Rough Energy Landscape Research Articles

Escaping Kinetic Traps Using Nonreciprocal Interactions.

Kinetic traps are a notorious problem in equilibrium statistical mechanics, where temperature quenches ultimately fail to bring the system to low energy configurations. Using multifarious self-assembly as a model system, we introduce a mechanism to escape kinetic traps by utilizing nonreciprocal interactions between components. Introducing nonequilibrium effects offered by broken action-reaction symmetry in the system pushes the trajectory of the system out of arrested dynamics. The dynamics of the model is studied using tools from the physics of interfaces and defects. Our proposal can find applications in self-assembly, glassy systems, and systems with arrested dynamics to facilitate escape from local minima in rough energy landscapes.

Scaffold Matcher: A CMA-ES based algorithm for identifying hotspot aligned peptidomimetic scaffolds.

The design of protein interaction inhibitors is a promising approach to address aberrant protein interactions that cause disease. One strategy in designing inhibitors is to use peptidomimetic scaffolds that mimic the natural interaction interface. A central challenge in using peptidomimetics as protein interaction inhibitors, however, is determining how best the molecular scaffold aligns to the residues of the interface it is attempting to mimic. Here we present the Scaffold Matcher algorithm that aligns a given molecular scaffold onto hotspot residues from a protein interaction interface. To optimize the degrees of freedom of the molecular scaffold we implement the covariance matrix adaptation evolution strategy (CMA-ES), a state-of-the-art derivative-free optimization algorithm in Rosetta. To evaluate the performance of the CMA-ES, we used 26 peptides from the FlexPepDock Benchmark and compared with three other algorithms in Rosetta, specifically, Rosetta's default minimizer, a Monte Carlo protocol of small backbone perturbations, and a Genetic algorithm. We test the algorithms' performance on their ability to align a molecular scaffold to a series of hotspot residues (i.e., constraints) along native peptides. Of the 4 methods, CMA-ES was able to find the lowest energy conformation for all 26 benchmark peptides. Additionally, as a proof of concept, we apply the Scaffold Match algorithm with CMA-ES to align a peptidomimetic oligooxopiperazine scaffold to the hotspot residues of the substrate of the main protease of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Our implementation of CMA-ES into Rosetta allows for an alternative optimization method to be used on macromolecular modeling problems with rough energy landscapes. Finally, our Scaffold Matcher algorithm allows for the identification of initial conformations of interaction inhibitors that can be further designed and optimized as high-affinity reagents.

Mechanism of sluggish diffusion under rough energy landscape

Mechanism of sluggish diffusion under rough energy landscape

Revealing the crucial role of rough energy landscape on self-diffusion in high-entropy alloys based on machine learning and kinetic Monte Carlo

Most of the outstanding functional and structural performance in high-entropy alloys (HEAs) relates to their sluggish diffusion properties under the rough potential energy landscape (PEL) induced by intrinsic chemical disorder. Due to the highly rugged and multi-dimensional nature of PEL, it is challenging to describe how the diffusion process is controlled by the PEL in HEAs. Here we develop machine learning (ML) models to accurately represent the local atomic environment dependence of PEL in HEAs. By combining the ML model with the kinetic Monte Carlo (kMC) method, we reveal that self-diffusion in HEAs is predominantly governed by the PEL roughness, as characterized by the elemental-specific site energies and migration barriers. Comparisons with previously-proposed simplified models for self-diffusion in HEAs elucidate that the models based on species-averaged migration barriers may be a suitable alternative method to rapidly assess diffusion properties, though the correlation effects may be underestimated. Aided by theoretical analysis, we show that the atomic concentrations of fast-diffusing elements and the differences in the averaged migration barriers for different species are the dominant factors influencing sluggish diffusion in HEAs.

Prediction of Fluorophore Brightness in Designed Mini Fluorescence Activating Proteins

The de novo computational design of proteins with predefined three-dimensional structure is becoming much more routine due to advancements both in force fields and algorithms. However, creating designs with functions beyond folding is more challenging. In that regard, the recent design of small beta barrel proteins that activate the fluorescence of an exogenous small molecule chromophore (DFHBI) is noteworthy. These proteins, termed mini fluorescence activating proteins (mFAPs), have been shown to increase the brightness of the chromophore more than 100-fold upon binding to the designed ligand pocket. The design process created a large library of variants with different brightness levels but gave no rational explanation for why one variant was brighter than another. Here, we use quantum mechanics and molecular dynamics simulations to investigate how molecular flexibility in the ground and excited states influences brightness. We show that the ability of the protein to resist dihedral angle rotation of the chromophore is critical for predicting brightness. Our simulations suggest that the mFAP/DFHBI complex has a rough energy landscape, requiring extensive ground-state sampling to achieve converged predictions of excited-state kinetics. While computationally demanding, this roughness suggests that mFAP protein function can be enhanced by reshaping the energy landscape toward conformations that better resist DFHBI bond rotation.

Resampling schemes in population annealing — numerical results

Population annealing (PA) is a population-based algorithm that is designed for equilibrium simulations of thermodynamic systems with a rough free energy landscape. It is known to be more efficient in doing so than standard Markov chain Monte Carlo alone. The algorithm has a number of parameters that can be fine-tuned to improve performance. While there is some theoretical and numerical work regarding most of these parameters, there appears to be a gap in the literature concerning the role of resampling in PA. Here, we present a numerical comparison of a number of resampling schemes for PA simulations of the 2D Ising model.

Irradiation damage versus lattice distortion in AlNbTiVCrx (x = 0, 0.5, 1) high-entropy alloys from first-principles calculations and irradiation experiments

Severe lattice distortions and rough energy landscapes were found in low-density, refractory AlNbTiVCrx (x = 0, 0.5, 1) high-entropy alloys (HEAs) via first-principles calculations. The corresponding samples were irradiated by 5 MeV Xe ions to fluences of 3.3 × 1014 ions/cm2 at 600 ℃. Fundamental irradiation behaviors were studied using transmission electron microscopy and nanoindentation. No phase decomposition was found after irradiation as well as void. The irradiation-induced dislocation loop size decreases, and the density increases with the increasing Cr content. This finding is attributed to the fact that Cr aggravates lattice distortion and roughens energy landscapes in the AlNbTiVCrx system, which will interrupt the long-distance migration of self-interstitials and delay their accumulation. Nanohardness increments exhibit good consistency with the product of dislocation loop size and density. This work provides an insight into the combination of atomic simulation and irradiation experiments for better understanding of the irradiation behaviors of refractory HEAs, and a possibility to modify defect evolution by regulating the compositional complexity.

Protein conformational transitions explored by a morphing approach based on normal mode analysis in internal coordinates.

Large-scale conformational changes are essential for proteins to function properly. Given that these transition events rarely occur, however, it is challenging to comprehend their underlying mechanisms through experimental and theoretical approaches. In this study, we propose a new computational methodology called internal coordinate normal mode-guided elastic network interpolation (ICONGENI) to predict conformational transition pathways in proteins. Its basic approach is to sample intermediate conformations by interpolating the interatomic distance between two end-point conformations with the degrees of freedom constrained by the low-frequency dynamics afforded by normal mode analysis in internal coordinates. For validation of ICONGENI, it is applied to proteins that undergo open-closed transitions, and the simulation results (i.e., simulated transition pathways) are compared with those of another technique, to demonstrate that ICONGENI can explore highly reliable pathways in terms of thermal and chemical stability. Furthermore, we generate an ensemble of transition pathways through ICONGENI and investigate the possibility of using this method to reveal the transition mechanisms even when there are unknown metastable states on rough energy landscapes.

Glass Dynamics Deep in the Energy Landscape.

When a liquid is cooled, progress down the energy landscape is arrested near the glass transition temperature Tg. In principle, lower energy states can be accessed by waiting for further equilibration, but the rough energy landscape of glasses quickly leads to kinetics on geologically slow time scales below Tg. Over the past decade, progress has been made probing deeper into the energy landscape via several techniques. By looking at bulk and surface diffusion, using layered deposition that promotes equilibration, imaging glass surfaces with faster dynamics below Tg, and optically exciting glasses, experiments have moved into a regime of ultrastable, low energy glasses that was difficult to access in the past. At the same time, both simulations and energy landscape theory based on a random first order transition (RFOT) have tackled systems that include surfaces, optical excitation, and interfacial dynamics. Here we review some of the recent experimental work, and how energy landscape theory illuminates glassy dynamics well below the glass transition temperature by making direct connections between configurational entropy, energy landscape barriers, and the resulting dynamics.

Structural Disorder and Collective Behavior of Two-Dimensional Magnetic Nanostructures.

Structural disorder has been shown to be responsible for profound changes of the interaction-energy landscapes and collective dynamics of two-dimensional (2D) magnetic nanostructures. Weakly-disordered 2D ensembles have a few particularly stable magnetic configurations with large basins of attraction from which the higher-energy metastable configurations are separated by only small downward barriers. In contrast, strongly-disordered ensembles have rough energy landscapes with a large number of low-energy local minima separated by relatively large energy barriers. Consequently, the former show good-structure-seeker behavior with an unhindered relaxation dynamics that is funnelled towards the global minimum, whereas the latter show a time evolution involving multiple time scales and trapping which is reminiscent of glasses. Although these general trends have been clearly established, a detailed assessment of the extent of these effects in specific nanostructure realizations remains elusive. The present study quantifies the disorder-induced changes in the interaction-energy landscape of two-dimensional dipole-coupled magnetic nanoparticles as a function of the magnetic configuration of the ensembles. Representative examples of weakly-disordered square-lattice arrangements, showing good structure-seeker behavior, and of strongly-disordered arrangements, showing spin-glass-like behavior, are considered. The topology of the kinetic networks of metastable magnetic configurations is analyzed. The consequences of disorder on the morphology of the interaction-energy landscapes are revealed by contrasting the corresponding disconnectivity graphs. The correlations between the characteristics of the energy landscapes and the Markovian dynamics of the various magnetic nanostructures are quantified by calculating the field-free relaxation time evolution after either magnetic saturation or thermal quenching and by comparing them with the corresponding averages over a large number of structural arrangements. Common trends and system-specific features are identified and discussed.

Forced deterministic dynamics on a random energy landscape: Implications for the physics of amorphous solids.

The dynamics of supercooled liquids and plastically deformed amorphous solids is known to be dominated by the structure of their rough energy landscapes. Recent experiments and simulations on amorphous solids subjected to oscillatory shear at athermal conditions have shown that for small strain amplitudes these systems reach limit cycles of different periodicities after a transient. However, for larger strain amplitudes the transients become longer and for strain amplitudes exceeding a critical value the system reaches a diffusive steady state. This behavior cannot be explained using the current mean-field models of amorphous plasticity. Here we show that this phenomenology can be described and explained using a simple model of forced dynamics on a multidimensional random energy landscape. In this model, the existence of limit cycles can be ascribed to confinement of the dynamics to a small part of the energy landscape which leads to self-intersection of state-space trajectories and the transition to the diffusive regime for larger forcing amplitudes occurs when the forcing overcomes this confinement.

Sampling from rough energy landscapes

We examine challenges to sampling from Boltzmann distributions associated with multiscale energy landscapes. The multiscale features, or roughness, corresponds to highly oscillatory, but bounded, perturbations of a smooth landscape. Through a combination of numerical experiments and analysis we demonstrate that the performance of Metropolis Adjusted Langevin Algorithm can be severely attenuated as the roughness increases. In contrast, we prove that Random Walk Metropolis is insensitive to such roughness. We also formulate two alternative sampling strategies that incorporate large scale features of the energy landscape, while resisting the impact of fine scale roughness; these also outperform Random Walk Metropolis. Numerical experiments on these landscapes are presented that confirm our predictions. Open questions and numerical challenges are also highlighted.

Nature of excitations and defects in structural glasses

The nature of defects in amorphous materials, analogous to vacancies and dislocations in crystals, remains elusive. Here, we explore their nature in a three-dimensional microscopic model glass-former that describes granular, colloidal, atomic and molecular glasses by changing the temperature and density. We find that all glasses evolve in a very rough energy landscape, with a hierarchy of barrier sizes corresponding to both localized and delocalized excitations. Collective excitations dominate in the jamming regime relevant for granular and colloidal glasses. By moving gradually to larger densities describing atomic and molecular glasses, the system crosses over to a regime dominated by localized defects and relatively simpler landscapes. We quantify the energy and temperature scales associated to these defects and their evolution with density. Our results pave the way to a systematic study of low-temperature physics in a broad range of physical conditions and glassy materials.

Glassy dynamics on networks: local spectra and return probabilities

The slow relaxation and aging of glassy systems can be modelled as a Markov process on a simplified rough energy landscape: energy minima where the system tends to get trapped are taken as nodes of a random network, and the dynamics are governed by the transition rates among these. In this work we consider the case of purely activated dynamics, where the transition rates only depend on the depth of the departing trap. The random connectivity and the disorder in the trap depths make it impossible to solve the model analytically, so we base our analysis on the spectrum of eigenvalues of the master operator. We compute the local density of states for traps with a fixed lifetime by means of the cavity method. This exhibits a power law behaviour in the regime of small relaxation rates , which we rationalize using a simple analytical approximation. In the time domain, we find that the probabilities of return to a starting node have a power law-tail that is determined by the distribution of excursion times . We show that these results arise only by the combination of finite configuration space connectivity and glassy disorder, and interpret them in a simple physical picture dominated by jumps to deep neighbouring traps.

Computing committor functions for the study of rare events using deep learning

The committor function is a central object of study in understanding transitions between metastable states in complex systems. However, computing the committor function for realistic systems at low temperatures is a challenging task due to the curse of dimensionality and the scarcity of transition data. In this paper, we introduce a computational approach that overcomes these issues and achieves good performance on complex benchmark problems with rough energy landscapes or in high dimensions. The new approach combines deep learning, data sampling, and feature engineering techniques. This establishes an alternative practical method for studying rare transition events between metastable states in complex, high dimensional systems.

REVO: Resampling of ensembles by variation optimization.

Conventional molecular dynamics simulations are incapable of sampling many important interactions in biomolecular systems due to their high dimensionality and rough energy landscapes. To observe rare events and calculate transition rates in these systems, enhanced sampling is a necessity. In particular, the study of ligand-protein interactions necessitates a diverse ensemble of protein conformations and transition states, and for many systems, this occurs on prohibitively long time scales. Previous strategies such as WExplore that can be used to determine these types of ensembles are hindered by problems related to the regioning of conformational space. Here, we propose a novel, regionless, enhanced sampling method that is based on the weighted ensemble framework. In this method, a value referred to as “trajectory variation” is optimized after each cycle through cloning and merging operations. This method allows for a more consistent measurement of observables and broader sampling resulting in the efficient exploration of previously unexplored conformations. We demonstrate the performance of this algorithm with the N-dimensional random walk and the unbinding of the trypsin-benzamidine system. The system is analyzed using conformation space networks, the residence time of benzamidine is confirmed, and a new unbinding pathway for the trypsin-benzamidine system is found. We expect that resampling of ensembles by variation optimization will be a useful general tool to broadly explore free energy landscapes.

Replica-Based Protein Structure Sampling Methods II: Advanced Hybrid Solvent TIGER2hs.

In many cases, native states of proteins may be predicted with sufficient accuracy by molecular dynamics simulations (MDSs) with modern force fields. Enhanced sampling methods based on MDS are applied for exploring the phase space of a protein sequence and to overcome barriers on rough conformational energy landscapes. The minimum free energy state is obtained with sampling algorithms providing sufficient convergence and accuracy. A reliable but computationally very expensive method is replica exchange molecular dynamics, with many modifications to this approach presented in the past. Recently, we demonstrated how our temperature intervals with global exchange of replicas hybrid (TIGER2h) solvent sampling algorithm made a good compromise between efficiency and accuracy. There, all states are sampled under full explicit solvent conditions with a freely chosen number of replicas, whereas an implicit solvent is used during the swap decisions. This hybrid method yielded a much better approximation to the agreement with calculations in an explicit solvent than fully implicit solvent simulations. Here, we present an extension of TIGER2h and add a few layers of explicit water molecules around the peptide for the energy calculations, whereas the dynamics in fully explicit water is maintained. We claim that these water layers better reproduce steric effects, the polarization of the solvent, and the resulting reaction field energy than typical implicit solvent models. By investigating the protein-solvent interactions across comprehensive thermodynamic state ensembles, we found a strong conformational dependence of this reaction field energy. All simulations were performed with nanoscale molecular dynamics on two peptides, the α-helical peptide (AAQAA)3 and the β-hairpin peptide HP7. A production-ready TIGER2hs implementation is supplied, approaching the accuracy of full explicit solvent sampling at a fraction of computational resources.

Thermodynamic Properties of Heisenberg Spin Systems

We present the simulation results of magnetic 2D and 3D structures with direct (for both of them) and Dzyaloshinskii-Moriya (DMI) (for 2D lattice) interactions in the frame of the Heisenberg model. We have adapted the multipath Metropolis algorithm for systems with complex types of exchange interactions and rough energy landscapes. We show the temperature behavior of magnetization, energy, and heat capacity, and reveal its critical temperatures and order parameter.

Statistical physics meets biochemistry: Wang-Landau sampling of the HP model of protein folding

We describe a minimalistic model of a protein undergoing folding into its native (ground) state. This HP protein model is composed of a lattice polymer with two types of monomers: hydrophobic (H) and polar (P). The specific sequence of the two types of monomers is often chosen to mimic real proteins. We explain how these kinds of minimalistic models are well suited to study fundamental phenomena of complex biological processes such as protein folding. We then show how Monte Carlo simulations and, in particular, Wang-Landau sampling, together with a properly chosen trial move set, provide a powerful computational means to investigate the rough energy landscape and native state(s) of this intriguingly simple system. We include the computer code which can be used to reproduce the results we discuss or as a template for further research.

Mapping Ligand Binding Landscapes using Weighted Ensembles of Trajectories

Protein-ligand interactions involve many conformational states. To bind the ligand must search across a rough free energy landscape that contains many metastable minima. A map of this “ligand binding landscape” would be useful during drug design, as it contains a set of possible binding modes that could be stabilized by modifications of the ligand. These maps would also include information about pose connectivity, both with each other and with the unbound state, which can provide insight into ligand binding kinetics, a useful parameter for drug design. Unfortunately, creating maps of the ligand binding landscape is a great challenge, as binding and release events typically occur on timescales that are beyond the reach of molecular simulation. The Dickson lab has developed simulation algorithms (e.g. WExplore, REVO) that can broadly and efficiently explore free-energy landscapes and are capable of simulating ligand release pathways that occur on timescales as long as minutes. Importantly, these methods employ only unbiased trajectory segments, allowing for the construction of Markov state models (MSM) and conformation space networks that combine the results of multiple simulations. In this talk I will describe how these methods have revealed new insights into long timescale ligand release pathways, as well as the interconversion pathways between ligand poses. In particular I will focus on the application of WExplore to map the binding landscapes of two bromodomain-inhibitor systems (Brd4-MS436 and Baz2B-ICR7), and the use of Markov state models to predict stable poses. In this work we find that transitions between deeply bound states convert through the unbound state 81% of the time, implying a trial-and-error approach to ligand binding. This finding has consequences for both the way we think about ligand binding processes, and how molecular dynamics should be used in drug design.