Dose–Response Analysis When There Is a Correlation Between Affinity and Efficacy
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1521-0111/89/2/297–302$25.00 http://dx.doi.org/10.1124/mol.115.102509 MOLECULAR PHARMACOLOGY Mol Pharmacol 89:297–302, February 2016 Copyright ª 2016 by The American Society for Pharmacology and Experimental Therapeutics Dose–Response Analysis When There Is a Correlation between Affinity and Efficacy Anthony Auerbach Department of Physiology and Biophysics, State University of New York at Buffalo, Buffalo, New York Received November 13, 2015; accepted December 10, 2015 ABSTRACT Downloaded from The shape of a concentration–response curve (CRC) is de- acetylcholine receptors there is an exponential relationship termined by underlying equilibrium constants for agonist between these two equilibrium dissociation constants. Conse- binding and receptor conformational change. Typically, ago- quently, knowledge of two receptor-specific, agonist-independent nists are characterized by the empirical CRC parameters constants—the activation equilibrium constant without agonists efficacy (the maximum response), EC50 (the concentration that (E0) and the affinity-correlation exponent (M)—allows an entire produces a half-maximum response), and the Hill coefficient CRC to be calculated from a measurement of either efficacy or (the maximum slope of the response). Ligands activate recep- affinity. I describe methods for estimating the CRCs of partial molpharm.aspetjournals.org tors because they bind with higher affinity to the active versus agonists in receptors that have a correlation between affinity resting conformation, and in skeletal muscle nicotinic and efficacy. 21 21 for binding (10,000 s /[L]×100 (mMs) ;Kd 5 100 mM) and forward/ Introduction 21 21 backward rate constants for activation (1000 s /2000 s ;E2 5 0.5). A simple reaction scheme describes the basic mechanism of In the multireceptor simulations, the sampling interval was 10 kHz, agonist action: (L 1 R) ↔ LR ↔ LR* (Del Castillo and Katz, and the peak macroscopic response as a function of the agonist 1957; Colquhoun, 1998), where L is the ligand, R is the resting concentration was fitted by the Hill equation. In the single-receptor (inactive) receptor, and R* is the active receptor. The equilib- simulations, the sampling interval was 100 kHz, and the rate at ASPET Journals on September 26, 2021 rium dissociation constant for the first, ligand-binding, step is constants were estimated by fitting interval durations globally by the two-site scheme (Purohit et al., 2015). 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.” Results The reaction can be extended to accommodate multiple agonist-binding steps. Together, the equilibrium constants Affinity-Efficacy Correlation. Figure 1 shows by simu- lation two ways to estimate Kd and E2. For this partial-agonist Kd and En determine the response as a function of ligand concentration (Fig. 1). example, the high-concentration asymptote of the macroscopic ∼ Nicotinic endplate acetylcholine receptors (AChRs) mediate peak response was 0.34 (a full agonist would be 1), and the m synaptic transmission between vertebrate nerve and skeletal EC50 was 173 M. From the equations in Fig. 1, we calculate 5 5 m muscle (Ashcroft, 2000; Changeux and Edelstein, 2005). The E2 0.52 and Kd 102 M. The rate constants obtained from CRC methods I describe are based on the experimental fitting single-receptor interval durations are given in the 5 m 5 observation that, in these receptors, affinity and efficacy are legend, with the result Kd 99 M and E2 0.50. Both not independent (4) (Jadey and Auerbach, 2012). In endplate methods give the same equilibrium constant estimates and require the measurement of responses to multiple [agonist]. AChRs, if one knows En then one also knows Kd (and vice versa). 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 Materials and Methods are truncated, especially at high [agonist] and possibly to different extents with different agonists. When desensitiza- Simulations and analyses were done using QUB software (www. tion and activation mix, the CRC parameters efficacy and qub.buffalo.edu). The model was a linear scheme with two equivalent EC50 do not provide accurate estimates of either En or Kd.In agonist-binding steps (Fig. 1), with backward/forward rate constants single-receptor responses, desensitization typically appears as long-duration gaps (inactive periods) with lifetimes that do This work was supported by the National Institutes of Health National Institute of Neurological Disorders and Stroke [Grant NS064969]. not change with [agonist] (Sakmann et al., 1980). These, too, dx.doi.org/10.1124/mol.115.102509. can foil rate and equilibrium constant estimation (Salamone 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. 297 298 Auerbach 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 1 R) ↔ LR (binding) and LR ↔ LR* (gating) are not independent but are related by a constant factor. Cycle. Although a linear reaction scheme is an excellent Downloaded from 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 pro- posed 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 molpharm.aspetjournals.org 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 26 allosteric constant) is E0 10 (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 Fig. 1. Example CRC. Top: A CRC simulation. Peak macroscopic conformation of the receptor. The extra, favorable agonist- responses (left inset) were fitted by the Hill equation (d and line) to binding energy that is generated when the receptor switches m estimate efficacy (0.34) and EC50 (173 M). Single-receptor interval spontaneously to the active conformation serves to increase at ASPET Journals on September 26, 2021 durations (right inset; R* is up) were fitted across concentrations to estimate rate constants (s21) 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. 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, re- ceptor heterogeneity, modal activity, inhibition by the agonist (for example, channel-block), nonstationary recep- tor 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 Fig. 2. Affinity-efficacy correlation. Kd and E2 were estimated from single- issues were either not present or were incorporated into the channel currents for different agonists of adult mouse AChR expressed analyses so that the activation equilibrium constants were in human embryonic kidney (HEK) cells (Jadey et al., 2011; Jadey and estimated accurately. Auerbach, 2012). On a log-log scale, Kd and E2 are correlated linearly. 2 Figure 2 shows K and E values for eight different agonists The y-intercept is 5.64, and the slope is = 1.95. ACh, acetylcholine; d 2 Ana, anabaseine;