Disentangling contact and ensemble epistasis in a riboswitch

Abstract

Mutations introduced into macromolecules often exhibit epistasis, where the effect of one mutation alters the effect of another. Knowing the mechanisms that lead to epistasis is important for understanding how macromolecules work and evolve, as well as for effective macromolecular engineering. Here, we investigate the interplay between “contact epistasis” (epistasis arising from physical interactions between mutated residues) and “ensemble epistasis” (epistasis that occurs when a mutation redistributes the conformational ensemble of a macromolecule, thus changing the effect of the second mutation). We argue that the two mechanisms can be distinguished in allosteric macromolecules by measuring epistasis at differing allosteric effector concentrations. Contact epistasis manifests as nonadditivity in the microscopic equilibrium constants describing the conformational ensemble. This epistatic effect is independent of allosteric effector concentration. Ensemble epistasis manifests as nonadditivity in thermodynamic observables—such as ligand binding—that are determined by the distribution of ensemble conformations. This epistatic effect strongly depends on allosteric effector concentration. Using this framework, we experimentally investigated the origins of epistasis in three pairwise mutant cycles introduced into the adenine riboswitch aptamer domain by measuring ligand binding as a function of allosteric effector concentration. We found evidence for both contact and ensemble epistasis in all cycles. Furthermore, we found that the two mechanisms of epistasis could interact with each other. For example, in one mutant cycle we observed 6 kcal/mol of contact epistasis in a microscopic equilibrium constant. In that same cycle, the maximum epistasis in ligand binding was only 1.5 kcal/mol: shifts in the ensemble masked the contribution of contact epistasis. Finally, our work yields simple heuristics for identifying contact and ensemble epistasis based on measurements of a biochemical observable as a function of allosteric effector concentration.

Significance

Mutations to protein or RNA molecules often have different effects when introduced individually versus together. To understand and engineer biological macromolecules, we must identify the mechanistic origins of this phenomenon. Here, we measured the interplay between direct, physical interactions between mutations (“contact epistasis”) and indirect interactions mediated by conformational ensembles (“ensemble epistasis”). We introduced pairs of mutations into an RNA molecule that transitions between several different conformations. We found epistasis arising from both contacts and the ensemble, and that the two mechanisms could synergize with one another. Our work reveals that one must consider the effects of mutations on multiple conformations to understand epistasis and suggests a few rules of thumb for disentangling contact and ensemble epistasis in other macromolecules.

Introduction

Mutations often exhibit epistasis, where the effect of one mutation alters the effect of another (1,2). Intramolecular epistasis—interactions between mutations within the same macromolecule—is useful for dissecting molecular function (3,4), can shape molecular evolution (2,5,6,7), and must be properly captured for macromolecular engineering (8,9,10). Knowledge of the mechanisms of epistasis is thus important for both understanding and designing macromolecules.

Broadly, there are two sources of epistasis in macromolecules. The most intuitive is mediated by structural contacts between the sites. Such contacts could be simple one-to-one interactions (e.g., ion pairs, basepairs, hydrogen bonds, etc.) (7,11,12) or larger networks of interactions (e.g., pathways of residues connecting distant sites) (4,13). The second mechanism arises from the ensemble of conformations populated by the macromolecule (14,15,16). This epistasis arises when a macromolecule populates several conformations that respond differently to the same mutation. A mutation at one site redistributes the relative populations of the conformations, thus altering the effect of a second mutation. The relative contributions of contact and ensemble mechanisms to epistasis across macromolecules are unknown.

We recently found that environment dependence can reveal ensemble epistasis (14). If one changes the environment such that the populations of the conformational ensemble are altered, one can change the contribution of ensemble epistasis. Many environmental factors—including pH, temperature, ionic strength, etc.—can, in principle, tune ensemble epistasis. For this study, we used the concentration of allosteric effector as our environmental perturbation. This is because the effector redistributes the ensemble of an allosteric macromolecule—and thus the measurable output of the system—in a predictable fashion. We recently demonstrated ensemble epistasis experimentally using the lac repressor, which exhibited epistasis in DNA binding as a function of its allosteric effector IPTG (17). Intriguingly, we also found that redistribution of the ensemble could not fully explain the epistasis we observed—suggesting that other mechanisms besides ensemble epistasis were at play in the protein.

Our previous work raised several questions. How can we distinguish between epistasis arising from contacts, the ensemble, or both simultaneously? What are the relative magnitudes of these two forms of epistasis? How do they influence one another? Is ensemble epistasis within the purview of proteins alone, or does it arise in other classes of macromolecules?

To address these questions, we investigated the contributions of contact and ensemble epistasis between pairs of mutations in the well-characterized adenine riboswitch aptamer domain. This RNA molecule has a three-state conformational ensemble that allows it to bind to adenine in a magnesium-dependent fashion (18,19,20,21,22,23,24,25). We measured magnesium-dependent epistasis between mutations and expected to have varying contributions of contact and ensemble epistasis, finding evidence for both mechanisms in all mutant cycles. Our work reveals that contact and ensemble epistasis can have comparable magnitudes, and that the two sources of epistasis can interact in highly nonlinear ways. Furthermore, our results suggest that one can use environment dependence as a useful heuristic for disentangling ensemble and contact epistasis in macromolecules.

Materials and methods

RNA sample preparation

We synthesized sequence variants of the Vibrio vulnificus adenine riboswitch aptamer domain by in vitro transcription of the corresponding DNA oligonucleotides (HiScribe T7 High Yield RNA Synthesis Kit, New England Biolabs, Ipswich, Massachusetts; oligonucleotides ordered from Eurofins, Luxembourg). The wild-type RNA sequence is shown with mutated positions in boldface: 3′GGGAAGAUAUAAUCCUAAUGAUAUGGUUUGGGAGUUUCUACCAAGAGCCUUAAACUCUUGAUUAUCUUCCC. The riboswitch product was purified by running the in vitro transcription reaction on a 12% denaturing polyacrylamide gel, identifying the 71-nucleotide product by UV shadowing, and extracting the product from the gel by electroelution. The purified RNA product was concentrated by ethanol precipitation, desalted, quantified by NanoDrop absorbance spectroscopy, and resuspended in 50 mM Tris-HCl before being stored at −80°C until further use.

Measurement of 2-aminopurine binding

We measured binding of 2-aminopurine (2AP), a fluorescent analog of adenine, to the riboswitch aptamer domain using a previously published protocol (19,25). Riboswitch aliquots were thermally denatured at 90°C for 1 min before refolding on ice for approximately 10 min. Increasing concentrations of RNA (0.125–8 μM) were combined with 50 nM 2AP, 100 mM KCl, and 0.1–100 mM MgCl₂ in 96-well black-bottomed plates. The assay plate was shaken in a Molecular Devices (San Jose, California) SpectraMax i3 fluorescence plate reader at 37°C for 5 min to allow the reaction mixtures to reach equilibrium. 2AP exhibits high fluorescence in water and quenched fluorescence when bound to the riboswitch. We excited each sample at 310 nm and measured emission between 335 and 450 nm. We integrated each emission spectrum and converted the area to fraction of 2AP bound using a negative control (0 nM RNA, 0 nM 2AP) and a positive control (0 nM RNA, 50 nM 2AP). Because 2AP fluorescence was negligible in samples where the fluorophore was completely bound by RNA, we used a negative control lacking 2AP entirely to universally mimic this fully quenched state. We averaged two technical replicates for each condition per experiment and collected at least three biological replicates for each experimental condition.

RNA ensemble modeling

As described in the results, we used a previously validated thermodynamic model of the riboswitch ensemble to extract information about the populated conformations (23). This model describes the riboswitch with two apo conformations—extended (E) and docked (D)—each of which can form a complex with 2AP (E·A and D·A). We tested three variants of the model: a two-state model with conformations E and D·A, a three-state model with conformations E, D, and D·A, and a four-state model with conformations E, E·A, D, and D·A. We also tried subvariants of these models in which we fixed the values of the different equilibrium constants. As described in the results, we found that the three-state model with K_dock set to 1.0 was able to reproduce our binding data without overfitting. A mathematical description of the three-state model follows; the two-state and four-state models are given in the supporting material (Fig. S1).

The three-state model is shown schematically in Fig. 1 A. The concentrations of the E, D, and D·A conformations are given by:

$$\left[E\right]=\frac{{\left[R\right]}_T-{\left[A\right]}_T+\left[A\right]}{1+K_{dock}{\left(K_{MG}{\left[M\right]}_T\right)}^{n_{MG}}}$$ (1)

$$\left[D\right]=\left[E\right]K_{dock}{\left(K_{MG}{\left[M\right]}_T\right)}^{n_{MG}}$$ (2)

$$\left[D·A\right]=\left[D\right]K_{2AP}\left[A\right]\text{,}$$ (3)

where K_dock is the equilibrium between E and D in the absence of Mg²⁺ or 2AP, K_2AP is the affinity of 2AP for the D conformation, K_MG is the relative affinity of the D and E conformations for Mg²⁺, n_MG is the difference in the number of Mg²⁺ ions bound by the D and E conformations, [R]_T is the total concentration of RNA, [M]_T is the total concentration of Mg²⁺, [A]_T is the total concentration of 2AP, and [A] is the concentration of free 2AP. Our experimental observable, the fraction of 2AP bound to RNA, is given by:

$$\theta =\frac{\left[D·A\right]}{{\left[A\right]}_T}=\frac{\left({\left[R\right]}_T-{\left[A\right]}_T-\left[A\right]\right)K_{dock}{\left(K_{MG}\left[M\right]\right)}^{n_{MG}}{K}_{2AP}\left[A\right]}{{\left[A\right]}_T\left(1+K_{dock}{\left(K_{MG}\left[M\right]\right)}^{n_{MG}}\right)}\text{.}$$ (4)

Figure 1.

Differentiating contact and ensemble epistasis in the adenine riboswitch. (A) A thermodynamic model of the riboswitch ensemble. The molecule can be extended (E), docked (D), or docked and bound to 2AP (D·A). The equilibria are defined by three parameters: K_MG, n_MG, and K_2AP. The equilibria are tuned by the concentration of Mg²⁺ ions (green circles) and 2AP (yellow star). (B) Epistasis arising from a contact between two sites. RNA molecules within braces represent an ensemble for a specific genotype (ab, Ab, aB, or AB). The equilibrium between the E and D forms is shown by arrows within the braces and measured by ΔG°′. The A and B mutations are indicated by bold gray arrows between ensembles. Bases at these sites can either be indented (black) or protruding (gray). Mutation A flips the left base from indented to protruding; mutation B flips the right base from protruding to indented. Mismatches are indicated with a red “×”. (C) Epistasis arising from the ensemble. The cartoon graphs show concentrations of E, D, and D·A for wild-type (top); ΔG_obs for four genotypes (xy, Xy, xY, and XY) taken from a hypothetical mutant cycle (middle); and epistasis in ΔG_obs calculated from the middle plot (bottom). These series are plotted as a function of Mg²⁺ concentration. The mathematical expressions for contact and ensemble epistasis are displayed at the bottom of the figure. To see this figure in color, go online.

This model has four explicit parameters: K_dock, K_MG, n_MG, and K_2AP. It also has two implicit parameters: [M] (the concentration of free Mg²⁺) and [A] (the concentration of free 2AP).

We implemented this model as a function that returns the concentrations of all relevant species ([A], [M], [E], [D], and [D·A]) given the values of the thermodynamic parameters (K_dock, K_2AP, K_MG, and n_MG) and total species concentrations ([R]_T, [M]_T, and [A]_T). This function encodes the thermodynamic relationships above (Eqs. 1, 2, 3, and 4) and enforces mass-balance relationships (e.g., [A]_T = [A] + [D·A]). Internally, this function guesses values for [A] and then iterates to self-consistency between the thermodynamic and mass-balance relationships. We derived the model allowing for any of the binding equilibria to be in a stoichiometric or nonstoichiometric regime. The full derivation is given in Appendix S1.

To estimate the values of the thermodynamic parameters consistent with our binding data, we used a Bayesian Markov chain Monte Carlo (MCMC) strategy to sample over parameter combinations. We analyzed all experimental conditions simultaneously for each genotype, globally estimating the thermodynamic parameters. We used the following likelihood function:

$$\mathcal{L}=-\frac{1}{2}\sum _i\left[\frac{{\left({\theta \left(\overrightarrow{p},\overrightarrow{c}\right)}_{i,calc}-{\theta }_{i,obs}\right)}^2}{{\sigma }_i^2}+\text{ln}\left(2\pi {\sigma }_i^2\right)\right]\text{,}$$ (5)

where θ_i,obs and σ_i are the mean and standard deviation of the measured 2AP fractional saturation at condition i; θ_i,calc is the calculated 2AP fractional saturation under these conditions given a vector of parameters ($\overrightarrow{p}$) and total concentrations ($\overrightarrow{c})$. If our experimental σ_i was <0.05, we set its value to 0.05, thus enforcing a minimum uncertainty on the mean of any observed data point. We performed MCMC sampling on the natural logs of all equilibrium constants. We generated 15,000 MCMC samples from 100 MCMC walkers for each genotype, discarding the first 1000 as burn in. We checked for convergence by comparing results from multiple independent runs. For the final three-state, three-parameter model we used in our analysis, we generated a million MCMC samples from six different sampling runs. We used the emcee 3.1.0 (26) and the “likelihood” Python libraries (https://github.com/harmslab/likelihood) for these calculations. We implemented the riboswitch binding models in Python 3.9 extended with numpy 1.21.1 (27), pandas 1.3.1 (28), and scipy 1.6.2 (29).

Results

Selecting the adenine riboswitch as a model system

We set out to measure the relative contributions of contact and ensemble epistasis using mutant cycles introduced into the adenine riboswitch aptamer domain, a 71-nucleotide RNA molecule composed of three helices surrounding an adenine binding pocket (30) (Figs. 1 A and 2 A). We selected this RNA molecule as our model system because it has a relatively simple, well-defined conformational ensemble consisting of two conformations: an extended form (E) and a compact, docked-loop form (D) (Fig. 1 A). Relative to the E conformation, the D conformation has extensive basepairs, hydrogen bonds, and hydrophobic packing between the two helical arms of the riboswitch. This forms an adenine binding pocket unique to the D conformation; therefore, the D conformation has much higher affinity for adenine than the E conformation. Mg²⁺ ions are key allosteric effectors for the riboswitch ensemble because they nonspecifically neutralize repulsion between backbone phosphate groups to promote formation of the compact D conformation, thus promoting adenine binding (23,31,32).

Figure 2.

2AP binding behavior can be captured by a thermodynamic model of the riboswitch ensemble. (A) Crystal structure of the adenine riboswitch aptamer domain (PDB: 5SWE) (35). The overall architecture is shown as a purple cartoon. Bases mutated in this study are indicated with colored spheres (labeled on structure). The bound adenine base is shown as yellow spheres; probable Mg²⁺ ion locations are shown as green spheres, taken from PDB: 1Y26 (36). (B) Each panel shows 2AP binding for the genotype indicated. Columns are (left to right) wild-type, a35u, g38c, and c50a. Rows are without (top) and with (bottom) the c60g mutation. The arrows between panels define three mutant cycles: a35u/c60g, g38c/c60g, and c50a/c60g. Each panel shows the fraction of 2AP bound, measured by relative 2AP fluorescence compared with free and fully quenched controls, as a function of riboswitch concentration. The green shading indicates the total Mg²⁺ concentration, labeled on each plot in millimolar. Each point is the average of at least three experimental replicates, with standard deviations indicated as error bars. Lines were calculated using parameters taken from 50 randomly selected MCMC samples of the three-state/three-parameter model shown in Fig. 1A. To see this figure in color, go online.

Quantifying epistasis between mutations to the riboswitch requires measuring how mutations affect a biological function when introduced individually versus together. As a proxy for the biological function of the riboswitch, we measured its binding to 2AP, a fluorescent analog of adenine. The 2AP binding energy is given by:

$$\Delta G_{obs}=-RTln\left(\frac{\left[D·A\right]}{\left(\left[D\right]+\left[E\right]\right)\left[A\right]}\right)\text{,}$$ (6)

where the conformations are those labeled in Fig. 1 A. We can calculate the concentrations of each of these species using Eqs. 1, 2, and 3, which require knowing the total concentrations of the molecular components (Mg²⁺, RNA, and 2AP), two apparent equilibrium constants (K_MG and K_2AP), and an ion uptake coefficient (n_MG). We can rewrite our 2AP binding energy in terms of these equilibrium constants by:

$$\Delta G_{obs}=-RTln\left(\frac{{\left(K_{MG}{\left[M\right]}_T\right)}^{n_{MG}}{K}_{2AP}}{1+{\left(K_{MG}{\left[M\right]}_T\right)}^{n_{MG}}}\right)\text{,}$$ (7)

where [M]_T is the total Mg²⁺ concentration. The 2AP binding energy can also be calculated directly from experimental measurements of the fraction of 2AP bound to RNA using:

$$\Delta G_{obs}=-RTln\left(\frac{\theta {\left[A\right]}_T}{{\left[R\right]}_T-\theta {\left[A\right]}_T}\right)\text{,}$$ (8)

where [A]_T is the total 2AP concentration, [R]_T is the total RNA concentration, and θ is the fraction of 2AP bound to RNA (Eq. 4). Describing ΔG_obs in terms of the system equilibrium constants (Eq. 7) or fraction of 2AP bound to RNA (Eq. 8) allows us to calculate a single thermodynamic observable that encapsulates the riboswitch’s conformational ensemble without needing to directly measure the population of each conformation spectroscopically.

Contact and ensemble epistasis will give different signals

We next asked how contact and ensemble epistasis would manifest conceptually and mathematically within the riboswitch system. Contact epistasis occurs when two mutations interact within a conformation, thus changing the energy of that conformation. Consider the predicted effects of a pair of mutations (a→A and b→B) at residues that form complementary contacts in the docked conformation (D) of the riboswitch (Fig. 1 B). When introduced individually, these mutations create noncomplementary interactions and shift the equilibrium toward the extended form (E). When introduced together, however, complementarity is restored, thus pushing the equilibrium back toward the docked conformation. Such contact epistasis will manifest in the microscopic equilibrium between these two conformations. This can be quantified using the biological standard state free energy for this equilibrium, ΔG°′ = –RTln(K_eq), yielding the following expression for contact epistasis:

$$\Delta \Delta G_{contact}=\left(\Delta G_{AB}^{{\circ }^{\prime }}-\Delta G_{aB}^{{\circ }^{\prime }}\right)-\left(\Delta G_{Ab}^{{\circ }^{\prime }}-\Delta G_{ab}^{{\circ }^{\prime }}\right)\text{.}$$ (9)

Contact epistasis is independent of the concentrations of other species in solution—such as that of Mg²⁺—because it arises from interactions within a conformation rather than a nonlinear redistribution of conformations. Thermodynamically, contact epistasis occurs in standard state free energies to reflect that the interactions are concentration independent.

Ensemble epistasis, by contrast, occurs when a mutation redistributes the relative populations within the ensemble, thus changing the effect of a second mutation on an ensemble-averaged observable. Consider another pair of mutations (x→X and y→Y) to the riboswitch (Fig. 1 C). If each mutation changes one or more of the equilibrium constants (Eq. 7), each of the xy, Xy, xY, and XY riboswitch variants will have different populations of the E, D, and D·A conformations at a given Mg²⁺ concentration. The altered equilibrium constants change the apparent 2AP affinity (ΔG_obs) for each variant in a nonlinear fashion. One can calculate ensemble epistasis as:

$$\Delta \Delta G_{ensemble}\left(\left[M{g}^{2+}\right]\right)=\left(\Delta G_{obs,XY}-\Delta G_{obs,xY}\right)-\left(\Delta G_{obs,Xy}-\Delta G_{obs,xy}\right)\text{.}$$ (10)

Ensemble epistasis depends on the Mg²⁺ concentration because the population of the ensemble—and thus ΔG_obs—is controlled by [Mg²⁺]. At low Mg²⁺ concentration, only the E conformation is populated. At high Mg²⁺ concentration, we expect ΔG_obs to plateau because 2AP becomes limiting. Additional Mg²⁺ can no longer change the relative population of D·A and thus can no longer alter ΔG_obs. At intermediate Mg²⁺ concentrations, multiple conformations are populated, thus maximizing the redistributive effects of each mutation and the magnitude of ensemble epistasis (shown schematically in Fig. 1 C).

In summary, contact epistasis appears in the standard state free energy of a specific conformation, while ensemble epistasis appears in an ensemble-averaged observable, such as a free energy of 2AP binding. Because contact epistasis could, in principle, strongly perturb the population of the conformational ensemble, we expect a complicated interplay between the two mechanisms. Without experimental knowledge of the relative magnitudes of each mechanism, however, we cannot predict what this synergy might look like for a real macromolecule.

Mutant cycle design

We next designed three mutant cycles for which we predicted different contributions of contact and ensemble epistasis. We built these cycles from four mutations that were reported previously to alter the riboswitch’s 2AP and Mg²⁺ affinity (Fig. 2 A) (19,20). All cycles involve c60g (Fig. 2 A, blue), paired with a35u, g38c, or c50a (Fig. 2 A, gray, orange, and pink). For consistency, we measured epistasis across all cycles as the difference between the effect of c60g introduced into a mutant riboswitch versus the wild-type riboswitch. In the wild-type background, c60g disrupts the complementarity of a Watson-Crick-Franklin basepair at the docked-loop interface of the D conformation, leading to a decrease in 2AP affinity (19,20).

We expected the a35u/c60g pair to exhibit minimal epistasis of either type. a35u has been reported to slightly increase ligand binding affinity by adding an extra hydrogen bond to the docked-loop interface (20). a35 is not in direct contact with c60, and the two sites are not in allosterically linked regions of the riboswitch (Fig. 2 A). We therefore anticipated little epistasis between these positions.

We expected the g38c/c60g pair to give strong contact epistasis (Fig. 1 B). g38 and c60 form a Watson-Crick-Franklin basepair at the docked-loop interface of the wild-type riboswitch’s D conformation (Fig. 2 A) (19,20). When introduced individually, these mutations disrupt this basepair and thus greatly decrease 2AP binding affinity. When both mutations are introduced together, c38/g60 form a reversed Watson-Crick-Franklin basepair, which has been shown to restore 2AP binding activity (19).

We expected the c50a/c60g pair to exhibit significant ensemble epistasis but little contact epistasis (Fig. 1 C). Positions 50 and 60 are ∼40 Å apart in the crystal structure (Fig. 2 A). c60 is at the docked-loop interface, while c50 is directly adjacent to the 2AP binding pocket. The c50a mutation has been shown previously to reduce 2AP binding affinity by destabilizing the ligand binding pocket (20). Because the docked-loop region and 2AP binding pocket are in allosteric communication, we anticipated epistasis between these two mutations; however, we predicted this would be mediated by the ensemble rather than a direct contact given the separation between the positions.

Defining the thermodynamic ensemble

With our system selected and our mutant cycles designed, we next set out to measure how the mutations perturbed the riboswitch ensemble. Previously, Leipply and Draper validated a thermodynamic model with two apo conformations of the adenine riboswitch—extended (E) and docked (D)—each of which can bind 2AP (E·A and D·A) (23). In a separate study (31), they found they could model RNA-Mg²⁺ interactions with an ion uptake coefficient that captured the number of excess Mg²⁺ ions bound by D relative to E. Put together, this results in a five-parameter thermodynamic model (Fig. S1) with the following terms: K_dock (the equilibrium between E and D in the absence of Mg²⁺), K_2AP (the affinity of the riboswitch for 2AP), K_link (the coupling between Mg²⁺/2AP binding and formation the D conformation), K_MG (the Mg²⁺ affinity of the D conformation relative to the E conformation), and n_MG (the difference in the number of Mg²⁺ ions bound by the D and E conformations).

To resolve the parameters in this model, we measured 2AP binding over 4 orders of magnitude in Mg²⁺ concentration: 0.1, 1, 10, and 100 mM. This covers the physiological range of Mg²⁺ concentrations (33). We varied RNA concentrations over ∼2 orders of magnitude, from 0.125 to 8 μM, while maintaining a constant, limiting 2AP concentration of 50 nM. The results of these experiments for all eight constructs are shown in Fig. 2 B. As expected, we observed increasing 2AP binding with increasing amounts of RNA for all variants. Furthermore, we observed a link between 2AP binding and Mg²⁺ concentration, with the midpoint of the 2AP binding curves decreasing with increasing Mg²⁺ concentration.

We used a Bayesian MCMC strategy to sample model parameters for each riboswitch variant that could reproduce our experimental observations. The four-state/five-parameter model reproduced our measured binding data well (Fig. S2). We noted, however, that the E·A conformation never accounted for more than 0.02% of the total RNA concentration for all datasets. We therefore asked if we could simplify our model by ignoring the E·A conformation, thus eliminating the model parameter K_link. This three-state/four-parameter model reproduced our experimental data as well as the original model (Fig. S2). We also noticed strong covariation between K_MG and K_dock across MCMC samples, so we tried a model in which we set K_dock to 1.0. In this formulation, K_MG is an apparent constant that accounts for both the energy of Mg²⁺ binding and the relative stability of the D versus E conformations. This three-state/three-parameter model still reproduced our experimental data well (Fig. 2 B).

Finally, we asked if we could simplify the model further by describing the system with only two conformations, E and D·A. This two-state model makes the riboswitch cooperative, meaning Mg²⁺ binding necessitates 2AP binding. This two-state model could not reproduce our measured 2AP binding data (Fig. S2). We used an Akaike information criterion (AIC) test to validate our visual inspection of the binding curves. The AIC test favors models with high likelihoods while penalizing models with excess parameters (34). Our AIC test favored the three-state/three-parameter model (Table S1). All fit parameters for the final three-state/three-parameter model are given in Table S2.

Mutant cycle no. 1: a35u/c60g

We next analyzed the epistasis within each mutant cycle, starting with a35u/c60g. These bases do not contact one another and are distant from both the Mg²⁺ and 2AP binding sites (Fig. 2 A). We therefore expected relatively weak contact and ensemble epistasis for this mutant pair.

We first looked for contact epistasis in the a35u/c60g mutant cycle. We extracted marginal distributions for the values of ΔG°′_2AP, ΔG°′_MG, and n_MG from the MCMC samples for the relevant variants (Fig. 3, A–C) and calculated epistasis in each parameter using Eq. 9. c60g had no detectable effect on ΔG°′_2AP when introduced alone, but increased ΔG°′_2AP by 0.6 [0.4, 0.8] kcal/mol in the a35u background. (In this notation, we write the median effect followed by the 95% credibility interval extracted from the marginal distribution in brackets.) If the mutations exhibited no epistasis, we would expect the a35u/g38c double mutant to have ΔG°′_2AP corresponding to the green point on Fig. 3 A. The actual change is larger, reflecting measurable epistasis in ΔG°′_2AP (0.52 [0.28, 0.73]). We saw little evidence of epistasis in ΔG°′_MG or n_MG: a35u and c60g both increased ΔG°′_MG by ∼0.5 kcal/mol (Fig. 3 B) and neither mutation had a measurable effect on n_MG (Fig. 3 C).

Figure 3.

Contact and ensemble epistasis in the a35u/c60g cycle. (A–C) Marginal distributions of ΔG°′_2AP, ΔG°′_MG, and n_MG taken from across all MCMC samples. The area of the violin plot encompasses all values seen. The central bar is the median; the top and bottom bars mark the 95% credibility interval. The black lines between series indicate the median effect of the c60g mutation in the wild-type (left) or a35u (right) backgrounds. Epistasis appears as a background-dependent difference in the slope of the line; the green lines and points indicate the expected value if the mutations do not exhibit epistasis. (D) ΔG_obs versus Mg²⁺ concentration calculated using Eq. 7 for wild-type (black), c60g (red), a35u (blue), and a35u/c60g (purple). Solid lines indicate the median of the MCMC samples; dashed lines indicate the standard deviation; areas indicate the 95% credibility region. (E) Epistasis in ΔG_obs calculated from the curves in (D). The green line indicates ΔG_obs calculated in the absence of contact epistasis (green points in (A–C)). The small arrows with cartoons indicate the Mg²⁺ concentrations where epistasis arises from the ensemble (1 mM) versus contacts (>10 mM). To see this figure in color, go online.

We next looked for evidence of ensemble epistasis in the a35u/c60g mutant cycle. We used Eq. 7 to calculate ΔG_obs versus Mg²⁺ concentration for all four variants (Fig. 3 D). Each variant gave decreasing ΔG_obs with increasing Mg²⁺ concentration, reflecting the shift in equilibrium from E to D·A (Fig. S3). ΔG_obs plateaued for all variants when the concentration of 2AP became limiting: additional Mg²⁺ could no longer change the relative concentration of D·A and thus could no longer alter ΔG_obs. We then calculated epistasis in ΔG_obs using Eq. 10 (Fig. 3 E). At low Mg²⁺ concentration, our uncertainty was high and the signal indistinguishable from zero; for Mg²⁺ above 1 mM, we observed approximately constant epistasis in ΔG_obs (0.31 [0.17, 0.43] kcal/mol).

We next investigated the interplay between contact and ensemble epistasis in the a35u/c60g mutant cycle by calculating magnesium-dependent epistasis in ΔG_obs after removing contact epistasis. We assumed that the parameter values for a35u/c60g double mutant were given by the sum of the median effects of a35u and c60g introduced alone, thereby eliminating epistasis in ΔG°′_2AP, ΔG°′_MG, and n_MG. These nonepistatic parameters are shown as the green points for a35u/c60g in Fig. 3, A–C. The epistasis in ΔG_obs we observe in the absence of contact epistasis is shown as a green line in Fig. 3 E. The curve behaves as expected for a system exhibiting ensemble epistasis. It peaks when the diversity of the ensemble is maximized and tails off to zero when the system populates a single conformation at high and low Mg²⁺ concentrations (Fig. S3). The ensemble contributes maximally to the epistasis in this mutant cycle around 1 mM Mg²⁺ (labeled arrow, Fig. 3 E). At higher Mg²⁺ concentrations, contact epistasis in 2AP binding (ΔG°′_2AP) is the source of epistasis between the mutations (labeled arrow, Fig. 3 E).

Mutant cycle no. 2: g38c/c60g

We next turned to the g38c/c60g mutant cycle. We expected these mutations to yield high-magnitude contact epistasis because the nucleotides form a Watson-Crick-Franklin basepair in the D conformation that inverts complementarity over the mutant cycle (Fig. 2 A).

As with the previous mutant cycle, we looked for evidence of contact epistasis in the g38c/c60g mutant cycle by considering the standard state free energies for the equilibria defining the system. We calculated the marginal distributions of ΔG°′_2AP, ΔG°′_MG, and n_MG for all four variants (Fig. 4, A–C). The parameters were well constrained for the wild-type, c60g, and g38c/c60g variants, but not g38c introduced alone. The g38c variant exhibits low 2AP binding, even at high Mg²⁺ and RNA concentrations (Fig. 2 B). Based on our MCMC samples, we are confident that g38c differs from the other variants; however, this could arise because the mutation increases ΔG°′_MG, increases ΔG°′_2AP, or some combination of the two (Fig. 4 D). This leads to a strong inverse correlation in estimates of ΔG°′_2AP and ΔG°′_MG (Fig. 4 D).

Figure 4.

Contact and ensemble epistasis in the g38c/c60g mutant cycle. (A–C) Marginal distributions of ΔG°′_2AP, ΔG°′_MG, and n_MG taken across MCMC samples. The area of the violin plot encompasses all values. The central bar is the median; the top and bottom bars mark the 95% credibility interval. The distributions of ΔG°′_2AP and ΔG°′_MG for the g38c variant are truncated at the dashed lines. The green lines and points indicate the expected value if the mutations exhibit no epistasis. (D) Estimates for ΔG°′_2AP and ΔG°′_MG across MCMC samples for the g38c variant (dark to light encodes low to high frequency). The median wild-type value is shown as a white circle. (E) Estimates of epistasis in ΔG°′_MG and ΔG°′_MG across MCMC samples. The red point indicates the centroid of the distribution. The concentric circles indicate where the total epistasis summed over ΔG°′_2AP and ΔG°′_MG is 1, 5, and 10 kcal/mol. (F) ΔG_obs versus Mg²⁺ concentration calculated using Eq. 7 for wild-type (black), c60g (red), g38c (blue), and g38c/c60g (purple). Solid lines indicate the median of the MCMC samples; dashed lines indicate the standard deviation; areas indicate the 95% credibility region. (G) Epistasis in ΔG_obs calculated from the curves in (F). The green line indicates ΔG_obs in the absence of contact epistasis (green points in (A–C)). The small arrows indicate the maximal contributions of ensemble and contact epistasis near 1 mM Mg²⁺. To see this figure in color, go online.

We calculated contact epistasis in ΔG°′_2AP, ΔG°′_MG, and n_MG for the g38c/c60g mutant cycle by drawing from each parameter’s MCMC distributions for each variant and applying Eq. 9. We observed strong covariation in epistasis between ΔG°′_2AP and ΔG°′_MG (Fig. 4 E). The centroid of the distribution yields estimates of the epistasis in each parameter as 1 and –6 kcal/mol, respectively (red point, Fig. 4 E). Only 0.4% of samples have epistasis <1 kcal/mol in both ΔG°′_2AP and ΔG°′_MG, strongly suggesting that contact epistasis is present in one or both parameters.

To look for ensemble epistasis in the g38c/c60g mutant cycle, we first calculated ΔG_obs versus Mg²⁺ concentration for all four variants (Fig. 4 F). Because of the strong covariation in our estimates of ΔG°′_2AP and ΔG°′_MG, our confidence in ΔG_obs for g38c is high despite low confidence in the absolute magnitudes of these parameters. We found that ΔG_obs plateaued for all variants except g38c. This reflects the fact that 2AP binding approaches saturation for all variants except g38c (Fig. S3).

We observed striking magnesium-dependent epistasis in ΔG_obs for the g38c/c60g mutant cycle (Fig. 4 G, Eq. 10). The value is indistinguishable from zero at low and high Mg²⁺ concentrations. We see epistasis of −1.5 [−1.9, −1.3] kcal/mol, peaking around the point where the riboswitch binds Mg²⁺ and 2AP (Fig. S3). The negative sign indicates the double mutant variant is 1.5 kcal/mol better at binding 2AP than we would expect based on the effects of the mutations alone. This is driven largely by the deleterious effect of g38c when introduced alone (blue curve, Fig. 4 F) and aligns with previous measurements of restored 2AP binding upon re-establishing a Watson-Crick-Franklin basepair between these sites in the D conformation (19,20). Previous work suggests that mutation g38c is more deleterious than c60g because g38 also coordinates an a33·a66 noncanonical basepair, forming a base quartet platform that further stabilizes the D conformation (20).

To decompose the interplay between ensemble and contact epistasis for the g38c/c60g mutant cycle, we eliminated contact epistasis from ΔG_obs by removing epistasis in ΔG°′_2AP, ΔG°′_MG, and n_MG as before. For the wild-type, c60g, and g38c/c60g variants, we used the median value for each parameter from its MCMC distribution (Fig. 4, A–C). To obtain ΔG°′_2AP and ΔG°′_MG for g38c, we subtracted the effect of c60g on each parameter in the wild-type background from the values for the g38c/c60g double mutant (green points, Fig. 4, A and B). This removal of contact epistasis led to the solid green curve in Fig. 4 G, which shows that ensemble redistribution alone contributes up to −0.5 kcal/mol of epistasis at low Mg²⁺ concentrations (labeled arrow, Fig. 4 G). Contact epistasis further amplifies this effect to yield the maximum epistasis of −1.5 kcal/mol (labeled arrow, Fig. 4 G). This contact epistasis is driven primarily by epistasis in Mg²⁺ affinity because eliminating epistasis in ΔG°′_MG gives a value entirely outside the MCMC samples for g38c (green point, Fig. 4 B).

Mutant cycle no. 3: c50a/c60g

We next turned to analyzing epistasis in the c50a/c60g mutant cycle. We expected this mutant cycle to yield minimal contact epistasis but significant ensemble epistasis between allosteric sites. These two positions are distant from one another but connect two allosterically linked regions of the protein: the docked helices (c60g) and the adenine binding site (c50a).

When introduced alone, we found that c50a had a deleterious effect of 1.1 [0.9, 1.3] kcal/mol on 2AP binding (Fig. 5 A), but no detectable effect on Mg²⁺ binding (Fig. 5, B and C). This matches previous observations and makes sense given its proximity to the adenine binding pocket (20). When c50a is combined with c60g, the riboswitch becomes almost unresponsive to Mg²⁺ as measured by 2AP binding. Even at the highest RNA and Mg²⁺ concentrations we studied, we saw only small shifts in 2AP binding (Fig. 2 B). As a result, the values of ΔG°′_2AP and ΔG°′_MG for the double mutant are poorly defined.

Figure 5.

Contact and ensemble epistasis in the c50a/c60g mutant cycle. (A–C) Marginal distributions of ΔG°′_2AP, ΔG°′_MG, and n_MG taken from across MCMC samples. The area of the violin plot encompasses all values. The central bar is the median; the top and bottom bars mark the 95% credibility interval. The distributions of ΔG°′_2AP and ΔG°′_MG for the c50a/c60g variant are truncated at the dashed lines. The green lines and points indicate the expected value if the mutations exhibit no epistasis. (D) Estimates of ΔG°′_2AP and ΔG°′_MG across MCMC samples for the c50a/c60g variant (dark to light encodes low to high frequency). The median wild-type value is shown as a white circle. (E) Estimates of epistasis in ΔG°′_2AP and ΔG°′_MG across MCMC samples. The concentric circles indicate combined epistasis between the two parameters of 1, 5, and 10 kcal/mol. (F) ΔG_obs versus Mg²⁺ concentration calculated using Eq. 7 for wild-type (black), c60g (red), and c50a (blue). Solid lines indicate the median of the MCMC samples; dashed lines indicate the standard deviation; areas indicate the 95% credibility region. The purple points indicate the mean and standard deviation of ΔG_obs for c50a/c60g calculated from the binding curves in Fig. 2 using Eq. 8. (G) Epistasis in ΔG_obs calculated from the curves in (F). The green line indicates ΔG_obs calculated assuming no contact epistasis (green points in (A–C)). The small arrows indicate the Mg²⁺ concentrations where the epistasis may arise from the ensemble versus contacts. To see this figure in color, go online.

To investigate contact epistasis for the c50a/c60g mutant cycle we looked at the joint distribution of ΔG°′_2AP and ΔG°′_MG for the double mutant (Fig. 5 D). Unlike the previous mutant cycle, we saw no covariation in our estimates of ΔG°′_2AP and ΔG°′_MG: ΔG°′_MG is essentially unconstrained by the data. There is some evidence for contact epistasis, even for these poorly defined parameters, as only 1.1% of MCMC samples had ΔG°′_2AP and ΔG°′_MG that both had epistasis less than 1 kcal/mol. This result is tentative, however, given the poor constraints on parameters given by the experimental data for c50a/g60c.

To look for ensemble epistasis, we next calculated ΔG_obs versus Mg²⁺ concentration for the wild-type, c50a, and c60g variants (Fig. 5 F). Because the value of ΔG°′_MG was unconstrained for the c50a/c60g variant, we could not use Eq. 7 to estimate ΔG_obs. We therefore turned to Eq. 8, which allows us to estimate ΔG_obs at each Mg²⁺ concentration directly from the 2AP binding curves (Fig. 2 B). ΔG_obs for c50a/c60g estimated using Eq. 8 is shown as points in Fig. 5 F. We observed no apparent change in ΔG_obs for c50a/c60g as a function of Mg²⁺ concentration, reflecting its unresponsiveness to Mg²⁺ ions. To calculate epistasis in ΔG_obs using Eq. 10, we used the average value of ΔG_obs for c50a/c60g across all measured Mg²⁺ concentrations (3.8 ± 0.5 kcal/mol; solid purple line, Fig. 5 F).

We observed magnesium-dependent epistasis in ΔG_obs for the c50a/c60g mutant cycle (Fig. 5 G): the value starts negative, then saturates just above 1 mM Mg²⁺. At low concentrations, the epistasis arises from the difference in the responsiveness of the variants to Mg²⁺. At high Mg²⁺ concentrations, the flat epistasis curve arises because the four variants reach different final populations of D·A (Fig. S3). The epistasis is positive because the double mutant is worse at binding 2AP than would be expected based on the individual effects of the mutations.

We next investigated the interplay between contact and ensemble epistasis in the c50a/c60g mutant cycle. As before, we calculated epistasis in ΔG_obs after removing epistasis in ΔG°′_2AP, ΔG°′_MG, and n_MG for all variants. For this analysis, we used the median values of our estimates for wild-type, c50a, and c60g variants and the green points for c50a/c60g (Fig. 5, A–C). The resulting ensemble epistasis in ΔG_obs is shown as a green line in Fig. 5 G. For this mutant cycle, the ensemble contributes a maximum of 0.7 kcal/mol of epistasis at Mg²⁺ concentrations below 1 mM (labeled arrow, Fig. 5 G). At higher Mg²⁺ concentrations, contact epistasis dominates (labeled arrow, Fig. 5 G). Given our uncertainty in the thermodynamic parameters, we cannot resolve what contact(s) and corresponding magnitudes lead to the observed epistasis.

General patterns of concentration dependence

We next asked if there were general rules that might allow one to make hypotheses about the source of epistasis given an experimental observable, even without detailed information about the thermodynamic ensemble under study. The riboswitch ensemble provides a useful framework for posing this question. We can rewrite our energetic observable as follows, pulling out the 2AP binding constant:

$$\Delta G_{obs}=-RTln\left(\frac{{\left(K_{MG}{\left[M\right]}_T\right)}^{n_{MG}}}{1+{\left(K_{MG}{\left[M\right]}_T\right)}^{n_{MG}}}\right)-RTln\left(K_{2AP}\right)\text{.}$$ (11)

When we do so, we see that ΔG_obs reduces to a Hill binding model (left term) with a Mg²⁺-independent offset controlled by 2AP affinity (right term). Such a model is generic and applicable to many systems. We can therefore think about the interplay of contact and ensemble epistasis in the adenine riboswitch as a model for many macromolecules of interest. What patterns arise if we introduce contact epistasis in K_2AP (target affinity), K_MG (allosteric effector affinity), and n_MG (allosteric effector cooperativity)?

In Fig. 6, we calculated epistasis in ΔG_obs for otherwise nonepistatic mutant cycles in which we injected varying amounts of epistasis into these equilibrium constants. We observe that each equilibrium constant gives a distinct effector-dependent epistatic signal. First, epistasis in 2AP affinity has no concentration dependence: it acts as a global offset to ΔG_obs (Fig. 6 A). We observed this in our experimental data: epistasis in ΔG°′_2AP led to a global offset (Fig. 3 E). The affinity for 2AP is independent of the population of the 2AP binding-competent conformation and thus does not depend on Mg²⁺ concentration.

Figure 6.

Contact epistasis alters epistasis in ΔG_obs. (A–C) Calculated effects of contact epistasis in K_2AP (A), K_MG (B), or n_MG (C) on epistasis in ΔG_obs versus Mg²⁺ concentration. The black curves show epistasis in ΔG_obs for two mutations that have no contact epistasis. Mutations were modeled as having an effect of +1 kcal/mol (A and B) or +0.5 ions (C). The colored curves are epistasis in ΔG_obs given the positive (blue) or negative (red) contact epistasis indicated next to each curve. Cartoon insets indicate which equilibrium is being perturbed. To see this figure in color, go online.

Contact epistasis in Mg²⁺ affinity gives more complex patterns. In the absence of contact epistasis in K_MG, we observe a peak in epistasis (black curve, Figs. 6 B and 1 C). The peak arises from the redistribution of conformations within the ensemble: there is no contact epistasis anywhere in the system. As we add contact epistasis, it modulates the signal for ensemble epistasis in a nonlinear fashion: the contact epistasis itself redistributes the ensemble, amplifying the ensemble epistasis. The largest effects of epistasis in K_MG occur at low Mg²⁺ concentrations where a change in Mg²⁺ binding can alter the relative population of the D·A conformation, and thus ΔG_obs (Fig. S3). This epistasis is lost at high Mg²⁺ because the molecule is saturated with Mg²⁺ and 2AP binding now controls ΔG_obs (Fig. S3). We observed this pattern for the g38c/c60g mutant cycle, where strong contact epistasis in ΔG°′_MG amplified ensemble epistasis at low and moderate Mg²⁺ concentrations, then dropped to zero at high Mg²⁺ concentrations (Fig. 4 G).

Finally, epistasis in ion uptake leads to a different pattern of magnesium-dependent epistasis. As with epistasis in K_MG, contact epistasis in n_MG plays a role at low Mg²⁺ concentration, then goes to zero when other factors outside Mg²⁺ binding are the most important determinants of the energetics of the system. Because the ion uptake coefficient controls the sensitivity to Mg²⁺—mathematically, it is equivalent to a Hill coefficient (31)—contact epistasis in n_MG manifests as a linear dependence between the log of Mg²⁺ concentration and epistasis. We saw no evidence for this particular form of epistasis in our riboswitch mutant cycles.

This rogues’ gallery of epistatic effects provides a useful lens to generate hypotheses about the origins of epistasis in a macromolecule, even without a detailed model of the thermodynamic ensemble for that system. If one were to observe a fixed epistatic signal across multiple concentrations of an allosteric effector, it would suggest contact epistasis on some feature of the macromolecule that does not depend on the effector concentration (Fig. 6 A). If one observed a peak in epistasis that dropped to zero at both high and low effector concentrations—concentration regimes in which few conformations are populated—the pattern would suggest ensemble epistasis and minimal contact epistasis (Fig. 6 B). If one observed high epistasis at low effector concentration that dropped to zero at high effector concentration, it would suggest contact epistasis in the parameters controlling effector binding (Fig. 6 B). And, finally, if one saw a monotonic change in epistasis with the logarithm of effector concentration, it would suggest contact epistasis in binding cooperativity (Fig. 6 C).

These rules of thumb are not definitive and require system-specific validation (to say nothing of the complexity that could arise if multiple classes of contact epistasis contribute to the observed signal). However, they do provide a useful starting point. One might imagine a high-throughput experiment conducted at a few allosteric effector concentrations. A simple analysis of the effector-dependent epistasis in the system would provide initial hints about the energetic couplings within the system. This could set up deeper biophysical investigation, possibly leading to fitting a thermodynamic model to the system (as demonstrated in this work) to extract the relevant thermodynamic parameters and epistatic mechanisms (16,17).

Discussion

We set out to investigate the relative contributions of contact and ensemble epistasis between mutations in the adenine riboswitch. We found evidence for both forms of epistasis in all three mutant cycles. Our work reveals that ensemble epistasis—previously measured in proteins (17)—also occurs in RNA and is likely a generic feature of allosteric macromolecules. Furthermore, we found that contact and ensemble epistasis interact in a nonlinear fashion. Even when the magnitude of ensemble epistasis was low (∼0.5 kcal/mol) in one mutant cycle, the ensemble could strongly attenuate the magnitude of contact epistasis (from ∼6 to ∼1.5 kcal/mol, Fig. 4, E and G). Thus, understanding intramolecular epistasis in this molecule requires accounting for both contact and ensemble mediated effects.

Interplay between ensemble and contact epistasis

In our previous theoretical and experimental work, we observed that mutations led to high-magnitude epistasis in thermodynamic observables defined by the ensemble (14,17), even without any epistasis between mutations at the level of equilibrium constants. This was not the case for adenine riboswitch. If we removed epistasis from equilibrium constants (ΔG°′_2AP, ΔG°′_MG, and n_MG), ensemble epistasis in ΔG_obs hovered around kT (0.6 kcal/mol). This can be seen in the green curves shown in Figs. 3 E, 4 G, and 5 G, which have peak magnitudes of 0.3 (a35u/c60g), −0.5 (g38c/c60g), and 0.7 (c50a/c60g) kcal/mol, respectively.

When we included contact epistasis, however, we observed much larger concentration-dependent epistasis in ΔG_obs for two of the three cycles: g38c/c60g (−1.5 kcal/mol) and c50a/c60g (1.5 kcal/mol). Concentration-dependent epistasis folds together contact and ensemble mechanisms in a nonlinear fashion. This is seen most clearly for g38c/c60g. This cycle had strong contact epistasis because it inverted the polarity of a Watson-Crick-Franklin G-C basepair. Contact epistasis in either ΔG°′_2AP or ΔG°′_MG is necessary to explain the observed binding curves (Fig. 4, B and C). Although we could not unambiguously determine which parameter exhibited epistasis due to strong covariance, the median estimate of contact epistasis was −6 kcal/mol in ΔG°′_MG and 1 kcal/mol in ΔG°′_2AP (red point, Fig. 4 E). This result is plausible, as the above value of ΔG°′_MG is consistent with a previous measurement of the g38/c60 interaction energy (19), as well as the expected energy for a G-C basepair (37). Furthermore, we would anticipate that epistasis from an inverted G-C basepair in the docked-loop conformations would manifest largely in ΔG°′_MG rather than indirectly in ΔG°′_2AP (Fig. 1 B).

Despite this strong contact epistasis (−6 kcal/mol) in the g38c/c60g mutant cycle, we observed epistasis of only −1.5 kcal/mol in our ensemble observable (ΔG_obs). The riboswitch’s conformational ensemble attenuates the strong contact epistasis, making epistasis in the observable much lower than the epistasis in the underlying equilibrium constant. We note that even MCMC samples with contact epistasis up to −45 kcal/mol in ΔG°′_MG and 10 kcal/mol in ΔG°′_2AP (distribution, Fig. 4 E) were accommodated with only ∼1.5 kcal/mol epistasis in ΔG_obs (gray area, Fig. 4 G). Although these massive energetic effects on ΔG°′_MG and ΔG°′_2AP are chemically implausible, they illustrate an important point. Epistasis in ΔG_obs arises due to shifts in the relative population of ensemble conformations—E, D, and D·A—rather than directly from the energetic effects of mutations on each conformation. Massive effects on the energy of each conformation due to physical contacts can be subsumed by the ensemble if these effects do not meaningfully redistribute the probabilities of the conformations.

The importance of multistate design

Our work points to the fundamental importance of conformational landscapes in modulating the effects of mutations and epistasis between them. Efforts to engineer or design macromolecules by focusing on individual conformations may face unexpected results if the macromolecule populates multiple conformations in solution.

As a hypothetical example, consider an attempt to optimize the adenine riboswitch’s ligand-bound conformation for stronger ligand binding affinity. Perhaps a pair of mutations are introduced to form another Watson-Crick-Franklin basepair to bridge the docked-loop interface. Structure-based calculations on the ligand-bound conformation might predict a certain favorable energetic interaction between the two mutations. They might incorrectly calculate the energetic effect on ligand binding; however, by failing to examine how the new basepair reshapes the conformational landscape. These considerations point to the value of multistate design to better account for the interplay between contact and ensemble sources of epistasis (38,39), especially since many biological macromolecules must toggle between multiple conformational states under physiological conditions for function.

Implications for allostery

This work also provides an intriguing lens for attempting to understand the molecular basis of allostery. Epistasis and allostery between sites have similar energetic origins: introduction of a new stabilizing or destabilizing interaction at one site alters the activity at another. Despite this basic equivalence, allostery and epistasis differ in their concentration dependence. Mutations are a fixed perturbation, independent of environment. In contrast, allosteric interactions depend on the concentrations of allosteric effectors. By combining measurements of epistasis (fixed perturbation to the probability of a conformation) with an allosteric perturbation (addition of allosteric effector) we gain insight into the thermodynamic basis for communication between sites in a macromolecule. In particular, we can separate epistatic effects into “contact” and “ensemble” categories, which may allow us to disentangle communication occurring through a molecule (13) from communication occurring through a redistribution of conformational states (15).

We see a hint of long-distance communication through a conformation rather than the ensemble for c50a/c60g. For this mutant cycle, the vast majority of MCMC samples exhibited epistasis in ΔG°′_MG (Fig. 5 E), even though these two mutations are ∼34 Å apart in the structure. One way to view this result is that communication is occurring via one of the conformations in the ensemble, rather than through redistributing the ensemble. The thermodynamic parameters for this cycle were poorly determined, so this observation remains tentative. We bring it up here, however, as an example of a promising way to disentangle through-conformation from ensemble-based mechanisms of long-distance communication in macromolecules: look for evidence of epistasis between distant mutations in the microscopic equilibria that define the ensemble (e.g., ΔG°′_MG) versus epistasis in the overall thermodynamic observable for the system (e.g., ΔG_obs).

Identifying sources of epistasis

Finally, our work suggests a few rules of thumb that researchers could use to make hypotheses about the roles of contact and ensemble mechanisms in a macromolecule of interest, even given limited measurements. This is particularly apropos for high-throughput studies, where one might be able to measure binding, or some other thermodynamic observable, as a function of allosteric effector. One might imagine using these rules to classify sites for further study, or to help develop a more rigorous mechanistic model of an ensemble for further characterization.

We observed a few basic patterns in our experimental (Figs. 3, 4, and 5) and model studies (Fig. 6). First, contact epistasis in the observed conformation (D·A, for the riboswitch) gives rise to epistasis independent of effector concentration (Fig. 6 A). Second, ensemble epistasis with no contact epistasis yields a peak in epistasis centered at the allosteric effector concentration that maximizes ensemble diversity (black line, Fig. 6 B) (14). Third, a convolution of contact and ensemble epistasis appears as a peak in epistasis, but with non-zero epistasis at high or low effector concentration (colored lines, Fig. 6 B). Fourth, contact epistasis in cooperativity (i.e., the ion uptake coefficient or a Hill coefficient) appears as epistasis that changes linearly as with the log of the allosteric effector concentration (Fig. 6 C). Intriguingly, we also observed very similar patterns of epistasis versus allosteric effector in our previous study of ensemble epistasis in the lac repressor (17).

Further work will be needed to generalize these results, with a particular focus on understanding how these patterns would change if several classes of contact epistasis were in play. It will also be important to explore different sorts of ensembles, with different numbers of conformations and linkages between allosteric effector and observable. This said, given the generic form of the mathematical model of the riboswitch ensemble (Eq. 11), we expect these findings to generalize well. Overall, our work highlights the importance of considering effects of mutations on multiple conformations to understand epistasis and suggests a few rules of thumb for disentangling contact and ensemble epistasis in macromolecules.

Data availability

All experimental data, software, and scripts to reproduce analyses reported in this manuscript are available on github (https://github.com/harmslab/riboswitch-epistasis).

Author contributions

D.R.W., J.R.W., and M.J.H. designed the research and edited the manuscript. D.R.W. performed the research and wrote the manuscript. D.R.W. and M.J.H. developed software and analyzed the data.

Editor: Doug Barrick.