in some cases stabilizing and in other cases destabilizing [84]. Reduced stabilization of the stacked conformation in nonaqueous solvents (DMSO, chloroform, methanol) was also observed [85].

PMF calculations have also been used to investigate the end-to-end extension of a duplex DNA 16mer, with the calculations designed to reproduce the conditions used in atomic force microscopy (AFM) experiments [86]. Results from these calculations were consistent with the barrierless extension of DNA observed in the AFM experiments. Detailed analysis of the simulated results yielded a model in which the unfavorable intramolecular mechanical energy of DNA associated with extension is compensated for by DNA–solvent interactions to yield the barrierless extension of the DNA. Alternative computational studies of the extension of DNA have been performed in vacuo with both atombased and internal coordinate based methods. Results from the internal coordinate based calculations, which have been developed to implicitly include solvent effects, yielded a qualitative picture of the structural changes in DNA as end-to-end extension is observed from which a new form of DNA, the S form, was identified [79]. Atom-based calculations of the stretching DNA in vacuum indicated that Watson–Crick hydrogen bonds remained intact throughout the extension of the DNA; however, the omission of solvent could significantly influence this conclusion [87]. In combination, the calculations to date on the energetics of oligonucleotides associated with structural perturbations strongly indicate that much information is to be gained from these types of computational studies.

D. DNA Phase Transitions

Molecular dynamics simulations have also been used to interpret phase behavior of DNA as a function of temperature. From a series of simulations on a fully solvated DNA hexamer duplex at temperatures ranging from 20 to 340 K, a ‘‘glass transition’’ was observed at 220–230 K in the dynamics of the DNA, as reflected in the RMS positional fluctuations of all the DNA atoms [88]. The effect was correlated with the number of hydrogen bonds between DNA and solvent, which had its maximum at the glass transition. Similar transitions have also been found in proteins.

E.Modified Oligonucleotides

Interest in chemically altered oligonucleotides has been generated by the possibility of using antisense technology in drug therapy [89] as well as to exploit chemically modified species in oligonucleotide structure–function studies, including nucleic acid–protein interactions [90]. Modifications studied to date via computational approaches include modifications of the phosphates, sugars, and bases. Modified phosphodiester backbones subjected to computational studies include phosphoramidate-modified species, peptide nucleic acids (PNAs), and 2′–5′ phosphodiester linkages [91–95]. Results from these studies yield insights into the influence of the backbone on the overall conformation of the oligonucleotide, including stabilization of triple helical structures. In another study, calculations were performed on DNA duplexes and triplexes, with guanidine groups replacing the normal phosphodiester linkages, yielding a polycation [96]. Results showed the triplex to be more stable than a polyanionic triplex. Examples of the study of modified sugar include calculations on oligonucleotides containing hexitol instead of the furanose rings [97]. These calculations indicated that oligonucleotides with hexitol sugars may form more stable duplexes with RNA than DNA due to alterations in solvation of the minor groove. Some

of the calculations involving modified bases included free energy perturbation calculations investigating the influence of bromination and methylation on the stability of Z-form oligonucleotides [98,99]. Other studies of modified oligonucleotides have been performed on the thymine dimers caused by UV photodamage of DNA showing the presence of bent structures consistent with experiment along with information on possible mechanisms by which repair enzymes may identify damaged sites [100]. Although these studies yielded useful insights, it should be emphasized that the quality of the results will be strongly influenced by the force field parameters and methods used in the calculations. Special care is required to ensure that the new parameters introduced into a force field to treat the novel chemical structures are (1) consistent with the force field and (2) modeling the modified structures correctly (see Chapter 2).

F.Alternative Secondary and Tertiary Motifs of Oligonucleotides

Beyond the helical structures and the RNA folded structures already discussed, both DNA and RNA are known to assume alternative structures, such as quadraplexes, that are of biological relevance. In addition, the nature of the hydrogen bonding patterns in oligonucleotides allows the design and construction of structures with the potential to significantly advance the field of nanotechnology [80,81]. To date the number of calculations on alternative oligonucleotide structures is not large; however, some good examples exist. MD simulations have been performed on four-stranded iDNA, which involves intercalated cytosines and have been shown to yield stable structures [101]. Quadraplexes are structures suspected to exist at the end of telomeres, stabilizing the terminal DNA in chromosomes. The quadraplexes themselves are stabilized by the presence of cations. Free energy perturbation calculations have been applied to investigate the influence of cation type on the extent of stabilization [102]. MD simulations have also been used to facilitate the determination of the structure and dynamics of quadraplexes based on NMR data [103]. Although only a few studies on alternative oligonucleotide folds have been studied to date, the ability of computational methods to investigate the stability of alternative structural motifs should facilitate the use of oligonucleotides in nanotechnology.

III. METHODOLOGICAL CONSIDERATIONS

Biological functions of nucleic acids occur at the level of single nucleosides up to chromosomes, with the majority of the functions intimately involving proteins as well as other biological molecules. Accordingly, computational methods for the study of nucleic acids must be able to access these variously sized systems. In this volume, the focus is on calculations at the atomistic level, and the present chapter remains consistent with this goal. Such a limitation, however, confines us to a discussion of computational studies of oligonucleotides that contain less than 50 basepairs. To overcome this limitation, a brief section on the different computational methods for the study of oligonucleotides, including approaches to the study of larger structures, follows.

A. Atomistic Models

Initial atomistic calculations on nucleic acids were performed in the absence of an explicit solvent representation, as discussed earlier. To compensate for this omission, various

changes in the energy functions were employed, including reduced charges in the phosphate groups, the inclusion of hydrated sodiums, and altered dielectric constants. These approaches did yield results that furthered our knowledge of the structure–function relationships of nucleic acids in a qualitative fashion. However, as computational methods matured, the demand for more quantitative results increased, requiring more realistic models in oligonucleotide calculations. These demands have been met by way of both improved algorithms for simulations and improved potential energy functions, not to mention the incredible increase in computational power that has occurred over the last two decades. These aspects are discussed in this section.

Central to the quality of any computational study is the mathematical model used to relate the structure of a system to its energy. General details of the empirical force fields used in the study of biologically relevant molecules are covered in Chapter 2, and only particular information relevant to nucleic acids is discussed in this chapter.

Initial applications of computational techniques to study nucleic acids used models adapted from protein force fields [104]. Other parameter work of note included early studies on the properties of the furanose ring in nucleic acids [105,106], which showed the importance of the proper treatment of the pseudorotation profile in potential energy functions. The first widely used force fields for the study of nucleic acids were associated with the programs CHARMM [107] and AMBER [108,109]. These force fields were developed for both extended atom and all-atom representations. Extended atom models, which have been used extensively for simulation studies of proteins, are of lesser value for calculations on nucleic acids because the smaller number of nonpolar hydrogens in nucleic acids makes the gain in computer efficiency for an extended atom model smaller for them than for proteins. Both force fields were also developed primarily for modeling and simulation studies in vacuum, based on the use of distance-dependent dielectric constants in the calculation of electrostatic interactions. Such an approach allows short-range electrostatic interactions to dominate while longer range interactions are damped in accord with the damping due to the dielectric constant of water. Both of these force fields were quite useful, and a number of the studies cited earlier were based on these works.

Computational studies of nucleic acids via atomistic models in the absence of solvent are expected to yield poor representations owing to the polyanionic nature of oligonucleotides. Although a number of efforts were made to circumvent the use of explicit solvent models, the results were generally unsatisfactory (see above). These failures motivated the inclusion of solvent models in MD calculations. Initial efforts, primarily based on the TIP3P water model [110], used the parameter sets developed for vacuum calculations; the only difference was the use of a dielectric constant of 1 for the electrostatic calculations rather than the distance-dependent dielectric. Results from these studies, along with work based on an earlier version of the GROMOS force field [111], generally led to improved agreement with experiment compared to the vacuum calculations. However, as MD simulations were significantly extending beyond 100 ps, it was observed that the structures deviated significantly from the canonical forms of DNA. It was not until the ‘‘second generation’’ force fields were developed that stable simulations of oligonucleotides in solution were achieved.

The second generation force fields for nucleic acids were designed to be used with an explicit solvent representation along with inclusion of the appropriate ions [28,29]. In addition, efforts were made to improve the representation of the conformational energetics of selected model compounds. For example, the availability of high level ab initio calculations on the conformational energetics of the model compound dimethylphosphate yielded

significant improvements in the models for the phosphodiester backbone. Similarly, new data on the interaction strengths for the Watson–Crick and Hoogsteen basepair interactions allowed for improvements in the nonbonded parameters. In the context of the nonbonded parameters is the need to properly balance the solute–solute interactions (e.g., Watson– Crick basepair interactions) with solute–solvent and solvent–solvent interactions. In the development of the CHARMM force field, special emphasis was placed on this balance [29]. On the basis of these force fields it was shown that stable MD simulations could be performed on a variety of sequences and structures; however, the results still had limitations [33].

Motivated by the need for further improvements, the AMBER [112] and CHARMM [36,113] force fields were optimized further. In addition, the BMS force field was developed, which incorporates features from the second generation CHARMM and AMBER force fields along with additional optimization of the parameters [38]. In all of these force fields, emphasis was placed on further optimization of the conformational energetics, yielding improved agreement with data on the canonical structures of DNA, surveys of dihedral distributions, and helical parameters of crystal structures from the Nucleic Acid Database (NAD) [114], along with a variety of other experimental and ab initio data. With the CHARMM force field, emphasis was placed on balancing local contributions, based on potential energy data for a series of model compounds, with global properties for duplexes in solution. This approach was designed to ensure that the proper contributions in the force field were combining to yield the desired properties of DNA and RNA in solution. It is expected that these improved energy functions will further facilitate the ability of MD simulations on nucleic acids to yield better quantitative agreement with experimental data.

Further extension of the atomistic models of nucleic acids will be achieved through additional optimization of the force fields using the present forms of the potential energy functions and extending the form of the potential energy function to include electronic polarizability along with other terms. Current work indicates that improvements in the revised CHARMM force field with respect to the treatment of sugar puckering are possible [113]. Concerning electronic polarizability, the polyanionic nature of nucleic acids and the influence of salt on their structure strongly indicate that gains in the quality of results from MD simulations will be made via its inclusion. To date, no calculations on nucleic acids using models that include electronic polarizability have been performed; however, QM studies have suggested that the polarization contribution to the solvation free energy of DNA is only 1–3% [115]. Thus, conclusions concerning the true gains to be made via the inclusion of electronic polarizability in nucleic acid simulations must wait until more detailed studies are performed. At this time one may speculate that some calculations addressing specific problems (e.g., energetics of phosphate–counterion interactions) will require polarizable models whereas other phenomena (e.g., conformational sampling of DNA and RNA duplexes) will see little improvement.

B. Alternative Models

Although the discussion thus far has concentrated on atomistic models of oligonucleotides in which the solvent is included explicitly, alternative models exist that allow for computational studies of larger oligonucleotides. For example, supercoiling of circular DNA has been studied using ribbonlike models [116]. This approach allows for the generation of trajectories using time steps of 100 fs, from which folding from the circular state into