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achieve electrical neutrality). In all cases, after the addition of ions it is essential that the solvent and ions be adequately equilibrated around the nucleic acid, as discussed in the following paragraph. The simulator should also be aware that ions strongly interacting at specific sites on the nucleic acid can strongly influence both the structural and dynamic properties.

Once the DNA or RNA has been overlaid with solvent and ions, the energy of the system is minimized, and the system is subjected to an equilibration MD simulation. These calculations should be performed with the nucleic acid atoms fixed or harmonically constrained to allow for the solvent to relax around the nucleic acid. Properties such as the potential energy and the solvent–solute interaction energy should be monitored as a function of simulation time to ensure that the system has relaxed to a satisfactory extent. In cases where the MD studies will be performed at constant volume (i.e., NVT or NVE simulations) it is suggested that the pressure of the system be monitored to ensure that it is in the vicinity of 1 atm. If the pressure differs significantly from 1 atm, the distances used for the deletion of solvent molecules can be increased or decreased accordingly (loop I of Fig. 1) [146]. Note that in simulations containing fixed or harmonically constrained atoms the calculated pressures may be incorrect, requiring that short MD simulations without constraints be performed to obtain accurate pressures. Once the solvent has adequately equilibrated around the solute, the entire system, including the nucleic acid, should be energy minimized. At this stage of the system preparation it is important that checks be performed to ensure that the system is still properly solvated. If problems in the simulation system are evident, the system should be resolvated in appropriate fashion to correct the problem and the equilibration redone, as shown in loop I of Figure 1.

C. Production MD Simulation

At this stage the production MD simulation can be initiated. It is generally preferable to perform simulations in the isobaric, isothermal (NPT) ensemble, which yields thermodynamic properties that correspond to the experimentally accessible Gibbs free energy [147]. In some cases initiation of the MD simulations can involve gradual heating of the system, although initiation of the simulation at the final desired temperature is often sufficient. Several algorithms for NPT simulations are available, including Berendsen’s method [148], the Langevin piston [149], and an extended method from Klein and coworkers [150]. It should be noted that Berendsen’s method does not correspond to a true ensemble, and problems associated with systematic oscillations in the system volume can occur [149]. When the simulation is initiated it is important to closely monitor both structural and energetic properties to ensure that significant perturbations of the solute do not initially occur due to the applied methodology. If such perturbations are present, the system preparation and equilibration approach should be evaluated for potential problems.

D. Convergence of MD Simulations

Rigorous proof of convergence of MD simulations cannot be performed [31]. Efforts can be made, however, to ensure that various properties calculated from a simulation have reached satisfactory levels of convergence. First, the simulations should be analyzed to determine if global (1) energetic and (2) structural properties have stabilized. Energetic stability is typically investigated by monitoring the potential energy versus time. During most simulations the potential energy initially relaxes, after which it fluctuates around a

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constant value for the remainder of the simulation. In NVE simulations the ratio of RMS fluctuations of the potential energy and kinetic energy can be monitored, with RMSPE/ RMSKE 0.01 indicating proper energy conservation. Analysis of structural properties versus time (e.g., RMS difference with respect to the starting structure) will also typically show an initial relaxation followed by fluctuations around an average value for the remainder of the simulation. In certain cases these fluctuations may be relatively large, indicating the sampling of alternative conformations that may be biologically relevant [16]. From the analysis of structural and energetic properties versus time, the initial portion of the production simulation during which relaxation of these properties occurs is discarded. Additional convergence tests are performed only on the remainder of the production simulation.

Tests of convergence on the remaining portion of the simulation can most readily be performed by calculating the desired property (i.e., the property that is of most interest to the simulator) over different simulation time lengths and monitoring the change in the average value as a function of simulation time. Convergence is indicated by the lack of significant change in the average value of the property as the simulation time increases. Another method involves separating the total trajectory into independent blocks (e.g., a 500 ps production simulation may be separated into five 100 ps blocks), calculating average values for each block, and comparing the average values from the individual blocks. These block averages can also be used to calculate overall averages and standard errors for individual properties [151]. This is an excellent method for obtaining the statistical significance of results from a simulation.

For an entire MD simulation, as well as for separate blocks from a simulation, determination of convergence of a property can be most rigorously carried out by calculating the time series of a property and determining its autocorrelation function and accompanying relaxation time. For adequate convergence the total MD simulation time or the block time should be approximately four times as long as the relaxation time. This test is also appropriate for determining the length of blocks that can be considered independent. It should be emphasized that the relaxation time and, accordingly, the amount of simulation time required for convergence are dependent on the property being investigated. Thus, the total required simulation time is dependent on the type of information that is to be obtained from a simulation. If it is determined that the properties of interest have not converged, then the production simulation should be extended, as indicated by loop II of Figure 1.

E.Analysis of MD Simulations

Molecular dynamics simulations at an atomic level of detail contain enormous amounts of information, making rational analysis of simulations essential. In cases where studies have been designed in close collaboration with experiment, the information to be obtained is that which corresponds to the experimental data. Alternatively, systematic analysis of the MD trajectory may be performed to identify specific properties of interest. Properties determined from simulations of nucleic acids can be separated into those associated with structure, hydration, and energetics. Structural properties include the dihedrals associated with the phosphodiester linkage, the sugar and the glycosidic linkage, Watson–Crick base pairing, and a variety of intramolecular distances such as the minor groove width based typically on interstrand P distances. Nucleic acid sugar puckering is typically not analyzed on the basis of the individual dihedrals, but rather with respect to the concept of pseudorotation, where the sugar conformation on is defined in terms of two variables, the pseudoro-

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tation angle and the amplitude [2,152]. An important class of structural properties comprises the helical or helicoidal parameters [153–155]. These terms define the orientations of the bases with respect to a global helical axis or locally with respect to adjacent bases or basepairs. Both methods have been used extensively, and a number of programs have been developed to perform this analysis, including CURVES [156], FREEHELIX [157], and a program by Babcock and Olson [158]. The CURVES method has been incorporated into an analysis and plotting package, DIALS and WINDOWS, which analyzes MD trajectories as well as individual coordinate files and presents data, including time series, in a highly compact fashion [159]. When using these programs the user should be aware of both the global and local definitions of the helicoidal parameters. Whereas in idealized helices the two definitions yield similar or identical results, the global method is preferable for standard helices and the local definition is more suited for distorted helices, such as those found in DNA–protein complexes, where a global helical axis cannot be rigorously defined. When performing structural analysis it is often best to initially calculate overall averages, use that information to identify terms that may be of interest, and then analyze those terms via time series or probability distributions. Probability distributions of DNA dihedrals and helicoidal parameters calculated via CURVES have been published [31], and those based on FREEHELIX are available (N Banavali, AD MacKerell Jr, unpublished).

Hydration properties, including interactions with ions, are strong determinants of DNA structure [2,160–162]. Hydration numbers can be determined on the basis of a variety of criteria [29,163,164] for comparison with individual crystal structures or with data from surveys of the NDB [165,166]. This type of analysis is often performed on different portions of the DNA such as the minor groove and is typically limited to first shell hydration. Complicating such analysis is the overlap of waters hydrating different sites. For example, a water molecule hydrating the sugar O4′ atom may also be hydrating atoms in the minor groove, and it has been shown that counterions can replace water molecules in the first shell of hydration [167]. Hydration can also be studied with respect to the dynamic properties of the water molecules. Examples include changes in water dynamics around DNA leading to local alterations of the dielectric environment [32] and results indicating that decreased water activity due to increased salt concentration is associated with changes in water mobility rather than hydration number [24]. Dynamics properties that have been analyzed include diffusion constants and rotational correlation times. Rotational correlation times may be analyzed on the basis of the water dipole axis, the axis perpendicular to the plane of the water, and the HEH axis. These motions are associated with molecular twisting (dipole axis), rocking (perpendicular), and wagging (HEH vector), respectively [168]. Difficulties in determining correlation times of individual waters occur with water molecules that undergo little motion on the time scale of the simulation, leading in some cases to undefined correlation times. A method of analysis based on the initial 4 ps of the decay of the autocorrelation function [169] can overcome some, but not all, of these problems.

Supplementing direct analysis of structural and hydration properties is the use of energetic analysis. For example, instead of monitoring individual Watson–Crick hydrogen bonds, interaction energies between basepairs can be monitored. Similarly, interaction energies between water and the nucleic acid can be determined to supplement the information discussed in the preceding paragraph. The advantage of energetics over direct structural or hydration information is the ability to more readily take into account all possible contributions, not just those envisioned by the simulator. Furthermore, it may be possible to better gauge the relative impact of different types of interactions on the trajectory obtained from an MD simulation. Such analysis has been used to identify energetic and, subsequently, structural phenomena allowing for the barrierless extension of DNA beyond

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its canonical B form and contributions to the barrier to strand separation [86]. Typically, energetic analysis should be performed using the same truncation scheme as that used in the MD simulation. This can be problematic in Ewald simulations, as it is currently not possible to use Ewald summations to determine interaction energies. One suggestion is to calculate interaction energies between atoms with the real-space truncation distance used in the simulation. An interesting alternative to direct analysis of interaction energies is the recalculation of solvation free energies using continuum models on individual nucleic acid structures from a solvated MD simulation. This approach has been used to understand the equilibrium between the A and B forms of DNA as a function of water activity [40,91]. Although the method successfully ordered the equilibrium when water activity was modified with ethanol, it was not successful at predicting the salt effect, which may be associated with atomic detail interactions not modeled in continuum approaches. Overall, energetic analysis of MD simulations offers an additional method to analyze simulations, often allowing for identification of structural contributions that are difficult to identify via direct structural analysis.

V.CONCLUSION

In this chapter we have attempted to give an overview of the types of nucleic acid systems that are accessible to computational study. These vary from nucleosides, through duplex DNA and RNA, up to nucleic acid–protein complexes. Accessibility to both longer time scales and larger systems is expected to increase as advances in both computational power and methods continue to occur. Although the work cited cannot be considered complete, it should allow the reader to access information required for moving into and obtaining a more general background of the field of nucleic acid simulations.

WEB SITES OF INTEREST

CHARMM program: yuri.harvard.edu

CHARMM force field: https://rxsecure.umaryland.edu/research/amackere/research.html

GROMOS: igc.ethz.ch/gromos/welcome.html AMBER: www.amber.ucsf.edu/amber/amber.html NAB: www.scripps.edu/case

Dials and Windows (via D. Beverige): www.wesleyan.edu/chem/faculty/beveridge

CURVES (also JUMNA): www.ibpc.fr/UPR9080/Curindex.html Protein Data Bank: www.rcsb.org/pdb

Nucleic Acid Data Bank (also FREEHELIX): ndbserver.rutgers.edu

National Partnership for Advanced Computational Infrastructure (NPACI): www.npaci.edu

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