I. SpinningTop
4.2. Dihedral Parameter Fitting
To limit computational expense, model dipeptides for each of the nine amino acids chosen were produced by capping the N- and C- terminus with an acetyl group and methyl amide group respectively. Structures of each of these model dipeptides are given in figure 4.1.
4.2.1. Quantum Chemical Calculations
A two-dimensional potential energy surface was produced for each of the eight amino acids which allow free rotation about bothΦandΨbackbone angles. A 15°×15° diagonal grid was
4.2. Dihedral Parameter Fitting
used, giving up to†288 points on the potential energy surface. Optimisations were performed at the B3LYP-D3(BJ)/def2-TZVP level of theory,8–12,15using a polarisable continuum solvation
model (PCM) to simulate bulk solvation effects,13,14using GAMESS-US version 18 AUG 2016 (R1).19,20The def2-TZVP basis set was downloaded from the Basis Set Exchange.16–18Calcula- tions were performed on the NIWA High Performance Computing Facility (HPCF) IBM Power6 cluster, with access provided by NeSI project 00170.
An initial conformational search using the def2-SVP basis set15was undertaken to find a low
energy starting conformation for the potential energy surface generation. At each gridpoint, one of the dihedrals across theΦandΨbonds was fixed at the given value, and the rest of the molecule allowed to relax. Due to the nature of how the input files were generated, only in the cases of glycine dipeptide and proline dipeptide were the fixed dihedrals defined by the atoms that define the backbone dihedral. In all other cases, the fixed dihedral involved the Cα carbon of the sidechain. Proline dipeptide followed the same scheme, but instead of two dimensional gridpoints,Ψwas fixed at 15° intervals andΦwas unrestrained. This gave 24 regularly spaced data points.
4.2.2. Molecular Dynamics Simulations
Using the same 15° diagonal grid as used to generate the QM potential energy surface, a free energy surface was generated using molecular dynamics simulations without the backbone pa- rameters. At each grid point, theΦandΨdihedral angles were constrained using SHAKE.28
Each constrained molecule was solvated in a periodic cubic water box in the absence of counter ions. The water boxes were initialised with a 15 Å distance of the solute to the box walls. Prior to the production simulations, the systems were equilibrated from 60 K to 300 K in five discrete steps with a simulation length of 10 ps each. All simulations were carried out at 300 K and a constant volume. A weak thermostat coupling with two baths for the solute and solvent was applied with a coupling constant of 0.1 ps. The SHAKE constraint algorithm was used to main- tain the bond distances at the energy minimum, and theΦandΨangles at their desired values. The 54A7 parameter set of the GROMOS force field was used,1 with the backbone dihedral parameters removed, except for proline dipeptide where only theΨparameters were removed. A time step of 2 fs was used, and the energy saved every 100 fs. Interactions within 0.8 nm were calculated at every time step. Interactions up to a distance of 1.4 nm were calculated along with the pairlist update every five steps and kept constant between updates. Long-range inter- actions were approximated with a reaction field contribution,29 accounting for a homogeneous
†some confirmations were unable to be minimised due to large initial gradients beyond the limit imposed by
4. Method
medium with relative dielectric constant of 61 beyond the 1.4 nm cut-off. Local elevation was used to gradually build up a bias potential and increase conformational space sampling. Each non-constrained dihedral was divided into thirty-six evenly spaced grid points, with each grid point having its own biasing potential applied. A magnitude of 100 J mol−1was utilised, ex- cept in the cases of glycine dipeptide, where the magnitude was 10 J mol−1, and the dihedrals within the five-membered ring of proline dipeptide, where the magnitude was 1 J mol−1. No bias potential was applied to the internal dihedrals of the aromatic ring in phenylalanine dipeptide. Each simulation was run for 1 ns using a locally modified‡version of the GROMOS molecular
dynamics engine.30–32All energies obtained were used to calculate the partition function, and thus the Helmholtz free energy of the constrained molecule.
4.2.3. Parameter Fitting
The difference between the QM potential energy surface and the free energy surface is used as the potential value for dihedral parameter fitting. To obtain this difference, each surface was in- terpolated onto a regular 5° rectangular grid using a piecewise cubic, continuously differentiable, and approximately curvature-minimising polynomial surface, as implemented in the cubic inter- polation option of SciPy.33,34Each interpolated point of the free energy surface was subtracted from the corresponding interpolated point of the QM potential energy surface. Simultaneous fitting of bothΦandΨterms was performed on these difference energies. Fitting multiplicity values were limited to one, two, three, and six. Fits included fitting phase, except for glycine dipeptide where the symmetry of the potential energy surface means that fit terms were limited to phase values of 0° or 180°. To follow the minimalism ethos of GROMOS force fields and align with chemical sensibilities, fit parameters were manually modified to have phase values limited to multiples of 30°.