Archetypal energy landscapes: Dynamical diagnosis

Memory effect and phase transition in a hierarchical trap model for spin glasses

We introduce an efficient dynamical tree method that enables us to explicitly demonstrate the thermoremanent magnetization memory effect in a hierarchical energy landscape. Our simulation nicely reproduces the nontrivial waiting-time and waiting-temperature dependences in this nonequilibrium phenomenon. We further investigate the condensation effect, in which a small set of microstates dominates the thermodynamic behavior in the multilayer trap model. Importantly, a structural phase transition of the multilayer tree model is shown to coincide with the onset of the condensation phenomenon. Our results underscore the importance of hierarchical structure and demonstrate the intimate relation between the glassy behavior and structure of barrier trees.

Perspective: new insights from loss function landscapes of neural networks

We investigate the structure of the loss function landscape for neural networks subject to dataset mislabelling, increased training set diversity, and reduced node connectivity, using various techniques developed for energy landscape exploration. The benchmarking models are classification problems for atomic geometry optimisation and hand-written digit prediction. We consider the effect of varying the size of the atomic configuration space used to generate initial geometries and find that the number of stationary points increases rapidly with the size of the training configuration space. We introduce a measure of node locality to limit network connectivity and perturb permutational weight symmetry, and examine how this parameter affects the resulting landscapes. We find that highly-reduced systems have low capacity and exhibit landscapes with very few minima. On the other hand, small amounts of reduced connectivity can enhance network expressibility and can yield more complex landscapes. Investigating the effect of deliberate classification errors in the training data, we find that the variance in testing AUC, computed over a sample of minima, grows significantly with the training error, providing new insight into the role of the variance-bias trade-off when training under noise. Finally, we illustrate how the number of local minima for networks with two and three hidden layers, but a comparable number of variable edge weights, increases significantly with the number of layers, and as the number of training data decreases. This work helps shed further light on neural network loss landscapes and provides guidance for future work on neural network training and optimisation.

Single Conformation Spectroscopy of Suberoylanilide Hydroxamic Acid: A Molecule Bites Its Tail

Suberoylanilide hydroxamic acid (SAHA) is a histone deacetylase inhibitor that causes growth arrest and differentiation of many tumor types and is an approved drug for the treatment of cancer. The chemical structure of SAHA consists of formanilide "head" and a hydroxamic acid "tail" separated by an n-hexyl chain, C₆H₅NH(C═O)-(CH₂)₆-(C═O)NHOH. The alkyl chain's preference for extended structures is in competition with tail-to-head (T-H) or head-to-tail (H-T) hydrogen bonds between the amide and hydroxamic acid groups. Laser desorption was used to bring SAHA into the gas phase and cool it in a supersonic expansion before interrogation with mass-resolved resonant two-photon ionization spectroscopy. Single conformation UV spectra in the S₀-S₁ region and infrared spectra in the hydride stretch and mid-IR regions were recorded using IR-UV hole-burning and resonant ion-dip infrared spectroscopy, respectively. Three conformers of SAHA were distinguished and spectroscopically characterized. Comparison of the experimental IR spectra with the predictions of density functional theory calculations (DFT, B3LYP D3BJ/6-31+G(d)) leads to assignments for the three conformers, all of which possess tightly folded alkyl chains that enable formation of a T-H (conformer A) or H-T (conformers B and C) hydrogen bonds. A modified version of the generalized Amber force field was developed to more accurately describe the hydroxamic acid OH internal rotor potential, leading to predictions for the relative energies in reasonable agreement with experiment. This force field was used to generate a disconnectivity graph for the low-energy portion of the potential energy landscape of SAHA. This disconnectivity graph contains more than one hundred minima and maps out the lowest-energy pathways between them, which could then be characterized via DFT calculations. This combination of force field and DFT calculations provides insight into the potential energy landscape and how population was funneled into the three observed conformers.

Transition Networks for the Comprehensive Characterization of Complex Conformational Change in Proteins

Functionally relevant transitions between native conformations of a protein can be complex, involving, for example, the reorganization of parts of the backbone fold, and may occur via a multitude of pathways. Such transitions can be characterized by a transition network (TN), in which the experimentally determined end state structures are connected by a dense network of subtransitions via low-energy intermediates. We show here how the computation of a TN can be achieved for a complex protein transition. First, an efficient hierarchical procedure is used to uniformly sample the conformational subspace relevant to the transition. Then, the best path which connects the end states is determined as well as the rate-limiting ridge on the energy surface which separates them. Graph-theoretical algorithms permit this to be achived by computing the barriers of only a small number out of the many subtransitions in the TN. These barriers are computed using the Conjugate Peak Refinement method. The approach is illustrated on the conformational switch of Ras p21. The best and the 12 next-best transition pathways, having rate-limiting barriers within a range of 10 kcal/mol, were identified. Two main energy ridges, which respectively involve rearrangements of the switch I and switch II loops, show that switch I must rearrange by threading Tyr32 underneath the protein backbone before the rate-limiting switch II rearrangement can occur, while the details of the switch II rearrangement differ significantly among the low-energy pathways.

Computer simulations of glasses: the potential energy landscape

We review the current state of research on glasses, discussing the theoretical background and computational models employed to describe them. This article focuses on the use of the potential energy landscape (PEL) paradigm to account for the phenomenology of glassy systems, and the way in which it can be applied in simulations and the interpretation of their results. This article provides a broad overview of the rich phenomenology of glasses, followed by a summary of the theoretical frameworks developed to describe this phenomonology. We discuss the background of the PEL in detail, the onerous task of how to generate computer models of glasses, various methods of analysing numerical simulations, and the literature on the most commonly used model systems. Finally, we tackle the problem of how to distinguish a good glass former from a good crystal former from an analysis of the PEL. In summarising the state of the potential energy landscape picture, we develop the foundations for new theoretical methods that allow the ab initio prediction of the glass-forming ability of new materials by analysis of the PEL.

Folding energy landscape and network dynamics of small globular proteins

The folding energy landscape of proteins has been suggested to be funnel-like with some degree of ruggedness on the slope. How complex the landscape, however, is still rather unclear. Many experiments for globular proteins suggested relative simplicity, whereas molecular simulations of shorter peptides implied more complexity. Here, by using complete conformational sampling of 2 globular proteins, protein G and src SH3 domain and 2 related random peptides, we investigated their energy landscapes, topological properties of folding networks, and folding dynamics. The projected energy surfaces of globular proteins were funneled in the vicinity of the native but also have other quite deep, accessible minima, whereas the randomized peptides have many local basins, including some leading to seriously misfolded forms. Dynamics in the denatured part of the network exhibited basin-hopping itinerancy among many conformations, whereas the protein reached relatively well-defined final stages that led to their native states. We also found that the folding network has the hierarchic nature characterized by the scale-free and the small-world properties.

Exploring the potential energy landscape of glass-forming systems: from inherent structures via metabasins to macroscopic transport

In this review a systematic analysis of the potential energy landscape (PEL) of glass-forming systems is presented. Starting from the thermodynamics, the route towards the dynamics is elucidated. A key step in this endeavor is the concept of metabasins. The relevant energy scales of the PEL can be characterized. Based on the simulation results for some glass-forming systems one can formulate a relevant model system (ideal Gaussian glass-former) which can be treated analytically. The macroscopic transport can be related to the microscopic hopping processes, using either the strong relation between energy (thermodynamics) and waiting times (dynamics) or, alternatively, the concepts of the continuous-time random walk. The relation to the geometric properties of the PEL is stressed. The emergence of length scales within the PEL approach as well as the nature of finite-size effects is discussed. Furthermore, the PEL view is compared to other approaches describing the glass transition.

Second derivatives in generalized Born theory

Generalized Born solvation models offer a popular method of including electrostatic aspects of solvation free energies within an analytical model that depends only upon atomic coordinates, charges, and dielectric radii. Here, we describe how second derivatives with respect to Cartesian coordinates can be computed in an efficient manner that can be distributed over multiple processors. This approach makes possible a variety of new methods of analysis for these implicit solvation models. We illustrate three of these methods here: the use of Newton-Raphson optimization to obtain precise minima in solution; normal mode analysis to compute solvation effects on the mechanical properties of DNA; and the calculation of configurational entropies in the MM/GBSA model. An implementation of these ideas, using the Amber generalized Born model, is available in the nucleic acid builder (NAB) code, and we present examples for proteins with up to 45,000 atoms. The code has been implemented for parallel computers using both the OpenMP and MPI environments, and good parallel scaling is seen with as many as 144 OpenMP processing threads or MPI processing tasks.

Hierarchical analysis of conformational dynamics in biomolecules: transition networks of metastable states

Molecular dynamics simulation generates large quantities of data that must be interpreted using physically meaningful analysis. A common approach is to describe the system dynamics in terms of transitions between coarse partitions of conformational space. In contrast to previous work that partitions the space according to geometric proximity, the authors examine here clustering based on kinetics, merging configurational microstates together so as to identify long-lived, i.e., dynamically metastable, states. As test systems microsecond molecular dynamics simulations of the polyalanines Ala(8) and Ala(12) are analyzed. Both systems clearly exhibit metastability, with some kinetically distinct metastable states being geometrically very similar. Using the backbone torsion rotamer pattern to define the microstates, a definition is obtained of metastable states whose lifetimes considerably exceed the memory associated with interstate dynamics, thus allowing the kinetics to be described by a Markov model. This model is shown to be valid by comparison of its predictions with the kinetics obtained directly from the molecular dynamics simulations. In contrast, clustering based on the hydrogen-bonding pattern fails to identify long-lived metastable states or a reliable Markov model. Finally, an approach is proposed to generate a hierarchical model of networks, each having a different number of metastable states. The model hierarchy yields a qualitative understanding of the multiple time and length scales in the dynamics of biomolecules.

Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation

The generalized master equation or the equivalent continuous time random walk equations can be used to compute the macroscopic first passage time distribution (FPTD) of a complex stochastic system from short-term microscopic simulation data. The computation of the mean first passage time and additional low-order FPTD moments can be simplified by directly relating the FPTD moment generating function to the moments of the local FPTD matrix. This relationship can be physically interpreted in terms of steady-state relaxation, an extension of steady-state flow. Moreover, it is amenable to a statistical error analysis that can be used to significantly increase computational efficiency. The efficiency improvement can be extended to the FPTD itself by modelling it using a gamma distribution or rational function approximation to its Laplace transform.

Characterizing Potential Surface Topographies through the Distribution of Saddles and Minima

Three related clusters of thirteen particles bound by pairwise Morse potentials with different ranges are the vehicles for relating the dynamics and kinetics of these clusters to the topographies of their energy landscapes. The analyses are based on the distributions of minima and saddles, on the asymmetries of the barriers and the kinetics of passage among the energy bands that the distributions of minima display. While all three of the examples are essentially structure-seekers, the extent of this character is clearly related to the range of the potential.

Analyzing energy landscapes for folding model proteins

A new benchmark 20-bead HP model protein sequence (on a square lattice), which has 17 distinct but degenerate global minimum (GM) energy structures, has been studied using a genetic algorithm (GA). The relative probabilities of finding particular GM conformations are determined and related to the theoretical probability of generating these structures using a recoil growth constructor operator. It is found that for longer successful GA runs, the GM probability distribution is generally very different from the constructor probability, as other GA operators have had time to overcome any initial bias in the originally generated population of structures. Structural and metric relationships (e.g., Hamming distances) between the 17 distinct GM are investigated and used, in conjunction with data on the connectivities of the GM and the pathways that link them, to explain the GM probability distributions obtained by the GA. A comparison is made of searches where the sequence is defined in the normal (forward) and reverse directions. The ease of finding mirror image solutions are also compared. Finally, this approach is applied to rationalize the ease or difficulty of finding the GM for a number of standard benchmark HP sequences on the square lattice. It is shown that the relative probabilities of finding particular members of a set of degenerate global minima depend critically on the topography of the energy landscape in the vicinity of the GM, the connections and distances between the GM, and the nature of the operators used in the chosen search method.

Equilibrium/nonequilibrium initial configurations in forward/reverse electron transfer within mixed-metal hemoglobin hybrids

In a protein-protein electron transfer (ET) photocycle, the "forward" ET reaction is initiated with the excited complex, [3DA], in an equilibrium ensemble of configurations, the majority of which exhibit less than the maximal ET matrix element. In contrast, the charge-separated intermediate complex is formed in a nonequilibrium set of configurations with maximal ET matrix elements and would be expected to return to the ground state with the largest rate constant possible unless conformational interconversion first "breaks the connection" and the complex converts to less-reactive substates. According to this analysis, the forward and back ET reactions should show a differential response to viscosity, and the latter could even show an increased rate constant under conditions which suppress departure from the reactive configuration(s). We now report that the viscosity dependences of forward and back ET rate constants for the photocycle within the [alpha2(Zn),beta2(Fe3+N3-)] mixed-metal hemoglobin hybrid at pH 7 show the anticipated behavior: kf decreases as viscosity increases, but, in sharp contrast, kb increases strongly.

Kinetics of model energy landscapes: an approach to complex systems

An idealized potential energy surface (PES), simply a PES-like network of stationary points, is demonstrated to be a useful tool to study kinetic relaxation of complex energy landscapes. Combined with a master equation, we show that if constructed with proper regularity, the kinetics of the PES is easy to predict and understand by carefully examining the eigenmodes of the master equation. By modifying the idealized PES model to make it more and more complicated, we demonstrate a systematic method to study the complex kinetics on large PES. The idealized PES model is used to explore the feasibility and the robustness of statistical sampling of large PES. We develop several sampling strategies, such as the "rough topography sampling" and the "low barrier saddle sampling" in the idealized PES model and find they are also applicable to a realistic PES of the 13-atom Morse cluster with range parameter rho= 6. To measure the robustness of the sampling methods, we compare the eigenvalue spectra, the eigenvector similarity and the relaxation times of the total energy of the full and sample PESs.