Molecular Simulations of RNA 2′- O -Transesterification Reaction Models in Solution

Molecular Simulations of RNA 2′‑O‑Transesterification ReactionModels in SolutionBrian K. Radak,†,‡ Michael E. Harris,§ and Darrin M. York*,†

†BioMaPS Institute and Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey 08854-8087,United States‡Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States§Department of Biochemistry, Case Western Reserve University School of Medicine, Cleveland, Ohio 44106, United States

*S Supporting Information

ABSTRACT: We employ quantum mechanical/molecular mechanical um-brella sampling simulations to probe the free energy surfaces of a series ofincreasingly complex reaction models of RNA 2′-O-transesterification inaqueous solution under alkaline conditions. Such models are valuable forunderstanding the uncatalyzed processes underlying catalytic cleavage of thephosphodiester backbone of RNA, a reaction of fundamental importance inbiology. The chemically reactive atoms are modeled by the AM1/d-PhoTquantum model for phosphoryl transfer, whereas the aqueous solvationenvironment is modeled with a molecular mechanics force field. Severalsimulation protocols were compared that used different ionic conditions andforce field models. The results provide insight into how variation of thestructural environment of the nucleophile and leaving group affects the freeenergy profile for the transesterification reaction. Results for a simple RNAbackbone model are compared with recent experiments by Harris et al. on thespecific base-catalyzed cleavage of a UpG dinucleotide. The calculated and measured free energies of activation match extremelywell (ΔF⧧ = 19.9−20.8 vs 19.9 kcal/mol). Solvation is seen to play a crucial role and is characterized by a network of hydrogenbonds that envelopes the pentacoordinate dianionic phosphorane transition state and provides preferential stabilization relativeto the reactant state.

■ INTRODUCTION

Cleavage of the phosphodiester backbone of RNA is anessential reaction in biology that is fundamental to manyimportant biological processes ranging from gene splicing andregulation to viral replication and cell signaling.1 It is thussignificant that several small RNA molecules, such as thehammerhead,2 hairpin,3 hepatitis δ virus,4 Varkud satellite,5 andglmS6 ribozymes, catalyze the phosphoryl cleavage of their ownbackbones. While the secondary and tertiary structures of theseribozymes are all distinct and their optimal ion identity andconcentration requirements differ significantly, they all catalyzethe same intramolecular 2′-O-transesterification reaction andform a 2′,3′-cyclic phosphate and 5′-hydroxyl as products.7−9As with all catalytic reactions, the mechanistic features of

ribozymes and protein enzymes are inherently related to theirrate enhancement relative to the background rate of the non-catalytic reaction, in this case the cleavage of an RNA backbonein aqueous solution. Hence, a logical starting point fordetermining the key mechanistic characteristics of self-cleavingribozymes would be first to determine the characteristics of theuncatalyzed mechanism. Indeed, there is an establishedliterature concerning model compounds for phosphate 2′-O-transesterification,10−14 but these studies frequently focus onnon-native, enhanced leaving groups,15−17 reactions perturbed

by chemical markers needed for spectroscopic analysis,18 ortemperature ranges far from normal biological conditions.15,19

Nonetheless, these studies provide a firm experimental baselinefor comparisons between native uncatalyzed and catalyzedreactions.Theoretical and computational approaches, particularly

molecular dynamics (MD), have emerged as a valuable toolin the study of chemical reactions because they allow access tofull atomistic detail. However, the degree to which meaningfulinsights into mechanism can be gained from simulations relieson the accuracy of the models that are employed for theparticular system under study. This leads to a natural synergisticrelationship between experiment and theory, with experimentproviding key benchmarks and theory providing detailedmolecular level interpretations and testable predictions. Thisapproach is well illustrated by the development of fast andaccurate quantum mechanical/molecular mechanical (QM/MM) methods calibrated to specific experimental and ab initiodata,20 which has opened the door for accurate tests of explicitreaction pathways, even for large biomolecules.21,22 The free

Received: August 24, 2012Revised: December 5, 2012Published: December 6, 2012

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© 2012 American Chemical Society 94 dx.doi.org/10.1021/jp3084277 | J. Phys. Chem. B 2013, 117, 94−103

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energy surfaces of such pathways can be rigorously compared toexperimental kinetics measurements and also have strongconnections to highly sensitive mechanistic probes such as thioand isotope23 as well as mutational24 effects.To rationally decompose the complexity of RNA backbone

cleavage, the present work focuses on mapping the free energyprofiles of a series of molecules undergoing transphosphor-ylation to form cyclic phosphates under basic conditions(Figure 1). The systems are akin to the simple UpG

dinucleotide recently studied experimentally by Harris et al.25

The dinucleotide UpG matches the sequence at the cleavagesite of the self-cleaving hepatitis δ virus ribozyme26 and is anactive substrate for ribonuclease A.27 Its cleavage mechanism insolution is therefore a valuable benchmark. In solution, theobserved first-order rate constant for UpG cleavage increaseslinearly from pH 10 to 13, becoming pH independent beyondthat point; extrapolating to “infinite” pH gives an intrinsic rateconstant of 0.06 s−1.25 Using novel techniques,25,28 Harris et al.also measured primary and secondary kinetic isotope effects(KIEs) for base-catalyzed UpG cleavage, as well as the solventD2O effect. The lack of any significant solvent D2O effectcombined with an estimated correction for equilibrium isotopeeffects on the nucleophile confirms the conventionally acceptedspecific base mechanism. Taken together, the KIEs suggest thatUpG undergoes a concerted mechanism with a “late”,productlike transition state.On the basis of QM/MM MD simulations, a theoretical free

energy of activation is calculated here that allows directcomparison to the experiments of Harris et al. The quality ofthe comparison begets significant confidence in also using thesimulations to characterize the reactant and transition statesstructurally, as well as analyze the role of water and ions in thereaction. Additionally, because of the wide range of modelsavailable for (and commonly used in) QM/MM studies, thesensitivity of the results to different water models andLennard−Jones parameters is compared and discussed.

■ COMPUTATIONAL METHODSMolecular Dynamics. MD simulations were performed

using the AMBER 1229 suite of programs. An integration step

of either 1 or 2 fs was used depending on whether or not theSHAKE30 algorithm (tolerance = 1.0 × 10−8) was used toconstrain bonds with hydrogen in the solute; the SETTLE31

algorithm was always used to constrain rigid water molecules.Temperature and pressure were regulated with the methods ofAndersen32 (310 K, “massive” collisions every 2000 steps) andBerendsen33 (1 bar, time constant of 5 ps, compressibility =44.6 × 10−6 bar−1), respectively. Long-range electrostatics weretreated using periodic boundary conditions in a rhombicdodecahedral cell and the particle mesh Ewald (PME) methodfor both MM34,35 and QM/MM36,37 calculations. PMEcalculations employed sixth-order B-spline interpolation with50 grid points (≈1 point/Å) along each axis; the Ewaldcoefficient was chosen such that the estimated error in thedirect space energy was on the order of 10−5 kcal/mol. QM/MM Ewald calculations used a reciprocal space defined by kmax= 7 and kmax

2 = 98; the QM/MM Ewald coefficient wasseparately chosen as 10V−1/3, where V is the cell volume in Å3

(see the Supporting Information for details). Lennard−Jonesand direct space Coulombic interactions were truncated at 10Å. For the QM/MM direct space, an atom-based switchingfunction was applied between 8 and 10 Å.All QM/MM simulations used the AM1/d-PhoT semi-

empirical Hamiltonian,20 with the QM region defined as theentire solute. Lennard−Jones parameters were taken fromeither the AMBER FF1038−41 or the CHARMM2742,43 nucleicacid force fields, with the exception of select interactions withsodium ions (see the Discussion and Supporting Information).The solvent environment was modeled using either theTIP3P44 or the TIP4P-Ew45 rigid water model and theassociated alkali metal and halide ion parameters of Joungand Cheatham.46 Simulations contained 2640 solvent mole-cules (e.g., water molecules or water molecules in approx-imately 140 mM NaCl; see the Supporting Information).The selected model is appropriate for several reasons, and

similar QM/MM models have been successfully used elsewherein studies of both enzymatic21,22,47 and nonenzymatic48,49

phosphoryl transfer. First, AM1/d-PhoT is specifically para-metrized to reproduce gas phase ab initio calculations of anextensive set of phosphate-containing compounds and reactions(see refs 20 and 50 and the Supporting Information) and hasalso been shown to be the best choice for reproducinggeometries and energies of penta-coordinated phosphorussystems among several common semiempirical methods.51

Second, a QM/MM approach to solvation (i.e., neglecting aQM description of the solvent) is well suited for the currentapplication since chemical participation of water (e.g., viahydrolysis or proton transfer) is not expected to occur.13,25 Anintermediate description including some water in the QMregion in an adaptive fashion (a necessary consequence ofdiffusion in a fully solvent-exposed reaction) could potentiallybe advantageous, but such an approach is difficult to implementwith smooth gradients suitable for dynamics and not yet widelyavailable.52 Therefore, at present, a QM/MM model is expectedto be preferable to a full QM description due to the vastlydecreased cost needed to obtain adequate sampling and sincethe bulk properties of MM water models are generally superiorto both semiempirical53 and even certain ab initio quantummodels.54

Umbrella Sampling. QM/MM MD umbrella sampling55

trajectories were performed along a mass-weighted atomtransfer coordinate, ξ = (1/2)(rP−O5′ − rP−O2′), where rA−B isthe distance between atoms A and B and the factor of one half

Figure 1. Reaction models for base-catalyzed phosphoryl transfer.Structural complexity increases from model 1 (2-hydroxy-ethyl ethylphosphate) to model 4 (an abasic RNA dinucleotide) to bridge thegap between the simple alkyl phosphates and the more complex RNAbackbone. Ribose ring-naming conventions are adopted for consis-tency such that all systems are said to undergo cleavage of the P−O5′bond, yielding a primary alkoxide (blue), and formation of a P−O2′bond, yielding a cyclic phosphate (red).

The Journal of Physical Chemistry B Article

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arises from the leaving group and nucleophile masses being thesame (it should be noted that, in the present case, this has noeffect on the thermodynamics). Although such a coordinate hasbeen widely used in the literature, especially with regard tophosphoryl transfer reactions,21−23,47−49,56 a recent study byRosta et al. has suggested that the calculated free energy barrieris potentially sensitive to this choice properly capturingorthogonal chemical events such as proton transfer.57 This isnot anticipated to be an issue here because the reactions takeplace in the high pH regime where the assumption of rapid,uncoupled deprotonation of the nucleophile is well-justified.25

The orthogonal events are thus entirely structural and notchemical (i.e., they involve solvent rearrangement).After an extensive initial equilibration protocol (see the

Supporting Information), production consisted of 1 ns (2 ns forthe dinucleotide system) for each window, with samplingomitting the first 250 ps for relaxation/equilibration within thewindow. The value of the progress coordinate was stored at 0.5ps intervals for analysis using the multistate Bennett acceptanceratio (MBAR).58 Approximately uncorrelated data sets wereobtained by subsampling configurations at intervals equal to thestatistical inefficiencies, which were estimated in eachsimulation by direct integration of the autocorrelation functionof the progress coordinate using the fast, adaptive integrationscheme of Chodera et al.59

MBAR Analysis and Free Energy Profiles. MBARprovides a general formalism for reweighting mechanicalobservables for estimation in arbitrary thermodynamic statesprovided that the relative statistical weight in those states isknown. The MBAR estimator for the expectation of anobservable, ⟨A⟩, that depends only on the configuration, x, isgiven by:58

∑ ∑

∑

=

≡

= =

=

− − −

A e w A

w N e

x x

x

( ) ( );

( ) [ ]

f

m

M

n

N

mn mn

mnl

M

lf u ux x

1 1

1

[ ( ) ( )] 1

m

l l mn mn

(1)

where M is the number of states, Nm is the number of samplesfrom state m, fl is the MBAR estimate of the free energy of statel relative to an arbitrary state (here, f1 ≡ 0), and ul(x) ≡ βUl(x)is the “reduced potential” characterizing state l.58,60 β and U(x)are the inverse temperature (for simplicity assumed to be thesame in all states) and potential energy. This expression is quitegeneral, but in the present context of umbrella sampling, theunindexed values refer to the unbiased state, and the sampleconfigurations, xmn, are drawn from M biased states.One method of estimating the free energy profile, F(ξ), is to

estimate the marginal distribution, ρ(ξ) = ⟨δ(ξ(x) − ξ)⟩; thefree energy profile, up to an additive constant, is then simplyF(ξ) = (−1/β)ln ρ(ξ). Because the δ function is only a functionin the distributional sense, an approximate estimator is neededfor finite sampling. A broad class of such estimators are knownin the statistics literature as kernel density estimators61−64 andmay take the following form:

δ ξ ξ ξ ξ⟨ − ⟩ = −→

⎛⎝⎜

⎞⎠⎟h

Kh

xx

( ( ) ) lim1 ( )

h 0 (2)

The function K is often referred to as a kernel and theparameter h as the bandwidth. A case more common to thechemical literature is when K is an indicator function;59,65,66

this returns the familiar histogram estimator, and h isrecognized as the bin width, with an additional parameterdefining the bin center. The results obtained with a histogramestimator are often qualitatively, and even quantitatively, similarto those obtained using a kernel density estimator. This generaltrend is confirmed in the present work (see the SupportingInformation). The marginal distribution could also becalculated using other estimators, such as those with aparametric form,67,68 although this may require slight variationof the MBAR formalism.To obtain a kernel-based estimator for F(ξ), eq 2 is

substituted into eq 1, and the logarithm is taken

∑ ∑β ξ ξ

ξξ ξ

− = +

≡−

= =

⎛⎝⎜

⎞⎠⎟

F f w K h

K hh

Kh

x x

xx

( ) ln ( ) ( ; , );

( ; , )1 ( )

m

M

n

N

mn mn m

mn mm

mn

m

1 1

m

(3)

As noted above, the additive constant f can be arbitrarily set tozero (or any other convenient value). This work employs astandard normal kernel density estimator with the bandwidthchosen in each window as twice that given by the data-basedalgorithm of Sheather and Jones69 (see the SupportingInformation); hence, the bandwidth is shown to vary amongstates.A more unusual class of observables can be defined as

expectations along a coordinate ξ:

δ ξ ξδ ξ ξ

⟨ ⟩ = ⟨ − ⟩⟨ − ⟩ξAA x

x( ( ) )

( ( ) ) (4)

Note that eq 4 is expressed as a ratio of expectations in anunconstrained ensemble, rather than as an expectation in aconstrained ensemble (i.e., the momentum conjugate to ξ isnonzero). Following a similar process as above and recognizingthe denominator as being related to eq 3, the followingestimator is obtained

∑ ∑ξ ξ = ×β ξ +

= =

A e w A K hx x x( ) ( ) ( ) ( ; , )F f

m

M

n

N

mn mn mn m( )

1 1

m

(5)

Note that in this context f is not arbitrary, although it could bemade zero in certain contexts.All of the estimators shown here were implemented in a

locally modified version of the Python MBAR implementationby Shirts and Chodera (pymbar v2.0).58 Visualization and otheranalyses were performed using Visual Molecular Dynamics(v1.8.7),70 particularly the VolMap plugin (default, unscaledradii, 0.5 Å resolution).

■ RESULTSValidation of QM/MM Model and Reaction Coordi-

nate. Both semiempirical quantum methods and approximatereaction coordinates require some degree of caution when usedin simulations; both can lead to significant deviation fromphysical behavior. In addition to the extensive validation anduse in the literature of AM1/d-PhoT in conjunction with thesimple atom transfer coordinate used here,21−23,47−49,56 webriefly present potential energy profiles comparing AM1/d-PhoT to standard B3LYP results as well as the reactioncoordinate paths with the optimized stationary points. Forsimplicity, the gas-phase 2′-O-transesterification reaction of 2-



hydroxy ethyl phosphate (similar to model 1 in Figure 1) isconsidered. As is clear from Figure 2 and Table 1, AM1/d-

PhoT provides excellent agreement with B3LYP/6-31+G(d), inboth the energy barrier and the geometry and location ofminima and saddle points. In all cases, the approximate reactioncoordinate correctly follows the reaction progress from reactantto product and predicts stationary points that are similar ingeometry. Interestingly, B3LYP with a slightly smaller basis set(as one might consider using in QM/MM simulations due tolower cost) yields a substantial underestimate of the reactionbarrier as compared to both the higher basis set and AM1/d-PhoT, although the geometries are still comparable. Lastly, it isworthwhile to note that AM1/d-PhoT was actually trained andtested on even higher level results [B3LYP/6-311++G-(3df,2p)//B3LYP/6-31++G(d,p)],20,50 which, at the veryleast, explains the existence of the minor deviations in energyand geometry visible in Figure 2.Free Energy Profiles and Mechanical Observables.

The principle results of the present work are the free energy

profiles, F(ξ), for each of the specific base-catalyzed reactionscalculated from umbrella sampling simulations. The values ofthe progress coordinate, ξ, corresponding to the reactant andtransition state (ξR and ξ⧧, respectively) are determined as:

ξ ξ

ξ ξ

=

=

ξ ξ

ξ

<

⧧

⧧F

F

arg min ( )

arg min ( )

R

(6)

The free energy barrier is then calculated as ΔF⧧ = F(ξ⧧) −F(ξR). Additionally, the averages of select mechanicalobservables, ⟨A⟩ξ, and their standard deviations, (⟨A2⟩ξ −⟨A⟩ξ

2)1/2, were estimated at fixed values of the reactioncoordinate (eq 5).Throughout this work, the term “reaction model” is used to

refer to a molecule that undergoes a reaction analogous to RNAtransesterification (i.e., contains a phosphodiester that reacts toform a cyclic phosphate). The reaction models used here areillustrated and numbered in Figure 1. In QM/MM simulationsof these reaction models, a “force field model” must also bechosen to describe the (nonbonded) MM and QM/MMinteractions between the (QM) solute and the solvent. Here,the force field models for the solutes take parameters fromeither the AMBER (A) or the CHARMM (C) force fields inconjunction with solvent (water or water + NaCl) defined bythe TIP3P (3) or TIP4P-Ew (4) water models. A fullsimulation model is then given by both a reaction model anda force field model. For example, an abasic RNA dinucleotide(reaction model 4 in Figure 1) with AMBER force fieldparameters in a simulation cell containing TIP4P-Ew water andsodium chloride is designated as 4-A4/NaCl. All models willhereafter be referred to with this nomenclature.

Abasic Dinucleotide Models. Solvent Environments. Webegin by examining several possible solvation models of anabasic RNA dinucleotide, for which a wealth of experimentaldata are available.10,11,13,25 In particular, we examine differencesin the free energy profile due to variations in the water modeland ion atmosphere. The results (Figure 3 and Table 2) showno statistically significant variation in the reaction barrier orgeometry when the water model is changed from TIP3P toTIP4P-Ew. The removal of ions (infinite dilution limit) appears

Figure 2. Gas-phase potential energy profiles for the 2′-O-trans-esterification of 2-hydroxy ethyl methyl phosphate (see the inset) atthe B3LYP/6-31+G(d) (red), B3LYP/6-31G(d) (blue), and AM1/d-PhoT (green) levels (as implemented in Gaussian 0986 andAMBER12/AmberTools12,29,37,87 respectively). The approximatereaction coordinate paths are obtained via constrained optimizationat different coordinate values (using the DL-FIND library88). Crossesdenote the location of optimized minima and transition states. Allenergies are relative to the optimized minimum at the relevant level oftheory.

Table 1. Potential Energy Barriers and Select GeometricQuantities at the Transition State at Different Levels ofTheory Using Multiple Reaction Coordinatesa

ΔE⧧ rP−O5′,TS rP−O2′,TS

B3LYP/6-31+G(d) IRC 42.6 2.49 1.84ARC 42.5 2.44 1.84

AM1/d-PhoT IRC 42.4 2.33 1.88ARC 42.3 2.27 1.87

B3LYP/6-31G(d) IRC 35.5 2.56 1.83ARC 35.5 2.63 1.83

aThe intrinsic reaction coordinate (IRC) is defined so as exactly toconnect the minimum and first-order saddle point. However, theapproximate reaction coordinate (ARC) used in simulations is notguaranteed to connect either point on the potential energy surface. Inthis case, the minimum and saddle point are defined by the reduced setof coordinates. It should be noted that the ARC values are obtainedunder the additional approximation that mΔξ = (1/2)(rP−O5′ −rP−O2′), with Δξ = 0.1 Å and m as some integer. Energies are in kcal/mol, and lengths are in Å.

Figure 3. Free energy profiles for reaction model 4 (inset) in differentsolvent environments. The experimental value is given for a UpGdinucleotide (ref 25, 310 K, and ionic strength of 1 M in NaOH/NaCl,dashed line). To aid visual comparison of barrier heights, the plots areshifted such that F(ξR) = 0.0. The average error bars (estimated 95%confidence interval, not shown for clarity) for all of the curves are lessthan 0.4 kcal/mol relative to the appropriate reactant state.



to slightly lower the barrier for both water models by the sameamount, although this difference is very similar in magnitude tothe estimated error (0.8 ± 0.6 kcal/mol).Solute−Solvent Interaction Models. A necessary aspect of

QM/MM simulations is to select a nonbonded, nonelectro-static interaction model for QM/MM interactions. Here, as isgenerally done, the choice is made from existing standard forcefield models. However, these models usually only aim todescribe a fixed valence chemical structure and are thus notnecessarily appropriate for describing chemical reactions.71 Asan investigation of the accuracy of the models used here, wecompare the free energy profiles calculated with different forcefield models. Three solute/solvent combinations were exam-ined using common parameters from the AMBER andCHARMM nucleic acid force fields and the TIP3P andTIP4P-Ew rigid water models.72 A summary of theseparameters is given in the Supporting Information. The freeenergy profiles largely display the same shape and only slightquantitative differences (Figure 4 and Table 3). The changefrom AMBER to CHARMM Lennard−Jones parameters leadsto a slight lowering of the profile between the reactant and thetransition states by roughly 1 kcal/mol but without changingthe reaction barrier to a statistically significant degree.

Varying the Structural Environment of the Nucleo-phile and Leaving Group. In our final analysis, a series ofreaction models that undergo phosphoryl transfer wereestablished to systematically dissect levels of model complexity(Table 1). In each case, the general reaction scheme is identicalto that of RNA cleavage. The nucleophile is either part of asimple alkyl chain or ribose ring, and the leaving group is eitherethoxide or 5′-deprotonated ribose. The calculated free energyprofiles (Figure 5) show the barrier magnitudes clustering intothree groups depending on whether or not the nucleophile ispart of a ribose ring and the size of the leaving group (ΔF1⧧ ≫ΔF2⧧ > Δ F3

⧧ ≫ ΔF4⧧, Table 3). A slightly differentclassification is obtained when comparing the location of theprofile minimum (i.e., the reactant state); in this case, thepresence of a ribose ring is the most obvious factor.

■ DISCUSSION

Comparison with Experiment: UpG Dinucleotide. Aprimary motivation of this work was the recent publication byHarris et al. of pH rate and KIE data for the specific base-catalyzed cleavage of a UpG dinucleotide.25 In that work, a rateconstant of 0.06 ± 0.002 s−1 was extrapolated at “infinite” pHnear biological conditions (310 K, ionic strength of 1 M inNaOH/NaCl).73 Applying transition state theory (and standarderror propagation) then gives a free energy of activation of 19.9± 0.02 kcal/mol. This is important because it provides optimalcomparison to the constant protonation state simulationsperformed here. Other experimental work has demonstratedsmall (<5.0 × 10−4 s−1), but detectable, variations of the rateconstant with respect to nucleobase sequence,14 but we do notconsider these effects in the present work.As seen in Figure 3 (and Figure 4), in all cases, the

agreement is very good (between 19.9 and 20.8 vs 19.9 kcal/mol) and within statistical errors. However, one must be warythat this agreement may, at least to some degree, beserendipitous. The model parameters related to solvation(water and solute Lennard−Jones parameters), which areknown to influence barriers in reactions where local changes incharge state occur along the reaction coordinate as in thepresent example, have not been tuned for the specific reactionconsidered here.The most prominent difference between the parameters of

our simulations and those of experiment is the ionic strength.By necessity, experiments at high pH require high concen-

Table 2. Free Energy Profile Extrema and Select Average Geometric Quantities at Fixed Values of the Reaction Coordinate forReaction Model 4 in Different Solvent Environmentsa

Reactant State

model ξR ΔF⧧ ⟨rP−O2′⟩ξR ⟨rP−O5′⟩ξR ⟨θO2′−P−O5′⟩ξR4-A3/NaCl −1.31 4.27 ± 0.11 1.65 ± 0.06 95 ± 284-A3 −1.31 4.27 ± 0.12 1.66 ± 0.06 97 ± 294-A4/NaCl −1.30 4.26 ± 0.12 1.66 ± 0.06 94 ± 264-A4 −1.31 4.28 ± 0.10 1.66 ± 0.06 90 ± 21

Transition State

model ξ⧧ ΔF⧧ ⟨rP−O2′⟩ξ⧧ ⟨rP−O5′⟩ξ⧧ ⟨θO2′−P−O5′⟩ξ⧧

4-A3/NaCl 0.20 20.8 ± 0.5 1.78 ± 0.08 2.20 ± 0.22 161 ± 84-A3 0.18 20.1 ± 0.4 1.78 ± 0.08 2.18 ± 0.22 162 ± 94-A4/NaCl 0.20 20.8 ± 0.4 1.78 ± 0.09 2.19 ± 0.22 161 ± 94-A4 0.18 19.9 ± 0.4 1.78 ± 0.09 2.15 ± 0.18 162 ± 8

aFor barrier heights, error bars represent approximate 95% confidence intervals; for all other quantities, they represent twice the population standarddeviation. Energies are in kcal/mol, lengths are in Å, and angles are in degrees.

Figure 4. Free energy profiles for reaction model 4 (inset) withdifferent force field models in the presence of sodium chloride. Theexperimental value is given for a UpG dinucleotide (ref 25, 310 K, andionic strength of 1 M in NaOH/NaCl, dashed line). To aid visualcomparison of barrier heights, the plots are shifted such that F(ξR) =0.0. Filled curves represent estimated 95% confidence intervals relativeto the appropriate reactant state.



trations of NaOH or some other base, usually buffered with asalt.12,14,25 Such conditions have been known to be problematicin periodic boundary simulations,74,75 and it was only recentlythat models (employed here) were developed that robustlyreproduce experimental bulk behavior.46 However, “local”,solvation properties, such as binding coefficients,76 are oftennot well reproduced by many ion models, at least not on thetime scales accessible in typical simulations. The modifiedsodium interactions used here (see the Supporting Informa-tion) were designed to prevent direct binding of sodium to thephosphate in a minimally perturbative fashion so at least toenforce consistency across all umbrella sampling simulation,which might otherwise sample different bound conformations.This is justifiable based on the low binding coefficient betweensodium and phosphates,76 but could be problematic at higherconcentrations where the fraction of bound sodium ions isexpected to be non-negligible. Li and Breaker have noted thatthe observed rate constant of base-catalyzed RNA cleavageincreased with increasing potassium concentration.14 However,they were only able to hypothesize that this effect was primarilydue to influence on the pKa of the 2′-hydroxyl group and noton the intrinsic rate constant, as they were unable to establishsimultaneously high pH and low potassium conditions.Although the system studied in that work was a DNA 22-merwith a single embedded RNA dinucleotide, the observed free

energy barriers (21.5−22.5 kcal/mol, 296 K, 3.16 M K+) arereasonably close to those measured and calculated for a simpledinculeotide. The results reported here are thus, at the veryleast, consistent with that hypothesis. Although it is tempting tosuggest that the hypothesis is supported by the lack of acatalytic effect when sodium is removed, additional data wouldbe required, for example (as suggested by a referee), examiningthe concentration dependence of the nucleophile pKa or morerigorously considering sodium binding.

Free Energy Barriers and Solvent Structure. While theeffects of solvation models on free energy profiles are obviousand easily compared, this does not necessarily make them anoptimal metric for assessing solvation model quality. That is,there is not necessarily a direct, or even one-to-one,correspondence between empirical parameters that give the“correct” free energy barrier and those that are physicallysensible. An ideal model would satisfy both criteria. As aqualitative check of the TIP3P and TIP4P-Ew water models, weexamine the radial and three-dimensional distribution of watermolecules (or rather the water oxygens) around thephosphorane transition state (Figure 6). The results for bothmodels are very similar, with distinct gaps of density aroundeach of the nonbridge oxygen bond axes as well as parallel tothe breaking and forming bonds (although a clear patch ofdensity appears along the nonbridge oxygen angle bisector).Viewing a representative transition statelike configurationshows tetrahedral coordination of water around each of thenonbridge oxygens. As would be expected, the water moleculesneatly reside on a density isosurface roughly corresponding tothe first peak of the radial distribution function.There have been some discussions in the literature

concerning the choice of Lennard−Jones parameters foratoms in the QM region.71,77,78 Mulholland and co-workersshowed that modified parameters for nucleobases in thequantum region can improve hydrogen bond geometries withMM water molecules79 and further demonstrated that changingbetween a point charge and ab initio electronic density forseveral common rigid water models can lead to unexpected(and potentially unsatisfying) results.80 By examining extremechoices of radii and well depths, Riccardi et al. demonstratedthat both a reduction potential and a proton transfer barrier, aswell as hydrogen-bonding interaction energies, display asystematic dependence on Lennard−Jones parameters.81

However, their conclusion was that physical accuracy (in thesense of properly balancing enthalpic and entropic effects,rather than agreement with experimental data or ab initio

Table 3. Free Energy Profile Extrema and Select Average Geometric Quantities at Fixed Values of the Reaction Coordinate forReaction Model 4 with Different Force Field Models in the Presence of Sodium Chloridea

Reactant State

model ξR ΔF⧧ ⟨rP−O2′⟩ξR ⟨rP−O5′⟩ξR ⟨θO2′−P−O5′⟩ξR4-A3/NaCl −1.31 4.27 ± 0.11 1.65 ± 0.06 95 ± 284-A4/NaCl −1.30 4.26 ± 0.12 1.66 ± 0.06 94 ± 264-C3/NaCl −1.32 4.29 ± 0.12 1.65 ± 0.06 95 ± 23

Transition State

model ξ‡ ΔF⧧ ⟨rP−O2′⟩ξ⧧ ⟨rP−O5′⟩ξ⧧ ⟨θO2′−P−O5′⟩ξ⧧

4-A3/NaCl 0.20 20.8 ± 0.5 1.78 ± 0.08 2.20 ± 0.22 161 ± 84-A4/NaCl 0.20 20.8 ± 0.4 1.78 ± 0.09 2.19 ± 0.22 161 ± 94-C3/NaCl 0.23 20.3 ± 0.4 1.77 ± 0.08 2.24 ± 0.14 164 ± 10

aFor barrier heights, error bars represent approximate 95% confidence intervals; for all other quantities, they represent twice the population standarddeviation. Energies are in kcal/mol, lengths are in Å, and angles are in degrees.

Figure 5. Free energy profiles for several reaction models (n-A3/NaCl,where n = 1−4 as in Figure 1). Experimental values are given for aUpG dinucleotide (ref 25, 310 K, and ionic strength of 1 M in NaOH/NaCl, dashed line, for comparison to model 4) and model 2 (ref 15,353 K, and 0.05 N in NaOH, dotted line). To aid visual comparison ofbarrier heights, the plots are shifted such that F(ξR) = 0.0. Filled curvesrepresent estimated 95% confidence intervals relative to theappropriate reactant state.



calculations) would be more profitably improved through otheraspects of the QM/MM model.The present work appears to support the view that simple

tuning of Lennard−Jones parameters is not a fruitful avenue toproducing more physically accurate free energy profiles in QM/MM simulations. However, that is not to say that theparameters cannot nontrivially affect the profile or that theexpected change cannot be predicted. In the present case,although not contrived to be so, the CHARMM potential forthe highly charged nonbridge oxygens has somewhat weakerrepulsive components (both Lennard−Jones ε and A = εRmin

12

coefficients) than the related AMBER parameters (see theSupporting Information). This would seem to correlate withthe near systematic lowering of the CHARMM free energyprofiles in comparison to the AMBER profiles (Figure 4). Thatis, with identical electrostatic interactions, the CHARMMparameters allow more preferential stabilization of the dianionictransition state relative to the monoanionic phosphate andnucleophile reactants since the weaker repulsion is over-whelmed by electrostatic attraction. It is not clear as to whetherthis systematic behavior is desirable. For example, a morephysically correct profile might possess a steeper approach to atransition state that is lower in energy, a scenario not obviouslyattainable by simple modification of the Lennard−Jonespotentials. This suggests the merit of a different approach.Developments in our group have sought to replace theempirical Lennard−Jones potential with a more physicalmodel that inserts directly into the QM/MM self-consistentfield calculation.82,83 This would allow for charge-dependentexchange and dispersion interactions and would not rely onstatic atom type-based parameters. The potential advantagewould arise from nonsystematic changes in the free energyprofile, thereby inserting more of the physical behavior thatRiccardi et al. found lacking.

Dependence of the Barrier on the Ground State.Among the free energy profiles of the four different reactionmodels studied here, the most obvious differences are thelocations of the reactant state and height of the free energybarrier (Figure 5). A comparison of the average geometries atall four reactant states (Table 4) shows nearly identical bondlengths between the phosphorus and the O5′ leaving group butquite different distances to the O2′ nucleophile. This issensible, as the ring structure effectively prevents the secondary

Figure 6. Snapshots from umbrella sampling simulations near thetransition state of model 4-A3 (top) and 4-A4 (bottom). Bothtransition state structures consist of a pentacoordinated phosphorane(orange) with advanced bond formation between the phosphorus andthe 2′-oxygen (upper black line) and bond cleavage between thephosphorus and the 5′-oxygen (lower black line). This is indicative ofa “late” transition state. Density maps of the water oxygens[transparent gray, isosurfaces correspond to 4πρbulkg(rmax)] indicatethree distinct areas of low solvation around the phosphorane.

Table 4. Free Energy Profile Extrema and Select Average Geometric Quantities at Fixed Values of the Reaction Coordinate forSeveral Reaction Models (n-A3/NaCl, Where n = 1−4 as in 0)a

Reactant State

model ξR ΔF⧧ ⟨rP−O2′⟩ξR ⟨rP−O5′⟩ξR ⟨θO2′−P−O5′⟩ξR1-A3/NaCl −1.47 4.58 ± 0.14 1.65 ± 0.06 130 ± 422-A3/NaCl −1.55 4.74 ± 0.14 1.65 ± 0.06 112 ± 603-A3/NaCl −1.32 4.29 ± 0.12 1.65 ± 0.06 100 ± 334-A3/NaCl −1.31 4.27 ± 0.11 1.65 ± 0.06 95 ± 28

Transition State

model ξ⧧ ΔF⧧ ⟨rP−O2′⟩ξ⧧ ⟨rP−O5′⟩ξ⧧ ⟨θO2′−P−O5′⟩ξ⧧

1-A3/NaCl 0.23 31.2 ± 0.5 1.78 ± 0.09 2.22 ± 0.24 166 ± 92-A3/NaCl 0.18 27.7b ± 0.6 1.77 ± 0.08 2.20 ± 0.24 162 ± 93-A3/NaCl 0.24 26.2 ± 0.4 1.77 ± 0.08 2.26 ± 0.33 164 ± 104-A3/NaCl 0.20 20.8c ± 0.5 1.78 ± 0.08 2.20 ± 0.22 161 ± 8

aFor barrier heights, error bars represent approximate 95% confidence intervals; for all other quantities, they represent twice the population standarddeviation. Energies are in kcal/mol, lengths are in Å, and angles are in degrees. bΔGexpt

⧧ = 28.5 kcal/mol (ref 15). cΔGexpt⧧ = 19.9 kcal/mol (ref 25).

See Figure 5 for details.



alkoxide from getting too far from the negatively chargedphosphate, even though it would be electrostatically favorableto do so. However, when no ring is present (as in models 1 and2), much larger separation is possible. In this case, the barrier tophosphoryl transfer is effectively increased by the addedconformational change. The sizable difference in the reactionbarriers of models 1 and 2 (3.5 ± 0.8 kcal/mol) is likelyexplained by differing solvation of the nucleophiles. That is, inprinciple, the additional methyl substituent on model 2 shouldstabilize the alkoxide more than in model 1, thereby loweringthe energy of the (deprotonated) reactant state and raising thebarrier. However, this is clearly not the case. The dominatingeffect is the increased solvent exposure of the primary alkoxidedue to the presence of fewer substituents. It thereforeexperiences a higher energetic cost when moving toward thetransition state and thus a higher reaction barrier.Near the transition state, all four models are strikingly similar

both in the shape of the free energy profile and in the averagegeometry (Table 4). The exception to this is model 3, whichhas a slightly longer bond breaking distance (rP−O5′), as well asa “looser” bonding environment, as indicated by increasedfluctuations. This variation is not enough, however, to changethe classification of the transition state; the progressed bondbreakage in all of the models clearly indicates a “late” transitionstate characterized by a nearly fully formed bond with thenucleophile and a nearly fully broken bond to the leavinggroup. However, unlike the difference between the reactionbarriers of models 1 and 2, the difference between models 3and 4 (5.4 ± 0.6 kcal/mol) seems more anomalous. Theobvious departure point for examining this is the nature of theleaving groups. In the current AM1/d-PhoT QM model, thegas-phase proton affinities of these compounds are quitedissimilar (ΔΔHPA = ΔHPA,EtOH − ΔHPA,5′‑ribose ≈ 18 kcal/mol)and indicative of a higher cost to remove the ethoxide leavinggroup (if one is willing to assume that the solvation propertiesare not too different). This rather dramatic difference alsoseems to be semiquantitatively in line with high-level densityfunctional theory (DFT) calculations84 (ΔΔHPA ≈ 10−16kcal/mol). However, a DFT model using continuum solvationsimilar to 3 was also recently shown to give a free energy barrierin very close agreement with that for an RNA dinucleotide,85

implying that the present result is in fact anomalous, but forreasons other than the proton affinity difference. Because of thereasonable agreement of models 2 and 4 with experimentalresults, we suspect that there is an imbalance of solvent effectswhen the nucleophile is large and the leaving group is small.This is in line with the clear importance of solute/solvent andQM/MM interactions and allows for the results for model 3 tobe seen as anomalous.

■ CONCLUSIONA significant aspect of understanding enyzmatic mechanism isunderstanding the nature of rate enhancement over the native(solution) mechanism. In this work, we have studied severalvariants of nonenzymatic phosphoryl transfer, a reactioncatalyzed by a wide range of proteins and the most ubiquitousamong known ribozymes. Using QM/MM MD umbrellasampling simulations, the free energy profiles were calculatedalong a simple atom transfer coordinate.The calculated barrier for an abasic dinucleotide agrees

almost exactly with the experimental result for the UpGdinucleotide under similar conditions. Analysis of the transitionstate structure indicates a “late” transition state, also in

agreement with inferences from the experimental KIEs.Although ionic conditions are a necessity of most high pHexperiments, removing all monovalent ions did not lead to anysignificant change in the barrier. However, it should be notedthat the model used here effectively precluded the possibility ofdirect coordination of cations to the phosphate. The obviouscorollary to this result is that changes in the rate constantobserved in conjunction with changes in ionic environment arelikely due to the types of coordination neglected here. Suchcoordination is likely to be electrostatically favorable but willcome at the cost of disrupting extensive hydrogen bondnetworks surrounding the dianionic transition state if it occursat the nonbridge oxygens.The conclusions made here are strengthened by extensive

testing of the available QM/MM models, including thecommonly used TIP3P and TIP4P-Ew water models, as wellas Lennard−Jones parameters from the AMBER FF10 andCHARMM27 force fields. Changing any one of these aspects ofthe model also does not lead to substantial change in thecalculated reaction barrier. This does not suggest that theseparameters are not important, as it has been demonstratedpreviously by our group and others that free energy profilesinvolving local changes in charge are highly sensitive tosolvation and Lennard−Jones parameters. Rather, the combi-nations of parameters examined here appear to be similarlybalanced in terms of overall solvation.The structural complexity of the leaving group and

nucleophile has a subtle effect on the structure of the transitionstate and a significant effect on the magnitude of the free energybarrier. This is most evident according to the presence orabsence of a ribose ring, which acts to constrain the degree ofseparation between the deprotonated nucleophile and themonoanionic phosphate in the reactant state. Lastly, improperbalancing of model pKa values for the nucleophile and leavinggroup can lead to abnormally disparate reaction barriers andshould be considered when embedding QM regions into largersystems such as ribozymes. Next-generation models that areable to consider adjustments of nonelectrostatic nonbondedinteractions as a function of local charge are likely toconsiderably improve the robustness of QM/MM simulationsfor chemical reactions or processes that involve chargemigration or change of local charge state.

■ ASSOCIATED CONTENT*S Supporting InformationMore detailed description of the umbrella sampling simulationprotocol and data analysis, the Lennard−Jones parameter setsused here, and a brief assessment of the kernel densityestimation approach along with comparisons to the moreconventional histogram approach. This material is available freeof charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected] authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe are grateful for financial support provided by the NationalInstitutes of Health (GM62248 to D.M.Y. and GM096000 toM.E.H.), as well as for the use of computational resources from



http://pubs.acs.org

mailto:[email protected]

the Minnesota Supercomputing Institute for AdvancedComputational Research. This research was also partiallysupported by the National Science Foundation throughTeraGrid resources provided by Ranger at TACC and Krakenat NICS under Grant Number TG-CHE100072.

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