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Molecular Dynamics Simulations of Peptides

Jeffrey Copps, Richard F. Murphy, and Sandor Lovas

Key Words: Force field; Molecular dynamics; simulations; replica exchange;GROMACS; GROMOS96; folding; secondary structure

1. IntroductionMolecular dynamics (MD) simulations fill a significant niche in the study

of chemical structure. While nuclear magnetic resonance (NMR) yieldsthe structure of a molecule in atomic detail, this structure is the time-averaged composite of several conformations. Electronic and vibrational circulardichroism spectroscopy and more general ultraviolet/visible and infrared (IR)spectroscopy yield the secondary structure of the molecule, but at low resolution.MD simulations, on the other hand, yield a large set of individual structuresin high detail and can describe the dynamic properties of these structures insolution. Movement and energy details of individual atoms can then be easilyobtained from these studies.

In MD simulations, trajectories (configurations as a function of time)of individual atoms are generated by simultaneous integration of Newton’sequation of motion. The forces acting on each atom are the negative derivativeof the potential energy and are termed the “force field.” Force fields areparametrized using physical data from x-ray crystallography, IR and Ramanspectroscopy, as well as high-level quantum mechanical calculations with modelcompounds, to reproduce the vibrational and conformational characteristics ofa wide variety of molecules. The potential energy is the sum of bond and angleenergies, the energy of bond rotations, and the energy of nonbonded van derWaals and electrostatic interactions, as in the general class I force field equation

From: Methods in Molecular Biology, vol. 494: Peptide-Based Drug DesignEdited by: L. Otvos, DOI: 10.1007/978-1-59745-419-3 7, © Humana Press, New York, NY

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Fig. 1. Potential energy function for general class I force field.

shown in Fig. 1(1). In class I empirical force fields, bond stretching and anglebending relative to equilibrium radius and angle values are described by theclassical, not quantum harmonic oscillator function, necessitating the use ofconstraints to approximate the quantum effects of vibrating bonds and bondangles. Bond rotation is described by a sinusoidal function that approximates thepeak energy from repulsion when a torsional angle is in the cis configuration, andthe minimum when it is in trans configuration. Finally, nonbonded interactionsare described by Lennard-Jones as well as Coulombic potentials, which approx-imate the long-range forces between uncharged and charged atoms, respectively.However, they overestimate the effect of molecular dipoles and cannot simulatemolecules with significantly different polar attributes simultaneously, and thisaffects real equilibrium distance values because of the movement of atoms basedon polarization. Class I force fields also cannot compute properties which are farfrom equilibrium or accurately predict vibrational spectra and they are temper-ature dependent. Class II empirical force fields approximate the forces uponmolecules using more descriptive, complicated functions, such as using a Morsepotential in place of the harmonic oscillator in describing bond stretching andangle bending, and higher-order terms in describing nonbonded interactions.This increases the accuracy of prediction, but also increases the computationtime, often prohibitively.

The choice of a particular force field depends on the type of system forwhich it has been designed. Several class I force fields have been designed fordescription of polypeptides, including AMBER (2), CHARMM (3), OPLS andOPLS-AA (4,5), and GROMOS96 (1). The computational study of peptidesand proteins can yield information regarding the importance of residues andfunctional groups in determining the structure, folding and solubility in variousenvironments. This information can then be applied to the study of the structure-activity relationships of those molecules in ligand–receptor complexes and aidin the design of new therapeutics.

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GROMOS96 is a united atom force field, modeling only polar hydrogenatoms explicitly, while aliphatic hydrogens (such as methyl hydrogens) aregrouped with attached carbons to form united atoms treated as single atoms.This small concession in modeling precision greatly accelerates computationaltime. GROMOS96 is implemented by the GROMACS MD simulation package(along with several other force fields) and was developed specifically for proteins(5). Due to its greatly optimized code, GROMACS is currently the fastestMD simulation program available. As a class I force field, the accuracy ofGROMOS96 is more limited than class II force fields. The speed of calculation,relatively accurate prediction of peptide conformation, ease of setup, and use ofthe GROMACS program and GROMOS96 force field have proven to be quiteuseful in conducting MD simulations, so both are used here to demonstrate aprocedure for conducting the MD simulation of the structure of a peptide in anexplicit solvent.

2. Methods for Standard MD Simulation (Fig. 2)1. The first step in running an MD simulation is the generation of an input structure.

Preferably, experimentally determined structures from NMR spectroscopy or x-ray crystallography studies should be available for the peptide/protein of interest.The RCSB Protein Databank (http://www.rcsb.org/) (6), for example, has manysuch structures available for download. Alternatively, energy-minimized struc-tures can be generated from the original sequence using theoretical methods suchas homology modeling and simulated annealing (7).

2. Once an input structure has been selected, it must be converted to the GROMACSfile format. The program pdb2gmx converts the initial structure file to theGROMACS structure file (.gro) format and generates a system topology filebased on predefined standard residue topologies describing the atoms, bonds,and torsional angles of the residues, as well as the force field (in this case,GROMOS96 with the 53a6 parameter set (8)) and protonation state of polarsidechains and of the N-terminus and C-terminus. If a standard topology is notavailable for a given residue, ab initio calculations may be required to generateone. A file that describes positional restraints on heavy atoms in the original .pdbfile is also generated (see Note 1).

3. Create a solvent box in which to solvate the protein and conduct the simulation.This can be accomplished using the editconf program and specifying a boxtype and box dimensions (see Note 2). For peptides/proteins, select the standardrectangular box and set box dimensions. This command generates a modified.gro file, which now includes the box dimensions, centering the solute in the boxunless otherwise specified.

4. The molecule can then be solvated with explicit solvent molecules by inputingthe modified structure file into the genbox program. Specify a solvent model in.gro file format (e.g., spc216.gro for water) consisting of a small box containing a

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Fig. 2. Flowchart for MD simulation method.

selected concentration of solvent molecules, and modify the system topology fileto include a topology file for the solvent molecules (see Note 3). With this infor-mation, GROMACS will solvate the protein by completely filling the simulationbox with duplicates of the solvent box, leaving a small area around the solute freeof solvent molecules, and generating modified system topology and .gro structurefiles to reflect these changes. If any atom of a solvent molecule is placed closerto any atom of the solute molecule(s) than the sum of the van der Waals radii ofthe two atoms, then that solvent molecule is removed (see Notes 4–6).

5. Create an index file using make ndx after the solvation of the protein. Defaultindex files are generated by and also required by most GROMACS commands,but it is generally desirable to create files with special index groups in order toanalyze the behavior of a particular subset of solute and/or solvent atoms.

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6. After the addition of solvent molecules, the energy of the system should beminimized to decrease high potential energy, as well as to bring the startingconfiguration close to an equilibrium state (see Note 7). Use the steepest descentmethod as the algorithm to minimize the system energy, because it is a goodapproximation of the local minimum and fairly expedient compared to othermethods (see Note 8). This method makes a user-defined number of stepstowards the negative gradient, always moving towards it, but without factoringin the history of previous steps, and constantly adjusting the size of the stepsto minimize the convergence time. An MD parameter file is needed, for theenergy minimization run and this should include the type of minimization methoddesired. It should also list the number of steps for the method to take, the initialstep size, and the tolerance (the user-defined value of the maximum force on anyparticular atom in the system under which the system is sufficiently minimized).The default initial step size of 0.01 nm and a tolerance of 0.05 kJ/mol/nm aretypical values, while a run of 2000 steps is usually sufficient to minimize theenergy of the system using the steepest descent method. Simple cut-off values forthe long range nonbonded potential functions (van der Waals, Coulombic) shouldbe included (see Note 6). This file, along with the .gro structure file and systemtopology file, are input into the grompp program, which combines the infor-mation and generates a binary run-input file. The run-input file can then be inputto the mdrun program, which will run the minimization and output trajectory,structure, and simulation log files.

7. If the system contains charged particles, an equal number of ions of oppositecharge must be introduced to neutralize the overall charge of the system (seeNote 9). The genion program can be used to replace individual solvent moleculeswith favorable electrostatic potential (or at random) with monoatomic ions,usually Na+ and Cl−. Once this is completed, the ion topology file, the numberof ions, and the ion charge must be included in the system topology file, whilethe replaced solvent molecules should be subtracted.

8. The system should then be minimized again, and a new index file should begenerated so that there is no discrepancy between the numbers of molecules inthe various system files.

9. After minimization, initial velocities must be generated for the solvent moleculesfor the final simulation, while the solute is held in place at the center of thesystem. This procedure is known as a positionally restrained MD simulation. Inthis case, the number of molecules, the volume of the system, and the temperatureof the system will be held constant, hence the description NVT dynamics. Asecond parameter file needs to be created for this preliminary simulation. Inaddition to including the position restraint file (generated during the conversionof the initial structure file to the .gro structure file) to fix the solute in place,as well as the long-range cut-off values, the reaction field method should bespecified as the cut-off algorithm type for nonbonded potentials. The file shouldinclude all bonds as constraints, and the LINCS constraint algorithm (9) shouldbe used (see Note 10). The MD algorithm should be specified as the integrator,and the user should make sure to enable periodic boundary conditions.

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10. In accordance with NVT conditions, the temperature of the system must becoupled to a virtual heat bath, with reference temperature (typically 300 K) setby the user. Solvent, solute, and ions should be coupled separately to a bath ofthe same reference temperature using Berendsen coupling (10), with the sametime constant (frequency of coupling, typically 0.1 ps) for each. Pressure is notcoupled and held constant during the NVT simulation, in order to keep the boxrigid and the volume constant. Dielectric constant and isothermal compressibilityof the solvent are also important parameters to set here. Finally, enable gener-ation of velocities for solvent atoms and ions. GROMACS will generate veloc-ities using a Maxwell distribution at a user-defined temperature, which should bethe same as the coupled temperature. Generate a run-input file using the gromppprogram as in the energy minimization step, input to the mdrun program, and startthe simulation. A positionally restrained simulation of 100 ps with a time step of2 fs is typical. Choose the number of processors on which to run the simulation,and if using more than one, use the MPI program described in the GROMACSmanual (11) (see Note 11).

11. For the full NPT (constant pressure, temperature, and number of molecules)simulation run, the parameter file is unchanged from the NVT simulation, exceptthat generation of velocities as well as position restraints should be turned off.Along with temperature coupling, pressure coupling should be enabled (and thesystem volume allowed to scale), with a reference pressure of typically 1 bar anda 1 ps time constant for coupling (10). Dispersion corrections for the cut-off ofthe long-range Lennard-Jones potential should also be enabled for both energyand pressure. Once again, generate the final run-input file using grompp, inputto mdrun and start the simulation. The user should specify a full simulation timeand a time step (again, typically 2 fs). The number of processors used for the fullsimulation should be the same as used for the NVT simulation and optimal forspeed and efficient use of computational resources (see Note 11).

12. When the simulation is complete, check the fidelity of the final trajectoryfile using the gmxcheck program and, if desired, convert to the less memory-consuming .xtc format. Subset group trajectories based on groups listed in theindex file can also be written.

3. Replica Exchange Molecular Dynamics (REMD)With standard MD simulations at low temperatures, an explicitly solvated

protein or peptide generally becomes trapped in any of many local energyminima, prohibiting a representative sampling of the entire range of conforma-tions. Of a few suggested solutions, REMD is least time-consuming, easiest toimplement, and theoretically sound (12–14). In REMD, multiple independentsimulations (“replicas”) are conducted, each at a different temperature in alimited range. At user-defined time steps, the trajectory coordinates of simula-tions of “neighboring” temperatures are either randomly exchanged or not

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exchanged depending on a probability equation, which is a function of thetemperatures and instantaneous potential energies of the two systems. Thealgorithm attempts only the exchange of “odd pairs” and “even pairs” everyother step to prevent the possibility of exchange being dependent on multiplesystems. The exchange of replicas allows the possibility for a system to gainenough energy to escape from a local minimum into which it may have fallenand ensures better sampling of possible conformations.

In an REMD study of Chingnolin using the GROMACS package, van derSpoel and Seibert (15) successfully folded, predicted the folding and unfoldingtime constants, derived folding energies, and calculated a melting curve for thedecapeptide.

4. Methods for REMD Simulation1. In practice, the user must first decide on the number of replicas to use, as well

as the temperature range. The smallest temperature replica should be low enoughto sample states of lowest energy, while the highest temperature should be highenough to overcome the various energy barriers of the system. In addition, sincethe probability of exchange is based partly on the difference in temperaturebetween the two systems being considered, and exchange probability falls offrapidly with difference (illustrating the importance of exchanging only neigh-boring replicas), temperatures should be chosen so as to facilitate the user’s desiredprobability of exchange (see Note 12).

2. Methods for the REMD simulation are the same as in the standard case, until theactual full NPT simulation. A separate run-input file must be generated for eachreplica using the grompp program. Then the simulation run (using mdrun) withreplica exchange enabled, the time step for exchange specified, and the number ofprocessors to be used is specified (see Note 13).

3. When the simulation is complete, the relevant frames of each trajectory file ofeach replica must be “demultiplexed” according to a replica index file, whichdescribes how the frames need to be ordered in the final composite trajectory file.The included GROMACS script demux.pl (16), which takes as input one of thestandard simulation log files, derives exchange data and generates the replica indexfile. This file then must be input to the trjcat program along with the trajectory filesto generate the final trajectory file.

4. As with the standard simulation, check the fidelity of the final trajectory file usinggmxcheck and convert to .xtc format if desired.

5. AnalysisSeveral analyses can be performed once the full simulation method has been

completed. One of the most useful is secondary structure analysis using do dssp.This program, in combination with the xpm2ps program, uses the Dictionary

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Fig. 3. DSSP for gastrin(1-6) in dimethyl sulfoxide.

of Protein Secondary Structure (DSSP) criteria for defining secondary struc-tural forms to graphically represent the structure of the molecule throughout thesimulation. An example the gastrin fragment G17(1-6) in dimethyl sulfoxide,showing a predominance of bend and turn structure is seen in Fig. 3.

A root-mean-square deviation (RMSD) analysis of the amide backbone atomsis often a strong indicator of conformational changes of a protein in solution. AnRMSD example is shown in Fig. 4, once again of G17(1-6) in DMSO. Periodsof conformational stability are indicated by stretches in which the RMSD does

Fig. 4. RMSD for G17(1-6) in dimethyl sulfoxide.

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not vary significantly, while variances of 1 A or more indicate a large change inthe fold of the protein.

Many other analyses can be made. The radius of gyration of a group ofatoms as a function of time can be calculated. The dihedral angles of residuescan be determined and Ramachandran plots made. The hydrogen bond or saltbridge propensity of atoms can be measured. As stated previously, by gener-ating index files for the simulation, groups of atoms for analysis can be defined.Cluster analysis can be performed with a user-defined RMSD criterion, groupingsimilar structures along the simulation trajectory into clusters. This feature isparticularly useful in determining which conformations of the protein are mostpredominant during the simulation.

6. Notes1. Because GROMOS is a united atom force field, either all nonpolar hydrogen

atoms must be edited out of the input structure or the user must specify that theybe ignored by GROMACS. Aromatic hydrogens are an exception, as GROMACScannot model aromatic–aromatic interactions with the united atom simplification.Hydrogens are then added to the structure topology by GROMACS as unitedatoms.

2. GROMACS offers a variety of box types: the standard cubical/rectangular box,the truncated octahedron, the hexagonal prism, and the rhombic dodecahedron.As the shape of the box can affect the simulation, a box type that best reflectsthe geometry of the solute should be used. For example, the truncated octahedronapproximates a sphere, so it is the optimal choice for globular proteins and otherroughly spherical molecules.

3. These solvent models are generally developed with a specific force field in mind,so using a solvent model with a force field for which it was not intended mayrequire the inclusion or exclusion of certain parameters.

4. The user should (1) check that the number of solvent molecules added is suitablefor the density of the solvent, (2) edit the output system topology file to includethe number of solvent molecules added, and (3) use a visualization program tocheck that the new structure file does not have any solvent molecules placed tooclose to the solute.

5. The effect of the surface of the box on the solute is of major importance in thesimulation of systems such as the one described here. The sudden cut-off of long-range nonbonded potentials at the box surface (beyond which is vacuum) wouldhave an unnatural effect on the dynamics of the simulation. Only an extremelylarge system size could ensure a small influence of this surface effect on thesolute. The computational cost of such a large system would be prohibitive. Forthis reason, periodic boundary conditions are used. The image of the simulationbox is translated repeatedly to form an “infinite” lattice. When a particle in thesimulation box moves, the image in all other translated boxes moves correspond-

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ingly with the same orientation. Thus, when a particle leaves the simulationbox, its translated image enters through the opposite face of the box. Whilethis solution is not strictly accurate for nonperiodic systems (i.e., liquids andsolutions), the errors generated are less likely to be as severe as those caused bythe sudden cut-off of potentials at the simulation box surface.

6. The long range of nonbonded interactions can also be a problem in periodicsystems. Interactions which do not decay faster than r-n, where n is thebox dimensionality, can often have a range greater than half the box length,which may allow a particle to interact with itself or allow it to interactwith another particle twice. A parameter must therefore be set to preventthese forces from extending past half the box (cubic) length. However, asimple cut-off is also problematic, especially for Coulombic interactions,both because of the discontinuity in the potential energy and force intro-duced (and the subsequent movement of ions), and the tendency of thepotential energy to diverge if it does not decay faster than r−3. GROMACShas implementable options to remedy this situation, including shift andswitch functions (17), the particle mesh Ewald method (18,19), and reaction-field methods (20), each of which modifies the simple cut-off scheme withvarying success. These options are implemented in the MD parameter file.

7. Forces on systems far from equilibrium may be extremely large and cause thesimulation to fail. Thus, it is important to bring the system as close to thenearest local minimum of the potential energy of the system as possible beforecommencing MD simulations. GROMACS cannot, in all likelihood, find theglobal minimum of the potential energy function, but it can find the nearest localminimum.

8. The conjugate gradient method and the L-BFGS minimizer, both of which incor-porate gradient information, can usually bring the potential energy closer to thelocal minimum than does steepest descent. However, convergence can take farlonger.

9. If this is not done, a large dipole moment will be induced on the system, causingincorrect energy calculations and other erroneous behavior.

10. As an alternative, the SHAKE algorithm (21) is used if angle constraints areimplemented or the LINCS algorithm otherwise fails.

11. Depending on available computational resources, it may be useful to do a shortfull simulation run to determine the optimal number of processors to use, sincebeyond a certain number of processors, speed of the simulation is not improvedand resources are wasted.

12. The GROMACS manual (11) suggests a relationship of T2 = (1 +1/sqrt(Natoms)T1, where Natoms is the number of atoms of the system, andT1 and T2 are neighboring temperatures for a protein/water system, giving anexchange probability of 13.5%. More detailed discussion is given in Nguyen andassociates (14).

13. REMD demands parallel computing. As GROMACS is currently implemented,each replica must be run on a separate processor.

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