SZYBKI Theory¶
MMFF94 Force Field¶
The MMFF potential expression is
where the seven terms describe bond stretching (b), angle bending (a), stretch-bending (s), out-of-plane bending (o), torsion (t), van der Waals (v) and electrostatic (c) interactions. Their functional forms are given below.
Bond Stretching¶
For a bond b between atoms i and j, the stretching potential is
where \(k_{ij}\) is the force constant, \(\Delta r_{ij}\) is the difference between the actual and reference bond lengths, and \(c_s\) is a so-called “cubic-stretch” constant for which the value is -2 Å\(^{-1}\).
Angle Bending¶
The bending potential of a bond angle a made by the bonds between the atoms i and j and the atoms j and k is given by
where \(k_{ijk}\) is the force constant, \(\Delta \vartheta_{ijk}\) is the difference between the actual and reference bond angles, and \(c_b\) is the so-called “cubic-bend” constant for which the value is \(-0.4 rad^{-1}\).
Stretch-Bend Interaction¶
The coupling between the stretching potential of two bonds forming a bond angle and bending that angle is described by
where \(k_{ijk}\) and \(k_{kji}\) are force constants which couple stretches of i-j and k-j to the i-j-k bend respectively. \(\Delta r\) and \(\Delta \vartheta\) are defined above.
Out-of-Plane Bending¶
For a trigonal center j, the potential of displacement for an atom l bonded to atom j out of plane i-j-k is
where \(k_{ijkl}\) is the force constant and \(\chi_{ijkl}\) is an angle formed by the bond j-l and the plane i-j-k.
Torsion Interactions¶
For every four bonded atoms i-j-k-l, the torsion interaction is described by the term
where \(V_1\), \(V_2\) and \(V_3\) are constants depending on atoms i, j, k, and l, and \(\Phi\) is the dihedral angle formed by bonds i-j and k-l.
Van der Waals Interactions¶
For a pair of atoms i and j separated by three or more bonds, where the distance between them is \(r_{ij}\), MMFF adopts the following van der Waals potential
where \(R_{ij}\) and \(\epsilon_{ij}\) are defined as follows:
where \(\alpha_i\) is the atomic polarizability of atom i, B is 0.2 or 0.0 if one of the atoms is polar hydrogen, \(N_i\) and \(N_j\) are the Slater-Kirkwood effective numbers of valence electrons, and \(G_i\), \(G_j\) and \(A_i\) are scale factors.
Electrostatic Interactions¶
The electrostatic interaction between two charged atoms i and j separated by at least three bonds is calculated from the standard Coulombic expression
where D is the dielectric constant for which the default value is 1, \(q_i\) and \(q_j\) are the MMFF partial charges on atoms i and j, \(r_{ij}\) is the interatomic distance, and \(\delta\) is the “electrostatic buffering” constant of 0.05 Å. The scaling factor f is 0.75 for 1,4 interactions and 1.0 otherwise.
Protein-Ligand Amber Potential¶
Protein-ligand interactions can be described by the Amber force field instead of MMFF potential,
where the summation is over protein-ligand atom pairs. \(r_{ij}\) is the interatomic distance, \(R_{ij}\) is the VdW distance for a pair of atoms, \(q_i\) and \(q_j\) are the Amber partial charges on atoms i and j, and \(\epsilon_0\) is the vacuum permittivity. VdW parameters \(R_{ij}\) and \(\epsilon_{ij}\) and partial charges are taken from work by Wang and coworkers [Wang-J-2000].
SMIRNOFF99Frosst and OPENFF¶
SMIRNOFF (SMIRKS Natural Open Force Field) is a kind of small molecule force field
springing from the Open Force Field Initiative (OFF Initiative or OFFI). The key feature
of this force field is its chemical
representation of parameters by using extended SMARTS patterns. “Extended”
here means that the SMARTS patterns also use atom assignment syntax from SMIRKS. For example,
[*:2]~[*:1]~[#1] specifies three atoms, where the first is assigned to be atom 2,
the second atom is assigned to be atom 1, and the third atom is not assigned. This format
frees the force field from using atom types; instead it uses the direct chemical perception
of the extended SMARTS patterns [Mobley-2018].
Compared to traditional atom typing approaches, this representation greatly reduces the
number of parameters needed to specify a general small molecule force field and makes
each parameter more independent.
While it currently has a traditional “Amber” type functional form as specified below, future versions expect to modify the functional form. The OPENFF Initiative release periodic revisions of the parameters which include changes to the chemical specifications (i.e., the extended SMARTS patterns) as well as the numerical parameters. The revision used by the API points will be given with the results.
SMIRNOFF99Frosst OPENFF1.0.0 (released in October 2019) force fields have six terms
describing bond stretching (b), angle bending (a), out-of-plane displacement (o), torsion (t), van der Waals (v), and electrostatic (c) interactions. Their functional forms are similar to those in the AMBER force field and are given below.
Bond Stretching¶
For a bond b the stretching potential has a simple harmonic form:
where \(k_b\) is the force constant and \(r\), and \(r_0\) are the actual and reference bond lengths, respectively.
Angle Bending¶
For bond angle a, the angle bending potential form is
where \(k_a\) is the force constant and \(\alpha\), and \(\alpha_0\) are the actual and reference bond angle values, respectively.
Torsion Interactions¶
Every four bonded atoms defines a torsion whose energy is represented with one to six terms of a Fourier series
where \(V_n\), \(d_n\), \(p_n\), and \(\gamma_n\) are the barrier height, degeneracy, periodicity and phase, respectively, for the n component of a torsion whose dihedral angle is \(\theta\).
Out-of-Plane Bending¶
It is defined as the potential of a displacement of the trigonal center atom a bonded to three other out-of-plane atoms, b, c, and d. Its functional form is similar to a regular torsion with a single term. Sometimes it is also called an improper torsion:
Since one can define three improper torsions of moving a trigonal atom a out of a plane defined by atoms b, c, and d, the energy of such a displacement is calculated as one third of the sum of those torsions.
Van der Waals Interactions¶
For a pair of nonbonded atoms i and j, a SMIRNOFF force field adopts Lennard-Jones potential
where \(\epsilon_{ij}\) is well depth and \(R_{ij}\) and \(r_{ij}\) are the equilibrium and actual distance between atoms i and j, respectively.
Electrostatic Interactions¶
A simple coulomb term
represents coulomb interactions, where \(q_i\) and \(q_j\) are partial atomic charges, \(\epsilon_0\) is the vacuum permittivity, and \(r_{ij}\) is the distance between atoms i and j.
Entropy Evaluation¶
Ligand entropy is evaluated as a sum of configurational entropy (\(S_c\)) and solvation entropy (\(\Delta S_s\)):
Configurational Entropy¶
Configurational entropy is calculated as
where q is the conformation dependent partition function:
Here \(q_t\), \(q_ir\), and \(q_iv\) are the translational, rotational, and vibrational partition functions, respectively, and \(n_c\) is the number of unique conformations in the ensemble. All three partition functions are calculated from the classical statistical mechanics expressions which could be found in Statistical Mechanics by Donald McQuarrie ([McQuarrie-1976]). Vibrational frequencies for each conformation, needed for evaluation of \(q_{ir}\) are derived from diagonalization of a Hessian matrix obtained from quasi-Newton optimization when convergence is achieved. Eigenvalues \(\lambda_i\) of the mass-weighted Hessian
are converted into wave numbers \(\tilde{\nu}_i\) according to
Solvation Entropy¶
Solvation entropy is split into electrostatic and hydrophobic parts:
The electrostatic part of solvation entropy is divided into the bulk component and tight electrostatic polar solute-water interactions (hydrogen bonds). The bulk contribution is estimated from the temperature dependence of the solvent dielectric constant as
The second term of the electrostatic solvation entropy is estimated as a constant of 28 J/(mol K).
The hydrophobic term, \(\Delta S_{s,hyd}\), is evaluated as
where \(\Delta G_{s,hyd}\) consists of three components
describing the free energy of cavitation, solute-solvent van der Waals, and inductive terms, respectively. The cavity formation term is calculated from Scaled Particle Theory [Pierotti-1976]. Analytical expressions for \(\Delta G_{VdW}\) and \(\Delta G_{Ind}\) terms are also taken from the 1976 Pierotti review.
Protein-Bound Ligand Entropy¶
The configurational entropy of a protein-bound ligand is calculated totally as vibrational entropy for 3N modes, assuming that three rotational and three translational degrees of freedom of a solution ligand are transformed into low vibrational degrees of freedom for the bound ligand.
Solvation entropy for a ligand in the active site is assumed to be a sum of its fractional value in solution determined by the percentage of the ligand surface exposed to the solvent, \(f\Delta S_s\), and a partial desolvation entropy of the protein active site, \(\Delta S_{des}\),
where f is the fraction of ligand surface exposed to the solvent. It is important to notice that \(S_{protein}\) is not an experimentally measurable value, and that only the difference between \(S_{protein}\) and \(S = S_c + \Delta S_s\) might be compared with experimental binding entropy.