Proline

Acetyl and N-methyl capped dipeptides of the natural amino acids, except proline, alanine, and glycine, were built using LEaP²⁹ at α (−60°, −45°) and β (−135°, 135°) backbone conformations.
$\overrightarrow{\chi }$ was explored by rotating in 10° increments, re-optimizing at each step, or by high temperature simulation (described in Results).
Quantum mechanics optimizations were performed with RHF/6-31G*. Scanned residues were optimized using GAMESS (US)³⁰ with default options. Optimization continued until the RMS gradient was less than 1.0 × 10⁻⁴ Hartree Bohr⁻¹, with an initial trust radius of 0.1 Bohr that could then adjust between 0.05 and 0.5 Bohr. Minimization proceeded by the quadratic approximation. Residues sampled by high temperature simulations were optimized using Gaussian98³¹ with VTight convergence criteria. Quantum mechanics energies for training data were calculated with MP2/6-31+G**. Molecular mechanics re-optimizations were performed in the gas phase with ff99SB for a maximum of 1.0 × 10^{7 (link)} cycles or until the RMS gradient was less than 1.0 × 10⁻⁴ kcal mol⁻¹ Å⁻¹, with a non-bonded cutoff of 99.0 Å and initial step size of 10⁻⁴. Dihedral restraint force constants were 2.0 × 10⁵ kcal mol⁻¹ rad⁻². Minimization employed 10 steps of steepest descent followed by conjugate gradient. Molecular mechanics energies were calculated from the last step of ff99SB minimization.

We set out to parameterize an energy function based on experimental thermodynamic data of small molecules, and high-resolution structural data of macromolecules (shortly “structural data”), with the broader aim of better recapitulating the large-scale energy landscape of protein folding or complex formation, high-resolution structural features, and the balance between natural amino acid preferences. The experimental thermodynamic data consists of the liquid properties of small molecules containing functional groups from natural amino acids¹² and vapor-to-water transfer free energies of protein side-chain analogs²⁵ . The structural data consists of large numbers (> 1000 cluster centers) of alternative conformations (decoys) for protein structures and complexes of known structure, and high-resolution crystallographic data. The agreement of an energy function with these data is represented by a target function F_total:
$${\mathbf{F}}_{\mathbf{\text{total}}}\left[E\left(\mathrm{\Theta }\right)\right]=w_{\mathit{\text{thermodynamic}}}{\mathbf{F}}_{\mathbf{\text{thermodynamic}}}\left[E\left(\mathrm{\Theta }\right)\right]+w_{\mathit{\text{structural}}}{\mathbf{F}}_{\mathbf{\text{structual}}}\left[E\left(\mathrm{\Theta }\right)\right]$$ where the target functions F_{thermodynamic} and F_structural are functionals of a biomolecular energy function E(Θ), which is a function of a set of parameters Θ (see the sections below for the details of E(Θ) in the study), and their relative contributions are adjusted by weights w. F_{thermodynamic}[E(Θ)] and F_structural[E(Θ)] are themselves a weighted linear sum of target functions evaluating performance on specific tasks; their exact composition varies depending on the aim of optimization, and is described in the following paragraphs and sections.
The energy parameters Θ subject to optimization consist of atom-type-dependent parameters, for example, the Lennard-Jones (LJ) radius and well-depth of each atom type. The total number of parameters simultaneously optimized in a single run is on the order of 100. A key advantage of the ability to simultaneously optimize a large number of parameters is that the introduction of significant changes to the physical models of energy terms (for example, an anisotropic solvation model, or change in LJ model of hydrogen atoms) may considerably shift the balance between the energy terms and require large-scale re-parameterization. Optimization of these large parameter sets, with respect to a wide range of thermodynamic and structural data, is performed by a newly developed parameter optimization protocol named dualOptE that uses Nelder-Mead simplex optimization²⁶ (Figure 1, details in following section).
We found several factors to be critical for energy function training to robustly transferrable to independent datasets. First, the training data need to be diverse; consequently, energy function performance is trained on a wide variety of sub-tasks, including recapitulation of sequence and side-chain rotamers, native monomeric structure discrimination, protein-protein docking, and the aforementioned thermodynamic recapitulation tasks. Second, the structure discrimination training sets must be dynamic; it is all too easy to train an energy function to consistently recognize the native structure in a sea of static decoys, but much more challenging when all structures are relaxed in the new energy function^{27 (link)}. In dualOptE, all tests involve some reoptimization against the current energy function: for example, the test measuring the ability to discriminate near-native monomeric conformations or protein-protein interfaces first optimizes a pre-generated set of structures against the current parameterization before assessing discrimination quality. Third, each cycle of parameter optimization must be carried out in a limited amount of computer time. Since we need to assess hundred or thousands of parameterizations in the course of an optimization trajectory, each test has to run on the order of several minutes. For example, during parameter optimization, a full liquid MC simulation to estimate liquid phase properties of small molecules at each step is not computationally tractable; we instead use static sets of snapshots from MC simulations. Following completion of a given parameter optimization run we carried out full liquid MC simulations and found that the static approximation was fairly accurate as long as there were not large changes in the energy function.
We employed multiple iterations of this dual energy function optimization approach. The first iteration, yielding the energy function opt-july15, introduced a new anisotropic implicit solvent model into the Rosetta energy function. Rosetta has previously used the Lazaridis-Karplus (LK) isotropic occlusion-based implicit solvation model^{28 (link)}, where the occluded volume of each atom is proportional to the fractional desolvation energy. The new anisotropic solvation model combines the isotropic part from the original LK model with a ne wly introduced anisotropic polar term^{29 (link)}, which accounts for anisotropic interactions between polar heavy atoms and solvent: occlusion of water binding sites is made more energetically disfavorable than occlusion away from such sites. A second series of optimizations follows introduction of attractive dispersion forces to hydrogens (originally pseudo-united-atom) as well as a reworked electrostatic model, yielded the energy function opt-nov15. For both energy function “snapshots”, following optimization, the resulting energy functions were validated on a set of independent structure prediction tasks too computationally intensive to be used in optimization^{30 (link)}. Details of energy functions (opt-july15 and opt-nov15) and energy terms, and a list of the tests used for optimization are described in following sections, a full list of atomic parameters determined by DualOptE appear in the Supplementary Tables S3–4, and details of the tests and datasets for optimization or independent validation in Supplementary Materials.
The resulting next-generation Rosetta energy function (opt-nov15) outperforms the previous energy function (talaris2014)^{13 (link)} on a wide range of structure prediction tests independent of the training set data. In contrast to opt-nov15, talaris2014 had been optimized solely relying on similar set of structural data we incorporate in the study without the use of small molecule data. We briefly summarize the energy function changes; full details are again provided below. First, there are changes in the physical models, notably the new anisotropic solvation model, a sigmoidal dielectric model, and explicit modeling of the effects of proline on the backbone torsion angles of the preceding residue. Second, there are changes in the representation; in previous Rosetta energy functions hydrogens are purely repulsive to speed computation (much shorter range distance cutoffs were required), whereas in the new energy function hydrogens make attractive LJ interactions. Third, there are changes in the overall balance of forces: compared to talaris2014, both solvation and electrostatic forces are considerably stronger relative to other non-bonded interactions. Fourth, there are changes in many energy function parameters: the attractive interactions of sulfur and aliphatic carbons are stronger (which bring better agreement with small-molecule liquid phase data), and the partial charges of charged chemical groups are more evenly distributed (rather than being primarily on the tip atoms).