ln )))))))))))))))))) (20) ([R t ] - [L t ]) [R t ]([L t ] - [B]) Thus, a plot of time versus the above complex function on the right side of the equation will give a straight line with a slope equal to the on rate (k +1 ) (Fig. 3A). In some cases one cannot utilize the above method, for instance if equilibrium is reached too quickly and the reverse (off) reaction becomes significant. In these cases a second rate equation making use of data collected until equilibrium is reached is used. Substituting the relationships in equations (3) and (6) into equation (16) (Note: [B e ] is the concentration of ligand bound at equilibrium, other terms have been defined previously) gives k +1 ([B max ] - [B e ])([L t ] - [B e ]) = k -1 [B e ] (21) at equilibrium. This can be rearranged to form ([B max ] - [B e ])([L t ] - [B e ]) k -1 = k +1 ))))))))))))))))))))))))) (22). [B e ] The true measured rate of formation of [B] is a combination of equations (15) and (16). BindingAnalysis2005.wpd - Page 5

d[b] )))) = k +1 [R][L] - k -1 [RL] (23) dt Substitution of (22) into (23) yields d[b] )))) = k +1 ([B max ] - [B])([L t ] - [B]) dt ([B max ] - [B e ])([L t ] - [B e ]) - k +1 [B]))))))))))))))))))))))))) (24). [B e ] )))))))))))))))))))))))))) TABLE III. Association data and analysis Time Functions (Min) [Bound Ligand] Short Term Equilibrium Other necessary info 0.0625 6.211e-12 6.231e+04 6.250e+04 0.125 1.235e-11 1.242e+05 1.250e+05 Ligand Conc = 1.000e-07 0.25 2.439e-11 2.469e+05 2.500e+05 Receptor Conc = 1.000e-09 0.5 4.758e-11 4.876e+05 5.001e+05 1 9.063e-11 9.505e+05 1.000e+06 2 1.648e-10 1.803e+06 2.001e+06 4 2.753e-10 3.225e+06 4.003e+06 8 3.991e-10 5.104e+06 8.010e+06 16 4.796e-10 6.549e+06 1.603e+07 32 4.992e-10 6.934e+06 3.207e+07 64 5.000e-10 6.951e+06 6.415e+07 128 5.000e-10 6.951e+06 1.283e+08 Figure 3. Association Rate Analysis: Panel A - short term method; Panel B - Equilibrium method. BindingAnalysis2005.wpd - Page 6

This second order rate equation can be integrated to give [B][B e ] [B e ] [L t ] - )))))) [B max ] [L t ][B max ] ln )))))))))))))))))))) = k +1 t )))))))))) - [B e ] (25) [L t ]([B e ] - [B]) [B e ] or [B][B e ] [B e ] [B e ] [L t ] - )))))) [B max ] )))))))))))))) ln )))))))))))))))))) = k +1 t (26) [B max ][L t ]-[B e ] 2 [L t ]([B e ] - [B]) which says that a plot of the complex function of [B] shown above on the left side of equation (26; y-axis) versus time (x-axis) will give a straight line with a slope of k +1 (Fig. 3B). As alluded to above, the concentration of ligand, [L], does not significantly change in many studies since the concentration of receptor, [B max ], is several orders of magnitude lower. Under these conditions, the reaction can be considered pseudo-first-order. Thus, equation (25) can be reduced to [B e ] [L t ]B max ln )))))))))) = k +1 t )))))) (27), [B e ] - [B] [B e ] and equation (26) to [B e ] [B e ] k +1 t = )))))) ln )))))))) (28). [L t ]B max [B e ] - [B] A plot of time versus the above complex function on the right side of the equation (similar to that discussed above) will give a straight line with a slope equal to the on rate (k +1 ). If pseudo-first-order kinetics are applicable, one can eliminate the need for determining B max as a condition of analyzing kinetic behavior. This method involves determining the slope (k obs ) of pseudo-first-order plots (f(b) from left hand side of equation (27), y-axis, versus time, x-axis) over a range of ligand concentrations. The observed rate constant (slope) is related to the ligand concentration k obs = k +1 [L] + k -1 (29). A plot of k obs versus ligand concentration will give a straight line with a slope of k +1 and a Y-intercept of k -1. BindingAnalysis2005.wpd - Page 7

Off Kinetics The rate of dissociation is measured by stopping the association of (radio)ligand with its receptor and following the amount of ligand that remains bound as a function of time. In practice, the interaction is allowed to proceed to equilibrium and the forward reaction is stopped by infinite dilution and/or addition of high concentrations of unlabeled ligand. The disappearance of bound ligand is defined by equation (15). Integration of this equation yields -k -1 t [B] = [B 0 ]e (30) where [B 0 ] is the amount (concentration) bound to the receptor at the start of the dissociation and the other terms have been defined previously. The equation (30) can be linearized by taking the logarithm of both sides )))))))))))))))))))))))))) TABLE IV. Dissociation data and Analysis Time (Min) [Bound Ligand] Ln(Bound Ligand) 0 5.000e-10-21.42 1 4.524e-10-21.52 2 4.094e-10-21.62 4 3.352e-10-21.82 8 2.247e-10-22.22 16 1.009e-10-23.02 32 2.038e-11-24.62 64 8.308e-13-27.82 128 1.380e-15-34.22 Figure 4. Dissociation Rate Analysis BindingAnalysis2005.wpd - Page 8

ln[b] = -k -1 t + ln[b 0 ] (31) Which says that a plot of the natural log of [B] versus time has a slope equal to k -1 and a Y-intercept of the natural log of [B 0 ] (Fig. 4B). Alternatively, one can plot the relationship between [B] (yaxis) and time (x-axis) on semi-log graph paper, read the [B 0 ] from the graph (Y-intercept) and obtain the off rate (k -1 ) from t ½, the time necessary to reduce the concentration of [B] to one-half of its initial value based on the relationship (note: 0.693 is ln(2.0)) k -1 = 0.693 / t ½ (32). BindingAnalysis2005.wpd - Page 9

Indirect Binding Analysis The interactions of unlabeled ligands with a receptor can be examined by studying their ability to compete with a (radio)labeled ligand for the receptor in question. This is especially important for ligands which have a low affinity for the receptor (high K d value) since non-specific binding becomes a very serious problem when the concentrations of ligand very high (i.e. those necessary to obtain significant binding when affinity is low). The other benefits of using unlabeled ligands in competition with (radio)labeled ones is that one does not have to prepare labeled version of every ligand of interest, and one can design a few labeled ligands which are highly specific for various receptor subtypes. When one talks about competing ligands or competitive inhibitors of receptor binding, one is dealing with two general subtypes, agonists - ligands which bind to the receptor and invoke a biological response, and antagonists - ligands which bind to the receptor and block the biological response. The simple model describing the competition in question between a (radio)ligand, [L], and a competitive inhibitor is shown below. k +1 R + L RL (33) + k -1 I k +1i k -1i RI According to the laws of mass action the the rates of formation of [RL], or [B], and of [RI], or [B i ] are d[b] )))) = k +1 [L](B max - [B] - [B i ]) - k -1 [B] (34) dt d[b i ] )))) = k +1i [I](B max - [B] - [B i ]) - k -1i [B i ] (35) dt At equilibrium, d[b]/dt = d[b i ]/dt = 0, and an equation can be derived for [B i ] from (34) and (35). B max [I] [B i ] = ))))))))))))))))))) (36) [I] + K i (1 + [L]/K d ) Here, K i is the equilibrium dissociation constant of the competitive inhibitor and the other terms have been defined previously. If K d is much greater than the concentration of receptors (i.e. [L] does not change significantly in the reaction), the above equation can be BindingAnalysis2005.wpd - Page 10

simplified to B max [I] [B i ] = )))))))))) (37) [I] + IC 50 where IC 50 is the concentration of inhibitor necessary to block 50% of the binding of [L] in the absence of inhibitor. In a typical competition experiment, the binding of a fixed amount of (radio)labeled ligand is inhibited by increasing amounts of a competing ligand. The amount of [L] that is bound to the receptor is described by [B 0 ][I] [B] = [B 0 ] - ))))))))) (38) [I] + IC 50 This equation can be rearranged to [B 0 ] [B] = ))))))))))) (39) 1 + [I]/IC 50 Non-linear regression analysis can be used to fit binding data to equation (39) to provide estimates of the IC 50 (fig. 5A), or the data can be linearized by using a log /logit plot (fig. 5B). The equilibrium dissociation constant of the competing (unlabeled) ligand, K i, is related to the IC 50 value by )))))))))))))))))))))))))) Figure 5. Cold Competition Curves. Panel A, Semi-log plot; Panel B, Log-logit plot BindingAnalysis2005.wpd - Page 11

IC 50 K i = ))))))))) (40) 1 + [L]/K d This calculation is only valid if the concentration of receptor is much less than K i and the concentration of free (radio)labeled ligand is much greater than that bound (i.e. the concentration of free (radio)ligand is not significantly affected by ligand binding). If the assay conditions are such that [L]/K d n 1, then K i. IC 50. BindingAnalysis2005.wpd - Page 12

COMPLEX MODELS - RECEPTOR SUBTYPES In many cases there are a large number of different receptors for each individual neurotransmitter. These subtypes arise directly from different gene products and/or from different combinations of various gene products. A result of this diversity is that a neurotransmitter may bind differently to each subtype and that the biological consequences of binding may be different. Investigation of the characteristics of various receptor subtypes can be undertaken in several ways. One can directly investigate the interaction(s) in question using a (radio)ligand which is specific for the receptor subtype under consideration or using a (radio)ligand which has different affinities for the various subtypes, or one can indirectly investigate the interaction by using a non-discriminating (radio)ligand and specific, unlabeled, competing ligands for each receptor subtype of interest. In the first instance, the methodology used in these investigations would be as described previously. If one has a (radio)ligand (from necessity a drug since a specific natural ligand would make it a receptor type, not a sub-type) for a specific subtype, one does not have to worry about competing reactions. In the second instance, there will be two or more competing reactions for the (radio)ligand [L]B max1 [L]B max2 [B] = )))))))))) + )))))))))) (41) [L] + K d1 [L] + K d2 Non-linear regression analysis can be used to fit this equation to )))))))))))))))))))))))))) Figure 6. Sample analysis of equilibrium binding data with 2 sites present, direct measurements. BindingAnalysis2005.wpd - Page 13

Figure 7. Sample analysis of equilibrium binding data with 2 sites present, indirect measurements. )))))))))))))))))))))))))) equilibrium binding data to provide estimates of the equilibrium dissociation constants and the B max for each receptor subtype (fig. 6A). The statistical validity of the fit is tested by comparing it to a one site fit of the data with an F-test. A Scatchard plot of 2-site binding data typically yields a curvilinear plot, provided that the affinities are different enough (fig. 6B). There are graphical methods (Rosenthal correction) to subtract one curve from the other and obtain estimates of the appropriate parameters (i.e. K d s and B max s). In the last instance, indirect investigation, non-linear curvefitting programs are necessary to obtain parameter estimates (fig. 7). This type of analysis is fraught with dangers. Depending on the selectivity of the ligands and the ratios of the two binding sites to each other, one may or may not see multiple sites all of the time, some of the time, or very infrequently. REFERENCES Segal, Irwin H., (1976) Biochemical Calculations, John Wiley and Sons, New York. McGonigle, P., and Molinoff, P.B. (1989) Quatitative Aspects of Drug- Receptor Interactions, in Basic Neurochemistry, 4 th Edition (Siegel, G.J., Agaranoff, B.W., Albers, R.W., and Moloniff, P.B, editors), Raven Press, New York, pp 183-201. BindingAnalysis2005.wpd - Page 14