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molcharge - Partial Charges

Introduction

The assignment of appropriate atomic partial charges, both to small molecule ligands and to biopolymers (such as proteins and nucleic acids) is essential to getting meaningful results from any electrostatics calculation.

A molecule may be considered a collection of atomic nuclei and the electrons that surround them. The number of protons in each nucleus defines its atomic number/element. If the number of electrons exactly matches the number of protons in these nuclei, the molecule is neutral and has no net charge. If there are more electrons than protons, the molecule has a net negative charge, and if there are less, the molecule has a net positive charge.

It is both the atomic nuclei and the net charge that define the identity of a molecule. Indeed, this is a representation common to quantum chemistry. Adding or removing electrons (or atoms) from a molecule produces a different molecule.

In the discrete world of cheminformatics, valence bond theory allows the electrons present in a system to be represented in terms of bonds with formal bond orders, and formal charges assigned to particular atoms. The sum of the formal charges is equal to the net charge on the molecule, but which atoms are assigned which formal charges can be to some extent arbitrary due to resonance delocalization. In such cases the same molecule may be represented by similar connection tables, but with formal charges assigned to different sets of atoms.

For example, guanidinium may be expressed as either N[C+](N)N with the formal charge assigned to the carbon, or as [NH2+]=C(N)N with the formal charge assigned arbitrarily to one of the otherwise equivalent nitrogens. A similar example is a thiocarboxylate group, where either C(=O)[S-] or C(=S)[O-] are both equally appropriate representations of the same chemical functionality.

A zwitterion is an electrically neutral molecule that is represented as containing atoms with positive formal charge as well as atoms with negative formal charge.

Perhaps the most important fact to appreciate when considering formal charges on atoms is that they are all artificial constructs by chemists to accommodate a particular chemical model. A figment of a chemist’s fevered imagination. Like valence bond theory, they are an exceptionally useful and powerful discretized model of the universe. But as with any model of reality, it has its limitations. Formal charges, for all their numerous benefits to mankind, unfortunately, are not localized on an atom.

The limitations of describing formal charges with valence bond theory is apparent even within cheminformatics. Sydnones, for example, are a class of heterocyclic compound that cannot be written using normal covalent bonds without introducing and arbitrarily assigning both positive and negative charges. Similarly, in inorganic chemistry, the ditechnetium cation, \(\mbox{Te}_{2}^{+5}\), causes similar problems where the +5 formal charge cannot be assigned to both technetium atoms without breaking symmetry.

A better model, or approximation, of the wave function describing the distribution of electron density around a molecule is the use of atomic partial charges. A partial charge is a floating-point value assigned to each atomic center intended to model the distribution of electrons over a molecule.

Atomic partial charges are yet another approximation, much like the formal charges described above. However, partial charges provide a much better model to describe the electric field, dipole moment and other observable properties of a molecule.

A common limitation of the use of partial charges is the assumption that they are conformationally invariant. Unfortunately, the distribution of electrons around a molecule depends upon the spatial configuration of its nuclei. Some partial charge assignment algorithms, such as the method of Goddard and Rappé, consider these conformational effects, whilst others that are based on quantum mechanics, such as the RESP and AM1BCC methods of Bayly et al., go to great lengths to eliminate conformational effects, for example, by restraining and symmetrizing symmetric atom positions. This is necessary in order to be able to properly handle multiple conformations and changes in geometry (e.g. geometry optimization) with a single set of atomic charges.

Theory

Marsili-Gasteiger Partial Charges

Marsili-Gasteiger partial charges are assigned using a two stage algorithm. In the first stage, seed charges are assigned to each atom in the molecule. For example, carboxylate oxygens are each assigned the value -0.5. During the second stage, these initial charges are then shared across bonds, moving a certain amount of charge from one atom to another. The partial charge moved and its direction is determined by difference in electronegativities of the atoms on each end of the bond. The relaxation algorithm is then iterated several times (by default eight passes), attenuating the charge moved with each iteration. OpenEye does not recommend use of this charge model for intermolecular interactions; it was never intended for this purpose. The author of the method (Johann Gasteiger) developed it to compare relative reactivity of related organic chemical functional groups within different molecular contexts. Here it is included for comparison purposes.

MMFF94 Partial Charges

The partial charges used by the MMFF94 and MMFF94s force fields are assigned using a four stage algorithm. In the first stage, each atom of the molecule is assigned an MMFF94 atom type. In the second stage, an initial seed partial charge is assigned to each atom based upon its atom type. For a few atom types, the initial partial charge also depends upon the local environment. In the third stage, the initial charges assigned to aromatic rings are shared between all atoms of the aromatic ring. Finally, in the fourth stage, a table of bond charge increments (BCI) is used to move charges across bonds based upon the bond type of the bond (single, double, triple) and the atom types of the atoms at each end. Developed for the electrostatic interactions within the above-mentioned force fields, they are the appropriate charges to use with these force fields most notably for intramolecular interactions of pharmaceutical and bio-organic small molecules. They are less well-suited (but still passable) for intermolecular interactions using the common two-body additive Coulomb interactions as used in Amber, Charmm, Gromacs. For these better choices would be amber99sb charges on proteins and peptides, and am1bccsym charges on the ligand.

AM1 Charges

AM1 charges are a set of Mulliken-type charges derived from a semi-empirical quantum-mechanical calculation. For further discussion of this method, please see Dewar et. al. These should not be used for intermolecular interactions of force fields.

AM1BCC Charges

AM1BCC charges start with Mulliken-type partial charges derived from the AM1 wave-function. In a second stage, bond-charge corrections (BCCs) are applied to the partial charges on each atom to generate new partial charges. Unlike “standard” AM1BCC charges, the default does not symmetrize charges (averaging those which are equivalent by bond topology, e.g. methyl hydrogens). This means that a multi-conformer molecule will not behave correctly with respect to conformational interconversion. For example, degenerate conformers will differ in energy, e.g. the 180-degree rotation of a carboxylate or amidinium. To get “standard” AM1BCC charges, specify the am1bccsym method.

It is important to avoid carrying out the AM1 calculation on a conformation with a strong short-range intramolecular polar interaction (e.g. a hydrogen bond or a salt bridge) because this significantly perturbs the AM1 starting charges used for AM1BCC, and the resulting AM1BCC charges would perform badly given a conformational change.

Due this issue, we incorporated into the recommended am1bccsym option the method proposed by Christopher Bayly and colleagues for assigning AM1BCC charges to a multiconformer molecule. It is based on the following procedure: the Coulomb electrostatic energy is calculated for every conformer using the absolute value of the MMFF94 partial charges (original negative charges are replaced with their absolute values). The standard AM1BCC calculation is then performed as above for the lowest electrostatic energy conformer determined in the previous step, and the AM1BCC charges obtained are assigned to all conformers.

The recommended am1bccsym option also carries out an AM1 geometry optimization lightly restrained to the starting coordinates is important to allow the relaxation of bond and bond angle degrees of freedom in AM1. Relatively small deviations of these from the AM1 optimum significantly affects the charges. Thus using the AM1 “single-point” option am1bccspt should be done with caution; bad input geometry will taint the charges. Thus, the partial charges produced from a single point calculation might, depending on the geometry, be similar to those from AM1 geometry optimized calculations but will not be as good quality.

OpenEye considers AM1BCC charges to be the best partial charge model currently available. For further discussion, please see the work of Christopher I. Bayly.

Amber ff94, ff96, ff99, ff99sb, and ff99sbc0 Partial Charges

The partial charges used by the AmberFF94 force field are based on fitting quantum mechanical electrostatic potentials (esp). They were developed to address two key issues with earlier esp-fit charge sets: unrealistically high charges on charge centers and the variation of atomic charges with conformation. While the latter should have some basis in electronic structure, numerical instability in the charge fitting process was the source of both these pathologies. AmberFF94 charge sets use restrained esp-fitting (RESP) to control the numerical instabilities and simultaneous multi-conformer fitting to lead to conformation-independent charges that are restricted to individual residues. Particular attention was given to ensure that backbone amides have consistent charges. The Amber force fields ff94, ff96, ff99, ff99sb, and ff99sbc0 all use the same set of RESP charges, they differ in other terms (mostly torsional).