Null method to estimate the maximal P_A at subsaturating concentrations of agonist

Agonist concentration–response relationships provide a variety of fundamental information about the properties and function of a receptor channel in the presence of a given agonist. The level at which the concentration–response relationship saturates can provide insight into the gating efficacy of the agonist. To that end, in macroscopic electrophysiological recordings, the standard, normalized concentration–response relationships are rescaled in units of probability of being in the active state (P_A units). This is done by comparing the amplitudes of peak responses to the test agonist to that of a reference agonist for which the P_A is known. For example, in the case of heteromeric GABA_A receptors, we and others have employed as a reference agonist the combination of saturating GABA and an allosteric agonist, e.g., propofol or etomidate, which is considered to activate all receptors and generate a peak response with P_A of 1 (Feng et al., 2014; Shin et al., 2017). The P_A at which the concentration–response curve for the test agonist saturates is its maximal P_A (P_A,max).

Estimation of P_A,max from a concentration–response curve requires the use of near-saturating concentrations of the test agonist. In practice, that is not optimal in some cases, for example, when there is limited availability of the compound or the compound has additional functional effects that selectively manifest at high concentrations. Here, we describe an approach to estimate P_A,max while employing only subsaturating concentrations of the agonist. In this approach, the test agonist is coapplied with a control agonist for which the P_A is known or can be determined. If the P_A of the control response is less than the P_A,max of the test agonist, then co-exposure to the test agonist is expected to increase the response to the control agonist. Conversely, coapplication of the test agonist reduces the response to the control agonist if the latter generates a response with a P_A greater than P_A,max of the test agonist. Importantly, the direction of the effect does not depend on the concentration of the test agonist, although the magnitude of the effect, and, hence, overall resolution are greater at higher concentrations. A caveat is the requirement that the test and control agonists act through the same binding sites.

The experiments were conducted on the rat α1β2γ2L wild-type and human β3(Q64R+G127T) mutant GABA_A receptors expressed in Xenopus laevis oocytes. The GenBank accession numbers are NM_183326 for α1, NM_012957 for β2, NM_183327 for γ2L, and NM_000814.5 for β3. The β3 mutant clone was purchased from Twist Bioscience. Synthesis of cRNA and details of oocyte handling and injection have been described in detail previously (Shin et al., 2017). The electrophysiological recordings were done using two-electrode voltage clamp at room temperature. The oocytes were voltage-clamped at −60 mV. Bath and agonist-containing solutions were gravity-applied (flow rate: 8 ml/min) from glass syringes through Teflon tubing, and switched manually using a perfusion system containing four-port bulkhead switching valves and medium pressure six-port bulkhead valves (IDEX Health and Science). The bath solution was ND96 (96 mM NaCl, 2 mM KCl, 1.8 mM CaCl₂, 1 mM MgCl₂, 5 mM HEPES; pH 7.4). Stock solutions of GABA (500 mM), piperidine-4-sulfonic acid (P4S; 3 mM), β-alanine (20 mM), and histamine (100 mM) were made in ND96 and stored at −20°C. Propofol stock solution was made in DMSO at 200 mM and stored at room temperature. Dilutions to final, working concentrations and adjustment of pH when needed were done on the day of the experiment.

The current responses were amplified with an OC-725C (Warner Instruments) or Axoclamp 900A amplifier (Molecular Devices), low-pass filtered at 40 Hz, digitized at 100 Hz with Digidata 1200 or 1320 series digitizers (Molecular Devices), and stored using Clampex (pClamp software package; Molecular Devices). Analyses to estimate current amplitudes were done using Clampfit (pClamp).

Two principal types of electrophysiological recordings were done. The first was measurement of standard concentration–response relationships to estimate the maximal level of activity from saturation of the curve. The second was measurement of the effect of coapplication of a low-efficacy agonist on the response to a high-efficacy agonist.

In P4S concentration–response measurements, oocytes expressing the α1β2γ2L receptor were exposed to 1 µM to 1 mM P4S. The durations of drug applications were 20–40 s, and the individual applications were separated by 2- to 3-min washes in ND96. For normalization purposes, to estimate the P_A of peak responses, each oocyte was additionally exposed to the combination of 1 mM GABA + 50 µM propofol.

To measure the modulatory effects of P4S on α1β2γ2L receptors activated by GABA or β-alanine, the receptors were activated by 20- to 40-s applications of 0.1–10 µM GABA or 0.2–1 mM β-alanine, and their combinations with 10–100 µM P4S. Each oocyte was exposed to a single concentration of GABA or β-alanine, and their combination with P4S. Each oocyte was also tested with 1 mM GABA + 50 µM propofol.

The histamine concentration–response curve for the mutated β3 receptor was measured by exposing oocytes expressing the β3(Q64R+G127T) receptor to 1–50 mM histamine. For normalization, each cell was also exposed to 1 mM GABA + 50 µM propofol. In separate experiments, we measured the ratio of peak responses to 1 mM GABA + 50 µM propofol and 1 mM pentobarbital. Pentobarbital is a highly efficacious agonist of β3 homomeric receptors, considered to activate all receptors and generate a peak P_A of 1 (Eaton et al., 2015). The calculated P_{A,1 mM GABA + 50 µM propofol} was 0.84 ± 0.07 (n = 6). Unlike the wild-type β3 receptor, the β3(Q64R+G127T) receptor is minimally constitutively active (P_A,const. = 0.004 ± 0.004; n = 12), and no correction for P_A,const. was needed to calculate P_A. The coapplication experiments were done by activating the β3(Q64R+G127T) receptor by 0.2–1 mM GABA in the absence and presence of 1 or 2 mM histamine.

The concentration–response data were fitted with the Hill equation or with Eq. 1 (below). The observations are presented as mean ± SD (number of oocytes). Experimenters were not blinded to experimental conditions, specifically to the drug concentrations employed. The oocytes were chosen randomly.

One prediction made by the coagonist model is that when a high-efficacy agonist (e.g., agonist X) is coapplied with a low-efficacy agonist (agonist Y) that binds to the same N sites, the response to the combination is greater than the response to agonist X alone when P_A,[X] < P_A,max,Y, and, conversely, the coapplication of Y reduces the response to X when P_A,[X] > P_A,max,Y. In other words, there is a “null point” at which the coapplication of Y has no effect on the response to X.

Note that the expression does not depend on [Y], so any concentration of Y can be used.

This result (Eqs. 14 and 7) shows that [x*] = [x′], and so the concentration of X at which the coapplication of any concentration of Y has no effect on the magnitude of the response also is the concentration of X that produces a response equal to P_A,max,Y.

This is illustrated in Fig. 1, A–C. Agonist X is the high-efficacy agonist with c_X = 0.004 (P_A,max,X = 0.89) and agonist Y is the low-efficacy agonist with c_Y = 0.02 (P_A,max,Y = 0.24). The concentration–response curve for X is right-shifted in the presence of Y (Fig. 1 A). Coapplication of Y enhances the response to low concentrations of X; accordingly, the response ratios (RR, calculated as I_X+Y/I_X) are >1 (Fig. 1 B). As the concentration of X is increased the RR are reduced, dropping below 1 due to inhibition of the response to high-efficacy agonist X by the low-efficacy agonist Y. The RR increases at very high [X], approaching 1 when X outcompetes Y for binding. RR plotted against P_A of the control response (Fig. 1 C) cross the RR = 1 line at P_A,X = P_A,max,Y. The value of the intercept can be estimated from linear regression fitting of the RR versus P_A,X relationship, although, due to the concave shape of the relationship the fitted P_A,max,Y is more accurate when using a narrow band of data points near the RR = 1 line.

For experimental validation, we first examined the activation of the α1β2γ2L GABA_A receptor by the high-efficacy agonists GABA and the low-efficacy agonist P4S (Thompson et al., 1999; O’Shea et al., 2000; Steinbach and Akk, 2001; Mortensen et al., 2002). The compounds activate the GABA_A receptor through interactions with the orthosteric binding sites (Krogsgaard-Larsen et al., 1980; Jones et al., 1998; Akk et al., 2011). We began by conducting P4S concentration–response measurements to estimate the P_A,max,P4S using a conventional approach. The oocytes were exposed to brief applications of 1 µM to 1 mM P4S. For normalization purposes, each oocyte was also exposed to 1 mM GABA + 50 µM propofol, which was assumed to generate a peak response with P_A of 1 (Shin et al., 2017). Sample current traces are given in Fig. 2 A. Fitting the concentration–response data with the Hill equation gave an EC₅₀ of 33 ± 12 µM (n = 5) and a Hill coefficient of 1.01 ± 0.27. The fitted maximal P_A was 0.18 ± 0.10. With L constrained to 8,000 (Shin et al., 2017) and N = 2, fitting Eq. 1 to the P_A data yielded a K_P4S of 14 ± 5 µM and a c_P4S of 0.027 ± 0.009 (Fig. 2 A). The P_A,max, K_P4S, and c_P4S estimates are similar to those reported previously for the concatemeric α1β2γ2L receptor (Shin et al., 2019).

We then measured the modulation of GABA responses by P4S. Each oocyte was exposed to a brief application of 0.1–10 µM GABA, GABA + P4S, and, for normalization, exposed to 1 mM GABA + 50 µM propofol. The concentration of P4S was held at 10, 30, or 100 µM. P4S increased the responses to low concentrations of GABA and inhibited the responses to high concentrations of GABA. We calculated the RR as I_{peak,GABA+P4S}/I_peak,GABA, and the P_A of the response to GABA as I_peak,GABA/I_{peak,GABA+propofol}. Sample current responses and the log-log relationships between the RR and P_A,GABA are given in Fig. 2 B. Fitting the RR-P_A,GABA data with y = x^slope × 10^intercept gave a y = 1 intercept, i.e., the estimated P_A,max,P4S, of 0.18 ± 0.05 (data from 21 oocytes) in the presence of 10 µM P4S, 0.18 ± 0.08 (n = 17) in the presence of 30 µM P4S, and 0.23 ± 0.08 (n = 20) in the presence of 100 µM P4S. These P_A,max,P4S estimates are similar to the one obtained from fitting the concentration–response curve (0.18 ± 0.10) despite using P4S at up to 100 times lower concentrations.

Next, we examined P4S-mediated effects on responses to the low-affinity, high-efficacy orthosteric agonist β-alanine. The α1β2γ2L GABA_A receptors were activated by 0.2–1 mM β-alanine in the absence and presence of 30 µM P4S. Coapplication of P4S affected the responses to β-alanine, similar to that observed with GABA as a control agonist. The data are summarized in Fig. 2 C. Fitting the RR-P_A,β-alanine data yielded a P_A,max,P4S of 0.24 ± 0.08 (n = 20). This is similar to the P_A,max,P4S estimates provided above.

Lastly, we measured the ability of histamine to affect GABA-elicited responses in the β3(Q64R+G127T) homomeric mutant receptor. The β3 receptor is normally efficiently activated by histamine but not by GABA due to unfavorable interactions of GABA with key residues in the orthosteric binding site (Saras et al., 2008; Masiulis et al., 2019; Sente et al., 2022). The Q64R+G127T mutations at the complementary side of the binding interface restore sensitivity to GABA, albeit at the expense of sensitivity to histamine (Gottschald Chiodi et al., 2019). For an initial independent estimate of gating efficacy in the presence of histamine, we measured the histamine concentration–response relationship. Oocytes expressing the β3(Q64R+G127T) receptor were exposed to 1–50 mM histamine, and, for normalization, to 1 mM GABA + 50 µM propofol. Fitting the concentration–response curve yielded a P_{A,max,histamine} of 0.50 ± 0.09 (n = 6). The EC₅₀ of the curve was at 9.0 ± 0.5 mM and the Hill coefficient was 1.45 ± 0.16 (Fig. 3 A). To test the null method, we measured the effects of 1 and 2 mM histamine on responses generated by 0.2–1 mM GABA. The extent and direction of effects scaled with P_A,GABA (Fig. 3 B). Fitting the RR-P_A,GABA data yielded a P_{A,max,histamine} of 0.33 ± 0.04 (n = 8) in the presence of 1 mM histamine, and 0.51 ± 0.12 (n = 15) in the presence of 2 mM histamine.

Here, we have described an approach to estimate P_A,max for low-efficacy agonists. The approach entails measurement of the effect of coapplication of a low-efficacy test agonist on the response to a high-efficacy control agonist and estimation of the P_A of the control agonist at which coapplication of the low-efficacy agonist has no effect on current amplitude. Model-based predictions were experimentally validated on heteromeric and homomeric GABA_A receptors using combinations of orthosteric agonists. The major benefit of the approach lies in the ability to employ low, subsaturating concentrations of the test agonist while the conventional approach to estimate P_A,max in macroscopic recordings through fitting the concentration–response relationship requires the use of near-saturating concentrations of agonist. Specifically, in our validation experiments, similar P_A,max values for P4S or histamine were obtained conventionally by fitting the concentration–response curves and the null method employing the low-efficacy agonists at up to 100 times less than a saturating concentration.

Colquhoun (1973) published a figure similar to our Fig. 1 A (his Fig. 7 c) and noted in the legend that the P_A at the point where the lines cross (i.e., the concentration of the higher efficacy agonist for which the coapplication of the lower efficacy agonist has no effect) is equal to the P_A,max for the lower efficacy agonist. However, the result is presented without derivation, and to the best of our knowledge was not tested experimentally. Steinbach and Akk (2019) also published a similar figure (their Fig. 4 B) and reported that the concentration of the higher efficacy agonist was given by the relationship in our Eq. 7. However, the result is again presented without derivation, and the P_A at that point is not given. Accordingly, elements of our approach were known but had not been fully developed and utilized in experimental work.

As noted above, the null method described here requires that the two agonists bind to the same sites. The combination of two agonists interacting with distinct sites results in allosteric potentiation. For example, coapplication of the low-efficacy allosteric agonist neurosteroid allopregnanolone with the orthosteric agonist GABA results in potentiation despite the P_{A,max,allopregnanolone} being very low (0.002; Shin et al., 2019).

While the approach is valid at any concentration of test agonist, the simulations shown in Fig. 1 C indicate that the slope of the RR versus P_A line becomes smaller as the concentration of test agonist is reduced. This is confirmed by observations summarized in Figs. 2 and 3. A smaller slope can reduce precision in real-life experiments, as evident, for example, when comparing modulation of GABA-activated receptors by 1 versus 2 mM histamine (Fig. 3 B). We also note that in our experiments, we estimated the P_A of the control response to GABA or β-alanine through normalization to the peak response to the combination of 1 mM GABA + 50 µM propofol in the same cell. While it is possible to base the P_A,control on previously reported activation properties of the control agonist, it is likely more accurate when comparing it to the reference response in the same cell.

We note that the approach utilizes pseudo steady-state peak responses. This can lead to inaccuracies if the amplitude of the peak is curtailed by rapid development of desensitization or block. In the case of these GABA_A receptors expressed in Xenopus oocytes, desensitization is relatively slow (c.f. the traces in Fig. 2) and the estimate of P_A,max for the agonist GABA (∼0.8) is similar in whole-oocyte responses (Feng et al., 2014; Shin et al., 2017) and single-channel clusters (Steinbach and Akk, 2001; Lema and Auerbach, 2006). A second concern is that the two compounds being applied need to have similar kinetics for their responses so that both effects will be at a pseudo steady-state at the peak. If these criteria are not met, the approach will not provide an accurate estimate.

P_A,max estimates can be directly obtained in single-channel recordings through measurements of intracluster P_A or kinetic properties of bursts of openings (e.g., Sine and Steinbach, 1986; Zhang et al., 1995; Steinbach and Akk, 2001). The potential drawbacks associated with these approaches, however, are low single-channel conductance in some receptor channels (e.g., sub-pS for homomeric ρ GABA_A and 5-HT_3A receptors [Brown et al., 1998; Wotring et al., 1999]) or that activity originating from a single ion channel cannot be reliably identified due to low efficacy of the agonist (e.g., many quaternary ammonium derivatives on the nicotinic receptor [Zhou et al., 1999; Akk and Steinbach, 2003]). An additional approach to measure P_A,max in macroscopic recordings is non-stationary noise analysis, but this method is best suited for P_A,max values of 0.5 and greater, and it also requires the use of saturating concentrations of agonist (Sigworth, 1980; Lingle, 2006). The null method described here is suitable for a nearly unlimited range of P_A values and, as we have shown, can produce independently verified P_A,max estimates at low concentrations of the low-efficacy agonist.

Our studies were conducted on the GABA_A receptor, which is well-characterized and has a rich pharmacology enabling the selection of agonists appropriate for the present study. There is at least qualitative published evidence that the model holds for related receptor channels. In the muscle-type nicotinic receptor, some non-depolarizing neuromuscular blocking agents potentiate the current response to low concentrations of acetylcholine and inhibit the response to high concentrations of acetylcholine (Steinbach and Chen, 1995; Fletcher and Steinbach, 1996). In the neuronal-type α4β4 nicotinic receptor, the low-efficacy orthosteric agonist choline enhances responses to low concentrations of acetylcholine and nicotine while reducing responses to high concentrations of acetylcholine (Zwart and Vijverberg, 2000).

In sum, we have presented an electrophysiological approach to estimate maximal P_A at low, subsaturating concentrations of a test agonist. Experimental validation was performed on heteromeric α1β2γ2L and homomeric β3 GABA_A receptors activated by several combinations of orthosteric agonists, demonstrating quantitative agreement between estimates of P_A,max from a traditional approach and the novel null method.

The two-state concerted-transition model was first proposed by Monod and coworkers (Monod et al., 1965). It was initially applied to transmitter-gated channels by Karlin (1967) and has subsequently been extensively used in studies of the GABA_A receptor (e.g., Chang and Weiss, 1999; Forman, 2012; Germann et al., 2019b). It is a remarkably simple model, originally formulated in terms of a homomultimeric protein with N subunits, although it has usually been implemented for proteins with N sites. The protein can exist in two states (Resting and Active), and all subunits (sites) undergo the state transition at the same time—there is never a mixture of R- and A-state subunits in a protein. In the absence of any agonist or antagonist, the ratio of R to A states is given by the parameter L. The value of L is typically high, reflecting a low level of constitutive activity (Shin et al., 2017). A chemical (X) binds to the R state with dissociation constant K_R,X and to the A state with K_A,X = c_XK_R,X. Again, an essential feature of the model is that all subunits (sites) have the same affinity for X, either K_R,X or K_A,X.. A more complete discussion of the points presented here is in Steinbach and Akk (2019).

A reaction scheme for the case that N = 3 is shown as Scheme 1:

A great advantage of the model is that only four parameters are needed to describe the effect of X: one describes a property of the receptor (L), which reflects the energy difference between the R and A states in the absence of agonist. Three describe properties of the chemical: N_X (the number of sites), K_R,X (the affinity for the resting state), and K_A,X (the affinity for the active state). Scheme 1 can be lengthened to any number of sites and the same equation is applicable with the appropriate N value.

When X and Y bind to the same sites there is then competition for occupancy. Scheme 2 shows a receptor with three sites that can be occupied by X or Y.

Here, active (A) states are on the upper plane, and resting (R) states are on the lower. The two sides show the Scheme 1 scheme for X (right side) and Y (left side), indicated by the dashed boxes. The cases when both chemicals are bound are shown in between the individual Scheme 1 sides. Note that cases when resting receptors have bound both chemicals are obscured (XYR, XY₂R, and X₂YR).

Karlin (1967) first proposed that the MWC model could be used to analyze macroscopic responses of nicotinic acetylcholine receptors, although the analysis presented was limited. Subsequently, several studies have used the model in studies of, for example, cyclic nucleotide-gated channels (Goulding et al., 1994) or TRPV1 receptors (Li and Zheng, 2024).

It has been most extensively studied for the muscle-type nicotinic receptor and the GABA_A receptor. Analysis of nicotinic receptor activation has focused on single-channel currents, and it was found that the individual rate constants can be well described by an MWC model (Auerbach, 2012). The study of GABA_A receptors was initiated by an analysis of the activation of a series of gain-of-function mutations and demonstrated a critical aspect of the model—that receptors could be active in the absence of an agonist (Chang and Weiss, 1999). Subsequently, S.A. Forman and our own laboratory have extensively analyzed interactions among drugs acting on the receptor. The initial report used interactions of the allosteric agonist etomidate with the orthosteric agonist (GABA) to quantitatively test the applicability of the MWC model (Rüsch et al., 2004). This work emphasized the observation that in the MWC model interactions among ligands can be analyzed using parameters obtained from single-compound experiments because the interactions are mediated through changes in the distribution of the receptor among states rather than specific interactions between agonists (for example on affinities or efficacies). The Forman group has continued work, extending to additional drugs and analysis of mutations (Ruesch et al., 2012; Szabo et al., 2019). The Akk laboratory has extended the work to multiple drug interactions and an increase in the number of states (Cao et al., 2018; Germann et al., 2019a).

It is clear that a simple two-state model cannot capture all the information from single-channel studies of multiple open states based on duration or conductance and multiple closed states of different mean durations. Indeed, early studies of nicotinic receptors in Torpedo electroplax (Sheridan and Lester, 1977) demonstrated that activation kinetics did not conform to the predictions for the MWC model (Colquhoun and Hawkes, 1977). More recent work has also found that some interactions among drugs are not fully consistent with the restrictions of the MWC model that all interactions occur through alterations of the distribution of the population of receptors between the R and A states (Szabo et al., 2019) and that the stabilization energy provided by a drug may not be constant for all levels of basal activity (Goldschen-Ohm et al., 2014; Pierce et al., 2024).

However, the model provides an accessible framework for analysis of receptor activation and, in particular, drug interactions. The basic utility and validity of the approach have been supported in studies, especially of GABA_A receptors, and predictions of the model (for instance the present manuscript) have led to some novel insights.

The data underlying Figs. 2 and 3 are available in the published article.

Eduardo Ríos served as editor.

We thank Professor Andrew Plested for helpful comments during the review of this manuscript.

This study was supported by the National Institutes of Health National Institute of General Medical Sciences grant R35GM140947 and funds from the Taylor Family Institute for Innovative Psychiatric Research.

Author contributions: A.L. Germann: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Validation, Visualization, Writing - review & editing, S.R. Pierce: Data curation, Formal analysis, Investigation, J.H. Steinbach: Conceptualization, Formal analysis, Methodology, Writing - original draft, Writing - review & editing, G. Akk: Conceptualization, Data curation, Formal analysis, Funding acquisition, Methodology, Project administration, Supervision, Visualization, Writing - original draft.