Grand canonical Monte Carlo and deep learning assisted enhanced sampling to characterize the distribution of Mg²⁺ and influence of the Drude polarizable force field on the stability of folded states of the twister ribozyme

Abstract

Molecular dynamics simulations are crucial for understanding the structural and dynamical behavior of biomolecular systems, including the impact of their environment. However, there is a gap between the time scale of these simulations and that of real-world experiments. To address this problem, various enhanced simulation methods have been developed. In addition, there has been a significant advancement of the force fields used for simulations associated with the explicit treatment of electronic polarizability. In this study, we apply oscillating chemical potential grand canonical Monte Carlo and machine learning methods to determine reaction coordinates combined with metadynamics simulations to explore the role of Mg²⁺ distribution and electronic polarizability in the context of the classical Drude oscillator polarizable force field on the stability of the twister ribozyme. The introduction of electronic polarizability along with the details of the distribution of Mg²⁺ significantly stabilizes the simulations with respect to sampling the crystallographic conformation. The introduction of electronic polarizability leads to increased stability over that obtained with the additive CHARMM36 FF reported in a previous study, allowing for a distribution of a wider range of ions to stabilize twister. Specific interactions contributing to stabilization are identified, including both those observed in the crystal structures and additional experimentally unobserved interactions. Interactions of Mg²⁺ with the bases are indicated to make important contributions to stabilization. Notably, the presence of specific interactions between the Mg²⁺ ions and bases or the non-bridging phosphate oxygens (NBPOs) leads to enhanced dipole moments of all three moieties. Mg²⁺–NBPO interactions led to enhanced dipoles of the phosphates but, interestingly, not in all the participating ions. The present results further indicate the importance of electronic polarizability in stabilizing RNA in molecular simulations and the complicated nature of the relationship of Mg²⁺–RNA interactions with the polarization response of the bases and phosphates.

I. INTRODUCTION

Molecular dynamics (MD) simulations depict the complicated behaviors of nucleic acids, proteins, and other biomolecules with a high level of accuracy at atomic level resolution for complex, heterogeneous systems. In atomistic MD simulations, the motion of every atom in a molecular system is propagated over time based on a general physical model, termed a force field (FF), from which detailed information on the conformational properties of the system may be obtained.¹ The method is a complimentary approach to experimental methods for characterizing the structure and dynamics and is especially useful in situations where experimental techniques face significant limitations.² Despite the success of MD simulations, there are still challenges as many biologically relevant conformational changes in biomolecules, such as the folding of RNA, occur on times in the range of microseconds to seconds, while current atomistic simulations are typically limited to the nanosecond to microsecond time range. In addition, the underlying FFs are inherently simple in the form of the potential energy function as required for computational efficiency, but this leads to limits in the accuracy of such models.

To overcome the time scale limitation, many enhanced sampling methods have been developed, which facilitate crossing high energy barriers to allow for molecular events that occur on time scales longer than those readily attainable in MD simulations to be accessed.^3,4 One category of enhanced sampling methods is that in which the overall probability of sampling a wide range of conformations is increased through general alterations of the potential energy function or through the addition of kinetic energy. These methods are useful in systems where the identification of proper reaction coordinates is challenging. Examples include simulated tempering,⁵ parallel tempering or replica exchange molecular dynamics (REMD),⁶ multicanonical simulation,⁷ accelerated molecular dynamics simulation,⁸ temperature-accelerated molecular dynamics,^9,10 statistical temperature sampling,¹¹ enveloping distribution sampling,^12,13 and integrated tempering sampling.^14,15 Meanwhile, there are simulation methods that are effectively guided by using predefined reaction coordinates (RCs) or collective variables (CVs) that accelerate conformational sampling by modifying the Hamiltonian of the system through biasing potentials defined by the RCs. Examples are umbrella sampling,¹⁶ potential smoothing methods,¹⁷ local elevation,¹⁸ conformational flooding,¹⁹ hyperdynamics,²⁰ conformational space annealing,²¹ adaptive biasing force method,²² local elevation umbrella sampling,²³ variational enhanced sampling,²⁴ and metadynamics.^25,26 Of these methods, metadynamics has been applied to a wide range of systems, including nucleic acids.^25,27,28 Notable is the utility of the method for the investigation of folding and unfolding events in the biomolecular systems.^26,29 The challenge when applying metadynamics, as well as the other methods, is the determination of proper RCs, which ultimately determines the effectiveness of the conformational sampling during the simulations.³⁰

The RCs are based on one or more order parameters (OPs) that define specific degrees of freedom in the molecular system under study. The OPs are typically selected based on the intuition of the researcher associated with their understanding of the system of interest. In some cases, such as the unbinding of a ligand from a protein with a solvent accessible binding site, the choice of the OPs is straightforward, while in cases where multiple degrees of freedom contribute to a molecular transition, the choice of OPs is less clear. To facilitate addressing this challenge, machine learning (ML) approaches have been developed to help define RCs. In the approach used in this study, the user selects a large number of possible order parameters, such as distances or dihedral angles between multiple groups of atoms. Unbiased MD simulations are then run on the system with the ML model used to identify those OPs that best define the conformational sampling of the system, followed by combining the selected OPs into a single RC for use in the enhanced sampling simulation. In the present study, we used the automatic mutual information noise omission (AMINO) approach to identify the most important OPs associated with the unfolding of the twister ribozyme.³¹ The contributions of the selected OPs are then weighted to form the RCs using the reweighted autoencoded variational Bayes for enhanced sampling (RAVE) method.³² These RCs act as the foundation for subsequent enhanced sampling simulations using metadynamics to drive the unfolding of the RNA riboswitch, twister.^33–36

Here, a detailed exploration focuses on characterizing the role of the distribution of Mg²⁺ ions in conjunction with the impact of the use of the Drude polarizable FF on the stabilization of the twister ribozyme building on our recent study of twister stabilization using the additive CHARMM36 (C36) FF.^37–39 That study used an oscillating chemical potential Grand Canonical Monte Carlo (GCMC) method^40–42 to generate multiple distributions of the ions Mg²⁺, K⁺, and Cl⁻ around twister yielding multiple simulation systems. The subsequent selection of RCs for each system followed by metadynamics simulations provides a detailed exploration of the impact of the ion distributions on the conformational stability of the system. Simulations with both the C36 additive and Drude polarizable FFs, including previously reported metadynamics simulations using C36,⁴³ are then analyzed to understand the impact of the explicit treatment of electronic polarization on system stability. This study offers insights into how the use of enhanced sampling methods for both ion distributions via GCMC and conformations through metadynamics in conjunction with the explicit treatment of electronic polarizability can advance our understanding of the factors stabilizing the folded states of biological systems, specifically RNA.

The model system for the present study is the twister ribozyme.^33–36 It is a pseudoknotted RNA, which is of particular interest due to the presence of complicated interactions such as non-canonical base pairs, triplexes, coaxial stackings, sharp turns, as well as both secondary and tertiary structures.⁴⁴ Such noncanonical interactions present significant challenges in the application of sampling algorithms. Knowledge of the factors stabilizing these pseudoknots can aid in improving the RNA tertiary structure predictions.^43,45 In addition, given the various biological functions of RNA, including protein synthesis, gene expression, and various cellular processes,^46–48 it is a promising medicinal target.^49,50 Accordingly, knowledge of the conformational properties of RNA is of therapeutic importance. In the present study, we build upon previous experimental and computational studies,^51–56 including enhanced sampling methods,^45,57–61 to improve our understanding of the environmental and intrinsic properties of RNA that contribute to the stabilization of its folded states.

We discussed this approach in our recent study using the CHARMM36 FF, where critical Mg²⁺ binding sites as well as the specific nucleotides that are favored by these ions toward the stabilization of twister were identified.⁴³ As shown below, the stabilization of the tertiary structure occurs with a wider range of ion distributions when an explicit electronic polarization of atoms is introduced in the system through the Drude force field. This study exemplifies the successful implementation of deep learning-facilitated enhanced sampling protocol combined with the use of a polarizable FF to investigate the role of ions in the stabilization of the tertiary structure of an RNA riboswitch.

II. METHODS

The present study examines the influence of ionic environment on the stability and folded states of twister ribozyme built on the methods applied in our previous studies using oscillating chemical potential GCMC, umbrella sampling, and metadynamics to investigate this system. The present calculations specifically extend the GCMC and metadynamics study using the C36 additive FF⁴³ to the explicit treatment of electronic polarizability using the classical Drude oscillator polarizable FF.^62,63 Briefly, the initial twister structure is from the 2.30 Å x-ray diffraction derived structure based on the Osa-1-4 sequence from Oryza sativa from Protein Data Bank (PDB: 4OJI).^35,64 The missing nucleotides (nucleotides 17 and 18) were added using CHARMM^65,66 and the internal coordinates in the CHARMM36 nucleic acid force field.^37–39 The protonation state of all bases corresponds to those associated with Watson–Crick interactions. In the catalytic mechanism, bases A7 and G45 assume alternate protonation states that are essential for catalysis.^67,68 We note that assigning these protonation states to those bases may stabilize the folded state. However, as we are studying the equilibrium between folded and unfolded states, where the latter may be assumed to have standard protonation states for all bases, the standard protonation states of the bases were used.

All systems were initially prepared using CHARMM-GUI⁶⁹ and were minimized and equilibrated in CHARMM⁶⁵ using harmonic restraints on the non-hydrogen atoms of the backbone and bases with force constants 1 and 0.1 kcal/mol/Ų, respectively. The modeled and minimized structure was used for the root-mean square difference (RMSD) analysis presented below. The twister ribozyme and ions were modeled using the CHARMM36 additive force field,^37–39 including the CHARMM36 default Mg²⁺, K⁺, and Cl⁻ parameters. Oscillating excess chemical potential (μ_ex) Grand Canonical Monte Carlo (GCMC) sampling was applied for insertion and deletion of ions to form different systems that differ in their ion distribution. The sampling involves MC insertion, deletion, rotation, and translation moves applied to the ions and water molecules. The ion distributions were resampled by using MC deletion and insertion moves on Mg²⁺ while simultaneously deleting or inserting K⁺ and Cl⁻ ions to maintain charge neutrality throughout the system. The resulting systems contain 40 magnesium ions (Mg²⁺), 42 potassium ions (K⁺), and 69 chloride ions (Cl^–), which correspond to the final concentrations of ∼100, 105, and 173 mM, respectively.

A. Drude system simulation setup

The nine additive systems generated in the previous study are further considered in this study using version 2020 of the Drude FF,^70,71 including specifically optimized Mg²⁺ parameters for use with nucleic acids.⁷² To convert the systems to the Drude polarizable model, the atom types were changed to that of the Drude FF. Drude oscillators were then added to all non-hydrogen atoms, and the lone pairs were added with CHARMM.^65,66 The ions were converted to their polarizable model, and the TIP3P water was converted to polarizable SWM4-NDP water.⁷³ Using the OpenMM simulation package,^74,75 polarizable systems were subjected to minimization for 5000 steps followed by 100 ps of equilibration with a time step of 0.5 fs under NPT conditions, during which the harmonic restraints on non-hydrogen backbone and bases were maintained with a force constant of 400 and 40 kJ/mol/nm², respectively. The production simulations were performed in the NPT ensemble using the Langevin integrator at 1 bar pressure and 300 K temperature with a time step of 1 fs. Constant pressure was maintained with the Monte Carlo barostat. Particle Mesh Ewald⁷⁶ was used to calculate the electrostatic interactions with a real space cutoff of 12 Å, and the Lennard-Jones potential was smoothed to zero between 10 and 12 Å by using the switch function, with an isotropic long-range correction applied to account for the truncated van der Waals interactions beyond the cutoff.

In the metadynamics simulations, the Watson–Crick (WC) interactions were maintained using harmonic restraints as performed in the previous study. This is due to the calculations being designed to investigate the forces stabilizing of the tertiary structure of twister. Accordingly, the WC interactions that stabilize the canonical secondary structure regions are not of direct interest, and, as discussed previously,⁴³ during metadynamics, those interactions were quickly lost and not regained disallowing investigation of the tertiary interactions. The WC interactions were maintained using a harmonic restraint defined as E = 1/2*k(x − x0)², where the force constant k is 40 kJ/mol/nm² and the equilibrium distance x₀ is 0.3 nm applied to N1–N3 distances in the WC pairs. These were maintained throughout the entire simulation process for all the metadynamics runs. We emphasize that interactions associated with the tertiary structure, for example the A28–A46 trans WC pair, the A8–G45 cis Hoogsteen–sugar edge pair, and the U24–A29 trans WC–Hoogsteen pair,³⁵ were not restrained as such restraints would stabilize the folded, tertiary structure of the ribozyme.

B. Deep learning order parameter selection and reaction coordinate determination

The determination of RCs for metadynamics in this work involves a two-stage machine learning protocol. The first stage uses a clustering-based dimensionality reduction method called AMINO,³¹ which determines the specific order parameters (OPs) that most define the dynamics of the system from a large set of manually selected candidate OPs. AMINO uses a K-medoids clustering with a mutual information-based distance metric to reduce the OP of each cluster by using rate-distortion theory. This reduces the high dimensional OP space to a few dimensions while conserving most of the information.³¹ In the present study, the information is associated with the conformational changes in the tertiary structure of twister. An unbiased 500 ns trajectory is used to compute the center of mass (COM) distance between all pairs of multinucleotide groups, with each group composed of four nucleotides that serves as a collection of candidate OPs. This generates 1275 OPs for 54 nucleotides. These OPs are input into AMINO, which reduces all the OPs to two representative, independent OPs for all the systems in this study.

The second stage of the protocol is the utilization of the most representative OPs selected in the first stage to optimize the reaction coordinate of the system. Reaction coordinate prediction uses the deep learning based reaction coordinate prediction method RAVE.³² This method employs a deep learning model known as a variational autoencoder (VAE)⁷⁷ that is designed to understand and capture a lower-dimensional latent space distribution from complex high-dimensional training data. OP trajectories selected in the AMINO clustering stage serve as input data for the VAE. The neural network architecture from our previous study⁴³ was replicated in this study as the initial molecular systems are identical. Small modifications on the neural network size may change the training time due to increased or decreased complexity, but similar predictions will be obtained once the training has converged. Therefore, the architecture was kept the same. It is composed of one input layer, five encoder hidden layers, five decoder hidden layers, and one output layer. The number of nodes in the input and output layers is the same as the number of OPs selected by the dimensionality reduction stage. These OPs are then used to parameterize a RC using the Kullback–Leibler divergence metric to replicate the same OP distribution.⁷⁷ In this study, the reaction coordinate is constructed as a linear combination of two OPs selected in the initial stage, with coefficients parameterized by RAVE. RAVE was applied iteratively through multiple rounds of metadynamics simulation runs to refine the RC throughout the metadynamics simulations.

C. Metadynamics simulations

Metadynamics simulations were performed to maximize the sampling of twister conformational space with the RCs optimized iteratively to maximize sampling. Well-tempered metadynamics²⁷ with a bias factor (γ) set to 15 and an initial hill height of 1.5 kJ is employed. The Gaussian width (σ) is determined as the standard deviation of the RC calculated from the initial unbiased MD trajectory. Biasing Gaussian energies are deposited every 1 ps, and each cycle of biased simulation spans 200 ns. Following each cycle, the RC is reassessed using RAVE based on the most recent 200 ns of dynamics, with the updated RC utilized for the subsequent 200 ns of metadynamics. The reevaluation of RCs involves solely the reweighting of the contribution of the OPs, with the same OPs predicted during first ML prediction throughout all cycles. Each metadynamics production run comprises 200 ns, resulting in a total simulation time of several microseconds depending upon the system (Table S1).

D. Metadynamics simulations in the absence of Mg²⁺ ions

Additional comparison of the influence of Mg²⁺ ions and the inclusion of explicit polarization in the FF on twister was made using metadynamics simulations in the absence of Mg²⁺ ions. The stable system from the previous study, system 8, was taken, and Mg²⁺ ions were removed, and additional K⁺ and Cl⁻ ions were added, yielding 122 K⁺ ions and 69 Cl⁻ ions. The metadynamics simulations were performed for these two systems with both additive and polarizable FFs, consistent with the method including the WC restraints used for the systems in the presence of Mg²⁺ ions. This allows for the impact of the presence and absence of Mg²⁺ and of explicit polarization on twister to be investigated. The production runs were performed for 500 ns with a time step of 1 fs for all the systems, and the metadynamics simulations were performed for a minimum of 4.6 μs.

E. Mg²⁺–RNA interactions

Analysis of interactions between the Mg²⁺ ions and the RNA involved direct coordination and outer sphere interactions. Direct coordination involved Mg²⁺ forming a direct interaction with an RNA non-hydrogen atom, including non-bridging phosphate oxygens (NBPOs) or base atoms. Outer-sphere interactions included those with Mg²⁺ within 4.2 Å of RNA non-hydrogen atoms. These included interactions with the NBPOs as well as the carbonyl oxygens and acceptor nitrogens of the bases. To identify Mg²⁺–RNA interactions present in the crystal structure, a cutoff of 5.0 Å was used rather than 4.2 Å in the analysis of the simulations to account for flexibility not present in the crystal structure that could allow for additional outer sphere coordination to occur.

F. Statistical analysis

Statistical analysis was based on blocks defined as the 200 ns metadynamics run between each reevaluation of the RCs. The block average and standard error were estimated discarding the first 200 ns of simulation. The averages were then calculated for each subsequent 200 ns segment (i.e., each metadynamics 200 ns run) of the remaining simulation, resulting in a set of averages for n blocks, depending on the total simulation length of each system. The overall average and standard error were then determined from these block averages, assuming independence between blocks.

III. RESULTS AND DISCUSSION

The crystal structure of twister, including the locations of the Mg²⁺ ions, as well as the general workflow used in this study is shown in Fig. 1. The components and durations of the metadynamics simulations in the systems studied are summarized in Table S1 of the supplementary material. Information included in Table S1 includes the number of 200 ns metadynamics simulation iterations performed on each system as they were not all performed for the same duration.

FIG. 1.

(a) Crystal structure of twister ribozyme with nucleotides numbered. The Mg²⁺ ions present in the crystal structure are shown as pink spheres. Tertiary contacts are colored red for T1 (C25–G49, C26–G48, C27–G47, and A28–A46), blue for T2 (G13–C31 and C14–G30), and green for T4 (adjacent NBPOs on U6 and C25) base–base interactions. (b) Flow chart representing the workflow of this study.

In our previous study, the RCs predicted by AMINO/RAVE in conjunction with metadynamics were able to sample a range of conformational states far from the folded state in the majority of simulations by using the C36 FF.⁴³ In a separate study on RNA hairpins using standard MD simulations with both the C36 additive and Drude polarizable FFs, the Drude FF sampled conformations closer to the experimental conformations for both the helical and non-canonical hairpin regions in systems without Mg²⁺ ions.⁷¹ Accordingly, in this work, metadynamics simulations were undertaken on the larger, more complex twister RNA using the nine different distributions of ions that were generated in the previous study. In addition, new RCs were generated in this study based on two OPs/RC. In the previous study, a range of 2–5 OPs were used to define the RC. In the two simulations using only two OPs in the RC with the additive FF (systems 7 and 9), the structure deviated significantly from the experimental crystal structure, indicating that the RCs based on two OPs are adequate to obtain sufficient conformational sampling. In addition to the nine systems, additional metadynamics simulations based on system 8 were run in the absence of Mg²⁺ to further investigate the ability of the polarizable FF to stabilize the structure of the twister RNA.

A. Stability of twister

The overall stability of the tertiary structure of the RNA in the metadynamics simulations was analyzed based on the root mean square deviation (RMSD) of the phosphodiester and sugar backbone non-hydrogen atoms with respect to the minimized crystal structure. The crystal structure was selected as reference as the present study is focused on the overall tertiary conformation of the RNA although we note that crystal contacts do impact the local regions of the structure, including around the active site of twister.^35,67,68 Results in Fig. 2 are presented as both time series and probability distributions for the nine systems including the results for C36 obtained from our previous study.⁴³ As seen in the previous study and further indicated in the new Drude simulations, the ML generated RCs allow the metadynamics simulations to sample a wide range of conformations despite the maintenance of the WC interactions in the simulations. Consistent with our previous studies of RNA hairpins, the Drude FF yielded conformations closer to the crystal conformation vs additive C36 in all the simulations, although large deviations are observed in three of the systems (systems 1, 6, and 7). In the remaining systems, the Drude simulations maintain tertiary conformations within 10 Å or less of the crystal structure. Of the systems, system 8 samples conformations closest to the crystal structure consistent with the maintenance of those conformations with the additive FF. These results support the ability of the polarizable model to maintain the experimental structures better than the additive FF, as seen with other biomolecules using both AMOEBA and Drude FFs,^71,78–82 and, based on the behavior of system 8, that Mg²⁺ plays an important role in the stabilization of the structure. However, our previous study showed that specific details of the distribution of the Mg²⁺ ions around the RNA were needed to stabilize the tertiary structure of twister. The present results with the polarizable model yielding stable structures in systems 2, 3, 4, 5, 8, and 9 indicate that a wider distribution of Mg²⁺ ions can stabilize the structure.

FIG. 2.

RMS difference (RMSD) for the RNA phosphodiester and sugar backbone non-hydrogen atoms for the nine systems as a function of time vs the x-ray crystal structure, 4OJI, following modeling of the missing two nucleotides and energy minimization. On the right of each panel, the probability distributions over the simulations are shown.

In addition to the overall tertiary structure, we examined the three tertiary interactions T1, T2, and T4 that define the overall twister structure [Fig. 1(a)]. The time series of the center of mass (COM) distances for the Drude metadynamics simulations are presented in Fig. S1 of the supplementary material. As expected, the evolution of the tertiary contacts closely correlates with the overall RMSD of the systems shown in Fig. 2. The largest deviations involve the loss of tertiary contacts T1 and T4 in systems 1, 6, and 7. Contact T2 is lost in systems 5 and 6, with that being the only contact substantially lost in system 5, leading to the upshift in the RMSD after 1 μs in that system. In system 4, periodic losses of T4 occur, corresponding to the larger RMSD values in Fig. 2. Notably, the tertiary contacts are all well maintained in system 8, indicating the high stability of that system.

B. Ion–RNA interactions

Initial analysis involved determining the overall number of ions in the vicinity of the RNA in the different systems. Presented in Table S2 are the average values of the total number of the individual ions within 9 Å of each RNA. The cutoff of 9 Å was selected as this corresponds to the distance commonly used when analyzing ions around linear DNA in the context of counterion condensation theory.^83,84 For all the systems, the number of ions is similar: ∼24 Mg²⁺, ∼8 K⁺, and ∼5 Cl⁻, yielding a total charge ranging from 51.4 to 52.0. This compares with the total RNA charge of −53 e, indicating that the total charge of the RNA was neutralized to an equal extent in all systems. Accordingly, more specific interactions of the Mg²⁺ ions with RNA were undertaken.

Subsequent analysis involved the average number of Mg²⁺ ions in the vicinity of different moieties in the RNA within a cutoff distance of 4.2 Å, and the results are presented in Table I. This cutoff includes the direct and outer-shell ions around the bases, the NBPOs, and the total RNA. Note that ions in the vicinity of both the bases and NBPOs were double counted such that the sums do not yield the total number around the full RNA. In general, the total number of Mg²⁺ ions is similar for all the systems, although smaller values occur with systems 1, 4, 6, and 7, with systems 1, 6, and 7 being unstable systems. Interestingly, when the number of ions interacting with the bases is examined, the values are smaller for the unstable systems, 1, 6, and 7 as well as for system 4. These results further indicate that Mg²⁺ contributions to stability are not simply a condensation effect but include contributions from specific interactions with the bases.

TABLE I.

Total number of Mg²⁺ ions in the vicinity (4.2 Å) of the bases, NBPOs, and the full RNA for the nine systems over the entire metadynamics simulations. Results for the unstable systems are shown in bold.

	Bases		NBPOs		Entire RNA
System	Block average	S.E.	Block average	S.E.	Block average	S.E.
System 1	7.23	0.14	12.89	0.12	16.55	0.09
System 2	9.10	0.12	12.06	0.17	16.65	0.07
System 3	10.15	0.11	11.68	0.23	17.10	0.1
System 4	8.46	0.16	10.88	0.12	15.50	0.14
System 5	10.85	0.17	10.29	0.14	17.20	0.11
System 6	8.51	0.13	11.49	0.14	16.15	0.13
System 7	8.44	0.15	11.58	0.1	16.52	0.11
System 8	10.23	0.13	12.81	0.11	17.23	0.09
System 9	9.46	0.16	11.98	0.12	16.97	0.09

To better understand the role of the interactions with the bases, analysis was performed to identify Mg²⁺ specific binding sites that may contribute to the stabilization of the systems. This was done by calculating the occurrence of the Mg²⁺ ions within 4.2 Å of each base, with the results shown in Fig. 3. In the stable systems (systems 2, 3, 4, 5, 8, and 9), high occupancy Mg²⁺ ions are distributed throughout the structure, leading to the overall stabilization. In the unstable systems (systems 1, 6, and 7), the distribution is more sparse, consistent with the data in Table I, with Mg²⁺ localized around a subset of bases. Two of the stable systems, 2 and 4, have less sampling around the bases in comparison with the other stable systems. The decreased number of Mg²⁺ ions in the vicinity of the bases in the unstable systems while the overall number of ions around the entire RNA molecule is similar in all systems (Tables I and S2) indicates that specific ion–base interactions are contributing to the stabilization of the RNA.

FIG. 3.

Heat map representing the occupancy probability of the individual Mg²⁺ ions with the bases over the metadynamics simulations. The probability by the ion at or less than 4.2 Å is indicated by the heat map intensity, where 1 indicates that the ion was present for 100% of the simulation time. The nucleotides directly coordinated with Mg²⁺ through the NBPOs in the crystal structure (U6, U24, and C26) are indicated by pink vertical bars.

Bases with any Mg²⁺ within 4.2 Å for a minimum of 70% of the full simulation time are listed in Table II, and the probabilities of any Mg²⁺ being within 4.2 Å of all the bases is shown in Fig. S2 of the supplementary material. Table II includes those nucleotides within 5 Å of Mg²⁺ ions in the crystal structure of PDB 4OJI (Table S3). In the unstable systems (1, 6, and 7), the number of bases within 4.2 Å of Mg²⁺ is the smallest of the systems, with the exception of system 4 that has the same number of bases close to Mg²⁺ as 6 and 7, consistent with the results in Table I. However, with stable system 4, the base on U24, which participates in direct coordination in the crystal structure, is interacting with Mg²⁺. Although the direct coordination in the crystal is through an NBPO, the O4 of the U24 base is 4.34 Å from a hydrated Mg²⁺ in the crystal structure. These results suggest that a Mg²⁺ ion at this location yields additional stability. In the case of system 2, which is also stable but has a lower number of bases adjacent to Mg²⁺ (Table I), there is a diverse collection of bases close to Mg²⁺ for greater than 70% of the simulation (Table II), indicating that non-specific stable ion–base interactions are contributing to stability. In addition, interactions with the base of U6 are occurring. This may impart additional stability as direct coordination of Mg²⁺ with the NBPO of U6 occurs although the U6 base is unpaired, and the solvent exposed in the crystal structure. A similar scenario occurs with system 9 where the number of bases close to Mg²⁺ is relatively low (Table I), but a diverse collection of stable ion–base interactions occurs based on the 70% occurrence during the simulation (Table II). These results indicate the importance of base–Mg²⁺ interactions in stabilizing the RNA structure where the number of such interactions, their probability, and the presence of specific interactions appear to be contributed to stability.

TABLE II.

Bases in the vicinity of Mg²⁺ based on interacting with Mg²⁺ for at least 70% of the total simulation time using a 4.2 Å cutoff and block average and standard error on the total count of these Mg²⁺ ions. The Mg²⁺ direct coordination sites in the crystal structure are underlined, and the Mg²⁺ binding nucleotides within 5 Å of RNA non-hydrogen atoms are bold (Table S3 of the supplementary material).

System	Bases interacting with Mg2+ for more than 70% of simulation time	Number of Mg2+ ions within 4.2 Å of bases
	Mg2+ direct coordination sites: U6, U24, C26
Crystal structure	Mg2+ binding sites: C5, A8, A10, G22, G23, U24, C25, C26, A29, A41, G42, A46, G47, G49, G50	Block average	S.E.
1	G22, G35, G45	2.59	0.05
2	U6, G22, G34, G35, A36, G44, G45, G49, G50	5.15	0.14
3	C1, C2, C9, G34, G35, A39, G44, G45, G50, G53, G54	5.58	0.06
4	G22, G23, U24, G48, G50	3.20	0.07
5	C5, A10, C11, G22, C26, G34, U37, G42, U43, G50	5.34	0.09
6	G35, G45, G48, G49, G50	3.61	0.07
7	G22, G23, G42, G48, G49	2.91	0.06
8	A7, C9, G13, G34, G35, A46, G49, G50	5.90	0.10
9	U6, C15, G23, C25, C26, G34, G35, G45, G48, G49, G50	5.48	0.09

Of the stable systems, system 8 was the most stable followed by systems 2 and 3 based on both the time series of the overall RMSD (Fig. 2) and the tertiary contact distances (Fig. S1). As shown in our previous study, with the additive FF, this was associated with the presence of Mg²⁺–NBPO interactions with U6, C25, and C26 for which direct coordination occurs in the crystal structure. The pattern is maintained in the polarizable simulations for system 8 as is evident in Fig. 4, which shows the high probability interactions between the NBPOs and Mg²⁺. One-dimensional probabilities of any Mg²⁺ being within 4.2 Å of all the NBPOs are shown in Fig. S3. Notably, those nucleotides are participating in tertiary contacts T1 and T4, thereby facilitating the stabilization. In systems 2 and 3, while strong interactions specifically with U24 or C26 are not occurring, interactions with the NBPOs of other nucleotides in that region are occurring. With system 2, interactions are occurring with G23 located between T1 and T4 and Mg²⁺ is close to G30 and C31 that are involved on T2 [Fig. 1(a)]. In the case of system 3, high probability interactions are occurring with C20, C21, and G23 (Fig. S3), which will contribute to the stabilization of T4. Interactions with NBPOs on nucleotides 30–35 are also occurring in system 3, with some interactions with U24 and C26 (Figs. 4 and S3). Given that these nucleotides are in the vicinity of the T1 and T4 tertiary contacts, they are suggested to contribute to the stabilization of this system. In contrast, with systems 4 and 5, where some minor loss of the T4 contact occurs (Fig. S1), the probability of Mg²⁺ ions close to U6 and U24 is relatively low (Fig. S3) and, similarly, Mg²⁺ probabilities are low around nucleotides 13–14 and 30–31 with system 5, which are in the vicinity of contact T2; the lack of Mg²⁺ ions around these nucleotides may contribute to the partial loss of that contact during the metadynamics simulations. Overall, these results again indicate the importance of specific interactions of Mg²⁺ with the RNA to stabilize the overall tertiary structure, with many of those interactions not observed in the crystal structure of twister.

FIG. 4.

Heat map representing the occupancy probability of specific Mg²⁺ ions with the NBPOs over the metadynamics simulations. The probability by the ion at or less than 4.2 Å is indicated by the heat map intensity, where 1 indicates that the ion was present for 100% of the simulation time. The nucleotides directly coordinated with Mg²⁺ through the NBPOs in the crystal structure (U6, U24, and C26) are indicated by pink vertical bars.

In addition to the Mg²⁺–RNA interactions noted above that are not observed in the crystal structure of twister, additional interactions are occurring with system 4, which appear to play a role in its stabilization. There are a number of high probability interactions with the bases of nucleotides 42–50 (Figs. 3 and S2), of which some have not been identified as Mg²⁺ binding sites based on the 5 Å cutoff distance (Table II). In addition, two separate Mg²⁺ ions have high probabilities of interaction with the base and phosphate of nucleotide 34 in system 4, which may contribute to the stabilization of the hairpin occurring in the region at the bottom of the structure shown in Fig. 1(a). These additional interactions may help explain the stability of system 4 despite the lower number of Mg²⁺ within 4.2 Å of the bases as compared to the other stable systems discussed above (Tables I and II). Thus, the use of the oscillating chemical potential GCMC method to sample a wide distribution of ions around the RNA can facilitate the identification of ion binding sites that contribute to stability and are not present in the crystal structure. This is consistent with work by Lemkul et al. where the location of Mg²⁺ ion in a crystal that was not observed in the crystal structure was identified.⁶¹

Visual analysis of representative distributions of the Mg²⁺ ions within 5 Å of the bases or NBPOs from snapshots at 2 µs of the simulations for all the systems is shown in Fig. 5. In the stable system, there is a tendency for the Mg²⁺–RNA interactions to be more distributed throughout the RNA than in the unstable systems (1, 6, and 7). In unstable system 1, the ions interacting with specific nucleotides are all located close to the lower portion of the RNA largely away from the tertiary contacts. In systems 6 and 7, the ions are more distributed although both the T1 and T4 contacts are lost (Fig. S1). With 6, no ions are in the vicinity of nucleotides 25–29, which contribute to those contacts (see the lower right portion of the image), while in 7, there are ions present around the nucleotides contributing to those contacts. This suggests the possibility that some of the ions may contribute to the disruption of the contacts. The anomalous stable system 4 contains a significant number of ions close to the bases with an ion in the vicinity of the bases involving the T1 tertiary contact as well as addition ions adjacent to the phosphate on one strand of the T1 contact. Such a distribution may contribute to the stability of that tertiary contact contributing to the overall stability of this system despite the lack of specific Mg²⁺–base interactions. With the highly stable system 8, Mg²⁺ ions with close contacts with both bases and NBPOs are distributed throughout the core region of twister in the vicinity of all three tertiary contacts. When considering all the stable systems, as discussed above, Mg²⁺ ions are not present in all the crystallographic identified positions and variations in the overall distributions of the Mg²⁺ ions are evident, indicating that it is not necessary to have a well-defined pattern of Mg²⁺–RNA interactions to yield stable conformations.

FIG. 5.

Conformations from the 2 µs time frame of the metadynamics simulations of the nine systems. All Mg²⁺ ions within 5 Å of bases or NBPOs are shown. Specific Mg²⁺ ions within 4.2 Å of a base for >70% of the simulation time are shown as purple spheres, with the respective bases labeled in black. Specific Mg²⁺ ions within 4.2 Å of an NBPO for >70% of the simulation time are shown as red spheres, with the respective NBPO labeled in orange. The remaining Mg²⁺ ions are shown as pink spheres with the Mg²⁺ ions identified in the crystal structure shown in the upper left panel. The base and NBPO atoms involved in interaction are highlighted in the CPK representation, with the bases colored by the base type [Gua (yellow), Ade (blue), Cyt (peach), and Ura (green)]. Nucleotides defining the tertiary contacts are in red for T1 (C25–G49, C26–G48, C27–G47, and A28–A46), blue for T2 (G13–C31 and C14–G30), and green for T4 (adjacent NBPOs on U6 and C25) with the nucleotides in parentheses.

C. Metadynamics simulations of twister in the presence and absence of Mg²⁺ and explicit polarization

To verify the impact of electronic polarization and the presence of Mg²⁺ on the stability of twister, two additional metadynamics simulations were performed using system 8. This yields a total of four simulations, including (1) CHARMM36 additive FF with Mg²⁺, (2) CHARMM36 additive FF in the absence of Mg²⁺, (3) Drude polarizable FF with Mg²⁺, and (4) Drude polarizable FF in the absence of Mg²⁺. Simulations 1 and 3 correspond to those presented above in Fig. 2. In simulations 2 and 4 without Mg²⁺, the K⁺ concentration was adjusted to maintain a neutral system. The comparative analysis of stability is done by the RMSD analysis of the non-hydrogen backbone atoms, and the results are shown in Fig. 6.

FIG. 6.

RMS difference for the RNA backbone nonhydrogen atoms as a function of time from metadynamics simulations under the C36 and Drude FFs and in the absence and presence of Mg²⁺. Probability distributions over the full simulations are shown in the right panel. RMSD values vs the x-ray crystal structure, 4OJI, following modeling of the missing two nucleotides and energy minimization.

The additional metadynamics simulations support the important roles of Mg²⁺ and of explicit polarizability in stabilizing twister. A comparison of the Drude and additive C36 results both in the presence and in the absence of Mg²⁺ shows that the explicit polarization included in the Drude FF stabilizes the system, with conformations being sampled much closer to the crystal structure than with the additive FF. In addition, the inclusion of Mg²⁺ significantly stabilizes the systems with both the additive and Drude polarizable FFs. With the C36 additive FF, the absence of Mg²⁺ leads to RMSD values that are as large as those in any of the unstable additive simulations presented in Fig. 2. Similarly, the Drude structure deviates significantly from the crystal conformation in the absence of Mg²⁺ although the inclusion of the explicit polarization leads to smaller changes vs the additive FF for this particular system and time scale of the metadynamics simulation. This analysis further supports that Mg²⁺ and the explicit treatment of electronic polarizability are critical in stabilizing the tertiary structure of the twister RNA.

D. Dipole moment analysis

The present results along with our previous study on RNA hairpins⁷¹ indicate the explicit treatment of electronic polarizability to lead to improved stability of the RNA in the simulations. The previous RNA study, which did not include Mg²⁺ ions, showed variations in the dipole moments of the bases throughout the hairpins that were indicated to contribute to the enhanced stability, especially in the loop regions in which non-canonical base–base interactions occur. In addition, variations in dipoles have been observed with the Drude FF in G-quadraplexes and peptides.^85,86 Accordingly, analysis of the dipole moments in the present simulations was undertaken. Emphasis was placed on the individual bases, the phosphate moieties, and the Mg²⁺ ions. For analysis of the phosphates, they were all aligned to a common molecular frame allowing a direct comparison of the magnitude of the dipoles despite the charged nature of the moiety.

Average dipole moments of the phosphate moieties, the Mg²⁺ ions, and the bases are presented in Figs. S4, S5, and S6, respectively, of the supplementary material. The majority of phosphates have dipole moments of ∼3 Debye. However, in each system, a subset of phosphates have average dipoles that deviate from the others, typically with larger dipole moments and larger RMS fluctuations (Fig. S4). With the Mg²⁺ ions, the dipole moments in the majority of cases fluctuate around zero, consistent with the symmetric environment of the ions hexacoordinated with water. As with the phosphates, a small subset of ions do show significant increases in their dipole moment and RMS fluctuations vs those of the majority of such ions (Fig. S5). The dipole moment analysis for the bases is shown in Fig. S6. The variations between the different base types are evident, as expected. In addition, in specific cases, there are variations in the average values and RMS fluctuations of a subset of bases of a given type from the values occurring for the majority of each base type, as discussed below. As is evident, the explicit inclusion of polarizability offers the potential for the electron distribution of individual moieties in the RNA as well as the surrounding ions to vary in response to their specific environments, a phenomenon that cannot occur with an additive FF.

To understand the cause of higher dipole moments in selected phosphates, we focus on systems 5 and 6, representing a stable and an unstable system, respectively. Presented in Fig. 7 are the average dipole moments of phosphate moieties in the upper panels [(a) and (b)] and the probabilities of Mg²⁺⁻NBPO interactions within 4.2 Å [(c) and (d)]. As is evident, the larger phosphate dipoles correspond strongly to high probability Mg²⁺–phosphate interactions. In stable system 5, the larger phosphate dipoles at A8, C9, and G35 correspond to high probability Mg²⁺ interactions. Similarly, in system 6, the large phosphate dipole moments at C21, G42, and G47 correspond to large phosphate dipole moments. A close examination of Figs. S3 and S4 shows this pattern to consistently occur in all the systems. In the majority of cases, the large phosphate dipoles involve a direct coordination of NBPOs with Mg²⁺ as evidenced by the analysis of Fig. S7, which shows the probability of bases NBPOs interacting with Mg²⁺ within 2.5 Å. Thus, the presence of explicit treatment of electronic polarization is indicated to lead to the polarization of the phosphates largely due to direct coordination with Mg²⁺ ions.

FIG. 7.

Dipole moments of PO4 moieties for (a) system 5 (i.e., stable system) and (b) system 6 (i.e., unstable system). The probabilities of Mg²⁺ being within 4.2 Å of each NBPO during the entire metadynamics simulations for (c) system 5 and (d) system 6.

Variations in the dipole moment of the bases were next analyzed. The inspection of Fig. S6 shows the majority of bases to have average dipoles close to those of the other bases of the same type. However, some perturbed dipoles and larger RMS fluctuations are evident upon a careful inspection of Fig. S6. For those bases, analysis of Fig. 3 allows for identification of the Mg²⁺ ions interacting with them, with the dipoles of those ions shown in Fig. S5. In all cases, the bases with perturbed dipoles are interacting with the Mg²⁺ ions with large dipoles (Fig. S5), indicated in parentheses in the following sentence. These include increased base dipole moments in U6 in system 2 (Mg29); terminal bases G53 and G54 in system 3 (Mg31); G42, U43 (Mg30), G50, and G51 (Mg21) in system 5; U48 and U49 in both systems 7 (Mg17) and 9 (Mg16); and G34 and G35 in system 8 (Mg27). Analysis of the probability of bases listed in the previous sentence being within 2.5 Å of Mg²⁺ in Fig. S7 shows that they all participated in direct Mg²⁺–base coordination. Thus, the direct coordination of the bases with Mg²⁺ ions leads to enhanced base dipoles.

To better understand the interactions leading to the larger dipole moments in the Mg²⁺ ions, the results in Fig. S5 may be compared with the results in Fig. S8, in which the probability of Mg²⁺ interactions within 2.5 Å of the bases and NBPOs are shown. From those figures, the Mg²⁺ ions with enhanced dipoles or RMS fluctuations and the bases and NBPOs within 2.5 Å of Mg²⁺ ions were extracted and are presented in Table III. For all the systems, the number of bases plus NBPOs in direct coordination with Mg²⁺ is much greater than the number of Mg²⁺ ions with enhanced dipoles. As the majority of these bases (Fig. S6) or NBPOs (Fig. S4) have enhanced dipoles or increased RMS fluctuations of their dipoles due to interactions with Mg²⁺, the lack of the enhanced dipoles in some of the associated ions indicates that direct interactions do not necessarily lead to altered Mg²⁺ dipoles.

TABLE III.

Mg²⁺ ions with enhanced dipoles and bases and NBPOs with direct interactions with Mg²⁺ ions in the studied systems. NP indicates not present.

System	Mg2+ with enhanced dipoles	Bases <2.5 Å of Mg2+	NBPOs <2.5 Å of Mg2+
1	18, 33	NP	G23, C26, C32, A36
2	29	U6	G23, C32
3	10, 27, 31	G44, G45, G53, G54	C20, C21, C33, G34
4	NP	NP	G34
5	11, 21, 30, 32	U37, G42, U43, G50, G51	A8, C9, G35, A36
6	26	G49	C21, G42, G47
7	17	A46, G48, G49	C26, G45, G48
8	20, 22, 27	A7, G34, G35	U6, A7, A8, C9, C25, G30
9	16, 28	U6, G45, G48, G49	A29, G35, A36, A40

To understand when enhanced Mg²⁺ dipoles are occurring, the data in Table III can be analyzed in conjunction with the results in Figs. 3 and 4. In addition, the data in Fig. S7 show the bases and NBPOs in direct contact with Mg²⁺, while Fig. S8 shows the Mg²⁺ ions in direct contact with bases or NBPOs. Numerous specific interactions of Mg²⁺ with bases or NBPOs are evident although only a small number of Mg²⁺ ions have enhanced dipoles (Table III). For example, in system 1, multiple Mg²⁺ ions are within 4.2 Å of NBPOs (Fig. 4), but enhanced dipoles only occur with Mg18 directly coordinated with the NBPO of G23 and Mg33 with A36 (Fig. S7). With system 2, the enhanced dipole of Mg29 is associated with direct coordination with the base of U6 (Fig. S7). Multiple direct interactions are observed with both bases and NBPOs in system 3 (Fig. S7) with Mg10 interacting with the NBPO of C33, Mg27 in direct coordination with the bases of G44 and G45, and Mg31 in direct coordination with the bases of G53 and G54 (Figs. S7 and S8). However, although direct interactions of the NBPOs of C20 and C21 occur with Mg16 and the NBPO of G34 with Mg8 (Fig. 4), those ions do not have enhanced dipoles (Fig. S5). In system 8, Mg²⁺ ions 20, 22, and 27 have enhanced dipoles, while enhancement does not occur with ions 3, 5, 9, and 16. Mg22 has direct coordination with the base and NBPO of A7 and NBPO of U6, while Mg27 has direct interactions with the bases of G34 and G35. However, Mg20, while interacting within 4.2 Å of G13 and C14, direct coordination is not occurring, representing an exception to the other Mg²⁺ ions with enhanced dipoles. A similar pattern is observed in the other systems, where enhanced Mg²⁺ dipoles are related to direct coordination with bases or NBPOs in the majority of cases. However, in many cases, the direct interactions of Mg²⁺ ions shown in Fig. S8 do not lead to enhanced ion dipoles as only a subset of those Mg²⁺ have perturbed dipoles (Fig. 5 and Table III).

While the above analysis focused on the effect of Mg²⁺ ions on the stability of twister including details of the impact of the presence of the ion on the electronic structure of specific moieties in the RNA, additional analysis was undertaken to determine if variations in the water dipole moments were occurring. Presented in Fig. 8 is the distribution of the dipole moment of waters within 3.5 Å of the NBPOs and for all waters in simulation system 8. As is evident, water being adjacent to the phosphates leads to a significant increase in the dipole moments of a subset of those water molecules. This observation is consistent with previous studies in which variations in dipole moments in regions around nucleic acids have been observed using a polarizable FF, including shifts in the dipole moments of waters interacting with the base flipping out of DNA and in peptide simulations.^87,88 Interestingly, shifts in “solvent-site dipole moments” associated with small clusters of waters have been observed adjacent to DNA using the additive FF where the dipole moment shift is associated with the alignment of the waters in the cluster due to their interactions with the DNA.⁸⁹ In general, alterations of the electronic structure of water, along with those shown above for the RNA and the ions, will all contribute to the forces driving the conformational properties of molecular systems, emphasizing the importance of the use of polarizable models that allow for induced dipoles in individual molecules along with more global solvent-site dipole changes to be explicitly treated in molecular simulations.

FIG. 8.

Distribution of the dipole moments of SWM4 waters within 3.5 Å of NBPOs based on non-hydrogen atoms as well as throughout the entire system based on the system 8 metadynamics simulation. The analysis was done by taking 100 snapshots from the 4.8 µs simulation.

IV. FORCE FIELD LIMITATIONS

Empirical force fields require continual improvements as additional target data with which to validate them become available as well as improved algorithms and more computational power allow for increases in conformational sampling. While the polarizable Drude FF has now reached a reasonable level of maturity, its widespread use is still limited. In particular, while the optimization of the Drude Mg²⁺ parameters includes the optimization of atom-pair specific NBFIX and NBTHOLE terms for various moieties in nucleic acids,⁷² MD simulations of nucleic acids that include Mg²⁺ have only been performed where the Mg²⁺ ions identified in the crystal structure were included.⁷⁰ Thus, the present study represents the first application in which excess Mg²⁺ ions are present. While the improvements over the additive C36 FF for the treatment of RNA by the Drude model are evident from the results presented above, as well as the previous study on RNA hairpins,⁷¹ in the present calculations, direct interactions of Mg²⁺ with the O2′ and O4′ sugar atoms were identified. These occurred in all but systems 4 and 7 as shown in Fig. S9, showing the probability of Mg²⁺ having interactions with O2′ and O4′ atoms as a function of nucleotide. As may be seen, one high probability interaction is present in most cases, and such interactions are present in both stable and unstable systems. As such interactions are not common in crystal structures in our experience and our parameter optimization study did not explicitly address interactions of Mg²⁺ with the sugar moieties, it appears that these interactions are a FF artifact. However, these interactions may impact the present results that their occurrence is not solely associated with stable or unstable systems, indicating that the broader observations from the present study are not impacted. Ongoing efforts in our lab to address this issue building upon methods developed in the recent optimization of group 1 monovalent ions⁹⁰ are ongoing.

V. SUMMARY AND CONCLUSION

The stable folding of RNA into tertiary structures is highly responsive to both the concentrations and varieties of cations present. As such, the understanding of the physical mechanisms behind the interactions between ions and RNA is essential for accurately assessing the RNA stability.⁹¹ The alkali and alkaline earth metals, such as Na⁺, K⁺, Mg²⁺, and Ca²⁺, play vital roles in maintaining the stability, proper folding, and functioning of nucleic acids. Monovalent ions such as Na⁺ and K⁺ create an ionic atmosphere surrounding nucleic acids, which facilitates the condensation of the polyanions, thereby contributing to their stabilization.⁹² Divalent ions can selectively bind to certain regions of the nucleic acid, thereby enhancing the stability of tertiary structures,⁹¹ that is typically not done by monovalent ions.⁹³ To better understand these roles on an atomistic scale, we have applied a novel GCMC approach to generate unbiased distributions of ions around RNA followed by molecular simulations. Alternate approaches to ion placement are available, including 3D-RISM, which has been specifically applied to the twister ribozyme.^94–96 However, 3D-RISM identifies ion-binding sites on the RNA itself, while the GCMC approach also allows for ion distributions in the aqueous environment remote from the nucleic acid to be generated. Using the GCMC approach with umbrella sampling in previous studies on twister showed that the folding of RNA is increased when specific NBPO pairs interact with Mg²⁺, while other specific nonsequential phosphate pairs were simultaneously destabilized, thereby facilitating the overall folding of the RNA, in what we referred to as a push–pull mechanism.^97,98 Building on this, we applied the GCMC approach and metadynamics along with the CHARMM36 additive FF to show that the presence of Mg²⁺ ions in the vicinity of specific phosphates corresponding to those observed in the crystal structure, along with additional interactions not observed experimentally, was important to stabilize the structure of twister.⁴³

Building on that previous study and motivated by a separate study showing the Drude polarizable FF to improve the stability of several RNA hairpins,⁷¹ we undertook metadynamics simulations with the polarizable Drude FF. Starting with the nine systems from our previous study in which independent ion distributions were generated around twister using oscillating µ_ex GCMC, new reaction coordinates were generated using AMINO and RAVE and polarizable metadynamics simulations performed. The resulting trajectories were then compared with the previously reported results using the additive FF. In contrast to the simulations with the additive FF, the majority of the systems maintained conformations close to the folded state. This additional stabilization of the tertiary structure was shown to be associated with the interactions of the Mg²⁺ ions with both the bases and the phosphates. These involved interactions present in the crystal structure of twister as well as many additional, previously unidentified Mg²⁺–RNA interactions.

Detailed analysis of the electron distributions in the context of the Drude FF of the bases, phosphates, and Mg²⁺ ions showed specific changes in the dipole moments of these species. This included increases in the dipoles of phosphates due to interactions with Mg²⁺ although, interestingly, the dipole moments of the ions were typically not enhanced. This suggests that the coordination of the ions by water and NBPOs does not yield a strongly anisotropic electronic environment around the ion in many cases. In contrast, enhanced dipole moments in the Mg²⁺ ions occurred in a number of cases due to direct coordination with the bases, with the dipoles of the bases involved in those interactions also enhanced. Overall, the perturbation of the dipoles of these species is hypothesized to contribute to the improved stability of the twister RNA using the polarizable Drude FF.

The outcomes from the present study speak to the utility of using GCMC to generate multiple unbiased ion distributions around a macromolecule in conjunction with automated ML approaches to direct individual RC selection for each system. This combination partially overcomes the severe time scale challenges of investigating the role of Mg²⁺ on RNA stability and folding. This is due to the lifetimes of RNA–Mg²⁺ interactions approaching the millisecond time scale, while the conformational sampling of RNA for a given ion distribution is, at minimum, on the order of microseconds. In the present study, we have, to a small extent, overcome this challenge but only in the context of understanding the nature of the interaction of Mg²⁺ with RNA, which stabilizes its structure and how perturbation of the electronic structure of the RNA and ions contributes to the stabilization. However, moving toward investigating the impact of Mg²⁺ on the actual folding of RNA at the time and length resolution accessed in the present study represents a formidable challenge.

SUPPLEMENTARY MATERIAL

The supplementary material encompasses the following: details of the systems studied; evolution of tertiary contacts (T1, T2, T4); average number of Mg²⁺, K⁺, and Cl⁻ ions within 9 Å of the full twister RNA; nucleotides in the crystal structure contributing to Mg²⁺ binding sites; probabilities of Mg²⁺ being within 4.2 Å of each base; probabilities of Mg²⁺ being within 4.2 Å of each NBPO; dipole moments of the PO4 moieties; dipole moments of the Mg²⁺ ions; dipole moments of the bases; probabilities of bases and NBPOs being within 2.5 Å of Mg²⁺; probabilities of Mg²⁺ ions being within 2.5 Å of bases and NBPOs; and probabilities of O2′ and O4’ atoms being within 2.5 Å of Mg²⁺.

AUTHOR DECLARATIONS

Conflict of Interest

A.D.M. is the cofounder and CSO of SilcsBio LLC.

Author Contributions

Prabin Baral: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Mert Y. Sengul: Formal analysis (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Alexander D. MacKerell, Jr.: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study and the input files required to perform the simulations are available upon request.