Elucidating the solution structure of the K-means cost function using energy landscape theory.

The K-means algorithm, routinely used in many scientific fields, generates clustering solutions that depend on the initial cluster coordinates. The number of solutions may be large, which can make locating the global minimum challenging. Hence, the topography of the cost function surface is crucial to understanding the performance of the algorithm.

Protein energy landscape exploration with structure-based models.

Exploring the multi-dimensional energy landscape of a large protein in detail is a computational challenge. Such investigations may include analysis of multiple folding pathways, rate constants for important conformational transitions, locating intermediate states populated during folding, estimating energetic and entropic barriers that separate populated basins, and visualising a high-dimensional surface. The complexity of the landscape can be simplified through coarse-grained structure-based models (SBMs).

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.

Morphological analysis of chiral rod clusters from a coarse-grained single-site chiral potential.

We present a coarse-grained single-site potential for simulating chiral interactions, with adjustable strength, handedness, and preferred twist angle. As an application, we perform basin-hopping global optimisation to predict the favoured geometries for clusters of chiral rods. The morphology phase diagram based upon these predictions has four distinct families, including previously reported structures for potentials that introduce chirality based on shape, such as membranes and helices.

Go-Kit: A Tool To Enable Energy Landscape Exploration of Proteins.

Coarse-grained Go̅-like models, based on the principle of minimal frustration, provide valuable insight into fundamental questions in the field of protein folding and dynamics. In conjunction with commonly used molecular dynamics (MD) simulations, energy landscape exploration methods like discrete path sampling (DPS) with Go̅-like models can provide quantitative details of the thermodynamics and kinetics of proteins. Here we present Go-kit, a software that facilitates the setup of MD and DPS simulations of several flavors of Go̅-like models.

Path Integral Energy Landscapes for Water Clusters.

The energy landscapes for a discretized path integral representation of the water dimer, trimer and pentamer are characterized in terms of the localized (classical) and delocalized minima and transition states. The transition states are finite-temperature approximations to the exact instanton path, and they are typically used to calculate the tunneling splittings or reaction rates. The features of the path integral landscape are explored, thus elucidating procedures that could usefully be automated when searching for instantons in larger systems.

Energy Landscape of the Designed Protein Top7.

To fold on biologically relevant time scales, proteins have evolved funnelled energy landscapes with minimal energetic trapping. However, the polymeric nature of proteins and the spatial arrangement of secondary structural elements can create topological traps and slow folding. It is challenging to identify, visualize, and quantify such topological trapping.

Effects of random pinning on the potential energy landscape of a supercooled liquid.

We use energy landscape methods to investigate the response of a supercooled liquid to random pinning. We classify the structural similarity of different energy minima using a measure of overlap. This analysis reveals a correspondence between distinct particle packings (which are characterised via the overlap) and funnels on the energy landscape (which are characterised via disconnectivity graphs).

Loss surface of XOR artificial neural networks.

Training an artificial neural network involves an optimization process over the landscape defined by the cost (loss) as a function of the network parameters. We explore these landscapes using optimization tools developed for potential energy landscapes in molecular science. The number of local minima and transition states (saddle points of index one), as well as the ratio of transition states to minima, grow rapidly with the number of nodes in the network.

From sticky-hard-sphere to Lennard-Jones-type clusters.

A relation M_{SHS→LJ} between the set of nonisomorphic sticky-hard-sphere clusters M_{SHS} and the sets of local energy minima M_{LJ} of the (m,n)-Lennard-Jones potential V_{mn}^{LJ}(r)=ɛ/n-m[mr^{-n}-nr^{-m}] is established. The number of nonisomorphic stable clusters depends strongly and nontrivially on both m and n and increases exponentially with increasing cluster size N for N≳10. While the map from M_{SHS}→M_{SHS→LJ} is noninjective and nonsurjective, the number of Lennard-Jones structures missing from the map is relatively small for cluster sizes up to N=13, and most of the missing structures correspond to energetically unfavorable minima even for fairly low (m,n).

Structure, thermodynamics, and rearrangement mechanisms in gold clusters-insights from the energy landscapes framework.

We consider finite-size and temperature effects on the structure of model Au clusters (30 ≤ N ≤ 147) bound by the Gupta potential. Equilibrium behaviour is examined in the harmonic superposition approximation, and the size-dependent melting temperature is also bracketed using molecular dynamics simulations. We identify structural transitions between distinctly different morphologies, characterised by various defect features.

Providing sustainable catalytic solutions for a rapidly changing world: a summary and recommendations for urgent future action.

In addition to summarizing the main thrusts of each paper presented at this Discussion, other urgent issues involving the role (and characterization) of new catalysts for eliminating oxides of nitrogen, for using CO liberated from steel mills, for fuel cells and the need for rapid decarbonization of fossil fuels are outlined.This article is part of a discussion meeting issue 'Providing sustainable catalytic solutions for a rapidly changing world'.

Multifunctional energy landscape for a DNA G-quadruplex: An evolved molecular switch.

We explore the energy landscape for a four-fold telomere repeat, obtaining interconversion pathways between six experimentally characterised G-quadruplex topologies. The results reveal a multi-funnel system, with a variety of intermediate configurations and misfolded states. This organisation is identified with the intrinsically multi-functional nature of the system, suggesting a new paradigm for the classification of such biomolecules and clarifying issues regarding apparently conflicting experimental results.

Pathways for diffusion in the potential energy landscape of the network glass former SiO.

We study the dynamical behaviour of a computer model for viscous silica, the archetypal strong glass former, and compare its diffusion mechanism with earlier studies of a fragile binary Lennard-Jones liquid. Three different methods of analysis are employed. First, the temperature and time scale dependence of the diffusion constant is analysed.

Structure and Thermodynamics of Metal Clusters on Atomically Smooth Substrates.

We analyze the structure of model Ni and Cu clusters (N = 55, 147) supported on a variety of atomically smooth van der Waals surfaces. The global minima are mapped in the space of two parameters: (i) the laterally averaged surface stickiness, γ, which controls the macroscopic wetting angle, and (ii) the surface microstructure, which produces more subtle but important templating via epitaxial stresses. We find that adjusting the substrate lattice (even at constant γ) can favor different crystal plane orientations in the cluster, stabilize hexagonal close-packed order, or induce various defects, such as stacking faults, twin boundaries, and five-fold disclinations.

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.

Machine learning landscapes and predictions for patient outcomes.

The theory and computational tools developed to interpret and explore energy landscapes in molecular science are applied to the landscapes defined by local minima for neural networks. These machine learning landscapes correspond to fits of training data, where the inputs are vital signs and laboratory measurements for a database of patients, and the objective is to predict a clinical outcome. In this contribution, we test the predictions obtained by fitting to single measurements, and then to combinations of between 2 and 10 different patient medical data items.

The RSC Faraday prize lecture of 1989 on platinum.

In 1861, Michael Faraday gave one of his last Friday Evening Discourses at the Royal Institution of Great Britain, London, on platinum, which he described as "this beautiful, magnificent and valuable metal". More than a hundred and twenty years later (in 1989), the author re-enacted, at the Royal Institution, many of the demonstrations that Faraday carried out in his memorable Discourse. This article outlines many of Faraday's views on, and experiments with, platinum.

The extraordinary impact of Michael Faraday on chemistry and related subjects.

Biographers of Michael Faraday, as well as many dictionaries of science, often describe him as a physicist, which he certainly was. But he was also an astonishingly effective chemist: in fact, he was the Fullerian Professor of Chemistry (at the Royal Institution, RI) from 1834 until the time of his death in August, 1867. To mark the sesquicentenary of his passing, this editorial, by one of his distant successors as Director and Fullerian Professor at the RI, focuses on Faraday's output and influence as a scientist.

Decoding heat capacity features from the energy landscape.

A general scheme is derived to connect transitions in configuration space with features in the heat capacity. A formulation in terms of occupation probabilities for local minima that define the potential energy landscape provides a quantitative description of how contributions arise from competition between different states. The theory does not rely on a structural interpretation for the local minima, so it is equally applicable to molecular energy landscapes and the landscapes defined by abstract functions.

Defining and quantifying frustration in the energy landscape: Applications to atomic and molecular clusters, biomolecules, jammed and glassy systems.

The emergence of observable properties from the organisation of the underlying potential energy landscape is analysed, spanning a full range of complexity from self-organising to glassy and jammed systems. The examples include atomic and molecular clusters, a β-barrel protein, the GNNQQNY peptide dimer, and models of condensed matter that exhibit structural glass formation and jamming. We have considered measures based on several different properties, namely, the Shannon entropy, an equilibrium thermodynamic measure that uses a sample of local minima, and indices that require additional information about the connections between local minima in the form of transition states.

Energy landscapes for machine learning.

Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions as local minima can be analysed in terms of the corresponding machine learning landscape. Methods to explore and visualise molecular potential energy landscapes can be applied to these machine learning landscapes to gain new insight into the solution space involved in training and the nature of the corresponding predictions.

Atomic clusters with addressable complexity.

A general formulation for constructing addressable atomic clusters is introduced, based on one or more reference structures. By modifying the well depths in a given interatomic potential in favour of nearest-neighbour interactions that are defined in the reference(s), the potential energy landscape can be biased to make a particular permutational isomer the global minimum. The magnitude of the bias changes the resulting potential energy landscape systematically, providing a framework to produce clusters that should self-organise efficiently into the target structure.

GPU-Accelerated Exploration of Biomolecular Energy Landscapes.

We present graphics processing unit (GPU)-acceleration of various computational energy landscape methods for biomolecular systems. Basin-hopping global optimization, the doubly nudged elastic band method (DNEB), hybrid eigenvector-following (EF), and a local rigid body framework are described, including details of GPU implementations. We analyze the results for eight different system sizes, and consider the effects of history size for minimization and local rigidification on the overall efficiency.