Reduced Model Captures Mg²⁺-RNA Interaction Free Energy of Riboswitches

Abstract

The stability of RNA tertiary structures depends heavily on Mg²⁺. The Mg²⁺-RNA interaction free energy that stabilizes an RNA structure can be computed experimentally through fluorescence-based assays that measure Γ₂₊, the number of excess Mg²⁺ associated with an RNA molecule. Previous explicit-solvent simulations predict that the majority of excess Mg²⁺ ions interact closely and strongly with the RNA, unlike monovalent ions such as K⁺, suggesting that an explicit treatment of Mg²⁺ is important for capturing RNA dynamics. Here we present a reduced model that accurately reproduces the thermodynamics of Mg²⁺-RNA interactions. This model is able to characterize long-timescale RNA dynamics coupled to Mg²⁺ through the explicit representation of Mg²⁺ ions. KCl is described by Debye-Hückel screening and a Manning condensation parameter, which represents condensed K⁺ and models its competition with condensed Mg²⁺. The model contains one fitted parameter, the number of condensed K⁺ ions in the absence of Mg²⁺. Values of Γ₂₊ computed from molecular dynamics simulations using the model show excellent agreement with both experimental data on the adenine riboswitch and previous explicit-solvent simulations of the SAM-I riboswitch. This agreement confirms the thermodynamic accuracy of the model via the direct relation of Γ₂₊ to the Mg²⁺-RNA interaction free energy, and provides further support for the predictions from explicit-solvent calculations. This reduced model will be useful for future studies of the interplay between Mg²⁺ and RNA dynamics.

Introduction

The thermodynamics and kinetics of biological macromolecules like RNA and proteins can be described by energy-landscape theory. In an energy-landscape description, each highly populated configuration of the system corresponds to a basin of attraction on the energy landscape. Various features of the landscape explain different processes. Macromolecular folding requires a native energetic bias to guide the search for the native basin to overcome Levinthal’s paradox and fold on biological timescales (1–3). This native bias implies that the free energy of a structure is determined largely by its similarity to the native structure. Such a landscape is well described by the principle of minimal frustration: native contacts are energetically stabilizing, and other contacts are less energetically significant (3).

Function occurs on the same landscape as folding, but frequently there are multiple basins of attraction within, and occasionally outside, the broader folded basin. To capture this functional frustration (4), inclusion of contacts and other information from these multiple basins is required (5). In some cases, the roughness of the landscape due to nonnative interactions is an important perturbation to the native-biased landscape (6,7). Nonspecific interactions, including electrostatics, are a source of frustration, because they stabilize interactions between both native and nonnative contacts. In proteins, electrostatic interactions can modulate the unfolded ensemble (8) and explain differences in thermodynamics and kinetics between homologous sequences (9). Although a minimally frustrated landscape is an apt baseline approximation for RNA with a native structure (10), frustration likely plays an even more important role in RNA than in proteins (11,12), including frustration and trapping that arise from electrostatic interactions (13,14). Indeed, the RNA energy landscape is quite electrostatically sensitive to the ionic environment (15,16), as the species and concentrations of ions in the ionic environment modulate the stability of competing structures. Modeling the ionic dependence of the RNA energy landscape is the focus of this work.

RNA is sensitive to the ionic environment because it is strongly negatively charged. For RNAs, such as riboswitches and ribosomal RNAs, to form compact structures that carry out biological function, counterbalancing RNA charge with cations such as Mg²⁺, K⁺, and Na⁺ is essential. Mg²⁺ is especially effective in competing with other ions to balance RNA charge, because it is divalent and small. Although monovalent ions are generally present in much greater concentrations, only half as many Mg²⁺ ions as monovalent ions need to be localized to the region surrounding the RNA, which leads to a comparable entropic cost for localization. The small size of Mg²⁺ allows it to outcompete and displace larger divalent ions by approaching the RNA more closely (17–19).

Mg²⁺ plays a ubiquitous role in RNA stability and dynamics, and it exhibits many complex behaviors in the vicinity of RNA. A description of the ionic dependence of the landscape should encompass these features. Mg²⁺ is particularly effective at stabilizing compact RNA structures by inducing effective attraction between otherwise repulsive phosphates (13,19), and as a result, it plays a dramatic role in energetically stabilizing RNA tertiary structure (15). In turn, higher concentrations of Mg²⁺ tend to slow RNA kinetics by stabilizing compact structures (14,20). Explicit-solvent studies have suggested that Mg²⁺ fluctuations couple with RNA dynamics (21). This study also found that most Mg²⁺ associated with RNA is in the outer-sphere regime, separated from the RNA by one hydration layer (21). Ion-ion correlations in the dense outer-sphere regime are important for obtaining accurate free energies (22). Recent experimental studies of the SAM-I riboswitch have revealed a complex interplay between Mg²⁺ and S-adenosyl methionine (SAM) in stabilizing the SAM-I riboswitch (23). Modeling these features requires a more detailed representation of Mg²⁺ than of other ions. In this article, we focus on extending a minimally frustrated description of the landscape to include electrostatic frustration and ionic effects with an explicit representation of Mg²⁺. The resulting model is able to accurately reproduce the thermodynamics of Mg²⁺-RNA interaction.

A structure-based model (SBM) is an ideal way to represent a minimally frustrated landscape. In the simplest form, only native contacts are stabilized energetically, resulting in a smooth, native-biased landscape. The usefulness of this approximation is shown by the ability of such models to capture many aspects of protein (5,24–27) and RNA (28–33) folding and function. SBMs are also attractive from a modeling perspective due to their simplicity and rapid sampling. Smoother landscapes, together with underdamping (34), result in enhanced sampling approximately 10⁴ times faster than explicit-solvent simulations (35,36). The simplicity of the SBM means that the effects of energetic perturbations like electrostatics on the overall native bias can be modeled and understood. In a pure SBM, electrostatic interactions and ions are only represented implicitly by their contribution to effective interactions of native contacts, and there is no ionic concentration dependence. To model the effects of ionic concentration dependence and of Mg²⁺ behavior in investigations of long-time dynamics of RNA, we design an SBM with explicit Mg²⁺ ions and an effective potential that implicitly represents KCl.

The energetic stabilization by Mg²⁺ is quantified by the Mg²⁺-RNA interaction free energy, ΔG_Mg²⁺. Theoretical calculations have shown that in the presence of excess KCl, ΔG_Mg²⁺ is related to the excess Mg²⁺ (Γ₂₊) according to

$$\text{Δ}{G}_{{\text{Mg}}^{2+}}=-k_{\text{B}}T{\int }_0^{C_{2+}}{\text{Γ}}_{2+}d\text{ln}{C}_{2+},$$ (1)

where both ΔG_Mg²⁺ and Γ₂₊ are functions of the Mg²⁺ concentration (C₂₊) (37). The excess ions of an ionic species i per RNA molecule is a thermodynamic quantity called the preferential interaction coefficient, Γ_i (37), and varies with ionic concentration. Ionic subscripts i of 2+, +, and − refer to Mg²⁺, K⁺, and Cl⁻, respectively. The charge of excess cations and depleted anions associated with the RNA must balance the RNA charge. Differences in ΔG_Mg²⁺ between conformational ensembles (ΔΔG_Mg²⁺) reveal preferential structural stabilization by Mg²⁺ that can be modulated by Mg²⁺ concentration.

Experimental curves for Γ₂₊ have been obtained for several systems using the fluorescent dye 8-hydroxyquinoline-5-sulfonic acid (HQS) (37). With these measurements, one can measure ΔΔG_Mg²⁺ between conformational ensembles (38,39) or characterize chelated Mg²⁺ binding in a specific site (40). Γ₂₊ has also been computed from microsecond-timescale explicit-solvent molecular dynamics simulations (21). Because the excess Mg²⁺ can be measured both experimentally and computationally, Γ₂₊ presents a powerful metric for uniting theory and experiment to build models of Mg²⁺-RNA interactions. Furthermore, because of the relation between Γ₂₊ and ΔG_Mg²⁺, such models will be well parameterized thermodynamically.

In this study, we focus on two well characterized riboswitches to calibrate the model. Riboswitches are attractive systems for the study of Mg²⁺-RNA interactions, because they are small and relatively simple but still possess many of the rich Mg²⁺-sensitive tertiary interactions seen in larger systems (Fig. 1, B and D). Riboswitches are responsible for regulating a wide range of metabolic pathways in bacteria. These noncoding RNAs reside in the 5′ untranslated region and often control transcription of their mRNA by binding metabolites directly related to their protein product. Riboswitches have proved to be excellent systems to investigate RNA structure, function, and dynamics using a wide range of experimental (23,41–43) and computational techniques (21,30,44–48). The adenine riboswitch (Fig. 1, A and B) has been extensively studied experimentally (39,49–51), and Γ₂₊ data obtained by HQS titrations are available for Mg²⁺ concentrations ranging from 1 μM to 1 mM (39). The SAM-I riboswitch (Fig. 1, C and D) binds the SAM metabolite, and has been extensively studied experimentally (23,52,53) and computationally (21,30,54). Γ₂₊ for the SAM-I riboswitch has been computed in explicit-solvent studies (21).

Figure 1.

Secondary and tertiary structures of the two riboswitches used in this study. Mg²⁺ further stabilizes tertiary interactions induced by the metabolites. Metabolites are shown in magenta, and binding pockets are shaded. (A and B) Secondary (A) and tertiary (B) structures of the adenine riboswitch aptamer. The adenine metabolite stabilizes the tertiary interaction between the P2 and P3 helices. (C and D) Secondary (C) and tertiary (D) structures of the SAM-I riboswitch aptamer. The SAM metabolite stabilizes the tertiary interaction between the P3 helix and the nonlocal P1 helix, stabilizing the P1 helix and preventing strand invasion by the competing antiterminator helix in the expression platform. To see this figure in color, go online.

Using thermodynamic data from these previous studies as a guide, we develop a model that incorporates ionic effects into SBMs. The model includes explicit Mg²⁺ ions that interact with the RNA through a Debye-Hückel potential, whereas KCl is represented implicitly through the screening length and a Mg²⁺-dependent Manning condensation factor that scales RNA charge. We calibrate the model to accurately reproduce the preferential interaction coefficient Γ₂₊ and the related Mg²⁺ interaction free energy, ΔG_Mg²⁺, in the native basin of these two riboswitches. The predictions for Γ₂₊ are consistent with experimental HQS measurements of the adenine riboswitch (39) and simulation results of the SAM-I riboswitch (21). With this hybrid structure-based model in hand, we are poised to address more sophisticated questions about the interplay of Mg²⁺, ligands, and RNA during folding and function.

Model and Methods

Structure-based models

SBMs have been used extensively in studies of proteins (24,25,55), and in several studies of RNA (28,30,31,48). Previous studies have included electrostatic effects in the SBM of RNA. The simplest models utilize a repulsive Debye-Hückel interaction between phosphates (28,29). In these models, all ions are implicit. We utilize Debye-Hückel interactions and include explicit Mg²⁺ ions, because explicit counterions are required for Debye-Hückel interactions to produce the attractive effective phosphate-phosphate interactions that Mg²⁺ induces.

Although they still lack explicit counterions, more recent SBMs with electrostatics have added a scaling of RNA charge to Debye-Hückel interactions to account for Manning condensed ions (56). Manning counterion condensation is a useful theoretical tool for understanding the behavior of polyelectrolyte chains like RNA in solution (57,58). Manning counterion condensation theory was originally conceived as the low-concentration limiting behavior of an infinite line of charge in solution. If the linear charge spacing, b, is shorter than the Bjerrum length, $l_{\text{B}}$, the length scale where the electrostatic and thermal energy scales cross (see Eq. 10), ions will condense out of solution onto the RNA and effectively rescale the charge. Although Manning condensation was assumed to be constant in previous SBMs (56), we have developed a model of Manning condensation that can vary with ionic concentration and allow separation of the condensed populations of Mg²⁺ and K⁺. This separation is critical for capturing Mg²⁺ concentration effects, since condensed Mg²⁺ replaces condensed K⁺, and hence the effective rescaling of RNA charge induced by K⁺ diminishes as Mg²⁺ concentration increases.

The competition between condensed Mg²⁺ and KCl provides the interface between discrete and continuum treatments, and if it is modeled well, treating Mg²⁺ as discrete ions and KCl as a continuum inducing an effective potential is a reasonable approximation. Continuum models can capture KCl behavior well, but cannot capture Mg²⁺ behavior because ion-ion correlations become important (59). Continuum approaches break down when the electrostatic coupling parameter (60) approaches unity (see Section S3 in the Supporting Material). The electrostatic coupling parameter varies like the surface charge density and the ion valency squared. Thus, the coupling parameter is four times higher for Mg²⁺ than for K⁺. Although the surface charge density of RNA is hard to estimate, the success of continuum models with KCl and their failure with Mg²⁺ suggests that RNA is in the intermediate regime where the electrostatic coupling parameter of Mg²⁺ is greater than one, whereas for K⁺ it is less than one. This is the regime to which a discrete Mg²⁺ and continuum KCl model is best suited.

We modify an all-atom SBM (30,55), which explicitly includes all heavy atoms, to include explicit Mg²⁺ ions and electrostatic interactions. Explicit Mg²⁺ ions are included to properly represent Mg²⁺-induced phosphate attraction, Mg²⁺-Mg²⁺ correlation, and the dense outer-sphere Mg²⁺ population. KCl screening is represented by a Debye-Hückel potential with a Mg²⁺-dependent KCl Manning condensation factor. The functional form of the potential is

$$V=V_{\text{SBM}}+V_{\text{Mg}\text{-}\text{Size}}+V_{\text{ES}},$$ (2)

where V_SBM is the all-atom SBM potential containing the minimally frustrated native bias, V_Mg-Size controls the size of the explicit Mg²⁺ ions, and V_ES contains the effective electrostatic interactions. Simulation input files can be obtained at http://smog-server.org/forcefields. V_SBM is given by

$$V_{\text{SBM}}=\sum _i^{\text{bonds}}\frac{{ϵ}_r}{2}{\left(r_i-r_{0i}\right)}^2+\sum _i^{\text{angles}}\frac{{ϵ}_{\theta }}{2}{\left({\theta }_i-{\theta }_{0i}\right)}^2+\sum _i^{\text{impropers}\text{or}\text{planar}}\frac{{ϵ}_{\chi }}{2}{\left({\chi }_i-{\chi }_{0i}\right)}^2+\sum _i^{\text{proper}\text{dihedrals}}{ϵ}_{\varphi }{F}_{\text{D}}\left({\varphi }_i-{\varphi }_{0i}\right)+\sum _{ij}^{\text{contacts}}{ϵ}_{\text{C}}\left({\left(\frac{{\sigma }_{ij}}{r_{ij}}\right)}^{12}-2{\left(\frac{{\sigma }_{ij}}{r_{ij}}\right)}^6\right)+\sum _{ij}^{\text{RNA}\text{-}\text{RNA}\text{non}\text{-}\text{contacts}}{ϵ}_{\text{NC}}{\left(\frac{{\sigma }_{\text{NC}}}{r_{ij}}\right)}^{12},$$ (3)

where

$$F_{\text{D}}\left(\varphi \right)=\left(1-\text{cos}\varphi \right)+\frac{1}{2}\left(1-\text{cos}3\varphi \right).$$ (4)

Geometric parameters ($r_{0i}$, ${\theta }_{0i}$, ${\chi }_{0i}$, ${\varphi }_{0i}$, and ${\sigma }_{ij}$) are set by their values in a crystal structure, so that the crystal structure is the global minimum in energy. Energetic parameters have been calibrated (30,55) and are listed in Section S1 in the Supporting Material. The shadow method (61) was used to obtain the contact map for the sum over contacts. The final term controls the excluded volume and enforces steric constraints. Excluded-volume parameters are given by σ_NC = 1.7 Å and ϵ_NC = ϵ_R, where ϵ_R = 1.5 k_BT is the reduced energy unit (see Section S1 in the Supporting Material), and T is the simulation temperature of 300 K.

The excluded volume of the Mg²⁺ ions is controlled by V_Mg-Size, which is given by

$$V_{\text{Mg-Size}}=\sum _{ij}^{\text{Mg-RNA}}{ϵ}_{\text{MgRNA}}{\left(\frac{{\sigma }_{\text{MgRNA}}}{r_{ij}}\right)}^{12}+\sum _{ij}^{\text{Mg-Mg}}{ϵ}_{\text{MgMg}}{\left(\frac{{\sigma }_{\text{MgMg}}}{r_{ij}}\right)}^{12}.$$ (5)

Parameters are calibrated in the first subsection under Results. For ϵ_MgRNA = ϵ_MgMg = ϵ_R, it is determined that σ_MgRNA = 3.4 Å and σ_MgMg = 5.6 Å.

The electrostatic potential is given by

$$V_{\text{ES}}=\sum _{ij}^{\text{Mg}\mathrm{-}\text{RNA}\text{Mg}\mathrm{-}\text{Mg}}\frac{1}{4\pi {ϵ}_0ϵ}\frac{{\alpha }_i{q}_i{\alpha }_j{q}_j}{r_{ij}}\text{exp}\left(-\frac{r_{ij}}{l_{\text{D}}}\right),$$ (6)

where the dielectric constant $ϵ=80$ mimics water, ${ϵ}_0$ is the permittivity of free space, $l_{\text{D}}$ is the Debye length (excluding Mg²⁺ screening), and α is the Manning condensation prefactor that rescales the charges q due to the presence of implicit condensed K⁺. Note that the sum includes only Mg²⁺-RNA and Mg²⁺-Mg²⁺ interactions. Since the SBM already implicitly includes native RNA-RNA interactions in V_SBM, such as hydrogen bonding and base-pairing interactions, RNA-RNA electrostatic interactions were excluded to avoid double counting. Since RNA-RNA electrostatic interactions are excluded, the model can only capture the indirect effect of KCl upon the Mg²⁺ dependence of the RNA energy landscape. This model is most appropriate at intermediate KCl concentrations where the native basin is stabilized by KCl. A different model of RNA-RNA electrostatic interactions that can overcome the native bias of V_SBM would be required to capture the full KCl dependence of the energy landscape, such as unfolding under low salt conditions.

Previous studies have shown that the Mg²⁺ distribution is controlled by many negatively charged atoms on the RNA, as opposed to only the phosphates (21,62). Since hydrogens are not represented in the SBM, united atom charges were placed on every heavy atom to reproduce the electrostatic potential and resultant Mg²⁺ distribution. United atom charges were obtained from Amber 99 (63) by adding all hydrogen charges to the nearest heavy atom. This approximation does not alter the charge of any atoms that interact strongly with Mg²⁺, with the possible exception of OH2′, which interacts moderately (21).

The Debye length in V_ES is given by

$$l_{\text{D}}=\sqrt{\frac{ϵ{ϵ}_0{k}_{\text{B}}T}{{\sum }_i{q}_i^2{n}_i}},$$ (7)

where $n_i$ and $q_i$ are the number density and charge of ionic species i, and $k_{\text{B}}$ is the Boltzmann constant. To compute $l_{\text{D}}$ for V_ES, Mg²⁺ is excluded from the sum over ionic species i, because Mg²⁺ is accounted for explicitly. The Manning condensation prefactor is given by

$${\alpha }_i=\{\begin{array}{ll}1-{\theta }_{+}\hfill & i=\text{RNA}\hfill \\ 1\hfill & i={\text{Mg}}^{2+}\hfill \end{array}.$$ (8)

θ₊ is the number of Manning condensed K⁺ per phosphate, which is explained in the second subsection under Results. Thus, α rescales the RNA charge due to the presence of implicit condensed K⁺, leaving the Mg²⁺ charge unchanged. Since SBMs use reduced units, V_ES is multiplied by the conversion factor in Eq. S3 to convert from kJ/mol to reduced energy units. The conversion factor was determined by noting that a temperature of 80 $T_{\text{R}}$ in simulation units corresponded to 300 K and then equating thermal energy scales. This conversion resulted in an ionic distribution appropriate to 300 K within our simulations. See Section S1 in the Supporting Material for more details.

Molecular dynamics simulations

Simulations were run using the Gromacs-4.6.1 molecular dynamics package (64). Langevin dynamics in the underdamped limit were used as a thermostat (see Section S1 in the Supporting Material for details). Implicit KCl concentrations of 50 mM (adenine riboswitch) and 100 mM (SAM-I riboswitch) were chosen for direct comparison with previous experiments (39) and simulations (21) on these systems. Simulations were run at varying concentrations of explicit Mg²⁺. Cubic boxes were chosen such that the box was at least 500 Å on a side, but also such that there was a minimum of 200 bulk Mg²⁺ ions in the box, in addition to the Γ₂₊ Mg²⁺ ions associated with the RNA. For the most dilute simulations (10⁻⁶ M Mg²⁺), this resulted in boxes 7000 Å on a side. Such low Mg²⁺ concentrations are completely inaccessible with explicit-solvent simulations, but have little effect on the computational expense of this model. Production simulations at each Mg²⁺ concentration were run for 2.5 × 10⁸ time steps.

Nonlinear Poisson-Boltzmann calculations

Two theoretical models have been used extensively in studies of RNA: nonlinear Poisson-Boltzmann (NLPB) theory (65,66) and Manning counterion condensation theory (57,67). Poisson-Boltzmann methods have played a critical role in shedding light on RNA electrostatics. These methods have been advanced by the seminal contribution of Baker and co-workers, whose methods have enabled rapid calculation of electrostatic potentials for very large molecules (68–70). NLPB treats ions in a mean-field manner, which neglects ion-ion correlations (22,71) and ion size effects (59,72,73). NLPB has been successfully applied to many systems (66,74) and predicts ΔΔG_Mg²⁺ within computational uncertainty (75,76). Although NLPB accurately predicts ΔΔG_Mg²⁺, values for Γ₂₊ and ΔG_Mg²⁺ are systematically low by ∼25%. We choose not to use NLPB directly in our SBM, partly because removing the double counting of RNA-RNA electrostatic interactions implicitly represented in the contact terms is not straightforward with NLPB and partly because of the systematically low values of Γ₂₊ and ΔG_Mg²⁺ given by NLPB calculations. NLPB calculations are used to corroborate the model and parameters obtained for Manning condensed K⁺ in the second subsection under Results. NLPB calculations were performed using the adaptive Poisson-Boltzmann solver (APBS) (68), with parameters listed in Section S2 in the Supporting Material.

The NLPB equation determines the potential by assuming that the mean ionic density fields are governed by the Boltzmann distribution. The linearized Poisson-Boltzmann (LPB) equation is obtained by linearizing the Boltzmann distribution around zero potential. The Debye-Hückel (DH) approximation is further obtained by neglecting the solute’s dielectric constant and ionic accessibility. Both LPB and DH are described by homogenous differential equations, and thus the solution may be multiplied by a coefficient α. By Gauss’ law, $\alpha =1-{\sum }_i{z}_i{\theta }_i$, where $z_i$ is the valence and ${\theta }_i$ is the Manning condensed fraction per phosphate of the ionic species i. This implies that the excess ions, Γ_i, may be split into a population that screens and a population that condenses.

The total condensed charge, ${\sum }_i{z}_i{\theta }_i$, may be obtained by fitting either LBP or DH to NLPB far from the RNA, where the potential is small and the linearized Boltzmann distribution is valid. In this case, fitting was performed at a set of points within a 240 Å box but outside a 120 Å box centered on the RNA. The condensed ions obtained by LPB and DH fits are identical at low concentrations but begin to differ near the 50 mM concentrations used in this study. Fitting to Debye-Hückel is used because that is more appropriate to the form of the potential used in Eq. 6, but for completeness, fits to both are shown in Fig. S1 in the Supporting Material.

To partition the Manning condensed ions by species, ion densities were computed from the full NLPB and scaled DH potentials. The difference in ion densities integrated over all space gives the condensed ions for that species. At high ionic concentrations, this also results in a nonzero condensation of Cl⁻. This occurs because the linearized potential predicts a nonphysical negative density in regions of strong potential, but NLPB gives approximately zero density.

Results

Calibration of Mg²⁺ ion size

Previous studies have shown that the affinity between an ion and RNA is strongly dependent on ion size (17–19,77). Given ${ϵ}_{\text{MgRNA}}={ϵ}_{\text{MgMg}}={ϵ}_{\text{R}}$, the effective size of Mg²⁺ is controlled by ${\sigma }_{\text{MgRNA}}$ and ${\sigma }_{\text{MgMg}}$ in the excluded-volume term. Because the ionic affinity for RNA is sensitive to ion size, these parameters must be carefully calibrated to accurately predict Γ₂₊.

Explicit-solvent simulations have shown that most excess Mg²⁺ resides in the outer-sphere peak between 3.5 and 5 Å from the RNA, separated by a single hydration layer (21). In this model, the position of this peak is controlled by the values of ${\sigma }_{\text{MgRNA}}$ in the excluded volume term (Eq. 5) and by the electrostatic potential (Eq. 6). To deconvolute the calibration of the Mg²⁺ excluded volume from the calibration of implicit Manning condensed KCl, excluded volume calibrations were performed in the high Mg²⁺ concentration regime (10 mM Mg²⁺ vs. 100 mM K⁺) where negligible KCl is condensed, and therefore $\alpha =1$. Within trial simulations, two metrics for the position of the peak were used: the position of the maximum of the peak and the the weighted average position between 3 and 5 Å. For ϵ_MgRNA = ϵ_R, both metrics suggest that σ_MgRNA = 3.4 Å yields a distribution most consistent with the outer-sphere peak observed in explicit-solvent simulations (Fig. 2).

Figure 2.

Comparison of the Mg²⁺ distribution about the SAM-I riboswitch in SBM and explicit-solvent Amber simulations. Distances to the closest RNA heavy atom were computed and binned. SBM captures well the general shape of the distribution, but misses the more subtle hydration features. The chelated ion at 2 Å could have been included in the SBM, but was omitted for simplicity. These curves were used to calibrate the excluded volume between Mg²⁺ and RNA atoms. The position of the maximum of the peak as well as the weighted average position between 3 and 5 Å both suggest that σ_MgRNA = 3.4 Å. To see this figure in color, go online.

To calibrate the Mg²⁺-Mg²⁺ excluded volume, the Mg²⁺-Mg²⁺ radial distribution function was used (Fig. S2). The radial distribution function rises sharply between 5 and 7 Å, and plateaus beyond this. σ_MgMg = 5.6 Å was found to most closely reproduce the midpoint of this rise.

Competition between Manning condensed Mg²⁺ and K⁺ modulates the effective electrostatic potential

The Debye-Hückel potential assumes a linearized dependence of the Boltzmann distribution of ions on the electrostatic potential. Such an assumption is good far from the RNA, but breaks down near the RNA where the potential is large. Equation 6 uses Manning counterion condensation to account for the nonlinear effects in this region, so that the far-field potential is correct. The additional ions near the RNA deviating from linearity may be regarded as the Manning condensed counterions. Thus, the excess ions Γ may be split into a diffuse screening population and a condensed population close to the RNA.

Manning counterion condensation theory was originally derived for an infinite line of charge under limiting law conditions: as the salt concentration approaches zero while taking the thermodynamic limit. If the linear charge spacing (b) is shorter than the Bjerrum length ($l_{\text{B}}$ – the length scale where the electrostatic and thermal energy scales cross), ions will condense out of solution onto the RNA. For an infinite line of charge, the number of ions per phosphate (θ) that condense is given by

$$\theta =\frac{1}{\left|Z\right|}\left(1-\frac{1}{\left|Z\right|\xi }\right),$$ (9)

where Z is the valency of the counterion, $\xi =l_{\text{B}}/b$, and

$$l_{\text{B}}=\frac{1}{4\pi {ϵ}_0ϵk_{\text{B}}T}.$$ (10)

Grouping phosphate and condensed counterion charges together results in an effective rescaling of phosphate charges by $\alpha =1-Z\theta$.

If explicit ions are used, bare RNA charges should be used, because condensation is captured by the ions. However, if ions are only represented implicitly through an effective potential, effective RNA charges that are rescaled by α must be used to account for Manning condensation. Because Mg²⁺ in the model is explicit and KCl is implicit, the condensed K⁺ per phosphate (θ₊) and condensed Mg²⁺ per phosphate (θ₂₊) must be separated so that θ₊ can be included in the effective potential, V_ES. θ₂₊ and θ₊ can be separated by solving the system of equations

$$\text{ln}\left(\frac{{\theta }_{2+}e}{C_{2+}V}\right)=2\text{ln}\left(\frac{{\theta }_{+}e}{C_{+}V}\right)$$ (11)

$$\text{ln}\left(\frac{{\theta }_{+}e}{C_{+}V}\right)=\left(1-{\theta }_{+}-2{\theta }_{2+}\right)F,$$ (12)

describing competition between condensed Mg²⁺ and K⁺ (78), where e is Euler's number. Equation 11 balances the entropic cost of localizing one Mg²⁺ ion with the cost of localizing two K⁺ ions. V is the effective volume into which a condensed ion is localized, and together with the K⁺ and Mg²⁺ concentrations, C₊ and C₂₊, determines the entropic cost of localization. Equation 12 balances the entropic cost of localizing a condensed ion with the energy benefit. $k_{\text{B}}TF$ is the free energy per phosphate to assemble the phosphate charges from infinity to the native geometry, originally approximated as an infinite line, and thus depends on screening length $l_{\text{D}}$ and RNA geometry. Computing F for an infinite line of charge (Eq. S11) and taking the appropriate limits in Eqs. 11 and 12 returns Eq. 9. The condensation is constant over a wide range of concentrations because both the entropic cost and F go like $\text{ln}\left(1/C\right)$.

Manning counterion condensation may be applied to finite systems with the caveat that whereas the entropic cost of condensation goes like $\text{ln}\left(1/C\right)$, F does not. Thus, as the concentration goes to zero, so does the condensation, and the condensation is not constant over a wide range of concentrations as it is for an infinite line of charge (Fig. 3 A). Equation 12 can be reexpressed as

$${\theta }_{2+}=\frac{1}{2}\left({\theta }_0-{\theta }_{+}\right)-A\text{ln}\left(\frac{{\theta }_{+}}{{\theta }_0}\right),$$ (13)

where θ₀ is the number of condensed K⁺ per phosphate in the absence of Mg²⁺ and A is $1/2F$. θ₀ and A in principle depend on RNA geometry, $l_{\text{D}}$, and $C_{+}$ through their dependence on F. Note that in the region where K⁺ and Mg²⁺ compete, $C_{+}\gg C_{2+}$. Thus, for constant $C_{+}$, $l_{\text{D}}$ is also approximately constant. Since our simulations remain in the native basin, RNA geometry is also constant, and θ₀ and A are constant as $C_{2+}$ varies at a fixed KCl concentration.

Figure 3.

NLPB calculations of Manning condensation for the adenine riboswitch. (A) The Manning condensed ions obtained from NLPB in the absence of Mg²⁺ over a wide range of KCl concentrations. The Manning condensed fraction is not constant with concentration, as predicted by Manning theory for an infinite line of charge. (B) The Manning condensed ions partitioned into the three ionic species. The Debye length is held constant at the value for 50 mM KCl, whereas the Mg²⁺ level is increased. θ₂₊ increases with increasing Mg²⁺ concentration. (C) Competition between condensed Mg²⁺ and condensed K⁺. Manning condensed ions from NLPB calculations are plotted along with fits to Eq. 13. Cl⁻ is a nonnegligible portion of the condensed fraction. The fit to Eq. 13 is better if Cl⁻ is ignored (green), but subtracting it from θ₊ (black) is more appropriate, so that the effect of condensed Cl⁻ will be included in the effective electrostatic potential. A schematic of the A = 0 assumption (Eq. 14) is plotted as a dashed line. To see this figure in color, go online.

Due to the difficulty of computing F, θ₀ and A are treated as fitting parameters, leaving only θ₊ and ${\theta }_{2+}$ free to vary with $C_{2+}$. NLPB calculations at fixed $l_{\text{D}}$ corroborate the form of Eq. 13, but there is also a nonnegligible population of condensed Cl⁻ (Fig. 3, B and C). This occurs because Debye-Hückel predicts a nonphysical negative concentration of Cl⁻ near the RNA. To account for their effect on electrostatic interactions, these condensed Cl⁻ must be subtracted from θ₊ in Eq. 6. Even if these condensed Cl⁻ ions (${\theta }_{-}$) are subtracted from θ₊, the fit with Eq. 13 is still quite good. A can be estimated by fits to NLPB calculations (Fig. 3 C), but for this work, $A=0$ is assumed for simplicity. Equation 13 then has the intuitive meaning that each additional condensed Mg²⁺ displaces two condensed K⁺. The effect of this assumption is most pronounced between medium and high $C_{2+}$, near ${\theta }_{2+}={\theta }_0/2$.

To compute θ₊ in Eq. 6, an estimate of ${\theta }_{2+}$ is required. For nearly all the Mg²⁺ and K⁺ concentrations considered, $C_{+}\gg C_{2+}$. Thus, the screening is almost exclusively due to K⁺ and Cl⁻. Since the excess Mg²⁺ can be divided into a screening population, which is approximately zero, and a condensed population, any excess Mg²⁺ is condensed. This means ${\text{Γ}}_{2+}$ is approximately equal to ${\theta }_{2+}$ (when both are expressed in units of ions per phosphate), and

$${\theta }_{+}=\{\begin{array}{ll}{\theta }_0-2{\text{Γ}}_{2+}\hfill & {\text{Γ}}_{2+}<\frac{1}{2}{\theta }_0\hfill \\ 0\hfill & {\text{Γ}}_{2+}>\frac{1}{2}{\theta }_0.\hfill \end{array}$$ (14)

The approximate equality of ${\text{Γ}}_{2+}$ and ${\theta }_{2+}$ is evidenced by the large population of Mg²⁺ in the outer-sphere regime and the near absence of diffuse Mg²⁺ observed in previous studies, even with a relatively high Mg²⁺ concentration (21).

To determine θ₊ for a particular Mg²⁺ concentration, an iterative, self-consistent approach is used. A value for θ₀ is chosen. After an initial guess for ${\text{Γ}}_{2+}$ and subsequent calculation of θ₊ with Eq. 14, the simulation is run and ${\text{Γ}}_{2+}$ is computed from the Mg²⁺ distribution and updated. The process is repeated until ${\text{Γ}}_{2+}$ converges. The single remaining adjustable parameter, θ₀, was used to fit the experimental data for the adenine riboswitch and was estimated from NLPB methods for the SAM-I riboswitch.

SBMs capture excess Mg²⁺ and Mg²⁺-RNA interaction free energy

With the Mg²⁺ excluded volume calibrated, and with a method for including Manning condensed KCl, we can compute preferential interaction coefficients and interaction free energies from simulation. The agreement with previous experimental data (39) and explicit-solvent data (21) is quite close (see Fig. 5), and suggests the model is well parameterized thermodynamically.

Figure 5.

Excess Mg²⁺ (Γ₂₊) and Mg²⁺-RNA interaction free energy (ΔG_Mg²⁺) in the hybrid SBM with explicit Mg²⁺ agree with previous experimental and computational findings. Uncertainties from bootstrap analysis are plotted as error bars, but are too small to see in most cases. (A) Comparison between Γ₂₊ for the adenine riboswitch obtained in this study with SBM and a previous experimental fluorescence study (39). The 71 base RNA has a net charge of −70. KCl is present at 50 mM. Even at comparatively low concentrations, Mg²⁺ is able to effectively compete with K⁺ and Cl⁻. (B) The Mg²⁺ interaction free energy (ΔG_Mg²⁺) obtained from Eq. 1 by integrating Γ₂₊. Agreement between the two curves suggests that SBMs with explicit Mg²⁺ are capable of capturing the energetic stabilization due to Mg²⁺. (C) Comparison between Γ₂₊ for the SAM-I riboswitch obtained in this study with SBM and a previous Amber 99 molecular dynamics study (21). The 94 base RNA has a net charge of −92. KCl is present at 100 mM. Since θ₀ is 18 ions, all values of Γ₂₊ above 9 ions are unaffected by K⁺ condensation. (D) Predicted ΔG_Mg²⁺ obtained by integrating Γ₂₊ of SAM-I. No data exist for comparison. To see this figure in color, go online.

For the adenine riboswitch, Leipply and Draper performed fluorescence measurements down to very low Mg²⁺ concentrations (39), where Mg²⁺ competition with K⁺ condensation is important. If condensed K⁺ is neglected in our model, the fit to experimental data is good for high Mg²⁺ concentration where ${\theta }_{+}=0$, but poor for low Mg²⁺ concentration (Fig. 4). NLPB calculations estimate ${\theta }_0=28.4$ ions for the adenine riboswitch in 50 mM KCl. We find that ${\theta }_0=20$ ions (0.29 ions per phosphate) produces a much closer fit (Fig. 4). With this fitted K⁺ condensation, it can be seen that the Mg²⁺-RNA interaction free energy is within 8% at 1 mM Mg²⁺, compared to the 25% typically observed with NLPB theory (75,76).

Figure 4.

Comparison of preferential interaction coefficients of the adenine riboswitch at 50 mM KCl obtained for several choices of θ₀ to experimental data. Although NLPB estimates suggest that θ₀ = 28.4 ions, θ₀ = 20 ions provides a closer fit to the experimental data. Manning condensed K⁺ cannot be neglected, as seen by the data for θ₀ = 0. To see this figure in color, go online.

It is unclear whether the NLPB estimation and experimental fit of θ₀ disagree because the approximations or parameters of NLPB break down or because the fit of θ₀ is compensating for other neglected effects, such as hydration or nonuniform K⁺ displacement. In either case, these discrepancies likely exist in the SAM-I system as well. Therefore, as an initial approximation, the NLPB estimate of θ₀ for the SAM-I riboswitch (25.7 ions) was scaled by the same ratio of 70% as observed for the adenine riboswitch to obtain ${\theta }_0=18$ ions (0.20 ions per phosphate).

An estimate of θ₀ was required for the SAM-I riboswitch because previous explicit solvent simulations could only be performed in a high Mg²⁺ concentration range (1–10 mM) where Mg²⁺ completely outcompetes K⁺ condensation (i.e., ${\theta }_{+}=0$). Since ${\theta }_0<{\text{Γ}}_{2+}$ for all three simulations, θ₀ cannot be constrained by a fit or adjusted to improve the fit. The agreement with previous explicit-solvent results for ${\text{Γ}}_{2+}$ for these high Mg²⁺ concentrations is quite close. The KCl condensation and choice of θ₀ is of importance for low Mg²⁺ concentrations and for the computation of the free energy. Although no data exist for comparison, it would be useful to devise new experiments to test predictions and constrain θ₀. It should be noted that there is a chelated ion in SAM-I that was neglected in this study. This ion would likely raise ${\text{Γ}}_{2+}$ near Mg²⁺ concentrations of ${10}^{-5}$ M, and would raise $\text{Δ}{G}_{{\text{Mg}}^{2+}}$ by a few $k_{\text{B}}T$.

Discussion

Model captures outer-sphere Mg²⁺ and can be extended for chelated Mg²⁺

Although the outer-sphere population accounts for the majority of excess Mg²⁺ (21), Mg²⁺ chelation sites can have an effect on ${\text{Γ}}_{2+}$ and $\text{Δ}{G}_{{\text{Mg}}^{2+}}$. A chelated Mg²⁺ ion raises ${\text{Γ}}_{2+}$ by nearly one ion at low Mg²⁺ concentrations, due to the typically high affinity of chelation sites. At higher Mg²⁺ concentrations, an absent chelated ion is largely replaced by less tightly associated ions, and ${\text{Γ}}_{2+}$ is independent of the presence of the chelated ion. A chelated ion affects $\text{Δ}{G}_{{\text{Mg}}^{2+}}$ over all concentrations, because the integral over ${\text{Γ}}_{2+}$ includes the lower concentrations, but the effect is generally small. For example, in one study, a chelation site became occupied near 2 μM and was shown to contribute several $k_{\text{B}}T$ to $\text{Δ}{G}_{{\text{Mg}}^{2+}}$ (40). This contribution was only a small fraction of $\text{Δ}{G}_{{\text{Mg}}^{2+}}$. Since most of $\text{Δ}{G}_{{\text{Mg}}^{2+}}$ is due to outer-sphere Mg²⁺, accurately representing outer-sphere Mg²⁺ is essential for a model of Mg²⁺-RNA interactions. Although sometimes important, especially for $\text{ΔΔ}{G}_{{\text{Mg}}^{2+}}$, energetic contributions from chelated ions are a perturbation in comparison.

This model focuses on modeling outer sphere and diffuse Mg²⁺. Previous work has demonstrated that for the SAM-I riboswitch, and likely for other RNAs of similar size, the outer-sphere population is dominant (21). That result is underscored here by the ability of this model to capture $\text{Δ}{G}_{{\text{Mg}}^{2+}}$ and ${\text{Γ}}_{2+}$ without chelated ions. For larger systems (62) or highly compact systems with buried phosphates (79), chelated ions are more important. Chelated ions have been included with harmonic restraints in the SBM without electrostatics (31). These techniques can be easily extended to this charged model by adding charge to the Mg²⁺ ion, and are quite adequate for native basin fluctuations. Further extensions to allow unbinding will be required for fluctuations outside the native basin or for computation of $\text{Δ}{G}_{{\text{Mg}}^{2+}}$ in systems where chelated Mg²⁺ is important.

Mg²⁺-RNA interactions are sensitive to Manning condensed KCl

Manning theory, as formulated for an infinite line of charge, suggests that condensation is constant over a wide range of concentrations. Although this is correct for infinite lines of charge, the riboswitch systems considered in this work show substantial variation of the Manning condensed ions over the small range of physiologically relevant KCl concentrations (Fig. 3 A). As a result, θ₀ varies with KCl concentration and RNA geometry. The excess Mg²⁺, and hence $\text{Δ}{G}_{{\text{Mg}}^{2+}}$, are sensitive to θ₀ (Fig. 4). Thus, accurate methods for estimating θ₀ for the structures and concentrations of interest are critical for the accuracy of the hybrid SBM. Two methods were used to estimate θ₀: a fitting of the experimental HQS data and NLPB calculations. These methods disagree by eight ions, which is about 10% of the RNA charge. As mentioned in the third subsection under Results, the reasons for this disagreement are unclear but likely stem from either approximations or parameter choices in the NLPB calculation or from the θ₀ fit compensating for other effects neglected by the SBM.

It could be that NLPB does not capture θ₀ well because of the parameters used in the calculations, specifically the K⁺ radius. Previous studies (75,76) have shown that standard NLPB parameters underestimate ${\text{Γ}}_{2+}$, and from our experimental fit, the parameters also seem to overestimate θ₀. This suggests that RNA affinity for K⁺ may be too high with these NLPB parameters. The K⁺ affinity can be lowered by increasing the radius of K⁺. Indeed, in NLPB theory, the ion radius is sometimes used as a fitting parameter (80). In the parameterization presented in this study, Mg²⁺ and K⁺ have the same radius of 2.0 Å. For Mg²⁺, this captures the hexahydrated size of Mg²⁺ well, but for K⁺ this is larger than the bare radius, and smaller than the hydrated radius. Other standard NLPB parameters use a larger K⁺ radius that corresponds to hydrated K⁺, and argue against using a smaller radius because NLPB cannot capture several effects relevant to dehydrated ions (81). Simulations have shown that some K⁺ dehydrates to associate with RNA (80), but this may be a force-field artifact, since NMR experiments suggest that K⁺ does not dehydrate in the presence of RNA (82). As a result, it is unclear what K⁺ radius is appropriate for NLPB calculations, but a larger radius may improve agreement with the fitted value for θ₀.

It is also possible that NLPB estimates of θ₀ differ from the experimental fit because the θ₀ fit compensates for other effects neglected by the hybrid SBM. θ₀ weakens Mg²⁺-RNA interactions at low Mg²⁺ concentrations. Thus, if the hybrid SBM underpredicts Mg²⁺-RNA affinity, especially at low Mg²⁺ concentrations, it will also underpredict θ₀ to compensate. Hydration is one possible neglected effect. In Fig. 2, the outer-sphere peak at 4 Å is higher for the explicit-solvent simulation (which includes hydration effects), than for the SBM. Although the area under the curves is nearly equal due to compensation by the width and height of the peak, the SBM slightly underpredicts Mg²⁺-RNA affinity at high concentrations, as seen in Fig. 5, A and C). If this trend continues to low concentrations, the model will underpredict θ₀ to compensate. Another possible neglected effect is the nonuniform displacement of K⁺. In the proposed SBM, condensed Mg²⁺ displaces K⁺ uniformly by lowering θ₊, whereas in reality the K⁺ is instead likely displaced from the immediate neighborhood of a Mg²⁺ ion, which could allow stronger interactions with the nearby phosphates.

The transferability of the model to other RNAs, other KCl concentrations, or other ionic species is of great interest. The form of the model is transferable, but results are sensitive to the specific value of the parameter θ₀. Transfer to other ionic species may be as simple as adjusting θ₀ for monovalent ions and ${\sigma }_{\text{MgRNA}}$ and ${\sigma }_{\text{MgMg}}$ for divalent ions (see Section S7 in the Supporting Material). A systematic way of predicting θ₀ is required to generalize the model to other KCl concentrations or other RNA systems where no experimental data exist for fitting. In this study, θ₀ could not be directly fit to experimental data for the SAM-I riboswitch. Since the experimental fit value for the adenine riboswitch was 70% of the NLPB estimate, the NLPB estimate of θ₀ was scaled by 70% in the SAM-I riboswitch as well. Explicit solvent simulations could not confirm or refute the choice of θ₀ because the low Mg²⁺ concentrations involved are inaccessible. Thus, SBM simulations of several systems must be compared with HQS fluorescence measurements to learn how to predict θ₀. NLPB estimates may turn out to be the best method for estimating θ₀. NLPB estimates might be improved by adjusting the NLPB K⁺ radius, or else the KCl concentration dependence NLPB estimate scaling factor, 70% here, could be probed. The tightly bound ion model (22) may give better results than NLPB. Alternatively, developing methods to compute the free energy, F, in Eq. 12 rather than fitting it may be fruitful. Models of θ₀ should be tested against fluorescent HQS titrations at several KCl concentrations. Such data already exist for an rRNA 58-mer (76), but new HQS titrations of other systems are of great interest.

Generalizing and applying the model

With the model of Mg²⁺-RNA interactions presented here corroborated by previous experimental and computational studies, one can investigate the effects of Mg²⁺ upon the RNA energy landscape. Such models can explore the relative stability of various basins on the energy landscape as a function of ionic concentration, from the intermediates of riboswitch folding to functional transitions in the ribosome. In this study, we have computed $\text{Δ}{G}_{{\text{Mg}}^{2+}}$ as a first step toward computing the differential Mg²⁺-RNA interaction free energy between two ensembles, $\text{ΔΔ}{G}_{{\text{Mg}}^{2+}}$. $\text{ΔΔ}{G}_{{\text{Mg}}^{2+}}$ is the relevant biological quantity because it determines the Mg²⁺-dependent free-energy difference between the two ensembles and thus the thermodynamic change in occupation probability as the ionic concentration changes. Several previous studies (75,76) have sought to compute $\text{ΔΔ}{G}_{{\text{Mg}}^{2+}}$ with NLPB, but the precision of these calculations is limited by large uncertainties in the partially unfolded ensemble, which was represented by a few static structural models. With the SBM approach, one can more fully sample the partially unfolded ensemble and probe how the ensemble changes with Mg²⁺ concentration, removing a large source of uncertainty in the NLPB approach.

As we move toward probing $\text{ΔΔ}{G}_{{\text{Mg}}^{2+}}$ between interconverting ensembles, care must be taken to ensure that the K⁺ condensation, θ₊, does not radically change between ensembles. Since we only examine ensembles close to the crystal structure, this investigation is only able to check the θ₊ dependence on Mg²⁺ concentration. However, as expressed in Eq. 13, θ₊ can also depend on Mg²⁺ condensation for which concentration is a proxy, and on geometry and screening length, $l_{\text{D}}$, through the parameters A and θ₀. Fortunately, changes in θ₀ and ${\theta }_{2+}$ due to structural changes will have opposite effects on θ₊ and may largely cancel. If it turns out that K⁺ condensation depends radically on structure, θ₊ can be made a time-dependent function of structural order parameters, likely through calculation of F in Eq. 12. Although this is undesirable from a modeling perspective, it is valid, since the relaxation time of the K⁺ condensation is much faster than the Mg²⁺ dynamics.

The hybrid SBM will be a useful tool for addressing questions beyond $\text{ΔΔ}{G}_{{\text{Mg}}^{2+}}$. Cooperativity in Mg²⁺-induced transitions can be quantified, and changes to transition states can be identified. Mg²⁺ is known to have a strong effect on RNA kinetics. This is due at least in part to higher barriers between states, but may also result from changes in diffusion due to coupling between Mg²⁺ and RNA dynamics. Extensive sampling of an SBM can partition the effect of Mg²⁺ on RNA kinetics into changing barrier height and diffusion. These effects will be quite interesting to examine in future investigations.

Conclusion

Structure-based models provide a baseline representation of the minimally frustrated landscape predicted by energy-landscape theory. Roughness, frustration, and nonspecific interactions such as electrostatics may be added to this baseline to determine their effects on RNA folding and function. To account for the ionic sensitivity and highly charged nature of RNA, we have extended an all-atom structure-based model to include electrostatic interactions with explicit Mg²⁺ and implicit KCl.

The excess Mg²⁺ is a powerful metric, which allows coupling of the SBM with fluorescence measurements and explicit-solvent simulations. The model was extensively parameterized and tested against previous explicit-solvent simulations and HQS fluorescence measurements of excess Mg²⁺. Because of the close connection between excess Mg²⁺ and the Mg²⁺-RNA interaction free energy, the agreement of the model with experiment suggests that the model is well parameterized to capture Mg²⁺-RNA interaction free energies. The success of the hybrid SBM model, which is designed to capture the dynamics of the outer-sphere Mg²⁺ population, further corroborates the dominance of this population suggested by explicit-solvent simulations.

NLPB calculations suggest that for compact RNA structures, Manning condensed ions are not constant over a wide range of ionic concentrations as they are for lone RNA helices approximated by infinite lines of charge. Manning counterion condensation and competition between condensed populations is seen to be a critical correction to the Debye-Hückel approximation implemented here. A model of Mg²⁺-RNA interactions must account for the effect of Manning condensed KCl.

The hybrid SBM opens up concentration regimes and timescales inaccessible to explicit-solvent simulation. With these models, one can explore the relative stability of basins of the energy landscape as a function of ion concentration. The effects of ions on kinetics through barrier height, transition states, and diffusion can also be investigated. SBMs are now poised to address complex questions about the interplay of Mg²⁺, ligands, and RNA in folding and function.