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. 2016 Feb;89(2):297–302. doi: 10.1124/mol.115.102509

Dose–Response Analysis When There Is a Correlation between Affinity and Efficacy

Anthony Auerbach 1,✉
PMCID: PMC4746484  PMID: 26655305

Abstract

The shape of a concentration–response curve (CRC) is determined by underlying equilibrium constants for agonist binding and receptor conformational change. Typically, agonists are characterized by the empirical CRC parameters efficacy (the maximum response), EC50 (the concentration that produces a half-maximum response), and the Hill coefficient (the maximum slope of the response). Ligands activate receptors because they bind with higher affinity to the active versus resting conformation, and in skeletal muscle nicotinic acetylcholine receptors there is an exponential relationship between these two equilibrium dissociation constants. Consequently, knowledge of two receptor-specific, agonist-independent constants—the activation equilibrium constant without agonists (E0) and the affinity-correlation exponent (M)—allows an entire CRC to be calculated from a measurement of either efficacy or affinity. I describe methods for estimating the CRCs of partial agonists in receptors that have a correlation between affinity and efficacy.

Introduction

A simple reaction scheme describes the basic mechanism of agonist action: (L + R) ↔ LR ↔ LR* (Del Castillo and Katz, 1957; Colquhoun, 1998), where L is the ligand, R is the resting (inactive) receptor, and R* is the active receptor. The equilibrium dissociation constant for the first, ligand-binding, step is Kd, which is the resting “affinity.” The equilibrium constant for the second, receptor-activation, step is E1 (where the subscript indicates the number of bound ligands), which relates to the maximum response for that ligand, or “efficacy.” The reaction can be extended to accommodate multiple agonist-binding steps. Together, the equilibrium constants Kd and En determine the response as a function of ligand concentration (Fig. 1).

Fig. 1.

Fig. 1.

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Example CRC. Top: A CRC simulation. Peak macroscopic responses (left inset) were fitted by the Hill equation (● and line) to estimate efficacy (0.34) and EC50 (173 μM). Single-receptor interval durations (right inset; R* is up) were fitted across concentrations to estimate rate constants (s−1) for agonist dissociation/association (1735/[L] • 88) and for the forward/backward activation conformational change (1003/2013). The Kd and E2 estimates were the same with either method. Bottom: CRC equations. L is the ligand, R is the resting receptor, and R* is the active receptor. Kd is the equilibrium dissociation constant for agonist binding to R, and En is the activation equilibrium constant with n bound ligands. The equations assume that activation of receptors with <n bound ligands is negligible, and that all sites are equivalent and independent. Response is the probability of a single receptor being active, EC50 is the [L] that gives a half-maximal response, and efficacy is the maximum response to the ligand.

Nicotinic endplate acetylcholine receptors (AChRs) mediate synaptic transmission between vertebrate nerve and skeletal muscle (Ashcroft, 2000; Changeux and Edelstein, 2005). The CRC methods I describe are based on the experimental observation that, in these receptors, affinity and efficacy are not independent (4) (Jadey and Auerbach, 2012). In endplate AChRs, if one knows En then one also knows Kd (and vice versa).

Materials and Methods

Simulations and analyses were done using QUB software (www.qub.buffalo.edu). The model was a linear scheme with two equivalent agonist-binding steps (Fig. 1), with backward/forward rate constants for binding (10,000 s−1/[L]·100 (μMs)−1; Kd = 100 μM) and forward/backward rate constants for activation (1000 s−1/2000 s−1; E2 = 0.5). In the multireceptor simulations, the sampling interval was 10 kHz, and the peak macroscopic response as a function of the agonist concentration was fitted by the Hill equation. In the single-receptor simulations, the sampling interval was 100 kHz, and the rate constants were estimated by fitting interval durations globally by the two-site scheme (Purohit et al., 2015).

Results

Affinity-Efficacy Correlation.

Figure 1 shows by simulation two ways to estimate Kd and E2. For this partial-agonist example, the high-concentration asymptote of the macroscopic peak response was 0.34 (a full agonist would be ∼1), and the EC50 was 173 μM. From the equations in Fig. 1, we calculate E2 = 0.52 and Kd = 102 μM. The rate constants obtained from fitting single-receptor interval durations are given in the legend, with the result Kd = 99 μM and E2 = 0.50. Both methods give the same equilibrium constant estimates and require the measurement of responses to multiple [agonist].

Receptor desensitization can alter the shape of the CRC and the equilibrium constant estimates. When desensitization is as fast (or faster) than activation, peak macroscopic responses are truncated, especially at high [agonist] and possibly to different extents with different agonists. When desensitization and activation mix, the CRC parameters efficacy and EC50 do not provide accurate estimates of either En or Kd. In single-receptor responses, desensitization typically appears as long-duration gaps (inactive periods) with lifetimes that do not change with [agonist] (Sakmann et al., 1980). These, too, can foil rate and equilibrium constant estimation (Salamone et al., 1999). In the experimental results shown below, desensitization events were excluded from the analysis.

There are many additional obfuscating factors in dose–response analyses, including solution exchange times, receptor heterogeneity, modal activity, inhibition by the agonist (for example, channel-block), nonstationary receptor number, unequal binding sites, activation of partially liganded receptors, sublevel responses of single receptors, kinetic complexity, and rate/equilibrium constants that are too fast/large or slow/small to be measured accurately. In the preparation that I consider here—single-channel currents from adult-type mouse skeletal muscle nicotinic AChRs expressed in human embryonic kidney (HEK) cells—these confounding issues were either not present or were incorporated into the analyses so that the activation equilibrium constants were estimated accurately.

Figure 2 shows Kd and E2 values for eight different agonists of endplate AChRs plotted on a log-log scale (Jadey et al., 2011; Jadey and Auerbach, 2012; Purohit et al., 2014). Remarkably, there is a linear relationship that holds over a ∼25,000-fold range in affinity. Partial agonists that do not have a ring (carbamylcholine, tetramethylammonium, choline) or that have a ring (anabasine, nicotine, dimethylthiopyrrolidine, and dimethylpyrrolidine) fall on the same line, as do those with a tertiary amino group (anabasine and nicotine) or a quaternary amino group (all of the others). The affinity-efficacy correlation in AChRs is robust. A correlation between affinity and efficacy has also been reported for partial agonists of GABAA (Jones and Westbrook, 1995) and N-methyl-d-aspartate (NMDA) receptors (Kalbaugh et al., 2004). The log of an equilibrium constant is proportional to the free energy difference between the end states, so the linear correlation in Fig. 2 indicates that in AChRs the energy changes in (L + R) ↔ LR (binding) and LR ↔ LR* (gating) are not independent but are related by a constant factor.

Fig. 2.

Fig. 2.

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Affinity-efficacy correlation. Kd and E2 were estimated from single-channel currents for different agonists of adult mouse AChR expressed in human embryonic kidney (HEK) cells (Jadey et al., 2011; Jadey and Auerbach, 2012). On a log-log scale, Kd and E2 are correlated linearly. The y-intercept is −5.64, and the slope is = 1.95. ACh, acetylcholine; Ana, anabaseine; CCh, carbamylcholine; Cho, choline; DMP, dimethylpyrrolidine; DMT, dimethylthiopyrrolidine; Nic, nicotine; TMA, tetramethylammonium.

Cycle.

Although a linear reaction scheme is an excellent approximation for most wild-type (WT) receptors, activation is more completely described by a cyclic mechanism called MWC (Monod, Wyman, and Changeux), after those who first proposed it for hemoglobin (Monod et al., 1965) which was applied to AChRs shortly thereafter (Karlin, 1967) (Fig. 3). Three aspects differentiate linear and cyclic schemes. First, in a cycle a receptor can activate spontaneously in the absence of any ligands, but typically with a low probability. In adult AChRs the unliganded, R ↔ R* activation equilibrium constant (the allosteric constant) is E0 ≈ 10−6 (Jackson, 1986; Nayak et al., 2012). Second, in a cycle agonists are seen to increase activity because they bind more tightly to the active versus resting conformation of the receptor. The extra, favorable agonist-binding energy that is generated when the receptor switches spontaneously to the active conformation serves to increase the relative stability of R* and hence increase En over E0 (Auerbach, 2015). I will use the symbols Jd and Kd for the higher (active) and lower (resting) affinities. For ACh and adult-type AChRs, Jd ≈ 35 nM and Kd ≈ 150 μM, to make Kd/Jd ≈ 4500. This low/high affinity ratio is called the coupling constant. For a given receptor, each agonist will have its own characteristic coupling constant.

Fig. 3.

Fig. 3.

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Cyclic model for binding and activation. Receptors can activate in the absence of ligands (R ↔ R*), and agonists can dissociate from active receptors (LnR* ↔ R* + nL). Linear activation schemes (Fig. 1) are good approximations when these two steps are uncommon. Dotted lines indicate that the binding of multiple agonists is sequential and not simultaneous; n is the number of (equivalent) agonist binding sites; E0 is the allosteric constant and is agonist independent; and En is the activation equilibrium constant of a fully occupied receptor. The low/high affinity ratio Kd/Jd is the coupling constant and is agonist specific. Ligands activate receptors when the coupling constant is >1 (eq. 1).

Agonist binding is sometimes described as “triggering” the activation conformational change. I think that this ballistic metaphor is misleading for two reasons. First, a diffusing ligand does not have sufficient momentum to activate mechanically. Rather, the passive, Kd-to-Jd increase in affinity provides chemical energy that serves to stabilize (relative to R) both the R ↔ R* transition state (to increase the activation rate constant) and the R* state (to increase the probability of being active). Once a ligand has bound with low affinity it behaves similarly to the other, covalently linked side chains, and changes position (energy) whenever called upon to do so within the global isomerization of the protein. Second, the AChR allosteric transition appears to occur by the same essential mechanism with or without agonists(Purohit and Auerbach, 2009). If the agonist is the trigger, what starts the process when no ligand is present? It appears that the AChR conformational change initiates, propagates, and terminates spontaneously (by thermal forces alone), by mechanisms that do not require the presence of an agonist. There is evidence suggesting that the AChR channel-opening conformational change starts at a linker in the transmembrane domain rather than at an agonist-binding site(Purohit et al., 2013).

The third important property of a cycle is that without input of external energy, the product of the equilibrium constants of steps connecting any two states does not depend on the pathway. Equating the products for the clockwise and counterclockwise routes between R and LnR* in Fig. 3 gives

graphic file with name mol.115.102509eq1.jpg (1)

Or efficacy is related to the product of the (receptor-specific) allosteric constant E0 and the (agonist-specific) coupling constant Kd/Jd at n equivalent agonist binding sites. In AChRs, E2ACh is ∼20, which is ∼20 million (45002) times greater than E0. The response without agonists (n = 0) and at full saturation (n = 2) is equal to (1 + 1/En)−1. The minimum and maximum asymptotes of the adult mouse endplate AChR CRC are ∼10−6 and ∼0.95, not 0 and 1. This distinction may seem unimportant, but from eq. 1 we calculate that if the allosteric constant was 10x smaller the maximum response would be only 0.67.

All of the constants in eq. 1 have been measured independently for AChRs, with much effort. E0 was estimated by extrapolating the effects of dozens of mutations on constitutive activity back to the zero-effect intercept (Nayak et al., 2012) or from the E2/E1 ratio (Jha and Auerbach, 2010). Kd was measured by using standard CRC methods and multiple [L] (Jadey et al., 2011; Jadey and Auerbach, 2012). Jd was estimated either by closing the cycle (microscopic reversibility) or by CRC methods using receptors that were active constitutively because of mutations (Grosman and Auerbach, 2001; Purohit and Auerbach, 2013). In WT AChR, the CRC equations for the simple linear reaction scheme are approximately correct because the unliganded, R* state is hardly visited. The unliganded activation pathway (R ↔ R*) is rarely taken because the allosteric constant is small, and dissociation from LnR* is infrequent because the active-state affinity is high because of a slow dissociation rate constant (Grosman and Auerbach, 2001). However, both of these uncommon routes have been revealed experimentally. For AChRs, the cyclic mechanism has been confirmed (Auerbach, 2012).

M.

The affinity-efficacy correlation in Fig. 2 arises because the low- and high-affinity binding constants (Kd and Jd) are correlated exponentially (Jadey and Auerbach, 2012):

graphic file with name mol.115.102509eq2.jpg (2)

Because of this relationship, the coupling constant in eq. 1 changes from Inline graphic to Inline graphic. M has been estimated for different agonists of AChR, and for all of these the extra energy of binding to the active receptor happens to be almost equal to the energy of binding to the resting receptor (M ≈ 1.92). M has been measured both for agonists (Jadey and Auerbach, 2012) and for dozens of mutations of aromatic binding site residues (Purohit et al., 2014), and is approximately the same. Like E0, M appears to be a fundamental constant of the receptor. Two nonaromatic residues at the binding site that do not contact the agonist have different M values (1.1 for αG153 and 3.3 for εP121). This suggests that the characteristic value of M = 1.9 pertains only to the agonist and to structural elements that interact with the ligand.

The low-high affinity correlation appears to be an intrinsic part of how the AChR operates. A priori, this makes sense. Establishing the low-affinity complex (Kd) involves diffusion and a local conformational change at the binding site (Jadey and Auerbach, 2012). If this structural change continues along the same local reaction coordinate to further increase the stability of the agonist molecule (Jd), then a correlation between the affinities will prevail. This is illustrated as an energy landscape in Fig. 4, which happens to look like the letter M. For each agonist, log Jd is larger by a constant factor than log Kd, with the factor in AChRs being M = 1.92. (In a previous report, the factor was called κ = 1/M [Purohit et al., 2014].) In AChR, establishment of the low-affinity complex is a stopover about halfway (energetically) between the apo and high-affinity conformations of the agonist binding site. It is possible that a correlation between low- and high-affinity binding is a common property of receptors.

Fig. 4.

Fig. 4.

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The M energy landscape. The black and gray lines pertain to a partial and a full agonist. For both, the energy change in high-affinity binding (proportional to log Jd) is M times greater than that in low-affinity binding (proportional to log Kd) (eq. 2). M is the ratio of the lengths of the vertical arrows, and it is the same for both agonists. In AChR, M = log(Jd)/log(Kd) = 1.92, and this constant is the basis for the affinity-efficacy correlation shown in Fig. 2. The dashed lines indicate that the full agonist “tilts” the unified energy landscape more than the partial one.

We can now modify eq. 1 by incorporating eq. 2:

graphic file with name mol.115.102509eq3.jpg (3)

It is relatively easy to measure Kd and En by a standard CRC. However, it took many years and hundreds of mutations to estimate the receptor-specific, agonist-independent constants E0 and M. By using eq. 3, these can be estimated readily from the affinity-efficacy log-log correlation plot for a WT receptor, as follows.

The y-intercept of the plot in Fig. 2 is log(E0). The fitted value of the intercept is −5.64 ± 0.48, which gives E0 in the range 0.7–7.6 × 10−7. The value of the allosteric constant estimated by the arduous extrapolation method was 7.4 × 10−7, which is within this range (Nayak et al., 2012). In AChR, there are two binding sites (n = 2), so the slope of the plot is 2(1 − M). The fitted value of the slope is −1.90 ± 0.16, so we calculate M = 1.95 ± 0.08, the same as the 1.92 value for agonists obtained by the more-difficult procedure. Simply by plotting Kd and E2 against each other on a log-log scale, it is possible to estimate the allosteric constant E0 and the low-high affinity correlation factor M. These two constants are the same for all of the agonists shown in Fig. 2.

Certainly, all ligands cannot have the same M. An antagonist is a ligand that binds but does not activate. For these, M ∼ 1, which means the same affinity for the resting and active state. An inverse agonist has an M < 1. The molecular mechanism for having a particular M value is not known. It is possible that for certain receptors and ligand families M will be constant but different than 1.9, or there could be a distribution of M values rather than a single, clear correlation.

Protocols.

For physiologists, measuring the efficacy of a partial agonist is easier than measuring affinity because all that is needed is an estimate of the maximum response at a high [L]. The following whole-cell protocol can be used to estimate affinity from efficacy, given the correlation.

  1. Apply a high concentration of a full agonist to estimate the maximum response. To minimize the effects of desensitization, the pulse onset should be as fast as possible. Try to minimize the effects of the other pitfalls in CRC construction, as listed earlier.

  2. In the same cell, measure EC50 and un-normalized efficacy values (as a fraction of the maximum response) for several partial agonists. From these, calculate the activation constants using appropriate equations for the receptor in question. It is possible to fit a plot of log (EC50) versus log (efficacy) directly to some nonlinear function, but transforming these values to equilibrium constants and fitting by a straight line is perhaps more convenient.

  3. Plot the Kd and En estimates obtained by using this standard multiple [L] approach on a log-log scale. A fit to a straight line will yield estimates of the fundamental constants E0 (the intercept) and M (the slope).

  4. Once the receptor has been calibrated in this manner, measure the efficacy (relative to the full agonist) of a novel ligand. This can be done by exposing the receptor to a very high [L]. Calculate En.

  5. Use eq. 3 to calculate Kd, EC50 (or whatever percentage), and the entire CRC, using the equations shown in Fig. 1.

For example, using the adult AChR values for the allosteric constant (E0 = 7.4 × 10−7), the Jd-Kd correlation (M = 1.92), and eq. 3, we calculate Kd = 1.9√(7.4 × 10−7/E2). Substituting the experimental estimate for ACh efficacy (0.96, or E2 ≈ 20), we calculate KdAch ≈ 123 μM, which is close to the value obtained by standard CRC analysis. Using this value of KdACh we now calculate EC50ACh = 33 μM (Fig. 1), which again is close to the value measured by CRC analysis of whole-cell currents. It is possible to approximate Kd from an E2 value measured at a single, saturating [ACh].

For both whole-cell and single-receptor responses, the typical measureable efficacy range is ∼0.05–0.90, which corresponds to 0.05 < En < 20. Below or above this range the responses saturate and become difficult to quantify accurately. For some agonists, it may be necessary to engineer E0 (and hence En; eq. 1) by adding a background mutation(s) that does not affect Kd so that the emergent En is in the suitable range (Jadey et al., 2011). The effect of the background on efficacy must be taken into account before making the log-log plot.

It is also possible to reverse the procedure, because knowledge of affinity can be used to estimate efficacy. Often, affinity is measured using a biochemical method such as isothermal calorimetry. For partial agonists, these measurements are only “effective” (not purely Kd) because the low and high affinities of resting, active, and other conformations are jumbled, with fractional contributions weighted by the interconversion equilibrium constant(s). If M is common for all partial agonists of a receptor and E0 is known, then it is possible to estimate efficacy from just a single measurement of Kd, as follows.

  1. Make a receptor mutation(s) that does not influence Kd or M, but that greatly reduces E0. This will increase the stability of the resting state and reduce the mixing of affinities caused by agonist activation. In AChRs, almost all mutations away from the agonist binding sites only affect the allosteric constant (Purohit et al., 2013), and one(s) that makes it substantially smaller than the WT is what is needed here.

  2. Use isothermal calorimetry (or another method) to characterize this receptor’s resting affinity for a weak partial agonist (Kd), which is no longer effective because the modified receptor has a low probability of changing affinity even when agonists are present.

  3. Calculate En (efficacy) by using eq. 3. With knowledge of Kd, of an agonist and the constants M and E0 for the receptor, one can estimate efficacy without expressing the receptor in a cell or making any response measurements.

There may be another way to estimate M. In mouse endplate AChR, the affinities of the active and desensitized states are similar. Hence, if Kd and the desensitized binding equilibrium dissociation constant can both be measured experimentally for a series of ligands, it might be possible to estimate M directly by using eq. 2.

Aside from the CRC, it is useful to know E0 for a receptor. To make a simple analogy, a ligand is like a deposit (of energy) into your bank account, and the response is the account balance after the deposit. The allosteric constant is the starting balance. One needs to know all three energies, including the starting one, to understand quantitatively the energy of the ligand action. In AChR, some genetic diseases are caused by mutations of residues that are distant from the agonist binding sites (Engel and Sine, 2005), and many of these do nothing more than change the allosteric constant E0. This may be true generally for other receptors. Such a change in the allosteric constant will influence efficacy, EC50, and the CRC without affecting affinity.

Equation 3 can be rearranged to solve for the allosteric constant. For the case of ACh and WT AChR, we calculate E0 = (20)(123 μM)1.9 = 6.2 × 10−7, which is similar to the E0 value estimated by the arduous mutation-extrapolation method. All one needs to know is M, En, and Kd for a given receptor to estimate the allosteric constant and the effect of mutations that change the level of constitutive activity.

Discussion

Scientists have been making CRCs for decades for many different receptors, including the muscle AChR. If the affinity-efficacy correlation is general, why has it not been observed previously? I think there are several reasons. The correlation is between En and Kd, whereas the corresponding CRC empirical parameters are efficacy and EC50. Often efficacy is not normalized to the response generated by a full agonist and therefore not even measured. The confounding factors mentioned previously can reduce the accuracy of the efficacy and EC50 estimates. Further, EC50 is a ratio of Kd and En, so it tends to hide a correlation between these equilibrium constants. Beyond these technical considerations, advances in our concept of receptor operation serve to rationalize the affinity-efficacy correlation. We are now certain that in AChR agonists activate receptors by binding more tightly to the active conformation, and that the low versus high affinities are correlated exponentially because the energy (conformation) changes that undergird each binding step are related linearly (Fig. 4).

The AChR is a typical receptor so it is possible that a correlation between affinity and efficacy is a more general phenomenon. If so, the potential impact on CRC analyses is significant. Once the allosteric constant E0 and the correlation constant M have been calibrated for a given receptor, then a complete dose–response profile can be calculated for an arbitrary partial agonist from an experimental estimate of either efficacy or affinity. A rapid screening system could be used in which a cell is exposed to a series of different drugs in rapid sequence to estimate only efficacy, rather than to a series of concentrations of one drug. In cases where affinity is easier to measure, it may be possible to estimate the efficacy of a ligand without having to express the receptor in cells and measure a physiologic response. The method also allows for estimates of the allosteric activation equilibrium constant in WT receptors and the effect of mutations on this constant.

M is the fundamental reason for the affinity-efficacy correlation. Although this correlation has been revealed in AChRs, it is important to establish whether it holds for more ligands and other receptors. Perhaps receptor modulators, too, will show a correlation. For a given receptor, E0 will definitely be the same for all ligands, and evidence from AChRs suggests that M will be the same for many. It may turn out that all receptors in a given family share a common M (set by structural changes only at the agonist binding site) but perhaps not E0 (set by many amino acids throughout the protein, by natural selection). Once a receptor has been calibrated, high-throughput CRC analysis can commence. If affinity-efficacy correlations turn out to be common, widespread knowledge of E0 and M values may impact significantly the future of dose–response analysis.

Acknowledgments

I thank all of the members of my laboratory, over the years.

Abbreviations

AChR

acetylcholine receptor

CRC

concentration–response curve

E0

activation equilibrium constant without agonists

L

ligand

M

affinity-correlation exponent

R

resting (inactive) receptor

R*

active receptor

WT

wild type

Authorship Contributions

Participated in research design: Auerbach.

Conducted experiments: Auerbach.

Contributed new reagents or analytic tools: Auerbach.

Performed data analysis: Auerbach.

Wrote or contributed to the writing of the manuscript: Auerbach.

Footnotes

This work was supported by the National Institutes of Health National Institute of Neurological Disorders and Stroke [Grant NS064969].

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