Molecular Modeling of Nucleic Acid Structure: Setup and Analysis

Abstract

The last in a set of units by these authors, this unit addresses some important remaining questions about molecular modeling of nucleic acids. It describes how to choose an appropriate molecular mechanics force field; how to set up and equilibrate the system for accurate simulation of a nucleic acid in an explicit solvent by molecular dynamics or Monte Carlo simulation; and provides some information about how to analyze molecular dynamics trajectories.

ISSUES WITH SIMULATING NUCLEIC ACIDS

From the information presented in previous units (UNITS 7.5, 7.8 & 7.9), one should have a reasonable understanding of the various trade-offs that are necessary to model nucleic acid structures. For instance, when choosing an energy representation and a means to sample relevant conformations, there is a tradeoff between detail, sampling, and computational cost. Although the discussions thus far have presented basic means for modeling nucleic acids, some important details have not been sufficiently addressed. Reasonable decisions that remain for the prospective modeler to address are: (1) which empirical molecular mechanical force field is appropriate; (2) if one wants to run an accurate simulation of a nucleic acid in an explicit solvent, how does one set up and equilibrate the system; (3) how long should the production time of the simulation be in order to see convergence in properties of interest; and (4) how does one analyze molecular dynamics trajectories? This unit provides the answers to some of these questions and outlines a protocol for accurate simulation of nucleic acids.

Which Molecular Mechanics Force Field Is Appropriate?

This is a difficult question. The answer is complicated, often contentious, and in part depends on what representation (i.e., implicit versus explicit solvent, internal coordinate versus all-atom) is applied. With an internal coordinate representation, where bonds and angles are fixed, the most widely used molecular mechanical force field for nucleic acids is the FLEX force field within JUMNA (Lavery et al., 1995; Harvey et al., 2003). These sets of parameters have been used to study DNA deformations and more local changes involving backbone conformations, base opening and flipping (Lebrun and Lavery, 1996; Zakrzewska, 2003).

For simulations where each atom is parameterized (known as all-atom simulation), force fields for nucleic acids have steadily and continually improved in recent years (Pérez et al., 2012; Šponer et al., 2012; T. E. Cheatham and Case, 2013; Šponer and Lankas, 2006). Realistic simulations of nucleic acids require the presence of solvent around the molecule, otherwise the separation of both chains of DNA is observed and a “melting effect” is produced. One can use an implicit model where the solvation effect is represented as a continuum instead of having each individual water (or other solvent) molecule present in the simulation. Advantages for using implicit solvation include faster conformational sampling, modeling larger systems, and simulating systems which undergo large conformational changes (Tsui and Case, 2000b, 2000a; Onufriev et al., 2004). Evaluations of several force fields applied to duplex DNA using implicit solvation show a reasonable general agreement with experiments and simulations using explicit solvation (Gaillard and Case, 2011), however further improvements in the generalized Born methods and effective radii are likely still needed to accurately model RNA conformational motifs.

For simulations including explicit solvent with particle mesh Ewald to treat long range electrostatic interactions, the most widely applied force fields for nucleic acids have been the AMBER ff9X (Cornell et al., 1995, www.ambermd.org) and CHARMM c27 and c36 (Mackerell et al., 1995, http://mackerell.umaryland.edu/CHARMM_ff_params.html) force fields. Less popular force fields include those from GROMOS (Van Gunsteren and Berendsen, 1987, www.gromos.net), and the older BMS (Bristol Meyer Squibb) force field for nucleic acids (Langley, 1998) which was explicitly parameterized in order to properly simulate A-DNA/B-DNA equilibrium under various conditions including water/ethanol and high salt. As discussed in UNIT 7.9, to treat the electrostatic interactions correctly, it is recommended that a smooth cut-off method be applied (such as an atom-based force-shifted cut-off) or that the electrostatic interactions be fully included via an Ewald treatment. Each of the currently available force fields for nucleic acids has various strengths and weaknesses. Direct comparison is difficult, but despite this there are rigorous studies using consistent simulation protocols that benchmark their relative performance (Wolf and Groenhof, 2012; Ricci et al., 2010; Reddy et al., 2003; Pérez et al., 2008).

The old CHARMM22 force field (Mackerell et al., 1995) did not accurately represent canonical B-form DNA structures since slow transitions to A-DNA were seen in solutions with low salt conditions (Feig and Pettitt, 1998; Mackerell, 1997a; Norberg and Nilsson, 1996; Mackerell, 1997b). This force field for nucleic acids was updated with the CHARMM27 release (Mackerell et al., 2001; Mackerell and Banavali, 2000), and this force field does a better job on B-DNA structures, albeit with less sequence-specific minor groove narrowing than expected. The latest CHARMM36 force field includes new parameters involving ε/ζ torsions and sugar pucker modifications that resulted in increased sampling of the B_I and B_II substate populations present in B-DNA (Hart et al., 2012), and also alterations in the 2′-hydroxyl parameters to improve modeling of RNA (Denning et al., 2011).

DNA simulations using the 43A1 and 45A3 GROMOS force field resulted in increased flexibility in the double helix due to limited Watson-Crick hydrogen bonding. A new set of parameters called 45A1 focusing on the charge distribution of the nucleotide bases and backbone torsion improved canonical hydrogen bonding and better reproduced experimental data (Soares et al., 2005). A more revised version called 53A6 showed more accurate performance in base pairing and energetics (Oostenbrink et al., 2005). Simulations run for more than 5ns using the 53A6 GROMOS force field were very unstable due to the fact that GROMOS does not have a specific topology description for the 5′ and 3′ terminal nucleotides. This force field also show strong geometrical deformation, limited BI→BII substate sampling, and overestimated groove widths (Ricci et al., 2010) and yields a poor RNA representation (Pechlaner et al., 2013).

More consistent behavior with nucleic acids is observed even with the older Cornell et al. force field (AMBER ff94) in conjunction with a particle mesh Ewald treatment of the electrostatic interactions. With this force field, spontaneous A-DNA to B-DNA transitions are seen with a variety of sequences as expected (T. E. Cheatham and Kollman, 1996, 1998), and B-DNA to A-DNA transitions are observed with phosphoramidate-modified backbones (Cieplak et al., 1997), consistent with results seen in experiment. This observation is very exciting since it suggests that conformational sampling is not overly inhibited under truly periodic boundary conditions for DNA in explicit solvents. The force field also predicts A-RNA to be stable, although B-RNA to A-RNA transitions do not occur spontaneously (T. E. Cheatham and Kollman, 1997b). This brings up the issue of poor sampling of RNA in nanosecond-length simulation, since B-RNA is stable for >10 ns unless concerted changes in the sugar pucker are forced.

This is likely due to the larger barriers to conformational transition due to 2′-hydroxyl interactions with the backbone and larger barriers to sugar re-puckering. Modelers should be aware that sampling is more limited in all-atom molecular dynamics simulations of RNA. The Cornell et al. force field also reproduced sequence specific structures, such as the expected TpG step bends in the major groove or the narrowing of the minor groove in poly adenine (A-tract) regions, and shows good agreement with crystal data (Young et al., 1997b). Expected structural differences between DNA-DNA, DNA-RNA, and RNA-RNA duplexes are well modeled, as are modified nucleic acids such as phosphoramidates (Cieplak et al., 1997) and photo-damaged DNA (Spector et al., 1997; Miaskiewicz et al., 1996). The Cornell et al. force field also demonstrated the ability to model changes in nucleic acid structure that result from changes in the solvent and ion environment. This includes the stabilization of A-DNA in a water and ethanol solution (T. E. Cheatham et al., 1997) and spontaneous B-DNA to A-DNA transitions in the presence of hexammine cobalt(III) (T. E. Cheatham and Kollman, 1997a). Limitations of this older force field include lower-than-expected sugar pucker, χ angles, and helical twist. These specific issues were addressed through tweaks in the sugar pucker and glycosidic torsion parameters (called AMBER ff98)(T. E. Cheatham et al., 1999) and further minor tweaks in the AMBER ff99 force field (Wang et al., 2000).

In the 1990s, computer power was a limiting factor, and simulations of nucleic acids rarely went beyond ~10 nanoseconds or so. With access to large-scale computational resources, including the porting of MD codes to specialized hardware, such as GPUs or Anton, now microsecond timescales are routine (Salomon-Ferrer et al., 2013; Shaw et al., 2008). In terms of the force field assessment and validation, a common theme that has emerged is that as the simulations push to longer timescales, deficiencies in the force fields can be separated from deficiencies due to incomplete sampling and identified (Bergonzo et al., 2013; Šponer et al., 2012; Yildirim et al., 2011). Numerous problems were found with the AMBER force fields. Incorrect torsions and deformation of the backbone were corrected by the parmbsc0 modification, which fixed the α/γ overpopulations present in parm99 (Pérez et al., 2007b; Svozil et al., 2008). RNA simulations using the parmbsc0 modifications exposed destabilization and ladder-like distortions, leading to many modifications focusing on the glycosidic torsion. Using high level Quantum Mechanical techniques, the OL modifications to the χ torsion improve the balance between syn and anti-syn conformations (Zgarbová et al., 2011). Yildirm, Chen and Garcia, 2013 have also provided modifications for the glycosidic torsion, which show improved simulation results compared to NMR observables (Chen and García, 2013; Yildirim et al., 2011). Extensive testing and benchmarking for the Cornell force fields, including the latest modifications, has been performed by the Ascona B-DNA Consortium (ABC consortium) using 136 DNA tetra nucleotide sequences with a total simulation time of several microseconds. The consortium exposes the sequence specific cases where the current force field modifications work and fail (Lavery et al., 2010; Beveridge et al., 2004; http://gbio-pbil.ibcp.fr/ABC/)

Simulations of nucleic acids using polarizable force fields were expected to give a better representation of the electrostatics of the system, although at higher computational costs. Results of a 25ns 10-mer DNA simulation using current polarizable force fields and fixed-charge force fields showed little to no improvement in the results (Babin et al., 2006). For more details about polarizable force fields refer to UNIT 7.08.

Given the constantly changing landscape, the limited application of some of these force fields, and variations in the applied methods, it is difficult to evaluate which force field is “best”. Experience plays a part in the selection of a force field and, therefore, careful evaluation of the published reports is important. A final point is that not all force fields are compatible with a given simulation code. The Cornell et al. force field is released with AMBER; the Mackerell force fields with CHARMM, the GROMOS force filed is released with GROMACS MD package and other force fields with other codes such as ACCELRYS Discovery Studio and NAMD. Since CHARMM and AMBER use a similar molecular mechanics potential and equivalent Lennard-Jones combining rules, these force fields can be interconverted; this is not easily possible for GROMOS, which uses geometric-mean combining rules and has a different form for the bond- and angle-stretching terms. The list for Molecular Dynamics software available is extensive, although, the authors present in this review the options best suited for nucleic acids simulations.

Besides choosing which force field to use, an additional question is what to do if parameters are missing for nucleic acid base modifications. This is a difficult question to answer in general terms, however, as most force fields have a specific protocol that should be followed to develop new parameters. For example, with the Cornell et al. force field, new intramolecular parameters are chosen by analogy to be consistent with existing parameters, general van der Waals parameters are obtained from simulations of neat liquids (e.g., OPLS; Jorgensen et al., 1996)), and restrained electrostatic potential (RESP; Bayly et al., 1993) fit charges from ab-initio calculations are obtained for each new nucleotide, residue, or substructure. For more information about adding missing parameters, one should read all of the relevant force field literature and search for guides or repositories of existing parameters on the Internet. For small organic molecules like ligands and residues one can use the General Amber Force Field (GAFF; Wang et al., 2005: http://ambermd.org/antechamber/gaff.html) or the CHARMM general force field CGenFF (Vanommeslaeghe et al., 2010). It is also possible to pose questions to the various email reflectors for each program to ask if other investigators have already developed parameters for the system of interest. Additionally, it may be appropriate to contact the corresponding author of the force field papers for more information on the specific protocol for developing new parameters.

Balance is also an important requirement for a given force field; this means that in addition to accurately modeling the intra-DNA interaction, it is important to have a balanced representation of the DNA with solvent. The current force fields (Cornell et al., Mackerell, and Langley) appear reasonably balanced. In addition to balance of the DNA with the explicit solvent, these force fields, in general, allow simulation of protein systems. Reasonable representation of DNA-drug binding and protein–nucleic acid structures has been seen with all three of the major force field derivatives (Cornell et al., Mackerell and GROMOS; Wang et al., 2001); Simulations of B-DNA in the microsecond range are now routine, exposing the force fields to a new set of problems that require attention, mainly involving B_I→B_II substate sampling, sugar puckering distribution and the presence and frequency of base-pair opening events (Pérez et al., 2007a). For A-RNA, long simulations using the latest AMBER bsc0χ(OL3) and CHARMM force fields shows sequence dependent distortions caused mainly by the internal geometrical parameters of the base pairs and lack of any long lived substate.

Setting Up a Nucleic Acid System with Explicit Water and Counter-Ions for a Molecular Dynamics or Monte Carlo Simulation

Setting up an initial in vacuo model, or a simulation with implicit solvent, is straightforward. Models can be generated in a variety of ways, usually based on known experimental structure, but can also be generated de-novo with the help of specialized software (there are multiple tools available to generate nucleic acid structures, for examples see Macke T. J. and Case, 1997; Hornus et al., 2013; Accelrys Software Inc., 2009). Issues related to model building are discussed in greater detail in UNIT 7.5.

A molecular dynamics (MD) or Monte Carlo (MC) simulation (see UNIT 7.8) is generally broken up into two loosely defined and sequential phases: equilibration and production. The initial part of the simulation is called the equilibration phase. The equilibration phase includes the beginning parts of the simulation, often including both minimization and dynamics, performed in order to obtain a starting structure which is in a local energy minimum and therefore a “stable” starting structure for simulation. The stability of a starting structure is measured by various metrics such as the root-mean-squared atomic deviation from the starting structure or an experimental reference structure, the temperature, pressure, density, and/or the total molecular mechanical energy. Often during the early parts of the equilibration phase, restraints or constraints are applied to the main system of interest (e.g. the nucleic acid) to maintain the initial structure while the environment (e.g. solvent, ions, and hydrogen atoms, typically) around the main system of interest relaxes. Generally, as the equilibration phase continues, restraint force constants are gradually reduced and eventually eliminated. The goal of the equilibration phase is to relax the system and its environment such that production dynamics can occur without restraints. Once the simulation environment is stable, the production phase of the simulation can be performed. The production phase is the part of the dynamics or sampling that is extensively analyzed. The omission of the equilibration phase from this detailed analysis is necessary to avoid bias. The precise definition of what the production and equilibration phases entail are somewhat ambiguous. For example, when one parameter (such as root-mean-squared deviation (RMSd) of the solute to its starting structure, or the temperature) has stabilized around an equilibrium value, this does not imply that all variables have equilibrated. Whether a given observable has fully equilibrated depends on how long that observable takes to relax from its initial value to its equilibrium value. Many observable properties, such as the distribution of ions or sampling of thermally accessible conformational sub states, take considerable amounts of time to equilibrate, although, with current high performance computing facilities with hundreds or even thousands of processors and multicore GPU computing, is possible to obtain a reasonable equilibrated structure as a starting point for longer production runs in which a detailed analysis will be made.

Equilibration is necessary in molecular dynamics simulations to relax structural distortions and remove any large forces present that may bias the dynamics and artificially move the nucleic acid structure away from its expected or desired structure. This means that equilibration is necessary to thermalize the system to put a comparable amount of kinetic energy into each degree of freedom. When this is not done, large forces may result at the distortions that, in turn, lead to large collisions on the local scale and create local “hot spots”. These hot spots may move the structure in unrealistic ways. Therefore, the goal of the equilibration procedure is to relax the system as much as possible to avoid biasing it away from the starting geometry. This is generally done through a series of minimization and molecular dynamics simulations where the temperature or kinetic energy is gradually increased.

In molecular dynamics simulation of a model in vacuo, limited equilibration is necessary. Small distortions in the structure can be relaxed by short minimizations (~100 to 1000 steps). Generally, a simple first-order method, such as steepest descent, is applied first to remove the largest forces, followed by a faster directed minimization method, such as conjugate gradient minimization. Minimization is performed until the change in energy or gradient between each minimization step is small (~0.1 to 0.0001 kcal/mol). Careful thermalization of the system, via a series of short molecular dynamics simulations where the temperature is gradually raised, is usually not necessary for in vacuo simulation since the system equilibrates rapidly (due to significantly fewer degrees of freedom than corresponding simulations with explicit solvent). If explicit ions are included in the in vacuo simulation, more careful equilibration may be necessary to relax the ion atmosphere often involving more steps and a combination of restraints on the solute to let the solvent relax, but this is less commonly done.

Adding explicit solvent significantly increases the complexity and computational cost, and also necessitates a more stringent equilibration protocol (see Basic Protocol). To perform a simulation with explicit solvent using an appropriate nucleic acid model starting structure (in vacuo), an initial configuration of the solvated model is necessary. Moreover, it is typically desirable to include at least enough explicit salt to neutralize the system, and any excess salt as desired, even though nucleic acid simulations are little affected by the ionic conditions or even the water model used (Beššeová et al., 2012; Faustino et al., 2010). Adding solvent is typically performed by first completely surrounding the desired system by a set of pre-equilibrated solvent “boxes” representing the coordinates of a unit cell of “bulk” water, and then deleting those waters that overlap the model or extend beyond the boundary of the system under consideration. The only problem with this approach is placing solvent inside interior cavities or at interfaces, such as a protein-DNA interface. Unfortunately, there is currently no clear consensus on how best to do this in the absence of experimental structural data. However, in the authors’ experience, the water structure relaxes rapidly with the commonly used water models and is able to diffuse efficiently even into tight interfaces where proteins bind DNA within short (100-psec) simulations. In the absence of a pre-equilibrated solvent box, it is possible to simply add a crystalline representation of the solvent equally spaced at approximately the correct density; in this case, longer equilibration of the solvent may be necessary to fully relax the system and remove the crystalline bias.

As discussed in UNIT 7.9, enough water should be added to not only fully hydrate the model but to represent bulk water some distance away from the model. However, adding water tremendously increases the cost of the simulation. Typically, when periodic boundary simulations are applied, ~8 to 12 Å of water surrounding the model in each direction is appropriate, although more adventurous souls could use more or less water. Given the issues and potential for artifacts with non-periodic systems, particularly for highly charged systems such as nucleic acids, no periodic boundary conditions are not recommended for nucleic acid simulations except under specific conditions. These include the desire to represent a minimally hydrated nucleic acid in vacuo or a very large nucleic acid where only a core part of the structure (surrounded by a blob of explicit water with stochastic boundary conditions) is of interest (Mazur, 2001). Non-periodic boundary conditions may also be appropriate for very large and irregularly shaped models where the regular shapes of periodic boundary conditions may require too much water or could, in principle, inhibit motion. However, in these cases, implicit solvent models are more appropriate. In the opinion of the authors, it is wise to avoid methods that include explicit solvent but also dampen the electrostatic interactions through the application of a distance dielectric function, since these methods tend to significantly dampen the conformational fluctuations.

For standard nucleic acid models representing a folded structure or 10 to 25 base pairs of a linear duplex, periodic boundary conditions are more appropriate. In principle, there is no reason to limit the shape of the periodic unit cell to a cubic shape, as any uniform space-filling shape is appropriate. Possible unit cell types are shown in Table 7.10.1. For long linear duplexes, the most appropriate unit cell might be the orthorhombic or hexagonal unit cells with one of the dimensions longer than the other two. This allows reasonable solvation of the duplex without large uninteresting regions that contain only solvent. However, an important consideration is that the rotational correlation time of small duplexes in solution is in the nanosecond time range. Therefore, during the dynamics in this type of orthorhombic box, the model may rotate to span the short edge of the box where the model can then interact directly with its periodic image (assuming a truly periodic method). This can also happen with constant pressure simulations where each box length is free to change. As shown in Figure 7.10.1, this can, in principle, lead to distortion of the model structure. This rotation can be removed, at the expense of adding an uncorrected net torque to the system, which can lead to artifacts, or be inhibited, by restraining the top and bottom of the duplex with weak restraints (which may inhibit bending). An additional issue is that interactions with periodic images in these long narrow boxes could, in principle, inhibit bending. In practice, inhibition of bending does not seem to be a major issue. Simulations on 24-mers reproduced expected sequence- and salt-specific bending in various phased A-tracts in long rectangular boxes (Young and Beveridge, 1998; McConnell and Beveridge, 2001). To avoid a possible bias from inhibited rotation, more voluminous cubic boxes can be applied. This, however, leads to much more water in the corners than is necessary. Therefore, more modelers have shifted towards using more “spherical” unit cells, such as the 14-sided truncated octahedron (Allen and Tildesley, 1987) and 12-sided rhombic dodecahedron (see Figure 7.10.2). These unit cells limit the volume while maintaining distance between periodic images. Of course, adding the solvent is a little trickier in non-orthorhombic unit cells since, when overlaying a larger solvent box, it is not a trivial procedure to remove waters outside the cell by simply checking if the water has coordinates larger than the box in a given dimension. A simple solution is to keep a set of the original coordinates with solvent, perform periodic imaging with the new unit cell type, and then, by comparison with the saved coordinates, delete any waters that have moved.

Table 7.10.1.

Standard Unit Cells Appropriate for Periodic Boundary Conditions

Restrictions on unit cell parameters, Volume

Cubic, a = b = c, α = β = γ = 90.0°, V = a3

Tetragonal, a = b, α = β = γ = 90.0°, V = ca2

Orthorhomic, α = β = γ = 90.0°, V= abc

Monoclinic, α = γ = 90°, V=abc × sin(β)

Triclinic, No restrictions, see legend

Hexagonal, a = b

Rhombohedral (trigonal), a = b = c, α = β = γ < 120.0°, V = a3 × [1 − cos(α)] × [1 + 2cos(α)]1/2

Octahedral (truncated octahedral), a = b = c, α = β = γ = 109.47122063449, V = (4(3)1/2/9)a3

Rhombic dodecahedral, a = b = c, α = γ = 60°, β = 90.0°, V = (1/2)1/2a3

^aRestrictions on the unit cells lengths (a, b, c) and angles (α, β, γ) are presented along with the volumes for a variety of simulation cells. The volume of a triclinic cell is V = abc × [1 − cos(α)² − cos(β)² − cos(γ)² + 2cos(α)cos(β)cos(γ)]^1/2.

Figure 7.10.1.

Orthorhombic unit cells and duplex rotation or unit cell size changes.

Figure 7.10.2.

A model of B-DNA in a solvated octahedral (truncated octahedral), unit cell after 1 μsec of molecular dynamics with particle mesh Ewald (PME) in AMBER.

Although in principle adding salt is as easy as adding solvent, it is slightly more complicated in practice (Auffinger et al., 2007). In an ideal case, random waters might be replaced by explicit salt ions up to the desired concentration, and then molecular dynamics or Monte Carlo simulation can be performed to equilibrate the salt. Before solvating the system, it is advisable to place the net-neutralizing counter-ions. If available, structural information regarding the placement of ions can be used as an initial guide. However, more often than not, this information is not available. The net-neutralizing counter-ions are added to balance the charge on the phosphates. Monovalent ions are typically the ion of choice because of their fairly rapid diffusion, small size and atomic number, and relatively rapid exchange times. Precise placement is not a major issue. One way to avoid any bias upon the placement of explicit salt ions is to add the ions and after this add the explicit solvent molecules. With the full solvated system, randomly swap ion molecules with water molecules and with this configuration, start the minimization steps. To achieve that, we can use the freely available analysis tool cpptraj (Roe and Cheatham, 2013). This program will swap an ion molecule with a randomly chosen water molecule. The input command of cpptraj looks like this (for AMBER type files):

## Considering a 12-base DNA system
## Load topology file
parm topology-name.parmtop
## Load initial coordinate file in restart format
trajin initial-coordinates.coords
## Specify the output name and type (AMBER restart)
trajout initial-coordinates-randomized.coords restart
## Run the randomizeions command
## This will swap Na+ ions with water getting
## no closer than 6.0 A from residues 1-12 and
## no closer than 4.0 A from any other Na+ ion.
Randomizeions :Na+ around :1-12 by 6.0 overlap 4.0
## Run the actual calculation
run

In the absence of any explicit information suggesting direct binding of an ion to the nucleic acid, placing ions that are directly bound (without bridging water) should be avoided, otherwise, the structure may distort under the influence of the bound ion. Placing ions within a hydration shell is reasonable since, in many cases, the interaction of a cation with a nucleic acid involves bridging water, such as with magnesium (Buckin et al., 1994) or barium (Sternglanz et al., 1976). In spite of this, specific interaction of ions with nucleic acids are observed (Auffinger et al., 2004). Direct interaction is also seen with divalent ions in RNA that are known to stabilize the tertiary structure.

Reasonable molecular mechanical potentials exist for treating these ions, and a variety of parameterizations are in common use (Li et al., 2013; Joung and T. E. Cheatham, 2009; Allnér et al., 2012; Smith and Dang, 1994; Aqvist, 1990).

After placing the net-neutralizing ions, excess salt can be added by replacing random waters some distance away from the nucleic acid and other ions. The key question is how much excess salt should be added. Since the system is being modeled at an atomic level with unit cell lengths of ~25 to 100 Å, very small changes in the unit cell size or the number of counter-ions have a large influence on the effective molarity. For a given ion, the molarity as reported in simulation literature is based on either the total amount of a specific ion present in the simulation (not just excess salt) or the total number of excess ions per the total volume (converted to moles/liter). Since the system is not a bulk macroscopic system, if the net-neutralizing salt is included in the calculation of molarity, this concentration may be much higher than expected under periodic boundary conditions. One can also refer to the ionic strength of the system; this includes all the ionized groups (including phosphates) and the effective ion concentration will be even higher! Best practice is to perform simulations of nucleic acids with at least net-neutralizing salt. DNA is known to denature in the absence of salt, although this is not readily seen in simulation with proper treatment of the long range electrostatic interactions. However, RNA duplex structure is rapidly degraded in MD simulations in the complete absence of salt. As most experiments are performed near physiological salt concentrations (~100–200 mM), adding in excess salt is also advisable. A general rule of thumb is to consider the molarity of water, 55 M. For 200 mM salt, this means one monovalent ion pair per 55*5 water molecules.

Given the small size of the unit cells and relatively large number of phosphates to added salt, these simulations are most often performed at high ionic strength. Despite the sensitivity of molarity to unit cell size, when looking at monovalent ions, there is very little salt dependence on dynamics or structure over the range of no salt (including no net-neutralizing salt) from 200 mM to 1 M salt in 1 μsec-length simulations with the Cornell et al. force field (T. E. Cheatham and Kollman, 1998 and unpublished results). It is not until high salt concentrations (>3 to 4 M) are reached that transitions in DNA duplex structure are seen besides global and local parameter modifications; these transitions have been observed in simulations with the Cornel et al., CHARMM and GROMOS nucleic acid force fields (Auffinger et al., 2004). Divalent and multivalent ions, on the other hand, have much more direct influence on structure. For example, only four Co(NH3)6 ions are necessary to observe B-DNA to A-DNA transitions with the Cornell et al. force field (T. E. Cheatham and Kollman, 1997a). Magnesium may also affect bending (Beveridge et al., 2004; Young and Beveridge, 1998).

Equilibrating Simulations with Explicit Solvent

After generating initial ion and solvent positions, it is necessary to equilibrate the system. This relaxes the system to the expected density and allows the water and ions to react to the presence of the nucleic acid. Minimization to remove unrealistic energies is an essential first step to this process; however, it is not sufficient given the multiple minima problem (UNIT 7.8). Moreover, the fact that minimized water or “ice” is not what is really desired, it is necessary to sample possible configurations via molecular dynamics or Monte Carlo simulation. Furthermore, the “pre-equilibrated” water will not have reacted to the presence of the nucleic acid. Thus, the system will most likely not be at the correct density. To remedy this, constant pressure equilibration under periodic boundary conditions is likely necessary. The initial solvent (and ion) equilibration is the most important part of any equilibration protocol prior to production MD. Given appropriate simulation methodologies, if the solvent and ionic atmosphere is well equilibrated, the simulation will likely be stable. In this case, the precise and intricate details of the remainder of the equilibration protocol are likely to be unimportant. This has been shown in molecular dynamics simulations of a DNA duplex where, after equilibration, there was little observable effect of varied ion placement when comparing three different mechanisms for placing sodium counter-ions (Young et al., 1997a). It should be noted that “equilibration” in this context refers to the generation of a more reasonable solvent structure and initial configuration that (1) does not contain local hot spots with unreasonable forces, (2) is at the correct pressure and density, and (3) has a reasonably stable potential energy. This equilibration does not refer to complete equilibration of the nucleic acid model, a process that takes significantly longer.

A standard procedure (see Basic Protocol) is to first perform minimization to remove any large energies (which will lead to initially large forces), and then perform ~25 to 100 psec of constant pressure dynamics, with the nucleic acid held fixed or restrained to the initial model in order to relax the surrounding water and ion environment. The progress of the equilibration procedure is typically monitored by plotting the potential energy, density, and pressure. Equilibration is thought to be complete when these (and other) values have stabilized (see Figure 7.10.3). After this phase, minimization is performed on the entire system with gradually reducing restraints on the initial nucleic acid model structure. Then, dynamics are performed on the entire system, slowly raising the temperature.

Figure 7.10.3.

Graphs monitoring the equilibration of a simulation of a DNA (Drew-Dickerson dodecamer) in explicit water. The DNA was held fixed and only the ions and water were allowed to move.

Various protocols are used (for examples see: Réblová et al., 2006; Spacková and Sponer, 2006; Henriksen et al., 2012; Oostenbrink et al., 2005), and in practice all these protocols seem to work consistently well. For more complicated systems, such as those involving high concentrations of salt or mixed solvent (such as ethanol and water), longer equilibration protocols are necessary. Note that this type of equilibration protocol tends to support water and ion conformations that stabilize the initial model structure. This can inhibit conformational transitions to other structures, such as B-DNA to A-DNA transitions in high salt, since the initial configurations are optimized to the “fixed” initial structure (Song et al., 2006; Yu and Fujimoto, 2013; Langley, 1998). A final note is that constant pressure simulation methodologies are plagued with pitfalls. If the forces restraining or fixing the initial conformation of the nucleic acid are not properly included with the calculated pressure, the pressure may be overestimated, leading to box expansion upon pressure scaling in longer equilibration simulations. For a more detailed discussion of this and other issues, see (T. E. Cheatham and Brooks, 1998).

EQUILIBRATION

This protocol describes the constraint/restraint of solute, relaxation of restraints, and equilibration (without restraints). Minimization is performed for ~50 to 10,000 steps, although less minimization (~200 to 500 steps) may be acceptable. Initial equilibration in molecular dynamics simulation takes ~10 to 100 psec. It is important to avoid large force constants when applying harmonic restraints in molecular dynamics, since these may lead to high frequencies and require shorter time steps for proper equilibration. Force constants in the range of 1.0 to 15.0 kcal/mol/Ų are reasonable.

This protocol is intended to serve only as a guide. The primary literature, program manuals, and available resources on the Internet should be consulted for more information. For AMBER, see http://ambermd.org/tutorials/.

Constrain or restrain solute (optional for in-vacuo simulation)

Delete

A) Minimization

• 1aRun initial minimization using a 25 kcal/mol restraint on DNA
• 2aIf necessary, turn off SHAKE constraints (that fix bond lengths involving hydrogen).

  The need for this step depends on the minimizer, force field, and SHAKE algorithm used.

• 3aPerform steepest descent minimization until energy change at each step is less than ~1.0 kcal/mol.
• 4aContinue with conjugate gradient until energy change is less than ~0.1 kcal/mol.
• 5aIf initial dynamics “blow up” (e.g., through SHAKE failures or sudden large increases in energy), perform additional minimization.

B) Molecular dynamics

• 1bApply an integration time step of 1 to 2 fsec for most available molecular mechanics force fields.
• 2bApply SHAKE on hydrogen atoms if using a rigid three-point water model.
• 3bApply constant pressure if using periodic boundaries in explicit solvent (UNIT 7.9).
• 4bMaintain desired kinetic energy or slowly raise (ramp up) the kinetic energy to desired values. Perform molecular dynamics simulation for ~10 to 100 psec.

  Longer equilibration times are likely necessary with high salt conditions, multivalent ions, mixed solvents, or slowly diffusing solvents.

Relax restraints (optional)

• 5Perform minimization only (~1000 to 5000 steps) with restraint force constants gradually moved to zero, or perform cycles of minimization (~1000 to 5000 steps) and dynamics (~1 to 50 psec) with restraint force constants gradually moved to zero (i.e., 15 to 10 to 5 to 2.5 to 0.0 kcal/mol/Å2).

Perform equilibration without restraints

• 6Perform minimization or molecular dynamics as described above.

  1. Make certain that the overall rotational and translational kinetic energy is removed after initial velocity assignment and at periodic intervals as necessary (Harvey et al., 1998).

  2. If molecular dynamics blow up due to SHAKE failure or large energy change, try more minimization and/or changing the coordinates of the initial geometry. If dynamics continue to fail, try decreasing the integration time step. If it continues to fail, look for a strong overlap of atoms and/or improper treatment of the electrostatic interactions.

  3. Monitor potential energy and the root-mean-squared deviation (RMSd) from starting structure, temperature, pressure, density, and volume in order to judge progress of equilibration. Begin production dynamics when these and other interesting observables appear to stabilize.

     At this point it is possible to change from an NPT (constant pressure, constant temperature) ensemble to an NVE (constant volume, constant energy) ensemble.

ENHANCED SAMPLING METHODS

Despite the advantages gained through advances in CPU and, more recently, GPU technology, long timescale dynamics remain limited to the microsecond timescale for most, and the millisecond timescale at the upper limits (Shaw et al., 2009). To access these conformational states we can move away from time-dependent dynamics and use techniques to sample the conformational space of the entire ensemble. These techniques include locally enhanced sampling, and replica exchange methods where temperature and/or another description of the Hamiltonian can be used to accelerate or increase the search of conformational space, as well as adaptively biased molecular dynamics and conformational flooding (Bergonzo et al., 2013; Henriksen et al., 2013). If the transition of interest is already known, and what is desired is the conformational change between two discrete states, path sampling methods can be used, such as the plain/nudged elastic band or string methods (Bergonzo et al., 2011). These are all advanced techniques, and each has different parameters for evaluating their success. For more accurate free energy differences between configurations, it is necessary to explicitly sample the configuration of accessible conformations connecting the end points, or states of interest, using a single simulation or series of simulations. This is done by using one or multiple sampling techniques, each with its own advantages and disadvantages and special care must be considered when using any of these methods. Some of the methods used for nucleic acids simulations are: umbrella sampling, thermodynamic integration, and steered molecular dynamics. These methods force the system to sample along a particular path using a variety of “tricks” like geometric restraints, thermodynamic exchange using replicates, Hamiltonian exchange, etc.

ANALYZING THE RESULTS

After equilibration, production simulations are run for as long as computationally feasible or necessary to sample a relevant number of conformational transitions of interest. Current state-of-the-art simulations of small solvated biomolecules (representing on the order of ~10,000 to 25,000 atoms) are performed for on the order of 1 to 5.0 μsec.

At regular intervals, the configuration of the system (including the values of various energy terms and the atomic coordinates) should be saved and recorded to file(s). There are a variety of means to analyze the results. Analysis is performed not only for the purpose of extracting useful information about the structure but also to check the simulations for any aberrant behavior. In general, in order to obtain meaningful statistics when monitoring a particular observable, the simulation should be run on a time scale that is at least an order of magnitude longer than the correlation time of that particular observable. Some properties relax very quickly, such as various equilibrium properties of water, i.e., density and average potential energy, which converge in short simulations (10 to 100 psec), as shown in Figure 7.10.3. Other properties, such as structural relaxation, substate populations, or folding, may occur on a very long time scale. As discussed earlier, not all properties of a given system may fully equilibrate within the time scale of the simulation. Figure 7.10.3 shows a representative graph of various properties during a molecular dynamics simulation. From the graphs it is clear that the simulation properties monitored fully converged during these simulations.

Common properties to monitor include the RMSd from the starting and average structures (created by performing a straight coordinate average of RMSd fit configurations over a stable portion of the trajectory) and helicoidal parameters (UNIT 7.5), among others. In addition to investigating the time dependence of properties, various correlation functions are also appropriate to extract information that can be compared more directly to experimental results. The major effort of any modeling project is often spent in analyzing the results (in recent years the huge increase in computational speed has made the data available much faster than the speed at which we can actually process it). In large part, the type of analysis methods applied depends on what one is trying to learn from the simulation and what experimental data is available for comparison. There is no one specific protocol that can be summarized here, although, one should always check the results with the use of a visualizer and ask basic questions, for example: Does the structure obtained from the simulation makes sense? Did it become deformed in an unexpected way? Did it “blow-up”? Sometimes it is hard to tell from energy values or RMSD alone if things have gone wrong. For more information and to get a better handle on various analysis tools, the primary literature should be consulted. A good source of information on modeling and analysis is provided by (Leach, 2001) and (Šponer et al., 2006). Additional information is available with each of the simulation programs and on their respective Web sites. For AMBER type trajectories (although it accepts as input other type of trajectory formats), the tool cpptraj provides a wide array of analysis tools. The basic workflow is to read a topology file which includes the parameters for each individual atom and read the trajectory files that match the loaded topology. Then, type the keywords for each analysis that we wish to perform, setting up atom masks, start and finish frames, time offset, and any type of special keyword required for each analysis and start the analysis. Cpptraj will read the frames into memory and execute each analysis generating output file if required. For more information refer to (Roe and Cheatham, 2013). Although it is beyond the scope of this unit to discuss all the means for analyzing molecular dynamic trajectories in detail, an important tool worthy of discussion is the means to judge the importance of sampled conformations from a molecular dynamics or Monte Carlo simulation.

INEXPENSIVE METHODS TO ESTIMATE CRUDE RELATIVE FREE ENERGY DIFFERENCES

Given two different conformational states of the same molecule (using the same force field) sampled in Monte Carlo or molecular dynamics simulation, an estimate of the relative free energy can be obtained either by characterizing the set of configurations that represents each sampled state or by characterizing the minimum energy conformation that best resembles each sampled state (Kollman et al., 2000; T. E. Cheatham et al., 1998). This characterization involves estimating the relative free energy. In this context, the free energy is the sum of the enthalpy and a temperature-weighted entropy term. Determining the energy or enthalpy for a given state is relatively straightforward; it comes directly from the molecular mechanics energy function, either as an average over the configurations or as the minimum energy of a representative conformation for each state. As discussed in UNIT 7.8, it is not directly possible to compare molecular mechanical energies among different molecules (due to different zero-point energies) or with different force fields (due to possible different scales and different zero-point energies). Therefore, in this unit, the reference to relative energy and free energy differences are for the same molecule. For different molecules, other techniques may be more appropriate, such as free energy perturbation (discussed briefly at the end of this section). Typically, in post-processing the energies the solvent is not included explicitly (as discussed in more detail below) but is represented implicitly.

In contrast to enthalpy, the entropy is less straightforward to estimate because it is an ensemble property. Although it can be calculated directly (at considerable cost), it is most often approximated and calculated independently for the solute and solvent (Andricioaei and Karplus, 2001; Baron et al., 2006; Karplus and Kushick, 1981). There are two basic methods for approximating the entropy; both comprise translational, rotational, and vibrational components. The translational and rotational components are calculated for a rigid rotor approximation or by some other means. The vibrational component can be estimated via two methods. The first involves the use of a “representative” minimum energy conformation. For this conformation, the normal modes of vibration are calculated using a harmonic approximation. These normal mode frequencies can then be used to estimate the vibrational components of the entropy based on the local fluctuations in the neighborhood of the minimum energy conformation. This can give crude estimates, assuming that (1) the conformation is truly at the energy minimum, (2) the single minimum-energy conformation represents the state of interest, and (3) the anharmonic effects are small. However, the entropy typically involves more than local fluctuations within a given potential energy well for macromolecules, such as the entropy from larger scale conformational rearrangements. If the state of interest is characterized by a number of substates, which is likely the case (Poncin et al., 1992; Madhumalar and Bansal, 2005), the approximation of a single representative state may break down. Therefore, a set of representative states may be more reliable. However, counting the number of effective states and estimation of the energy based on a complete enumeration of the partition function necessitates reasonable sampling that may, in practice, not be feasible. The alternative procedure to calculate the vibrational entropy uses a quasi-harmonic approach with vibrational frequencies estimated from the fluctuations observed during molecular dynamics. This allows estimation based only on the relatively important fluctuations in the representative set of states (Karplus and Kushick, 1981; Baron et al., 2006; Andricioaei and Karplus, 2001). Given the sampling difficulties, entropic effects are difficult to estimate and lead to the calculation of “crude” relative free energies. Note that these approaches are only valid (in practice) for estimation of the entropy of the solute. In spite of the difficulties in estimating entropy, several approaches show good results in estimating if not the approximate values, but the magnitudes of change among similar systems (Brice and Dominy, 2011).

As mentioned, the entropy of the solute is a little more problematic to estimate. In general, a clear consensus on how best to estimate the entropic component has not emerged. One might consider making the assumption that the differences in entropy are largely represented in the solvation terms and not due to differences in the configurational entropy of the solute; however, this is likely invalid in most cases. With the A/B-RNA case study, in principle and as a first approximation, an estimation of the entropy can be obtained using the vibrational partition function and a harmonic approximation to the normal modes to estimate the frequencies in representative minimum-energy conformations. A problem with this approach is that in-vacuo minimization of nucleic acids (without the water) leads to distortion of the structure away from the structure represented in solution; therefore, the calculated entropy may not accurately represent the entropy as estimated from the various snapshots in the respective trajectories. It will, however, reasonably estimate the entropy of the gas phase model structure. For the model average structures of [CCAACGTTGG]₂ A-RNA and B-RNA (averaged over nanosecond portions of the respective trajectories taken at 1-psec intervals), minimization moved the models 1.9 Å and 2.4 Å, respectively, from the average structure. Estimation of the entropy difference using the normal modes favors A-RNA by ~3.0 kcal/mol (at 300 K); although this is not a large difference, it is significant (T. E. Cheatham and Kollman, 1997b).

In general, the ability to rank the various “models” can be used to judge the utility or importance of a given model. Since the same sequence and force field is used, it is possible to directly compare the molecular mechanical energies (although, note that the solvent- solvent energies may have to be normalized if the two different simulations contain differing numbers of waters). The easiest way to estimate the relative free energy is to break up the total into contributions from the solvent (typically done implicitly) and the solute from the MD or MC simulation. Given a series of representative configurations from the dynamics, it is possible to determine the average intra solute energy (or enthalpy).

The free energy of solvation can be estimated more directly. Under the assumption of linear response, a simple approximation to the solvation free energy in explicitly solvated simulations (also assuming that the bulk of this energy is represented by “close” waters, so that normalization for the total number of waters is not necessary) equates this free energy with half the solute-solvent interaction energy, 1/2 E_{solute-solvent}. A better estimate might be obtained by stripping the explicit water from each configuration and then performing a quick calculation on this conformation with an implicit solvent treatment. This will give an estimate of the solvation free energy. Recall that the implicit water models are typically parameterized to reproduce the free energy of solvation directly with a polarization component from Poisson-Boltzmann or a generalized Born treatment of the electrostatics and nonpolar contributions from a surface area term. This type of treatment has been applied to investigate a small turn-forming peptide based on long solvated MD trajectories (Bashford et al., 1997), and has been applied by various groups not only on the A/B-RNA case study but also the A/B-DNA equilibrium under various conditions, as well as a variety of other applications (Kollman et al., 2000). These “MM-PBSA” techniques are very useful tools for postprocessing MD or MC trajectories to give further insight (Miller et al., 2012). Note also that since water and ions are often integral parts of nucleic acid structure, when using these methods it may be necessary to include an explicit representation of a subset of the bound waters or ions. For example, in order to more accurately estimate the binding affinity of DAPI to DNA (Spacková et al., 2003) and the stability of G-DNA models (Šponer et al., 2013; Fadrná et al., 2004), a small number of waters near the DAPI and the bound ions, respectively, needed to be included in the MM-PBSA analysis.

Typically, multiple simulations are run using biasing potentials along a given reaction coordinate, and the results are accumulated and unbiased through a procedure such as the weighted histogram method (Kumar et al., 1995, 1992; Roux, 1995). This procedure has been used, for example, to characterize protein folding (Boczko and Brooks, 1995), protein-DNA interactions (Bergonzo et al., 2011), intra-base pair distortions (Banavali and Mackerell, 2002), groove distortions (Zacharias, 2006), etc. The use of these biasing potentials requires some understanding of the reaction path between the two states of interest, and therefore is not straightforward and is very computationally demanding.

Calculating relative free energies of different molecules upon small chemical changes, free energy perturbation or thermodynamic integration techniques can be applied; for detailed reviews about the origins of some of these sampling methods see (Beveridge and DiCapua, 1989) and (Kollman, 1993).

SUMMARY

The methods and tools for accurate simulation of small nucleic acids in solution have advanced considerably in recent years. A summary of the highlights is presented in recent reviews (Laughton and Orozco, 2009; Pérez et al., 2012; Hashem and Auffinger, 2009). In general, when simulating a poly-ionic system such as nucleic acids, it is necessary to not only provide a proper representation of the long-range electrostatic interactions through atom-based force-shifted cutoffs or an Ewald treatment, but also include some representation of the surrounding environment (i.e., water and salt). Tremendous strides have been made in recent years, including accurate representation of A-tract bending, specific ion association, and sequence-specific structure and dynamics. The current generation of force fields still retains some systematic errors and clearly more computer power is necessary to begin to tackle larger-scale problems and longer simulation times. However, the future holds tremendous promise.