Rate-equilibrium Free Energy Relationship Research Articles

Timing of CFTR Pore Opening and Structure of Its Transition State

In CFTR, the chloride ion channel mutated in cystic fibrosis (CF) patients, pore opening is coupled to ATP-binding-induced dimerization of two cytosolic nucleotide binding domains (NBDs) and closure to dimer disruption following ATP hydrolysis. CFTR opening rate, unusually slow because of its high-energy transition state, is further slowed by CF mutation ΔF508. Here, we exploit equilibrium gating of hydrolysis-deficient CFTR mutant D1370N and apply rate-equilibrium free-energy relationship analysis to estimate relative timing of opening movements in distinct protein regions. We find clear directionality of motion along the longitudinal protein axis and identify an opening transition-state structure with the NBD dimer formed but the pore still closed. Thus, strain at the NBD/pore-domain interface, the ΔF508 mutation locus, underlies the energetic barrierfor opening. Our findings suggest a therapeutic opportunity to stabilize this transition-state structurepharmacologically in ΔF508-CFTR to correct its opening defect, an essential step toward restoring CFTR function.

Energetic Contributions to Channel Gating of Residues in the Muscle Nicotinic Receptor β1 Subunit

In the pentameric ligand-gated ion channel family, transmitter binds in the extracellular domain and conformational changes result in channel opening in the transmembrane domain. In the muscle nicotinic receptor and other heteromeric members of the family one subunit does not contribute to the canonical agonist binding site for transmitter. A fundamental question is whether conformational changes occur in this subunit. We used records of single channel activity and rate-equilibrium free energy relationships to examine the β1 (non-ACh-binding) subunit of the muscle nicotinic receptor. Mutations to residues in the extracellular domain have minimal effects on the gating equilibrium constant. Positions in the channel lining (M2 transmembrane) domain contribute strongly and relatively late during gating. Positions thought to be important in other subunits in coupling the transmitter-binding to the channel domains have minimal effects on gating. We conclude that the conformational changes involved in channel gating propagate from the binding-site to the channel in the ACh-binding subunits and subsequently spread to the non-binding subunit.

Hydrophobic interactions between the voltage sensor and pore mediate inactivation in Kv11.1 channels

Kv11.1 channels are critical for the maintenance of a normal heart rhythm. The flow of potassium ions through these channels is controlled by two voltage-regulated gates, termed “activation” and “inactivation,” located at opposite ends of the pore. Crucially in Kv11.1 channels, inactivation gating occurs much more rapidly, and over a distinct range of voltages, compared with activation gating. Although it is clear that the fourth transmembrane segments (S4), within each subunit of the tetrameric channel, are important for controlling the opening and closing of the activation gate, their role during inactivation gating is much less clear. Here, we use rate equilibrium free energy relationship (REFER) analysis to probe the contribution of the S4 “voltage-sensor” helix during inactivation of Kv11.1 channels. Contrary to the important role that charged residues play during activation gating, it is the hydrophobic residues (Leu529, Leu530, Leu532, and Val535) that are the key molecular determinants of inactivation gating. Within the context of an interconnected multi-domain model of Kv11.1 inactivation gating, our REFER analysis indicates that the S4 helix and the S4–S5 linker undergo a conformational rearrangement shortly after that of the S5 helix and S5P linker, but before the S6 helix. Combining REFER analysis with double mutant cycle analysis, we provide evidence for a hydrophobic interaction between residues on the S4 and S5 helices. Based on a Kv11.1 channel homology model, we propose that this hydrophobic interaction forms the basis of an intersubunit coupling between the voltage sensor and pore domain that is an important mediator of inactivation gating.

Pore Helices Play a Dynamic Role as Integrators of Domain Motion during Kv11.1 Channel Inactivation Gating

Proteins that form ion-selective pores in the membrane of cells are integral to many rapid signaling processes, including regulating the rhythm of the heartbeat. In potassium channels, the selectivity filter is critical for both endowing an exquisite selectivity for potassium ions, as well as for controlling the flow of ions through the pore. Subtle rearrangements in the complex hydrogen-bond network that link the selectivity filter to the surrounding pore helices differentiate conducting (open) from nonconducting (inactivated) conformations of the channel. Recent studies suggest that beyond the selectivity filter, inactivation involves widespread rearrangements of the channel protein. Here, we use rate equilibrium free energy relationship analysis to probe the structural changes that occur during selectivity filter gating in Kv11.1 channels, at near atomic resolution. We show that the pore helix plays a crucial dynamic role as a bidirectional interface during selectivity filter gating. We also define the molecular bases of the energetic coupling between the pore helix and outer helix of the pore domain that occurs early in the transition from open to inactivated states, as well as the coupling between the pore helix and inner helix late in the transition. Our data demonstrate that the pore helices are more than just static structural elements supporting the integrity of the selectivity filter; instead they play a crucial dynamic role during selectivity filter gating.

Biophysical Coupling Mechanisms in NMDA Receptor Gating

NMDA receptors (NMDARs) are heterotetrameric glutamate-gated ion channels, typically composed of 2 GluN1 and 2 GluN2 subunits. The central function of NMDARs is gating - the process of converting agonist binding energy into pore opening by repositioning M3, the major pore-lining transmembrane helix. For glutamate receptors, it is largely assumed, though lacking direct evidence, that gating occurs through tension: ligand-binding domain (LBD) closure around an agonist generates tension in the linkers connecting the LBD to M3 (the M3-S2 linker), shifting M3 away from the central pore axis. To directly test this model, we altered the tension of the GluN1 and GluN2A M3-S2 linkers through residue insertions (to decrease tension) or deletions (to increase tension). Based on single-channel analyses, we find that these manipulations specifically affect pore opening (as opposed to ligand binding) and that they alter gating more dramatically in GluN2A than in GluN1. All-atom molecular dynamics simulations on a modeled GluN1/GluN2A receptor showed that this subunit-specific difference may arise, in part, from the disparate extensions and orientations of the GluN1 vs GluN2A M3-S2 linkers. Our functional data also suggests that the GluN1 M3-S2 linker gates primarily through tension. using rate equilibrium free energy relationship (REFER) analysis and length-tension analysis, we find that for the GluN1 M3-S2 linker, tension arises primarily during the C1-O1 transition (assuming a linear kinetic scheme) with a spring constant of ∼7.2 pN/nm, agreeing well with other biological springs. In contrast, our functional data suggests that mechanisms other than tension mainly mediate the role of the GluN2A M3-S2 linker in gating. Alternative mechanisms may include twisting and changes in electrostatic interactions.

Mapping Heat Exchange in an Allosteric Protein

Nicotinic acetylcholine receptors (AChRs) are synaptic ion channels that spontaneously isomerize (i.e., gate) between resting and active conformations. We used single-molecule electrophysiology to measure the temperature dependencies of mouse neuromuscular AChR gating rate and equilibrium constants. From these we estimated free energy, enthalpy, and entropy changes caused by mutations of amino acids located between the transmitter binding sites and the middle of the membrane domain. The range of equilibrium enthalpy change (13.4 kcal/mol) was larger than for free energy change (5.5 kcal/mol at 25°C). For two residues, the slope of the rate-equilibrium free energy relationship (Φ) was approximately constant with temperature. Mutant cycle analysis showed that both free energies and enthalpies are additive for energetically independent mutations. We hypothesize that changes in energy associated with changes in structure mainly occur close to the site of the mutation, and, hence, that it is possible to make a residue-by-residue map of heat exchange in the AChR gating isomerization. The structural correlates of enthalpy changes are discussed for 12 different mutations in the protein.

Molecular Thermography: Mapping Enthalpy and Entropy Changes in Acetylcholine Receptor Gating

We studied enthalpy and entropy changes in AChR gating from the temperature dependence of the gating rate and equilibrium constants estimated from the single-channel currents from wild type and mutant AChRs activated by different agonists (acetylcholine , choline, carbachol or none) (5-35oC, HEK cells, cell attached , −110 mV , mouse α[[Unsupported Character - ]]βδe). The mutations were between the binding sites and the gate, in the α-subunit [W149, G153, A96, Y127, E45, P272, P265 and C418] or the δ-subunit [V269 and L265]. The forward and reverse isomerization rate constants were temperature-dependent (activation enthalpy ∼21 kcal/mol; Q10∼3.8). For αG153S, αW149T, αP272G and three different agonists the free energy change (ααGo, at 25oC) and the enthalpy change (ααHo) were approximately equal. These values differed by >2 kcal/mol for αY127C/E, αW149F/N/R and δV269C/A. The slope of the rate-equilibrium free energy relationship (Φ) for residues αW149 and δV269 was constant with temperature, and the slope for the enthalpy component of this relationship was equal to Φ. The results suggest that the map of gating enthalpy changes consequent to mutation are different than that for free energy changes. We speculate that the changes in enthalpy are mainly local to the site of perturbation and that we can make a residue-by-residue map of heat absorption and emission in AChR gating (a ‘molecular thermogram’).

Temperature Dependence of Acetylcholine Receptor Channels Activated by Different Agonists

The temperature dependence of agonist binding and channel gating were measured for wild-type adult neuromuscular acetylcholine receptors activated by acetylcholine, carbamylcholine, or choline. With acetylcholine, temperature changed the gating rate constants (Q 10 ≈ 3.2) but had almost no effect on the equilibrium constant. The enthalpy change associated with gating was agonist-dependent, but for all three ligands it was approximately equal to the corresponding free-energy change. The equilibrium dissociation constant of the resting conformation (K d), the slope of the rate-equilibrium free-energy relationship (Φ), and the acetylcholine association and dissociation rate constants were approximately temperature-independent. In the mutant αG153S, the choline association and dissociation rate constants were temperature-dependent (Q 10 ≈ 7.4) but K d was not. By combining two independent mutations, we were able to compensate for the catalytic effect of temperature on the decay time constant of a synaptic current. At mouse body temperature, the channel-opening and -closing rate constants are ∼400 and 16 ms −1. We hypothesize that the agonist dependence of the gating enthalpy change is associated with differences in ligand binding, specifically to the open-channel conformation of the protein.

Application of rate-equilibrium free energy relationship analysis to nonequilibrium ion channel gating mechanisms

Rate-equilibrium free energy relationship (REFER) analysis provides information on transition-state structures and has been applied to reveal the temporal sequence in which the different regions of an ion channel protein move during a closed–open conformational transition. To date, the theory used to interpret REFER relationships has been developed only for equilibrium mechanisms. Gating of most ion channels is an equilibrium process, but recently several ion channels have been identified to have retained nonequilibrium traits in their gating cycles, inherited from transporter-like ancestors. So far it has not been examined to what extent REFER analysis is applicable to such systems. By deriving the REFER relationships for a simple nonequilibrium mechanism, this paper addresses whether an equilibrium mechanism can be distinguished from a nonequilibrium one by the characteristics of their REFER plots, and whether information on the transition-state structures can be obtained from REFER plots for gating mechanisms that are known to be nonequilibrium cycles. The results show that REFER plots do not carry information on the equilibrium nature of the underlying gating mechanism. Both equilibrium and nonequilibrium mechanisms can result in linear or nonlinear REFER plots, and complementarity of REFER slopes for opening and closing transitions is a trivial feature true for any mechanism. Additionally, REFER analysis provides limited information about the transition-state structures for gating schemes that are known to be nonequilibrium cycles.

Relationship between nucleotide binding and ion channel gating in cystic fibrosis transmembrane conductance regulator.

We have employed rate-equilibrium free energy relationship (REFER) analysis to characterize the dynamic events involved in the allosteric regulation of cystic fibrosis transmembrane conductance regulator (CFTR) function. A wide range of different hydrolysable and poorly hydrolysable nucleoside triphosphates were used to elucidate the role of ATP hydrolysis in CFTR function. The linearity of the REFER plots and Phi values near unity for all ligands tested implies that CFTR channel gating is a reversible thermally driven process with all structural reorganization in the binding site(s) completed prior to channel opening. This is consistent with the requirement for nucleotide binding for channel opening. However, the channel structural transition from the open to the closed state occurs independently of any events in the binding sites. Similar results were obtained on substitution of amino acids at coupling joints between both nucleotide binding domains (NBD) and cytoplasmic loops (CL) in opposite halves of the protein, indicating that any structural reorganization there also had occurred in the channel closed state. The fact that fractional Phi values were not observed in either of these distant sites suggests that there may not be a deterministic 'lever-arm' mechanism acting between nucleotide binding sites and the channel gate. These findings favour a stochastic coupling between binding and gating in which all structural transitions are thermally driven processes. We speculate that increase of channel open state probability is due to reduction of the number of the closed state configurations available after physical interaction between ligand bound NBDs and the channel.

Toward CFTR Structural Dynamics During Ion Channel Gating

While CFTR catalyses the irreversible hydrolysis of its physiological ligand (ATP), there is disagreement about whether CFTR channel gating is a thermally driven process (Aleksandrov & Riordan 1998 FEBS Lett. 431:97, Csanady et al., 2006 JGP128:509). Knowing the answer to this question is of fundamental importance to understand problems leading to CFTR functional failure in cystic fibrosis, an inherited disease of high morbidity and mortality. Although CFTR single channel recordings do not provide a direct measure of enzymatic activity, they do provide an opportunity to establish the relationship between ATP binding and hydrolysis at the nucleotide binding domains (NBDs) and channel gating. The use of Rate Equilibrium Free Energy Relationship (REFER) is a possible way to probe structural dynamics in an ion channel gating (Auerbach, 2005 PNAS 102:1408). We use this conceptual framework to reveal the role of ATP hydrolysis in CFTR ion channel function. The REFER graphs for the set of nucleotide ligands have slope Φ ≈1 for the openings and Φ ≈ 0 for the closings. This means that the nucleotide triphosphate interaction with the binding sites is required for channel opening. In contrast, the ion channel transition from the open to the closed state occurs independently of any events in the binding sites. This could not be interpreted by conventional mechanical models of molecular devices where the mobile species are levered from the initial to the final position by force generated in the binding sites. However this behavior fits very well to Brownian ratchet models where internal thermal diffusion is a key element of the process, and specific interactions induce no mechanical movements directly but instead are used to trap favorable fluctuations.

The Unliganded Gating Mechanism Of Nicotinic Acetylcholine Receptors

The nicotinic acetylcholine receptor (AChR) switches between C (low agonist affinity and low conductance) and O (high agonist affinity and high conductance) conformations (‘gating’). The probability of channel opening is very low in the absence of agonist, but when agonists are present at the two transmitter-binding-sites opening increases rapidly (∼20μs), transiently to a high probability (∼0.95). We observe that ‘gain-of-function’ mutants that increase the diliganded gating equilibrium constant (without affecting agonist binding to C) also increase the frequency of spontaneous openings. Unliganded openings occur in clusters in AChRs having several of such mutations. We analyzed the intra-cluster interval durations to estimate that the unliganded gating equilibrium constant is ∼1.15 × 10-7 (mouse, α2βδε, -100 mV). The agonist affinity ratios (C vs. O) for acetylcholine, carbamylcholine, tetramethylammonium and choline are ∼15,600, ∼6700, ∼6700 and ∼600. The monoliganded (with ACh) gating equilibrium constant is ∼1.7 × 10-3. Acetylcholine provides only ∼0.9 kBT more binding energy per site than tetramethylammonium, but ∼3.1 kBT more than choline. Mutations of binding site residue αW149 increase unliganded gating, and the mutation αW149F reduces the ACh affinity of C only by 13-fold, but of O by 190-fold. Rate-equilibrium free energy relationships for different regions of the protein show similar slopes (Φ-values) for un- vs. diliganded gating. The mechanisms of the gating conformational change and of desensitization are similar with and without ligands at the transmitter binding sites.

Unliganded gating of acetylcholine receptor channels

We estimated the unliganded opening and closing rate constants of neuromuscular acetylcholine receptor-channels (AChRs) having mutations that increased the gating equilibrium constant. For some mutant combinations, spontaneous openings occurred in clusters. For 25 different constructs, the unliganded gating equilibrium constant (E(0)) was correlated with the product of the predicted fold-increase in the diliganded gating equilibrium constant caused by each mutation alone. We estimate that (i) E(0) for mouse, wild-type alpha(2)beta delta epsilon AChRs is approximately 1.15 x 10(-7); (ii) unliganded AChRs open for approximately 80 micros, once every approximately 15 min; (iii) the affinity for ACh of the O(pen) conformation is approximately 10 nM, or approximately 15,600 times greater than for the C(losed) conformation; (iv) the ACh-monoliganded gating equilibrium constant is approximately 1.7 x 10(-3); (v) the C-->O isomerization reduces substantially ACh dissociation, but only slightly increases association; and (vi) ACh provides only approximately 0.9 k(B)T more binding energy per site than carbamylcholine but approximately 3.1 k(B)T more than choline, mainly because of a low O conformation affinity. Most mutations of binding site residue alphaW149 increase E(0). We estimate that the mutation alphaW149F reduces the ACh affinity of C only by 13-fold, but of O by 190-fold. Rate-equilibrium free-energy relationships for different regions of the protein show similar slopes (Phi values) for un- vs. diliganded gating, which suggests that the conformational pathway of the gating structural change is fundamentally the same with and without agonists. Agonist binding is a perturbation that (like most mutations) changes the energy, but not the mechanism, of the gating conformational change.

Protein folding transition states probed by loop extension

We propose a new way to characterize protein folding transition states by (1) insertion of one or more residues into an unstructured protein loop, (2) measurement of the effect on protein folding kinetics and thermodynamics, and (3) analysis of the results in terms of a rate-equilibrium free energy relationship, alpha(Loop). alpha(Loop) reports on the fraction of molecules that form the perturbed loop in the transition state. Interpretation of the changes in equilibrium free energy using standard polymer theory can help detect residual structure in the unfolded state. We illustrate our approach with data for the model proteins CI2 and the alpha spectrin SH3 domain.

Ionic-Strength-Dependent Effects in Protein Folding: Analysis of Rate Equilibrium Free-Energy Relationships and Their Interpretation

The traditional approach to studying protein folding involves applying a perturbation, usually denaturant or mutation, and determining the effect upon the free energy of folding, DeltaG0, and the activation free energy, DeltaG(not equal). Data collected as a function of the perturbation can be used to construct rate equilibrium free-energy relationships, which report on the development of interactions in the transition state for folding. We examine the use of the ionic-strength-dependent rate equilibrium free-energy relationship in protein folding using the N-terminal domain of L9, a small alpha-beta protein, as a model system. Folding is two-state for the range of ionic strength examined, 0.045-1.52 M. The plot of DeltaG(not equal) versus DeltaG0 is linear (r2= 0.918), with a slope equal to 0.45. The relatively low value of the slope indicates that the ionic-strength-dependent interactions are modestly developed in the transition state. The slope is, however, greater than that of a plot of DeltaG(not equal) versus DeltaG0 constructed by varying pH, thus demonstrating directly that ionic-strength-dependent studies probe more than simple electrostatic interactions. Potential transition movement was probed by analysis of the denaturant, ionic strength cross-interaction parameters. The values are small but nonzero and positive, suggesting a small shift of the transition state toward the native state as the protein is destabilized, i.e., Hammond behavior. The complications that arise in the interpretation of ionic-strength-dependent rate equilibrium free-energy relationships are discussed, and it is concluded that the ionic-strength-dependent studies do not provide a reliable indicator of the role of electrostatic interactions. Complications include incomplete screening of electrostatic interactions, specific ion binding, Hofmeister effects, and the potential presence of electrostatic interactions in the denatured state ensemble.

Chimeric constructs endow the human CFTR Cl − channel with the gating behavior of murine CFTR

The cystic fibrosis transmembrane conductance regulator (CFTR) is a Cl(-) channel gated by ATP-driven nucleotide-binding domain (NBD) dimerization. Here we exploit species differences between human and murine CFTR to investigate CFTR channel gating. Using homologous recombination, we constructed human-murine CFTR (hmCFTR) chimeras with sequences from NBD1, NBD2, or the regulatory domain (RD) of human CFTR replaced by the equivalent regions of murine CFTR. The gating behavior of hmRD and human CFTR were indistinguishable, whereas hmNBD1 and hmNBD2 had subtle effects on channel gating, prolonging both burst duration and interburst interval. By contrast, hmNBD1+2, containing both NBDs of murine CFTR, reproduced the gating behavior of the subconductance state of murine CFTR, which has dramatically prolonged channel openings. The CFTR potentiator pyrophosphate (PP(i)) enhanced human, hmRD, and hmNBD1 CFTR Cl(-) currents, but not those of hmNBD2, hmNBD1+2, and murine CFTR. By analyzing the rate-equilibrium free-energy relationships of chimeric channels, we obtained snapshots of the conformation of the NBDs during ATP-driven dimerization. Our data demonstrate that the conformation of NBD1 changes before that of NBD2 during channel opening. This finding suggests that NBD dimerization does not proceed by a symmetric tweezer-like motion, but instead in an asymmetric fashion led by NBD1. We conclude that the NBDs of murine CFTR determine the unique gating behavior of its subconductance state, whereas NBD2 controls channel potentiation by PP(i).

Electrostatic Interactions in the Denatured State and in the Transition State for Protein Folding: Effects of Denatured State Interactions on the Analysis of Transition State Structure

The development of electrostatic interactions during the folding of the N-terminal domain of the ribosomal protein L9 (NTL9) is investigated by pH-dependent rate equilibrium free energy relationships. We show that Asp8, among six acidic residues, is involved in non-native, electrostatic interactions with K12 in the transition state for folding as well as in the denatured state. The perturbed native state p K a of D8 (p K a = 3.0) appears to be maintained through non-native interactions in both the transition state and the denatured state. Mutational effects on the stability of the transition state for protein (un)folding are often analyzed in respect to change in ground states. Thus, the interpretation of transition state analysis critically depends on an understanding of mutational effects on both the native and denatured state. Increasing evidence for structurally biased denatured states under physiological conditions raises concerns about possible denatured state effects on folding studies. We show that the structural interpretation of transition state analysis can be altered dramatically by denatured state effects.

Engineering Diverse Changes in β-Turn Propensities in the N-Terminal β-Hairpin of Ubiquitin Reveals Significant Effects on Stability and Kinetics but a Robust Folding Transition State

Using the N-terminal 17-residue beta-hairpin of ubiquitin as a "host" for mutational studies, we have investigated the influence of the beta-turn sequence on protein stability and folding kinetics by replacing the native G-bulged turn (TLTGK) with more flexible analogues (TG3K and TG5K) and a series of four-residue type I' beta-turn sequences, commonly found in beta-hairpins. Although a statistical analysis of type I' turns demonstrates residue preferences at specific sites, the frequency of occurrence appears to only broadly correlate with experimentally determined protein stabilities. The subsequent engineering of context-dependent non-native tertiary contacts involving turn residues is shown to produce large changes in stability. Relatively few point mutations have been described that probe secondary structure formation in ubiquitin in a manner that is independent of tertiary contacts. To this end, we have used the more rigorous rate-equilibrium free energy relationship (Leffler analysis), rather than the two-point phi value analysis, to show for a family of engineered beta-turn mutants that stability (range of approximately 20 kJ/mol) and folding kinetics (190-fold variation in refolding rate) are linearly correlated (alpha(f) = 0.74 +/- 0.08). The data are consistent with a transition state that is robust with regard to a wide range of statistically favored and disfavored beta-turn mutations and implicate a loosely assembled beta-hairpin as a key template in transition state stabilization with the beta-turn playing a central role.

Φ-Value Analysis of a Linear, Sequential Reaction Mechanism: Theory and Application to Ion Channel Gating

We derive the analytical form of a rate-equilibrium free-energy relationship (with slope Φ) for a bounded, linear chain of coupled reactions having arbitrary connecting rate constants. The results confirm previous simulation studies showing that Φ-values reflect the position of the perturbed reaction within the chain, with reactions occurring earlier in the sequence producing higher Φ-values than those occurring later in the sequence. The derivation includes an expression for the transmission coefficients of the overall reaction based on the rate constants of an arbitrary, discrete, finite Markov chain. The results indicate that experimental Φ-values can be used to calculate the relative heights of the energy barriers between intermediate states of the chain but provide no information about the energies of the wells along the reaction path. Application of the equations to the case of diliganded acetylcholine receptor channel gating suggests that the transition-state ensemble for this reaction is nearly flat. Although this mechanism accounts for many of the basic features of diliganded and unliganded acetylcholine receptor channel gating, the experimental rate-equilibrium free-energy relationships appear to be more linear than those predicted by the theory.

Plasticity of Acetylcholine Receptor Gating Motions via Rate-Energy Relationships

Like other protein conformational changes, ion channel gating requires the protein to achieve a high-energy transition-state structure. It is not known whether ion channel gating takes place on a broad energy landscape on which many alternative transition state structures are accessible, or on a narrow energy landscape where only a few transition-state structures are possible. To address this question, we measured how rate-equilibrium free energy relationships (REFERs) for di-liganded and unliganded acetylcholine receptor gating vary as a function of the gating equilibrium constant. A large slope for the REFER plot indicates an openlike transition state, whereas a small slope indicates a closedlike transition state. Due to this relationship between REFERs and transition-state structure, the sensitivity of the REFER slope to mutation-induced energetic perturbations allows estimation of the breadth of the energy landscape underlying gating. The relatively large sensitivity of di-liganded REFER slopes to energetic perturbations suggests that the motions underlying di-liganded gating take place on a broad, shallow energy landscape where many alternative transition-state structures are accessible.