Main results of the thesis

The papers included in this thesis are organized as follows:

• Method development for QM/MM-FEP (Papers I, II, III, IV)

• Application of other QM/MM methods to estimate ligand-binding affinities (Papers IV).

• Method development for MM FEP (Papers V).

• Application of binding entropy calculation in the study of protein–ligand binding free energies (Papers VI).

In the following, I will discuss the main results of each paper.

Paper I. Converging ligand-binding free energies obtained with free-energy perturbations at the quantum mechanical level.

In two earlier studies in our research group, QM/MM-FEP was attempted with the

ssEA and NBB approaches, but no converged results could be obtained

. In my

first paper, the convergence of QM/MM-FEP was studied for host–guest binding free

energies. We showed that QM/MM-FEP can be converged to 1 kJ/mol by using 700

000 QM calculations. The precision of the method follows the expected

dependence, where N is the number of snapshots (Figure 7.1). We compared several

methods to calculate the free energy, viz. NBB4, ssEA and ssEAc, and showed that

ssEAc was the most cost-efficient method. However, the accuracy of the ssEAc

method was not better than MM FEP (Figure 7.2).

Figure 7.1. (a) Convergence of the ssEAc predictions of ΔΔG^MM→QM with respect to the number of considered snapshots for the eight transformations (b) standard error of the calculations based on 1000

bootstraps.

Figure 7.2. Comparison of the MM and SQM/MM energies using the ssEAc method compared to the experimental relative affinitiesfor the eight transformations. The black line shows the perfect correlation.

Paper II. Comparison of methods to obtain ligand-binding free energies with QM/MM methods.

In the second paper, the convergence of the QM/MM energies are studied when explicit QM/MM MD simulations are performed and the MM→QM/MM perturbation is subdivided into several steps with another coupling parameter Λ in QM/MM method space, i.e. the RPQS method. The paper is the first to explicitly verify that the reference-potential method is in agreement with direct alchemical QM/MM free-energy perturbation and that the two methods give identical results.

Furthermore, this paper demonstrated that the reference-potential method has a four times lower computational cost than the direct QM/MM-FEP (Table 7.1, Figure 7.3).

Table 7.1. Comparison of the reference-potential method and the direct QM/MM-FEP sampling. Unit: kJ/mol.

RPQS 4 + 4 Λ

Direct QM/MM-FEP

17-18 λ Difference

pClBz → Bz 23.3 ± 0.5 23.2 ± 0.9 0.1

mClBz → Bz 9.4 ± 0.5 11.3 ± 0.8 1.9

Figure 7.3. Comparison of the experimental and calculated affinities obtained with either MM or RPQS with nine Λ-values. The line shows the perfect correlation.

Paper III. Relative ligand-binding free energies calculated from multiple short QM/MM MD Simulations.

In Paper III, the computational efficiency of the RPQS approach developed in Paper

II was investigated and it was examined whether it could be sped up by using

multiple short FEP calculations, employing the fact that the MM simulations already

thoroughly sample the phase space (the RPQS-MSS method). The paper shows that

eight free-energy perturbations for the octa-acid guest molecules converges to within

1 kJ/mol in less than 50 ps of sampling using an ensemble of 100 independent

simulations per perturbation (Figure 7.4). For the ninth ligand (Figure 7.5), longer

simulations were needed (~70 ps), owing to a mismatch between the preferred

structures of the MM and QM energy functions.

Figure 7.4. Convergence profiles for the nine ligands in this study as a function of simulation time per window.

Figure 7.5. Convergence profiles for the ninth ligand in this study as a function of simulation time per window for a 150 ps trajectory.

Paper IV. Binding free energies in the SAMPL6 octa-acid host–guest challenge calculated with MM and QM methods.

In Paper IV, we used a standard MM-FEP protocol

, RPQS and binding free energies from QM/MM-minimised structures

at the PM6-DH+ and TPSS-D3 levels of theory for the blind-prediction challenge SAMPL6. The best method was found to be the RPQS method (Paper II), which gave a MAD of 2.4–5.0 kJ/mol, r

= 0.81–0.93 and τ

= 0.84–1.00. It was (together with standard MM FEP) one of the best five methods in the competition (Figure 7.6).

a b

Figure 7.6. Comparison of the experimental and calculated absolute affinities obtained with the (a) MM-FEP and (b) QM/MM-MM-FEP methods. The black line shows the perfect correlation. OAH and OAM are two variants of the octa-acid deep-cavity host.

Paper V. Binding affinities of the farnesoid X receptor in the D3R Grand Challenge 2 estimated by free-energy perturbation and docking.

In this paper, the charge correction scheme by Rocklin et al.

2013 was implemented together with the recent free-energy perturbation protocol in the AMBER software for blind-prediction in the D3R Grand Challenge 2. The results gave a MAD of 7.5 kJ/mol compared to the experimental estimates and a squared correlation coefficient r

= 0.1. The results suggested that including a charge correction for free-energy perturbation involving a change in the net charge improves the experimental agreement with experimental data significantly (~8 kJ/mol) on average and in the correct direction. These results were among the four best in the competition out of 22 submissions (Figure 7.7).

a b

Figure 7.7. Comparison between the experimental and calculated binding free energies for the two FEP sets in the D3R Grand Challenge 3, (a) FE set 1 (b) FE set 2.

Paper VI. Detailed characterization of the binding of diastereomeric ligands to galectin-3.

Previously, binding entropy calculation has been applied to protein–ligand binding by several groups

. Here, we study two diastereomeric ligands binding to the protein galectin-3. We compare small structural differences in the ligands, which affect the conformational entropy of protein–ligand complexes. One of the two ligands, show two conformations in the crystal structure. We calculate entropies with the method developed by Genheden et al.

2009, using a windowing scheme to calculate average conformational entropies by dihedral histogramming. The results indicate that –T∆∆S

is 9 ± 5 kJ/mol between the two complexes. The calculated relative conformational entropy agrees with the experimental conformational entropy (from backbone and methyl groups) of 12 ± 8 kJ/mol both in sign and magnitude (Figure 7.8, Table 7.2).

Figure 7.8. Conformational entropy contributions to ligand binding reported per residue. TΔ∆Sconf is color coded onto the galectin-3 structure with blue hues indicating positive values and red hues indicating negative values, with the color intensity ranging from weak (white) for T∆ΔSconf = 0 to intense (maximally blue or red) for |TΔ∆Sconf| = 3 kJ/mol.

Table 7.2. Conformational entropy differences between the various R– and S–galectin-3 complexes and the apo protein, obtained from the MD simulations^a.

R–apo S–apo S2–apo – TΔ∆Sconfb 43 ± 5 33 ± 5 32 ±5 – TΔ∆Sconfc 67 ± 5 57 ± 4 58 ± 5

aS2 is the second conformation of the S ligand in complex with galectin-3.

b Includes all protein dihedrals.

c Includes all protein and ligand dihedrals.