Estimating binding free energy

The binding free energy of each protein-ligand complex was estimated using the HTS (computational High Throughput Screening) scoring function. This empirical scoring function, which can rapidly predict the affinity of a ligand for a protein binding site, assumes that the total free energy of binding can be decomposed into a linear combination of physically meaningful terms. Experimental data for protein-ligand complexes, combined with theoretical models of the underlying physics, were used to develop and optimize the functional form and parameters.

The HTS function can be written as:

Individual terms will be discussed in detail in the following.

The terms in the HTS function were calibrated separately whenever possible. The solvation terms were fitted to experimental solvation free energies and the entropy terms were derived from sublimation thermodynamic data. Hydrogen bond and metal center bond terms were adopted from the paramet-rization of the Ludi scoring function [19]. The HTS function has been validated on a set of proprietary protein-ligand complexes without further calibration: HlV-1 Protease (34 structures) (a = 1.70 kcal/mol, Glycinamide Ribonucleotide Transformylase (35 structures) a = 1.14 kcal/mol, Thymidylate Synthase (39 structures) a = 1.5 1 kcal/mol, Stromelysin/Matrilysin/Collage-nase (3 1 structures) a = 1.47 kcal/mol, FKBP (54 structures) a = 1.30 kcal/mol

Interactions between the ligand and protein are weak at the surface of the protein due to solvent screening and the flexibility of surface side chains. The effect is taken into account by expressing the interaction terms as function of the burial. The burial factor, b, depends on the distance, d, of a particular atom from the nearest point on the molecular surface of the complex. The burial factor is linearly scaled from 0 at the surface to a value of 1 at a distance of 4 A below the molecular surface.

The hydrogen bond energy, AGh-b, is a function of the burial, the hydrogen bond geometry, fgeometry, and the type of donor and acceptor atom, fdonor/acceptor. The standard hydrogen bond parameters are -2.2 kcal/mol for chb and 0.3 for cb.

The terms fAP and fAa are taken from the Ludi scoring function [19]. AR is the deviation of the H... Acceptor distance from the ideal hydrogen bond distance (1.9 A), and Aa is the deviation of the donor-hydrogen-acceptor angle from linearity. 0 is the angle of the donor out of the acceptor plane. fdonor/acceptor is a factor describing the hydrogen bond strength of the donor and acceptor type, and A Gdes-hb accounts for desolvation of the donor and acceptor atoms involved in hydrogen bonds. Hydrogen bonds are defined by the following criteria: H...A distance < = 3 A, donor-H-acceptor angle > = 90 degrees, donor-acceptor-acceptor-antecedent angle > = 85 degrees, and the H-acceptor-acceptor-antecedent angle > 75 degrees. Interactions that are outside of these ranges are included in the electrostatic and desolvation terms of Equation 1. C-H bonds in aromatic systems are treated as weak hydrogen bond donors and aromatic carbons are treated as weak acceptor atoms.

A Ges is an estimate of the free energy of electrostatic interaction between ligand and protein atoms not directly involved in hydrogen bonds. These interactions are described as dipole-dipole, dipole-charge or charge-charge interactions. To mimic solvent screening of the electrostatic interactions, the effective dielectric is set to 4 in the interior of the protein and is linearly scaled to a value of 10 at the molecular surface using the burial factor.

A GM is an estimate of the electrostatic free energy of interaction between the ligand and any metal centers within the protein involved in ligand binding, such as the zinc that can interact with hydroxamic acids in stromelysin.

fAR = 1.0 A R <= 0.2A, fAR =1.0 - (A R - 0.2)/0.4 0.2A < A R < 0.6A, (7)

This energy is a function of the burial, fb, the metal to acceptor distance, fA R, and the identity of the acceptor, facceptor. It is also a function of the angle made between the metal center and the plane of the acceptor, 0. The term fA R was taken from the Ludi scoring function [19]. AR is the deviation of the Metal ...Acceptor distance from the ideal distance.

A desolvation penalty is applied to polar atoms of the ligand and the binding site that do not form hydrogen bonds. The total desolvation penalty, A Gdes-p, is a sum of atomic contributions. Every atom's contribution depends on the change in its solvent-accessible surface area, AA, on a function of the burial factor, fb, and a solvation parameter, a, that is specific for the atom type. The solvation parameters for a number of atom types have been derived from experimental data [20]. Solvent-accessible surface areas are calculated with the Amber 94 radii [ 14] and a water probe radius of 1.4 A.

Burial of nonpolar surface contributes favorably to the binding free energy. The total contribution, AGdes-np, is written as a sum over the change in surface area of each non-polar atom, AA, modulated by a function of the burial factor, fb, which enhances the hydrophobic interactions in deep pockets and attenuates hydrophobic interactions at flat protein surfaces:

The solvation parameter, a alkane = 0.007 kcal mol-1 A-2, was obtained from hydration free energies of alkanes [ 19].

A GVdw is an estimate of contribution of the van der Waals interaction to complex formation. The van der Waals interaction is estimated using a Lennard-Jones type potential with a soft-core as described by Beutler et al. [16]. The soft-coring of the potential allows for small overlaps of the ligand and protein atoms, implicitly accounting for some mobility of the atoms while making the function more robust computationally.

The change in free energy due to inducing strain into a ligand upon complex formation, A Gstrain, is difficult to estimate, since the energy of only a single configuration is used in the calculation. Currently, the only strain recognized by the program is the syn-pentane interaction [21]. An example of a syn-pentane or g+g- interaction is the 1,3-diaxial interaction in cyclohexane derivatives, where two substituents in a 1,5 arrangement are in close proximity. An energy penalty, AGsyn-pentme = 2 kcal/mol, is added to the strain energy term for each syn-pentane interaction.

A Ginternal is an estimate of the loss in free energy due to the immobilizing internal rotations in the protein and ligand when complexation occurs. In the current form of HTS, only the freezing of internal rotations in the ligand is taken into account. This loss in conformational entropy is calculated from the estimated change in the number of conformational states between the bound and unbound ligand. To estimate the number of states in the unbound state each rotatable bond in the ligand is analyzed. Bonds to terminal groups, e.g. CH3, NH3, and bonds that rotate symmetric groups, e.g. carboxylate or phenyl, are not considered in the calculation. The rotation around an sp2-sp2 bond generates two states, m = 2, and the rotation around an sp2-sp3 and sp3-sp3 bond generates three states, m = 3, in the unbound ligand. We calculate the number of states in the bound form by formally splitting the ligand into two fragments at the rotatable bond and evaluating the interaction energy of each fragment with the protein. Suppose both fragments interact strongly with the protein, in this case the ligand is frozen into one state for the rotat-able bond under consideration. Now suppose a situation where one fragment interacts strongly with the protein and the other fragment does not interact with the protein, but is exposed to solvent. In this case, the rotation about the bond is not restricted. The loss in mobility is then related to the energy of the fragment that has the lowest interaction energy, Elow = Maximum(Efnigment1, Efnigment2). To estimate the number of states available in the bound state we assume that we have one state with the energy, E1 = Elow and that the other m-1 states are not occupied. The unoccupied states are assigned an energy,

E2 ... Em = 0.0 kcal/mol, and we evaluate the number of states in the bound form, Z by summing over all states, m.

The free energy due to entropy loss, A Gonial, is calculated as a sum over all rotatable bonds:

The binding of a ligand to a protein reduces the number of translational and rotational degrees of freedom in the system and therefore provides a contribution, A Gtrans/rot, to the binding free energy:

A Gtrans/rot AGcratic °.6 A H-interaction (12)

Here we model this factor as the sum of terms which represents the change in cratic entropy [22] upon complexation and a term that accounts for enthalpy-entropy compensation. At a concentration of 1.0 mol/1 the cratic entropy contributes -8 cal/mol/K, corresponding to a free energy of 2.4 kcal/mol at 300 K.

The second term in A Gtrans/rot is based on the phenomenon of enthalpy-entropy compensation. A strong interaction between the protein and the lig-and results in a strong reduction in the mobility of the ligand and is associated with a large loss in entropy. We use the enthalpy-entropy relationship between the enthalpy and entropy of sublimation to calibrate the entropy term. The analysis of the sublimation thermodynamics of rigid molecules by Searle and Williams [23] gave the following relationship: -TA Ssublmation = -0.6 A Hsublimation. The same scaling factor is applied to the interaction energy to get an estimate of the entropy loss upon ligand binding.

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