CHARMm Principles

3       Performing Energy and Force Calculations

The potential energy for any molecular structure can be computed in CHARMm given a PSF, a full parameter set with appropriate force constants and equilibrium geometries, and Cartesian coordinates. In addition to the energy, forces on each of the atoms in the molecular system can also be computed. Force calculations are particularly important during molecular dynamics simulations.

This chapter describes the CHARMm energy function and energy minimization algorithms that can be used to refine structures or locate stationary points for conformational searching for dynamics simulations.

This chapter explains

• Applying the CHARMm energy function

• Minimizing energy

Applying the CHARMm energy function

The total energy is expressed by the following equation:

Potential Energy = E_bond + E_theta + E_phi + E_impr + (internal)
E_elec + E_vdw + (external)
E_cons + E_user + E_other (special)

In order to perform all of these functions, you are afforded a great deal of flexibility in using the empirical energy function. Indeed, almost everything can be customized, from parameters and the specification of individual energy terms to user-supplied energy routines.

Internal energy terms

• Ebond -- Bond potential:

Eq. 1     

• Etheta -- Bond angle potential:

Eq. 2     

• Ephi -- Dihedral angle (torsion) potential:

Eq. 3     

Where n = 1, 2, 3, 4, 6

• Eimpr -- Improper torsions:

Eq. 4     

The dihedral angle (torsion) energy term is a four-atom potential based on the dihedral angle about an axis defined by the middle pair of atoms.

The improper torsional potential is designed to maintain chirality about a tetrahedral extended heavy atom, such as an amino acid alpha carbon without an explicit hydrogen, and to maintain planarity about sp² hybridized atoms, such as carbonyl carbons with a quadratic distortion potential. Without such a term, out-of-plane potentials tend to be quartic. Additionally, this term provides a better force field near the minimum energy geometry, a consideration that is important for dynamics calculations and vibrational analysis.

External energy terms

• Eelec -- Electrostatic. The constant dielectric equation is:

Eq. 5     

• Evdw -- Van der Waals:

Eq. 6     

Nonbonded energy terms

Computing electrostatic potential

Electrostatic potential is computed using the partial atomic charges provided in the RTF. Several different approaches are included in CHARMm for treating the electrostatic interactions:

• Option one -- Applies regular Coulombic electrostatic interaction terms.

• Option two -- Introduces an approximate solvent screening term in the dielectric constant by setting the dielectric constant proportional to R, that is,

  ,

resulting in the electrostatic interaction of the form

.

This option is intended as an approximate way to deal with solvent affects without explicitly including solvent.

• Option three (extended electrostatics) -- Uses a multipole approximation to permit efficient (that is, time-saving) evaluation of all of the electrostatic interactions in the system. The approach treats short-range interactions in the usual way, but expresses long-range, group-group interactions in terms of multipoles. Multipole expansion is truncated after quadrupole moments. A constant dielectric is used.

This approach is designed particularly for simulations where solvent is being included explicitly and the details of solvation and solvent polarization are of interest.

The van der Waals energy is approximated in the CHARMm empirical energy function by the Lennard-Jones potential energy function. This function is often referred to as a 6-12 potential because the attractive force is proportional to R^-6, while the repulsive force is modeled by R^-12.

Nonbonded list

External terms are defined by nonbonded and hydrogen-bonded lists specified by the user. Nonbonded interactions refer to van der Waals terms and electrostatic interactions between atom pairs.

To maximize the efficiency of the nonbonded calculations, a list is created that contains all pair interactions to be considered. Atom pairs are not included in the list if they are too far apart (beyond the long-range cutoffs), or if they are directly connected (either through a bond or a bond angle). In the latter case, the atoms will be present in the excluded list.

The list generation approach was chosen over other alternatives (such as a pairwise search) for two reasons:

• During minimization or dynamics, the relative positions between atoms do not change radically between one step and the next. Also, a pairwise search through the list of atoms is relatively costly in computational terms.

The list of atoms is updated periodically during minimization or dynamics to account for changes in atomic positions. If electrostatic groups are used, the list is stored in terms of group pairs. This leads to improved efficiency and a smaller list, as well as the ability to handle long-range group interactions.

• Nonbonded terms are defined on atom pairs and, for a first approximation, would require N² calculations if N is the total number of atoms. To avoid this computational burden, various truncation and approximation schemes can be employed in constructing the nonbonded lists.

The nonbonded list is primarily generated with the UPDATE command, although it can be generated or modified by providing nonbond-list specifications with other commands that include the following:

• NBONDS

• ENERGY

• MINIMIZE

• DYNAMICS

During an energy minimization or molecular dynamics simulation, the nonbonded list can be updated according to a frequency that you set. The INBFRQ keyword is used to indicate the frequency for updating the nonbonded list. Three choices are available:

• =0 -- In this case, the nonbonded list is not updated at all during minimization or dynamics

• =n -- In this case, the nonbonded list is updated every n steps of minimization or dynamics

• =-1 -- In this case, a heuristic test is performed at each step of minimization or dynamics, and an update is performed, if necessary

• Every time energy is called, displacement is computed for each atom since the list update.

• If any atom has moved by more than (CUTNB-CTOFNB)/2.0, it is possible that some atom pairs in which the two atoms are now separated by less than CTOFNB are not in the atom pairs list. Therefore, a list update is done.

• If all atoms have moved by less than (CUTNB-CTOFNB)/2.0, then all atom pairs within the CTOFNB distance are already accounted for in the nonbonded list and no update is necessary.

For a typical macromolecule, nonbonded interactions represent the bulk of the energy evaluation time. The efficiency of the nonbonded calculation is increased by including only atom pairs that are closer than the cutoff distance in a nonbonded list. The cutoff distance criterion, while computationally efficient, has some disadvantages. The possibility always exists that the overall nonbonded interactions at long range are not negligible because such interactions are very numerous. However, the slope of the van der Waals and electrostatic potentials in this region is very small. Because the nonbonded interaction vanishes at a greater rate than the total energy, the cutoff makes very little difference for most calculations.

A more significant disadvantage is that a discontinuity in the energy function results at the cutoff distance. For energy minimization of molecular dynamics calculations, this discontinuity can significantly distort computational results. For example, the total energy of the molecular system may not be conserved under such circumstances.

To resolve these difficulties, both the van der Waals and electrostatic energy terms can be modified by techniques that smooth the function around the cutoff distance, as described in the following paragraphs:

• Switching function -- The size of the nonbonded lists can be made smaller by indicating a distance to be applied to a switching function beyond which no interaction is computed.

Two variables corresponding to the start and finish of the switching regions (r_on and r_off, respectively) are required to define the switching function. Depending on the value of r_ij, the following values are used to multiply individual electrostatic or van der Waals energy terms:

• If r_ij < r_on, then use 1.0

• If r_on < r_ij < r_off, then use:

  

• If r_ij > r_off, then use 0.0

However, energy can still be significant at the cut-off distance and this can result in artificially large forces at long range. This is especially true for relatively short (that is, less than 12 Å) cutoff distances and small buffer regions (that is, when r_off - r_on < 3 Å).

• Shifted potential -- The shifted potential modifies the radial function so that energy and forces go to zero at some cut-off distance. Eq. 7 defines the functional form of the shifted potential. This term is simply multiplied with the individual electrostatic or van der Waals terms in the empirical energy function:

Eq. 7     

One disadvantage to this function is that it has a discontinuity in the second derivatives at the cutoff distance.

• Groups -- Another choice is to compute the potential energy based on groups rather than individual atoms. If two groups are neutral with non-zero dipoles, their long range interaction is proportional to 1/R³, which decays faster than the individual atom interactions 1/R.

Dielectric constant

The dielectric constant is an experimentally derived property of the bulk solvent that reflects the polarizability of the solvent molecules. A polarizable solvent such as water has a greater dielectric constant than less polar liquids. Electrostatic interactions in such high dielectric polarizable solvents are greatly attenuated.

A dielectric constant of 1.0 is sufficient for solvated simulations that include an atomic polarizability term. Because most calculations do not include these factors, a number of models have been used to simulate the dielectric behavior for such systems.

The EPS keyword is used to specify an explicit value for the dielectric constant. A relatively large dielectric constant can be used for simulating the aqueous environment of small systems. However, most calculations on molecular systems use a smaller dielectric constant. For example, a dielectric constant between 2.0 and 10.0 has been employed for simulations in the interior of a protein.

For electrostatic interactions in closely packed molecules, the number of solvent molecules between two interacting charges is usually less than that which would be experienced in the bulk solvent. Thus, the dielectric constant for the micro solvent environment is often not as great as for the bulk solvent. To simulate this effect, a distance-dependent dielectric constant can be used. This constant also acts as an approximate solvent screening term in which the electrostatic interaction experiences a greater and greater attenuation as the two charges are separated. The RDIE keyword is used to specify the use of this distance-dependent dielectric constant in energy calculations.

Hydrogen bonds

The selection of hydrogen bonds involves three checks:

• Any good hydrogen bond has a length less than a defined cutoff.

• Angle linearity between the donor and acceptor has a value less than a defined cutoff. Typically, the best hydrogen bond has an angle D-H-A of 180.

• If a hydrogen donor has more than one acceptor that satisfies the above constraints, the routine selects the acceptor with the lowest energy.

However, certain choices for the hydrogen boned selection process will never conserve energy in a dynamics run. This is true, for example, if you use the extended-atom representation for your calculations.

The function used in CHARMm to calculate the hydrogen bond energy is:

Note that hydrogen bond energy is not included as a default energy term. The current CHARMm parameter set has been derived in such a way that hydrogen bond effects are described by the combination of electrostatic and van der Waals forces.

Other energy terms

• Econs -- For some purposes, it is useful to restrict the changes that occur in a structure. For this reason, CHARMm provides the option of including different types of constraints in the energy when manipulating the structure through minimization or dynamics. An energy constraint term can be included when the following constraints are applied:

Atom constraint -- Used to rigidly maintain the position of certain atoms and delete the energy terms involving only these atoms. This can serve as an effective way of simulating a small part of a very large system with relative ease and increased efficiency.

Atom harmonic constraint -- Used primarily to avoid large displacements of atoms when minimizing, often when bad contacts are present, while still allowing the structure to relax.

Dihedral constraint -- Used to maintain certain local conformations or to examine a series of different conformations when making potential energy maps.

NOE constraint -- Used to incorporate NOE constraints (representing experimentally derived inter-proton distances) into the energy function. This is an especially important technique for generating structures for molecules in aqueous solution of lipid membranes as well as for gas phase studies.

Euser -- CHARMm also allows the addition of a user-supplied routine to the energy function that calculates an arbitrary energy term.

This script begins by generating a PSF and coordinates for enkephalin. The coordinates are copied to the COMP array. The energy of the system is then calculated in a number of different ways:

• ENERGY -- Calculates the energy of the system.

• ENERGY COMP -- Calculates the energy using the coordinates in the COMP array. Note that this uses the same PSF. Only the Cartesian coordinates have changed. This can be very useful. For example, after copying the coordinates to the COMP array, you can request an energy minimization. You can then compare the starting energy ENERGY COMP (the COMP array contains the starting coordinates) with the final ENERGY and make a decision based on the energy difference.

• SKIPE -- Skips certain energy terms and examines individual energy terms (for example, only the van der Waals component).

• Input file: energy.inp

• Other required files:

AMINO.BIN

PARM.BIN

*...
* Copyright (c) 1994
* Accelrys Inc.
* All Rights Reserved
*
! Open and read an all hydrogen amino acid topology file
OPEN READ UNIT 11 FILE NAME "$CHM_DATA/AMINOH.BIN"
READ RTF UNIT 11 FILE
CLOSE UNIT 11
! Open and read the parameter file
OPEN READ UNIT 12 FILE NAME "$CHM_DATA/PARM.BIN"
READ PARA UNIT 12 FILE
! Read the enkephalin sequence
READ SEQUENCE CARD
* Pentapeptide sequence for met-enkephalin
*
TYR GLY GLY PHE MET 
! Generate the PSF with a segment identifier of ENKP.
GENERATE ENKP SETUP
! Construct initial Cartesian coordinates
IC PARAMETERS
IC SEED 1 N 1 CA 1 C
! Copy coordinates to comparison set
COOR COPY COMP
! Update nonbonded lists
UPDATE RDIE SHIFT VSHIFT
! Get energy using standard routines
FAST 0
ENERGY
ENERGY COMP
GETE PRINT
FAST 1
! Get energy using optimized routines
ENERGY
! Get nonbonded energy only
SKIPE ALL EXCL VDW ELEC
ENERGY
! Restore default energy terms
SKIPE EXCL ALL
ENERGY
STOP

CHARMm has five different minimization methods, all working in Cartesian space, that provide a flexible array of iterative methods to facilitate energy minimization. Although the resulting conformation may only represent a local minimum, even macromolecules can be energy minimized efficiently using a number of these techniques.

All of the minimization methods take a molecular structure to a local minimum in the potential energy surface. There is no guarantee that this will be a global minimum. Small molecular systems can be minimized to a global minimum, but multiple runs from different starting points should be done to confirm that a global minimum has indeed been found.

With macromolecules, a very low probability exists that a local minimum will be the global minimum. In fact, a global minimum may never be found because of the complexity of the potential energy surface.

Using minimization

Minimization used as part of a programmed conformational search can yield many unique minima that can be useful in understanding large parts of the molecule's conformational space.

For example, minimization is an important tool in analyzing proteins that are generated through site-directed mutagenesis. After substituting, inserting, or deleting residues in a sequence, minimization, along with side-chain conformation scanning, can be used to determine whether the resulting mutuant structure is very much perturbed with respect to the wild type. If the perturbation is minimal, it is possible to model the structure of the mutant protein without resorting to X-ray diffraction studies.

Minimization methods

• Steepest descents (SD) -- A very simple, first derivative method. Uses only first derivative information and saves only the current location of the coordinates from iteration to iteration. In general, SD converges very slowly to a local minimum in a complex potential energy surface. This method is very useful for small changes, such as the removal of unfavorable steric contacts.

• Conjugate gradient (CONJ) -- Exhibits better convergence than the steepest descents method. The method is iterative and makes use of the previous history of minimization steps and the current gradient to determine the next step.

• Powell (POWE) -- A variation of the conjugate gradient method with improved efficiency. Use whenever the ABNR method (described below) is not possible.

• Newton-Raphson (NRAP) -- Implementation in CHARMm involves diagonalization of the second derivative matrix, then finding the optimum step size along each eigenvector.

When one or more negative eigenvalues exist, a blind application of the equations will find a saddle point in the potential. To overcome this problem, a single additional energy and gradient determination is performed along the eigenvector displacement for each small or negative eigenvalue. From this additional data, the energy function is approximated by a cubic potential and the step size that minimizes this function is adopted.

The advantages of this algorithm are that it avoids saddle points in the potential energy surface and converges rapidly when the potential is nearly quadratic.

The major disadvantage is that large computational requirements makes this technique time consuming and memory demanding for large molecules. Additionally, it is currently not possible to use SHAKE (described in Chapter 5, Setting Constraints and Periodic Boundaries) or images with this method.

• Adopted Basis-set Newton-Raphson (ABNR) -- Similar to conjugate gradients, but fewer energy evaluations are usually necessary because the linear interpolation phase of conjugate gradients is avoided.

This method performs energy minimization using a Newton-Raphson algorithm applied to a subspace of the coordinate vector spanned by the displacement coordinates of the last positions. The second derivative matrix is constructed numerically from the change in the gradient vectors, and is inverted by an eigenvector analysis that allows the routine to recognize and avoid saddle points in the energy surface. At each step, the residual gradient vector is calculated and used to add a steepest descent step, incorporating new direction into the basis set.

This method is the method of choice for most applications. Because it avoids the large storage requirements of the full NRAP second derivative method, larger systems can be minimized more efficiently.

• Truncated-Newton (TN) minimization package (TNPACK) -- This sixth method was developed by T. Schlick and A. Fogelson. TNPACK is based on the preconditioned linear conjugate-gradient technique for solving the Newton equations. The structure of the problem (sparsity of the Hessian) is exploited for preconditioning. Thorough experience with the new version of TNPACK in CHARMM has been described in a paper in the Journal of Computational Chemistry 15, 532 (1994). Through comparisons among the minimization algorithms available in CHARMM, we find that TNPACK compares favorably to ABNR in terms of CPU time when curvature information is calculated by a finite-difference of gradients (the "numeric" option of TNPACK). With the analytic option, TNPACK can converge more rapidly than ABNR for small and medium systems (up to 400 atoms) as well as large molecules that have reasonably good starting conformations; for large systems that are poorly relaxed (i.e., the initial Brookhaven Protein Data Bank structures are poor approximations to the minimum), TNPACK performs similarly to ABNR. SHAKE is currently unavailable with this method.

Convergence criteria

• RMS gradient

• Step size

• Energy change

Note that although a zero RMS gradient is a necessary condition for a minimum, it is not a satisfying condition (that is, saddle points on the potential energy surface have zero gradient but are not minima).

Minimization requirements

Because all energy minimizations involve calculating the potential energy of the system, you must have a PSF, coordinates, and a parameter file available. Hydrogen bonded and nonbonded lists must also be created prior to any energy evaluation and subsequent minimization.

Example: Minimizing enkephalin

The following files are used or generated in this example:

• Input file: enkpmin.inp

• Other required files:

AMINO.BIN

PARM.BIN

ENKPINI.CRD

• Created file: ENKPMIN.CRD

*...
* Copyright (c) 1994
* Accelrys Inc.
* All Rights Reserved
*
! Open and read the topology and parameter files (binary)
OPEN READ UNIT 11 FILE NAME "$CHM_DATA/AMINOH.BIN"
READ RTF UNIT 11 FILE
CLOSE UNIT 11
OPEN READ UNIT 12 FILE NAME "$CHM_DATA/PARM.BIN"
READ PARAMETERS UNIT 12 FILE
CLOSE UNIT 12
! Read the sequence information and generate the PSF
READ SEQUENCE CARDS
* Enkephalin sequence
*
5
TYR GLY GLY PHE MET
GENERATE ENKP SETUP
OPEN READ UNIT 08 CARD NAME ENKPINI.CRD
READ COOR CARD UNIT 08
CLOSE UNIT 08
FAST 1
UPDATE RDIE SHIFT VSHIFT CUTNB 16.0 INBFRQ 10 IHBFRQ 0
! First set of minimization
MINIMIZE CONJ NSTEP 100 NPRINT 10
! Second set of minimization
MINIMIZE ABNR NSTEP 200 NPRINT 10
! Write out the minimize coordinates
OPEN WRITE UNIT 04 CARD NAME ENKPMIN.CRD
WRITE COORDINATES CARD UNIT 04
* Minimized enkephalin coordinates
*
STOP

The following files are used or generated in this example:

• Input file: crnmin.inp

• Other required files:

AMINO.BIN

PARM.BIN

• Created files:

CRNINI.CRD

CRNMIN.CRD

*...
* Copyright (c) 1994
* Accelrys Inc.
* All Rights Reserved
*...
* This CHARMm command file generates a PSF for crambin
* and performs several minimizations on the structure
* and computes the changes relative to the starting 
* coordinates.
*...
*
! Open and read an all hydrogen amino acid topology file (binary)
! and parameter file
OPEN READ UNIT 11 FILE NAME "$CHM_DATA/AMINOH.BIN"
READ RTF UNIT 11 FILE
CLOSE UNIT 11
! Open and read the parameter file (binary)
OPEN READ UNIT 12 FILE NAME "$CHM_DATA/PARM.BIN"
READ PARA UNIT 12 FILE
CLOSE UNIT 12
! Read the sequence information and generate the PSF
READ SEQUENCE CARDS
* Sequence for crambin as taken from PDB file
*
46
THR THR CYS CYS PRO SER ILE VAL ALA ARG SER ASN PHE
ASN VAL CYS ARG LEU PRO GLY THR PRO GLU ALA ILE CYS
ALA THR TYR THR GLY CYS ILE ILE ILE PRO GLY ALA THR
CYS PRO GLY ASP TYR ALA ASN

! Generate the PSF with a segment identifier of 1CRN.
! and set up the internal coordinate tables.
GENERATE 1CRN SETUP
! Patch disulfide bonds in to the PSF
PATCH DISU 1CRN 3 1CRN 40 SETUP
PATCH DISU 1CRN 4 1CRN 32 SETUP
PATCH DISU 1CRN 16 1CRN 26 SETUP
! Read the PDB coordinates
OPEN READ UNIT 13 CARD NAME 1CRN.PDB
READ COORDINATES PDB UNIT 13
CLOSE UNIT 13
! Build and optimize all hydrogen positions
HBUILD
! Build any remaining coordinates
IC FILL PRESERVE
IC PARAMETERS
IC BUILD
! Setting up comparison coordinate set for future analysis
COOR COPY COMP
! Write the coordinates to disk
OPEN WRITE UNIT 13 CARD NAME CRNINI.CRD
WRITE COORDINATES CARD UNIT 13
* Crambin initial coordinates 
*
UPDATE RDIE SHIFT VSHIFT
! First set of minimization
MINIMIZE SD NSTEP 100 INBFRQ -1 CUTNB 30 STEP 0.010 NPRINT 10
! Comparisons based on RMS
COOR RMS MASS
! Second set of minimization
MINIMIZE SD NSTEP 150 INBFRQ -1 CUTNB 30 STEP 0.0050 NPRINT 10
! Comparisons based on RMS
COOR RMS MASS
! Internal coordinate comparisons; select and print
! only the phi/psi internal coordinates
IC FILL
IC DIFF
IC KEEP SECO SELE ATOM * * N .OR. ATOM * * CA END
IC KEEP THIR SELE ATOM * * CA .OR. ATOM * * C END
IC DELE FOUR SELE ATOM * * O END
IC PURG
IC PRIN
! Write out crambin minimized coordinates
OPEN WRITE UNIT 04 CARD NAME CRNMIN.CRD
WRITE COORDINATES CARD UNIT 04

* Crambin coordinates after 250 steps of SD minimization
*
STOP

Fletcher, R. (ed.), Optimization, Academic Press, New York and London, 1969.

Flectcher, R. and Reeves, C. M., The Computer Journal, 7 149 (1964).

Powell, M. J. D., "Restart Procedure for the Conjugate Gradient Method," Mathematical Programming, 12 241 (1977).

Press, W. H., Flannery, B. P., Teukolshy, S. A., and Vetrrling, W. T., Numerical Recipes: The Art of Scientific Computing, University Press, Cambridge, 1987.

Gunsteren, W. F. and Karplus, M., J. Comp. Chem., 1 266 (1980).