# SZYBKI Theory¶

## Force Field¶

Theory documentation for forcefields is available from OEFF theory.

## Entropy evaluation¶

Ligand entropy is evaluated as a sum of configurational entropy ($$S_c$$) and solvation entropy ($$\Delta S_s$$):

$S = S_c + \Delta S_s$

### Configurational entropy¶

Configurational entropy is calculated as:

$S_c = kN \Bigg[ 1 + \ln \Bigg(\frac{q}{N} \Bigg) + \frac{T}{q} \frac{\partial q}{\partial T} \Bigg]$

where q is the conformation dependent partition function:

$q = q_t \sum_{i=1}^{n_c} e^{-\frac{\epsilon_i}{kT}} q_{iv} q_{ir}$

Here $$q_t$$, $$q_ir$$ and $$q_iv$$ are the translational, rotational and vibrational partition functions respectively, $$n_c$$ is the number of unique conformations in the ensemble. All 3 partition functions are calculated from the classical statistical mechanics expressions which could be found in [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:

$\mathbf{H}^{m} = \mathbf{M^{-1/2}HM^{-1/2}}$

are converted into wavenumbers $$\tilde{\nu}_i$$ according to:

$\tilde{\nu}_i = \frac{1}{2\pi c} \sqrt{\lambda_i}$

### Solvation entropy¶

Solvation entropy is split into electrostatic and hydrophobic parts:

$\Delta S_s = \Delta S_{s,elec} + \Delta S_{s,hyd}$

The electrostatic part of solvation entropy is divided in to 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:

$\Delta S_{s,elec\_bulk} = -\bigg(\frac{\partial \Delta G_s}{\partial \epsilon_{solv}}\bigg) \bigg(\frac{\partial \epsilon_{solv}}{\partial T}\bigg)$

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:

$\Delta S_{s,hyd} = -\bigg(\frac{\partial \Delta G_{s,hyd}}{\partial T}\bigg)$

where $$\Delta G_{s,hyd}$$ consists of 3 components:

$\Delta G_{s,hyd} = \Delta G_{cav} + \Delta G_{VdW} + \Delta G_{Ind}$

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¶

Configurational entropy of a protein bound ligand is calculated totally as vibrational entropy for 3N modes, assuming that 3 rotational and 3 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, $$\Delta S_{des}$$

$S_{protein} = S_v + f\Delta S_s + \Delta S_{des}$

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.