Energy landscapes for proteins described by the UNRES coarse-grained potential.

The self-assembly of proteins is encoded in the underlying potential energy surface (PES), from which we can predict structure, dynamics, and thermodynamic properties. However, the corresponding analysis becomes increasingly challenging with larger protein sizes, due to the computational time required, which grows significantly with the number of atoms. Coarse-grained models offer an attractive approach to reduce the computational cost. In this Feature Article, we describe our implementation of the UNited RESidue (UNRES) coarse-grained potential in the Cambridge energy landscapes software. We have applied this framework to explore the energy landscapes of four proteins that exhibit native states involving different secondary structures. Here we have tested the ability of the UNRES potential to represent the global energy landscape of proteins containing up to 100 amino acid residues. The resulting potential energy landscapes exhibit good agreement with experiment, with low-lying minima close to the PDB geometries and to results obtained using the all-atom AMBER force field. The new program interfaces will allow us to investigate larger biomolecules in future work, using the UNRES potential in combination with all the methodology available in the computational energy landscapes framework.

Global analysis of energy landscapes for materials modeling: A test case for C60.

In this contribution, we employ computational tools from the energy landscape approach to test Gaussian Approximation Potentials (GAPs) for C60. In particular, we apply basin-hopping global optimization and explore the landscape starting from the low-lying minima using discrete path sampling. We exploit existing databases of minima and transition states harvested from previous work using tight-binding potentials. We explore the energy landscape for the full range of structures and pathways spanning from the buckminsterfullerene global minimum up to buckybowls. In the initial GAP model, the fullerene part of the landscape is reproduced quite well. However, there are extensive families of C1@C59 and C2@C58 structures that lie lower in energy. We succeeded in refining the potential to remove these artifacts by simply including two minima from the C2@C58 families found by global landscape exploration. We suggest that the energy landscape approach could be used systematically to test and improve machine learning interatomic potentials.

Exploring energy landscapes for solid-state systems with variable cells at the extended tight-binding level.

The design of novel materials requires a theoretical understanding of dynamical processes in the solid state, including polymorphic transitions and associated pathways. The organization of the potential energy landscape plays a crucial role in such processes, which may involve changes in the periodic boundaries. This study reports the implementation of a general framework for periodic condensed matter systems in our energy landscape analysis software, allowing for variation in both the unit cell and atomic positions. This implementation provides access to basin-hopping global optimization, the doubly nudged elastic band procedure for identifying transition state candidates, the missing connection approach for multi-step pathways, and general tools for the construction and analysis of kinetic transition networks. The computational efficacy of the procedures is explored using the state-of-the-art semiempirical method GFN1-xTB for the first time in this solid-state context. We investigate the effectiveness of this level of theory by characterizing the potential energy and enthalpy landscapes of several systems, including silicon, CdSe, ZnS, and NaCl, and discuss further technical challenges, such as translational permutation of the cell. Despite the expected limitations of the semiempirical method, we find that the resulting energy landscapes provide useful insight into solid-state simulations, which will facilitate detailed analysis of processes such as defect and ion migration, including refinement at higher levels of theory.

Enhancing Biomolecular Simulations with Hybrid Potentials Incorporating NMR Data.

Some recent advances in biomolecular simulation and global optimization have used hybrid restraint potentials, where harmonic restraints that penalize conformations inconsistent with experimental data are combined with molecular mechanics force fields. These hybrid potentials can be used to improve the performance of molecular dynamics, structure prediction, energy landscape sampling, and other computational methods that rely on the accuracy of the underlying force field. Here, we develop a hybrid restraint potential based on NapShift, an artificial neural network trained to predict protein nuclear magnetic resonance (NMR) chemical shifts from sequence and structure. In addition to providing accurate predictions of experimental chemical shifts, NapShift is fully differentiable with respect to atomic coordinates, which allows us to use it for structural refinement. By employing NapShift to predict chemical shifts from the protein conformation at each simulation step, we can compute an energy penalty and the corresponding hybrid restraint forces based on the difference between the predicted values and the experimental chemical shifts. The performance of the hybrid restraint potential was benchmarked using both basin-hopping global optimization and molecular dynamics simulations. In each case, the NapShift hybrid potential improved the accuracy, leading to better structure prediction via basin-hopping and increased local stability in molecular dynamics simulations. Our results suggest that neural network hybrid potentials based on NMR observables can enhance a broad range of molecular simulation methods, and the prediction accuracy will improve as more experimental training data become available.

The alchemical energy landscape for a pentameric cluster.

We investigate the energy landscape of an alchemical system of point particles in which the parameters of the interparticle potential are treated as degrees of freedom. Using geometrical optimization, we locate minima and transition states on the landscape for pentamers. We show that it is easy to find the parameters that give the lowest energy minimum and that the distribution of minima on the alchemical landscape is concentrated in particular areas. In contrast to the usual changes to an energy landscape when adding more degrees of freedom, we find that introducing alchemical degrees of freedom can reduce the number of minima. Moreover, compared to landscapes of the same system with fixed parameters, these minima on the alchemical landscape are separated by high barriers. We classify transition states on the alchemical landscape by whether they become minima or remain transition states when the potential parameters are fixed at the stationary point value. We show that those that become minima have a significant alchemical component in the direction of the pathway, while those that remain as transition states can be characterized mainly in terms of atomic displacements.

Properties of kinetic transition networks for atomic clusters and glassy solids.

A database of minima and transition states corresponds to a network where the minima represent nodes and the transition states correspond to edges between the pairs of minima they connect via steepest-descent paths. Here we construct networks for small clusters bound by the Morse potential for a selection of physically relevant parameters, in two and three dimensions. The properties of these unweighted and undirected networks are analysed to examine two features: whether they are small-world, where the shortest path between nodes involves only a small number or edges; and whether they are scale-free, having a degree distribution that follows a power law. Small-world character is present, but statistical tests show that a power law is not a good fit, so the networks are not scale-free. These results for clusters are compared with the corresponding properties for the molecular and atomic structural glass formers ortho-terphenyl and binary Lennard-Jones. These glassy systems do not show small-world properties, suggesting that such behaviour is linked to the structure-seeking landscapes of the Morse clusters.

Exploring the potential energy landscape of the Thomson problem via Newton homotopies.

Locating the stationary points of a real-valued multivariate potential energy function is an important problem in many areas of science. This task generally amounts to solving simultaneous nonlinear systems of equations. While there are several numerical methods that can find many or all stationary points, they each exhibit characteristic problems. Moreover, traditional methods tend to perform poorly near degenerate stationary points with additional zero Hessian eigenvalues. We propose an efficient and robust implementation of the Newton homotopy method, which is capable of quickly sampling a large number of stationary points of a wide range of indices, as well as degenerate stationary points. We demonstrate our approach by applying it to the Thomson problem. We also briefly discuss a possible connection between the present work and Smale's 7th problem.

Energy landscapes of planar colloidal clusters.

A short-ranged pairwise Morse potential is used to model colloidal clusters with planar morphologies. Potential and free energy global minima as well as rearrangement paths, obtained by basin-hopping global optimisation and discrete path sampling, are characterised. The potential and free energy landscapes are visualised using disconnectivity graphs. The short-ranged potential is found to favour close-packed structures, with the potential energy primarily controlled by the number of nearest neighbour contacts. In the case of quasi-degeneracy the free energy global minimum may differ from the potential energy global minimum. This difference is due to symmetry effects, which result in a higher entropy for structures with lower symmetry.

Designing a Bernal spiral from patchy colloids.

A model potential for colloidal building blocks is defined with two different types of attractive surface sites, described as complementary patches and antipatches. A Bernal spiral is identified as the global minimum for clusters with appropriate arrangements of three patch-antipatch pairs. We further derive a minimalist design rule with only one patch and antipatch, which also produces a Bernal spiral. Monte Carlo simulations of these patchy colloidal building blocks in the bulk are generally found to corroborate the global optimization results.