Integral equation models for solvent in macromolecular crystals.

The solvent can occupy up to ∼70% of macromolecular crystals, and hence, having models that predict solvent distributions in periodic systems could improve the interpretation of crystallographic data. Yet, there are few implicit solvent models applicable to periodic solutes, and crystallographic structures are commonly solved assuming a flat solvent model. Here, we present a newly developed periodic version of the 3D-reference interaction site model (RISM) integral equation method that is able to solve efficiently and describe accurately water and ion distributions in periodic systems; the code can compute accurate gradients that can be used in minimizations or molecular dynamics simulations. The new method includes an extension of the Ornstein-Zernike equation needed to yield charge neutrality for charged solutes, which requires an additional contribution to the excess chemical potential that has not been previously identified; this is an important consideration for nucleic acids or any other charged system where most or all the counter- and co-ions are part of the "disordered" solvent. We present several calculations of proteins, RNAs, and small molecule crystals to show that x-ray scattering intensities and the solvent structure predicted by the periodic 3D-RISM solvent model are in closer agreement with the experiment than are intensities computed using the default flat solvent model in the refmac5 or phenix refinement programs, with the greatest improvement in the 2 to 4 Šrange. Prospects for incorporating integral equation models into crystallographic refinement are discussed.

Predicting Site-Binding Modes of Ions and Water to Nucleic Acids Using Molecular Solvation Theory.

Site binding of ions and water shapes nucleic acids folding, dynamics, and biological function, complementing the more diffuse, nonspecific "territorial" ion binding. Unlike territorial binding, prediction of site-specific binding to nucleic acids remains an unsolved challenge in computational biophysics. This work presents a new toolset based on the 3D-RISM molecular solvation theory and topological analysis that predicts cation and water site binding to nucleic acids. 3D-RISM is shown to accurately capture alkali cations and water binding to the central channel, transversal loops, and grooves of the Oxytricha nova's telomeres' G-quadruplex ( Oxy-GQ), in agreement with high-resolution crystallographic data. To improve the computed cation occupancy along the Oxy-GQ central channel, it was necessary to refine and validate new cation-oxygen parameters using structural and thermodynamic data available for crown ethers and ion channels. This single set of parameters that describes both localized and delocalized binding to various biological systems is used to gain insight into cation occupancy along the Oxy-GQ channel under various salt conditions. The paper concludes with prospects for extending the method to predict divalent cation binding to nucleic acids. This work advances the forefront of theoretical methods able to provide predictive insight into ion atmosphere effects on nucleic acids function.

Force Field for Mg(2+), Mn(2+), Zn(2+), and Cd(2+) Ions That Have Balanced Interactions with Nucleic Acids.

Divalent metal ions are of fundamental importance to the function and folding of nucleic acids. Divalent metal ion-nucleic acid interactions are complex in nature and include both territorial and site specific binding. Commonly employed nonbonded divalent ion models, however, are often parametrized against bulk ion properties and are subsequently utilized in biomolecular simulations without considering any data related to interactions at specific nucleic acid sites. Previously, we assessed the ability of 17 different nonbonded Mg(2+) ion models to reproduce different properties of Mg(2+) in aqueous solution including radial distribution functions, solvation free energies, water exchange rates, and translational diffusion coefficients. In the present work, we depart from the recently developed 12-6-4 potential models for divalent metal ions developed by Li and Merz and tune the pairwise parameters for Mg(2+), Mn(2+), Zn(2+), and Cd(2+) binding dimethyl phosphate, adenosine, and guanosine in order to reproduce experimental site specific binding free energies derived from potentiometric pH titration data. We further apply these parameters to investigate a metal ion migration previously proposed to occur during the catalytic reaction of the hammerhead ribozyme. The new parameters are shown to be accurate and balanced for nucleic acid binding in comparison with available experimental data and provide an important tool for molecular dynamics and free energy simulations of nucleic acids where these ions may exhibit different binding modes.

Cation-Anion Interactions within the Nucleic Acid Ion Atmosphere Revealed by Ion Counting.

The ion atmosphere is a critical structural, dynamic, and energetic component of nucleic acids that profoundly affects their interactions with proteins and ligands. Experimental methods that "count" the number of ions thermodynamically associated with the ion atmosphere allow dissection of energetic properties of the ion atmosphere, and thus provide direct comparison to theoretical results. Previous experiments have focused primarily on the cations that are attracted to nucleic acid polyanions, but have also showed that anions are excluded from the ion atmosphere. Herein, we have systematically explored the properties of anion exclusion, testing the zeroth-order model that anions of different identity are equally excluded due to electrostatic repulsion. Using a series of monovalent salts, we find, surprisingly, that the extent of anion exclusion and cation inclusion significantly depends on salt identity. The differences are prominent at higher concentrations and mirror trends in mean activity coefficients of the electrolyte solutions. Salts with lower activity coefficients exhibit greater accumulation of both cations and anions within the ion atmosphere, strongly suggesting that cation-anion correlation effects are present in the ion atmosphere and need to be accounted for to understand electrostatic interactions of nucleic acids. To test whether the effects of cation-anion correlations extend to nucleic acid kinetics and thermodynamics, we followed the folding of P4-P6, a domain of the Tetrahymena group I ribozyme, via single-molecule fluorescence resonance energy transfer in solutions with different salts. Solutions of identical concentration but lower activity gave slower and less favorable folding. Our results reveal hitherto unknown properties of the ion atmosphere and suggest possible roles of oriented ion pairs or anion-bridged cations in the ion atmosphere for electrolyte solutions of salts with reduced activity. Consideration of these new results leads to a reevaluation of the strengths and limitations of Poisson-Boltzmann theory and highlights the need for next-generation atomic-level models of the ion atmosphere.

Modulating RNA Alignment Using Directional Dynamic Kinks: Application in Determining an Atomic-Resolution Ensemble for a Hairpin using NMR Residual Dipolar Couplings.

Approaches that combine experimental data and computational molecular dynamics (MD) to determine atomic resolution ensembles of biomolecules require the measurement of abundant experimental data. NMR residual dipolar couplings (RDCs) carry rich dynamics information, however, difficulties in modulating overall alignment of nucleic acids have limited the ability to fully extract this information. We present a strategy for modulating RNA alignment that is based on introducing variable dynamic kinks in terminal helices. With this strategy, we measured seven sets of RDCs in a cUUCGg apical loop and used this rich data set to test the accuracy of an 0.8 µs MD simulation computed using the Amber ff10 force field as well as to determine an atomic resolution ensemble. The MD-generated ensemble quantitatively reproduces the measured RDCs, but selection of a sub-ensemble was required to satisfy the RDCs within error. The largest discrepancies between the RDC-selected and MD-generated ensembles are observed for the most flexible loop residues and backbone angles connecting the loop to the helix, with the RDC-selected ensemble resulting in more uniform dynamics. Comparison of the RDC-selected ensemble with NMR spin relaxation data suggests that the dynamics occurs on the ps-ns time scales as verified by measurements of R(1ρ) relaxation-dispersion data. The RDC-satisfying ensemble samples many conformations adopted by the hairpin in crystal structures indicating that intrinsic plasticity may play important roles in conformational adaptation. The approach presented here can be applied to test nucleic acid force fields and to characterize dynamics in diverse RNA motifs at atomic resolution.

Competitive interaction of monovalent cations with DNA from 3D-RISM.

The composition of the ion atmosphere surrounding nucleic acids affects their folding, condensation and binding to other molecules. It is thus of fundamental importance to gain predictive insight into the formation of the ion atmosphere and thermodynamic consequences when varying ionic conditions. An early step toward this goal is to benchmark computational models against quantitative experimental measurements. Herein, we test the ability of the three dimensional reference interaction site model (3D-RISM) to reproduce preferential interaction parameters determined from ion counting (IC) experiments for mixed alkali chlorides and dsDNA. Calculations agree well with experiment with slight deviations for salt concentrations >200 mM and capture the observed trend where the extent of cation accumulation around the DNA varies inversely with its ionic size. Ion distributions indicate that the smaller, more competitive cations accumulate to a greater extent near the phosphoryl groups, penetrating deeper into the grooves. In accord with experiment, calculated IC profiles do not vary with sequence, although the predicted ion distributions in the grooves are sequence and ion size dependent. Calculations on other nucleic acid conformations predict that the variation in linear charge density has a minor effect on the extent of cation competition.

Structural fidelity and NMR relaxation analysis in a prototype RNA hairpin.

RNA hairpins are widespread and very stable motifs that contribute decisively to RNA folding and biological function. The GTP1G2C3A4C5U6U7C8G9G10U11G12C13C14 construct (with a central UUCG tetraloop) has been extensively studied by solution NMR, and offers and excellent opportunity to evaluate the structure and dynamical description afforded by molecular dynamics (MD) simulations. Here, we compare average structural parameters and NMR relaxation rates estimated from a series of multiple independent explicit solvent MD simulations using the two most recent RNA AMBER force fields (ff99 and ff10). Predicted overall tumbling times are ∼20% faster than those inferred from analysis of NMR data and follow the same trend when temperature and ionic strength is varied. The Watson-Crick stem and the "canonical" UUCG loop structure are maintained in most simulations including the characteristic syn conformation along the glycosidic bond of G9, although some key hydrogen bonds in the loop are partially disrupted. Our analysis pinpoints G9-G10 backbone conformations as a locus of discrepancies between experiment and simulation. In general the results for the more recent force-field parameters (ff10) are closer to experiment than those for the older ones (ff99). This work provides a comprehensive and detailed comparison of state of the art MD simulations against a wide variety of solution NMR measurements.

Comparison of structural, thermodynamic, kinetic and mass transport properties of Mg(2+) ion models commonly used in biomolecular simulations.

The prevalence of Mg(2+) ions in biology and their essential role in nucleic acid structure and function has motivated the development of various Mg(2+) ion models for use in molecular simulations. Currently, the most widely used models in biomolecular simulations represent a nonbonded metal ion as an ion-centered point charge surrounded by a nonelectrostatic pairwise potential that takes into account dispersion interactions and exchange effects that give rise to the ion's excluded volume. One strategy toward developing improved models for biomolecular simulations is to first identify a Mg(2+) model that is consistent with the simulation force fields that closely reproduces a range of properties in aqueous solution, and then, in a second step, balance the ion-water and ion-solute interactions by tuning parameters in a pairwise fashion where necessary. The present work addresses the first step in which we compare 17 different nonbonded single-site Mg(2+) ion models with respect to their ability to simultaneously reproduce structural, thermodynamic, kinetic and mass transport properties in aqueous solution. None of the models based on a 12-6 nonelectrostatic nonbonded potential was able to reproduce the experimental radial distribution function, solvation free energy, exchange barrier and diffusion constant. The models based on a 12-6-4 potential offered improvement, and one model in particular, in conjunction with the SPC/E water model, performed exceptionally well for all properties. The results reported here establish useful benchmark calculations for Mg(2+) ion models that provide insight into the origin of the behavior in aqueous solution, and may aid in the development of next-generation models that target specific binding sites in biomolecules.

Multiscale methods for computational RNA enzymology.

RNA catalysis is of fundamental importance to biology and yet remains ill-understood due to its complex nature. The multidimensional "problem space" of RNA catalysis includes both local and global conformational rearrangements, changes in the ion atmosphere around nucleic acids and metal ion binding, dependence on potentially correlated protonation states of key residues, and bond breaking/forming in the chemical steps of the reaction. The goal of this chapter is to summarize and apply multiscale modeling methods in an effort to target the different parts of the RNA catalysis problem space while also addressing the limitations and pitfalls of these methods. Classical molecular dynamics simulations, reference interaction site model calculations, constant pH molecular dynamics (CpHMD) simulations, Hamiltonian replica exchange molecular dynamics, and quantum mechanical/molecular mechanical simulations will be discussed in the context of the study of RNA backbone cleavage transesterification. This reaction is catalyzed by both RNA and protein enzymes, and here we examine the different mechanistic strategies taken by the hepatitis delta virus ribozyme and RNase A.

Ion counting from explicit-solvent simulations and 3D-RISM.

The ionic atmosphere around nucleic acids remains only partially understood at atomic-level detail. Ion counting (IC) experiments provide a quantitative measure of the ionic atmosphere around nucleic acids and, as such, are a natural route for testing quantitative theoretical approaches. In this article, we replicate IC experiments involving duplex DNA in NaCl(aq) using molecular dynamics (MD) simulation, the three-dimensional reference interaction site model (3D-RISM), and nonlinear Poisson-Boltzmann (NLPB) calculations and test against recent buffer-equilibration atomic emission spectroscopy measurements. Further, we outline the statistical mechanical basis for interpreting IC experiments and clarify the use of specific concentration scales. Near physiological concentrations, MD simulation and 3D-RISM estimates are close to experimental results, but at higher concentrations (>0.7 M), both methods underestimate the number of condensed cations and overestimate the number of excluded anions. The effect of DNA charge on ion and water atmosphere extends 20-25 Å from its surface, yielding layered density profiles. Overall, ion distributions from 3D-RISMs are relatively close to those from corresponding MD simulations, but with less Na(+) binding in grooves and tighter binding to phosphates. NLPB calculations, on the other hand, systematically underestimate the number of condensed cations at almost all concentrations and yield nearly structureless ion distributions that are qualitatively distinct from those generated by both MD simulation and 3D-RISM. These results suggest that MD simulation and 3D-RISM may be further developed to provide quantitative insight into the characterization of the ion atmosphere around nucleic acids and their effect on structure and stability.

Bridging the gap between theory and experiment to derive a detailed understanding of hammerhead ribozyme catalysis.

Herein we summarize our progress toward the understanding of hammerhead ribozyme (HHR) catalysis through a multiscale simulation strategy. Simulation results collectively paint a picture of HHR catalysis: HHR first folds to form an electronegative active site pocket to recruit a threshold occupation of cationic charges, either a Mg(2+) ion or multiple monovalent cations. Catalytically active conformations that have good in-line fitness are supported by specific metal ion coordination patterns that involve either a bridging Mg(2+) ion or multiple Na(+) ions, one of which is also in a bridging coordination pattern. In the case of a single Mg(2+) ion bound in the active site, the Mg(2+) ion undergoes a migration that is coupled with deprotonation of the nucleophile (C17:O2'). As the reaction proceeds, the Mg(2+) ion stabilizes the accumulating charge of the leaving group and significantly increases the general acid ability of G8:O2'. Further computational mutagenesis simulations suggest that the disruptions due to mutations may severely impact HHR catalysis at different stages of the reaction. Catalytic mechanisms supported by the simulation results are consistent with available structural and biochemical experiments, and together they advance our understanding of HHR catalysis.

A variational linear-scaling framework to build practical, efficient next-generation orbital-based quantum force fields.

We introduce a new hybrid molecular orbital/density-functional modified divide-and-conquer (mDC) approach that allows the linear-scaling calculation of very large quantum systems. The method provides a powerful framework from which linear-scaling force fields for molecular simulations can be developed. The method is variational in the energy, and has simple, analytic gradients and essentially no break-even point with respect to the corresponding full electronic structure calculation. Furthermore, the new approach allows intermolecular forces to be properly balanced such that non-bonded interactions can be treated, in some cases, to much higher accuracy than the full calculation. The approach is illustrated using the second-order self-consistent charge density-functional tight-binding model (DFTB2). Using this model as a base Hamiltonian, the new mDC approach is applied to a series of water systems, where results show that geometries and interaction energies between water molecules are greatly improved relative to full DFTB2. In order to achieve substantial improvement in the accuracy of intermolecular binding energies and hydrogen bonded cluster geometries, it was necessary to extend the DFTB2 model to higher-order atom-centered multipoles for the second-order self-consistent intermolecular electrostatic term. Using generalized, linear-scaling electrostatic methods, timings demonstrate that the method is able to calculate a water system of 3000 atoms in less than half of a second, and systems of up to one million atoms in only a few minutes using a conventional desktop workstation.

Mapping L1 ligase ribozyme conformational switch.

L1 ligase (L1L) molecular switch is an in vitro optimized synthetic allosteric ribozyme that catalyzes the regioselective formation of a 5'-to-3' phosphodiester bond, a reaction for which there is no known naturally occurring RNA catalyst. L1L serves as a proof of principle that RNA can catalyze a critical reaction for prebiotic RNA self-replication according to the RNA world hypothesis. L1L crystal structure captures two distinct conformations that differ by a reorientation of one of the stems by around 80Å and are presumed to correspond to the active and inactive state, respectively. It is of great interest to understand the nature of these two states in solution and the pathway for their interconversion. In this study, we use explicit solvent molecular simulation together with a novel enhanced sampling method that utilizes concepts from network theory to map out the conformational transition between active and inactive states of L1L. We find that the overall switching mechanism can be described as a three-state/two-step process. The first step involves a large-amplitude swing that reorients stem C. The second step involves the allosteric activation of the catalytic site through distant contacts with stem C. Using a conformational space network representation of the L1L switch transition, it is shown that the connection between the three states follows different topographical patterns: the stem C swing step passes through a narrow region of the conformational space network, whereas the allosteric activation step covers a much wider region and a more diverse set of pathways through the network.

Identification of dynamical hinge points of the L1 ligase molecular switch.

The L1 ligase is an in vitro selected ribozyme that uses a noncanonically base-paired ligation site to catalyze regioselectively and regiospecifically the 5' to 3' phosphodiester bond ligation, a reaction relevant to origin of life hypotheses that invoke an RNA world scenario. The L1 ligase crystal structure revealed two different conformational states that were proposed to represent the active and inactive forms. It remains an open question as to what degree these two conformers persist as stable conformational intermediates in solution, and along what pathway are they able to interconvert. To explore these questions, we have performed a series of molecular dynamics simulations in explicit solvent of the inactive-active conformational switch in L1 ligase. Four simulations were performed departing from both conformers in both the reactant and product states, in addition to a simulation where local unfolding in the active state was induced. From these simulations, along with crystallographic data, a set of four virtual torsion angles that span two evolutionarily conserved and restricted regions were identified as dynamical hinge points in the conformational switch transition. The ligation site visits three distinct states characterized by hydrogen bond patterns that are correlated with the formation of specific contacts that may promote catalysis. The insights gained from these simulations contribute to a more detailed understanding of the coupled catalytic/conformational switch mechanism of L1 ligase that may facilitate the design and engineering of new catalytic riboswitches.

Threshold occupancy and specific cation binding modes in the hammerhead ribozyme active site are required for active conformation.

The relationship between formation of active in-line attack conformations and monovalent (Na(+)) and divalent (Mg(2+)) metal ion binding in hammerhead ribozyme (HHR) has been explored with molecular dynamics simulations. To stabilize repulsions between negatively charged groups, different requirements of the threshold occupancy of metal ions were observed in the reactant and activated precursor states both in the presence and in the absence of a Mg(2+) in the active site. Specific bridging coordination patterns of the ions are correlated with the formation of active in-line attack conformations and can be accommodated in both cases. Furthermore, simulation results suggest that the HHR folds to form an electronegative recruiting pocket that attracts high local concentrations of positive charge. The present simulations help to reconcile experiments that probe the metal ion sensitivity of HHR catalysis and support the supposition that Mg(2+), in addition to stabilizing active conformations, plays a specific chemical role in catalysis.

Role of Mg2+ in hammerhead ribozyme catalysis from molecular simulation.

Molecular dynamics simulations have been performed to investigate the role of Mg2+ in the full-length hammerhead ribozyme cleavage reaction. In particular, the aim of this work is to characterize the binding mode and conformational events that give rise to catalytically active conformations and stabilization of the transition state. Toward this end, a series of eight 12 ns molecular dynamics simulations have been performed with different divalent metal binding occupations for the reactant, early and late transition state using recently developed force field parameters for metal ions and reactive intermediates in RNA catalysis. In addition, hybrid QM/MM calculations of the early and late transition state were performed to study the proton-transfer step in general acid catalysis that is facilitated by the catalytic Mg2+ ion. The simulations suggest that Mg2+ is profoundly involved in the hammerhead ribozyme mechanism both at structural and catalytic levels. Binding of Mg2+ in the active site plays a key structural role in the stabilization of stem I and II and to facilitate formation of near attack conformations and interactions between the nucleophile and G12, the implicated general base catalyst. In the transition state, Mg2+ binds in a bridging position where it stabilizes the accumulated charge of the leaving group while interacting with the 2'OH of G8, the implicated general acid catalyst. The QM/MM simulations provide support that, in the late transition state, the 2'OH of G8 can transfer a proton to the leaving group while directly coordinating the bridging Mg2+ ion. The present study provides evidence for the role of Mg2+ in hammerhead ribozyme catalysis. The proposed simulation model reconciles the interpretation of available experimental structural and biochemical data, and provides a starting point for more detailed investigation of the chemical reaction path with combined QM/MM methods.