Beyond static structures: Putting forth REMD as a tool to solve problems in computational organic chemistry.

In strained monoatomic chains with Lennard-Jones interactions, we revealed a stable static non-homogeneous structure appearing as a result of a certain phase transition. Positions of individual particles in this structure form an exact arithmetic progression whose difference depends on the value of the strain. For N-particle chain, this structure is characterized by one long and N-1 short interatomic distances (bonds). In the vicinity of the static structure, we found discrete breathers of new type which essentially differ from the traditional breathers in the form of Sievers-Takeno and Page modes. It is well known that these modes possess some staggered structures and demonstrate exponential decay of the particle amplitudes from the core to their tails. In contrast to such properties, our breathers are characterised by smooth decay and amplitudes of the particles form approximately a decreasing arithmetic progression. Core of these breathers is located on two particles with long bond in static structure. Our breathers demonstrate soft type of nonlinearity (the frequency decreases with increasing of amplitudes) and they are stable dynamical objects for amplitudes up to 20%-30% of interparticle distance of the strained equidistant chain. For infinitely small amplitudes these breathers tend to the above described static non-homogeneous structure. We studied dependence of their properties on amplitude, strain and the number of particles in the chain. There exist a reason to suppose that the above static and dynamical structures can exist in real monoatomic chains consisting of carbon, boron, and other atoms.

This special issue of PSS is in honor of Thomas Frauenheims 60th birthday. His central theme always was the understanding of what he called real materials properties, meaning systems of a size which makes a comparison of their computed properties with experimental observables meaningful. In the 1990's he started to use a DFT based tight-binding (TB) method, and he very quickly became a driving force in developing the applications and capabilities of this methodology. While being originally an approximation to DFT-LDA, Thomas has realized its formal similarity to tight-binding methods used in solid state physics. These TB methods were known to be quite accurate for the materials classes they have been parameterized for, but the parameterization itself could be quite tedious and time-consuming. The approximate DFT method could be viewed as a simple way of getting TB parameters, therefore Thomas introduced the name Density-Functional Tight-Binding (DFTB) for this methodology. Starting out with studies on complex carbon systems, studying their structure and spectroscopic properties, he quickly became interested in a broad range of materials, from molecules, clusters, surface problems, over voids and vacancies in semi-conductor materials to biological molecules. The atomistic description of nano-scale materials was and is in the center of his interest. DFTB has been the workhorse in all these years, although he has never been limited to a single methodology. His science was always rather problem than methodology driven, therefore he augmented the semi-empirical DFTB calculations at both ends with other methods, by empirical force fields on the one hand or plane wave DFT and even post-Hartree Fock methods, on the other, when very accurate calculations have been required. And of course, the nature of those systems often asked for so called ‘multi-scale’ solutions, i.e. a combination of various methods for the different scales in one methodology, like in the QM/MM approaches. Beside that, however, much of his efforts was directed into the extension of DFTB's capabilities in order to cover various materials properties, as there are, for example, vibrational spectra of molecules and solids, optical properties of molecules and clusters within a DFTB adapted time-dependent DFT (TD-DFT) approach, non-adiabatic molecular dynamics simulations to model pump-probe processes, calculations of hyperfine coupling constants for radicals and magnetic properties of clusters with a spin-polarized extension of DFTB, calculations of scanning probe (STM) images of surfaces and the calculation of electronic transport properties within the non-equilibrium Green's function technique, to mention only some of the applications and methodological extensions of the DFTB. All this of course was not possible with the capabilities of a single work group. One of Thomas' most outstanding qualities is his openness and ability to bring co-operations into a family-like setting. Over the years, he has been closely working with various groups in several countries on different continents. These co-operations contributed significantly to the expertise, which is now assembled in the computational machinery, partly manifested in the DFTB+ program suite, partly in many other pieces of code being available now for DFTB. His attitude was never one of competition but rather of cooperation. We hope that he will continue in this spirit to further inspire materials science research in the future. Gotthard Seifert, Marcus Elstner, and Peter Deák (Guest Editors)

The biological function of proteins is determined not only by their static structures but also by the dynamic properties of their conformational ensembles. Numerous high-accuracy static structure prediction tools have been recently developed based on deep learning; however, there remains a lack of efficient and accurate methods for exploring protein dynamic conformations. Traditionally, studies concerning protein dynamics have relied on molecular dynamics (MD) simulations, which incur significant computational costs for all-atom precision and struggle to adequately sample conformational spaces with high energy barriers. To overcome these limitations, various enhanced sampling techniques have been developed to accelerate sampling in MD. Traditional enhanced sampling approaches like replica exchange molecular dynamics (REMD) and frontier expansion sampling (FEXS) often follow the MD simulation approach and still cost a lot of computational resources and time. Variational autoencoders (VAEs), as a classic deep generative model, are not restricted by potential energy landscapes and can explore conformational spaces more efficiently than traditional methods. However, VAEs often face challenges in generating reasonable conformations for complex proteins, especially intrinsically disordered proteins (IDPs), which limits their application as an enhanced sampling method. In this study, we presented a novel deep learning model (named Phanto-IDP) that utilizes a graph-based encoder to extract protein features and a transformer-based decoder combined with variational sampling to generate highly accurate protein backbones. Ten IDPs and four structured proteins were used to evaluate the sampling ability of Phanto-IDP. The results demonstrate that Phanto-IDP has high fidelity and diversity in the generated conformation ensembles, making it a suitable tool for enhancing the efficiency of MD simulation, generating broader protein conformational space and a continuous protein transition path.

STATIC STRUCTURE, DYNAMIC FORM: AN ANALYSIS OF ELLIOTT CARTER’S CONCERTO FOR ORCHESTRA KLAAS COULEMBIER INTRODUCTION LLIOTT CARTER’S CONCERTO FOR ORCHESTRA is one of his most intriguing, complex, and fascinating compositions. It is a tour de force in the organization and arrangement of different musical materials, in the domains of both pitch and rhythm. Several authors have expressed their admiration for the composition, while acknowledging that it is very difficult to penetrate. In 1989, David Harvey opened his chapter on Carter’s Concerto for Orchestra with the following statement: The Concerto for Orchestra is Carter’s richest and most complex work to date in every respect. A complete account of its material, techniques, and their realisation in the textures of the work is beyond the scope of the present study; indeed, it may be doubted that such a project is at all feasible, given the size of the work, the density of the orchestral writing, the richness of the harmonic elaborations generated by Carter’s intervallic techniques of composition , and the limitations of analysis at the present time.1 E 98 Perspectives of New Music That this work is attractive to analysts is beyond question; the trepidation with which they have approached is, however, not only the result of its multi-layered musical surface, but also, no doubt, stems from the startling number of sketches Carter generated in producing it, conveying the impression that the construction of the composition may be even more puzzling than its form. Nevertheless, to gain a deeper insight into the workings of this music, the sketches appear to be as necessary as they are daunting. Jonathan Bernard concluded his 1983 article in Music Analysis with the remark that it would be impossible to make a fully comprehensive analysis of this composition. Dealing with the huge expanses of Carter’s scores . . . is still not easy. To retrace the steps of a composer who produces thousands of pages of sketches and works for thousands of hours in the course of writing a piece is likely to be a formidable undertaking, to say the least. . . . The prospect of reading, much less writing, a so-called “complete analysis” carried out according to the methods presented here is truly fearsome to contemplate. Eventually, perhaps, someone will have a bright idea that will make everything seem much simpler. Until then we can only have faith in the music, continue to analyse, and hope for the best.2 Rather than try to be that person with the bright idea, I want to contribute to the understanding of this composition by focusing on its overall temporal and dramatic organization. Therefore, in dealing with the more than 3000 pages of sketches that I studied during a weeklong stay at the Paul Sacher Foundation, I have kept my focus here exclusively on rhythmic sketches and temporal calculations. In that respect the following analysis complements existing literature in which pitch organization has often been the focal point. Despite the dependence on sketches, this analysis is not aimed at a mere reconstruction of the compositional process.3 By revealing the intricate relation between the rigid and static background structures and their more supple and subtle surface manifestations, this analysis tries to show which strategies and methods Carter applied to achieve such a compelling dramatic and dynamic musical discourse. THE CONCERTO FOR ORCHESTRA IN LITERATURE: AN OVERVIEW Apart from a chapter in David Harvey’s dissertation,4 there are no exhaustive, start-to-finish analyses of the Concerto for Orchestra. Static Structure, Dynamic Form 99 David Schiff gives a comprehensible overview of the composition in The Music of Elliott Carter.5 Jonathan Bernard has returned to the composition on several occasions, dealing with pitch structures in his article on “spatial sets,”6 or with the relationship between the literary source of the composition in “Poem as Non-Verbal Text.”7 Several publications of the Paul Sacher Foundation also include descriptions of the composition’s elaborate sketch resources available in their collection.8 Scholarly literature on Carter’s music in general has regained new energy in the last decade, particularly since the composer’s centennial celebration in 2008. Recent publications often consolidate important studies (such as Jonathan Bernard...

The emergence of deep learning, particularly AlphaFold, has revolutionized static protein structure prediction, marking a transformative milestone in structural biology. However, protein function is not solely determined by static three-dimensional structures but is fundamentally governed by dynamic transitions between multiple conformational states. This shift from static to multi-state representations is crucial for understanding the mechanistic basis of protein function and regulation. This review outlines the fundamental concepts of protein dynamic conformations, surveys recent computational advances in modeling these dynamics in the post-AlphaFold era, and highlights key challenges, including data limitations, methodological constraints, and evaluation metrics. We also discuss potential strategies to address these challenges and explore future research directions to deepen our understanding of protein dynamics and their functional implications. This work aims to provide insights and perspectives to facilitate the ongoing development of protein conformation studies in the era of artificial intelligence-driven structural biology.

Dynamic hydrogen-bond networks provide proteins with structural plasticity required to translate signals such as ligand binding into a cellular response or to transport ions and larger solutes across membranes and, thus, are of central interest to understand protein reaction mechanisms. Here, we present C-Graphs, an efficient tool with graphical user interface that analyzes data sets of static protein structures or of independent numerical simulations to identify conserved, vs unique, hydrogen bonds and hydrogen-bond networks. For static structures, which may belong to the same protein or to proteins with different sequences, C-Graphs uses a clustering algorithm to identify sites of the hydrogen-bond network where waters are conserved among the structures. Using C-Graphs, we identify an internal protein-water hydrogen-bond network common to static structures of visual rhodopsins and adenosine A2A G protein-coupled receptors (GPCRs). Molecular dynamics simulations of a visual rhodopsin indicate that the conserved hydrogen-bond network from static structure can recruit dynamic hydrogen bonds and extend throughout most of the receptor. We release with this work the code for C-Graphs and its graphical user interface.

The Density-Functional Tight Binding (DFTB) method is a popular semiempirical approximation to Density Functional Theory (DFT). In many cases, DFTB can provide comparable accuracy to DFT at a fraction of the cost, enabling simulations on length and time scales that are unfeasible with first-principles DFT. At the same time (and in contrast to empirical interatomic potentials and force fields), DFTB still offers direct access to electronic properties such as the band structure. These advantages come at the cost of introducing empirical parameters to the method, leading to a reduced transferability compared to true first-principle approaches. Consequently, it would be very useful if the parameter sets could be routinely adjusted for a given project. While fairly robust and transferable parametrization workflows exist for the electronic structure part of DFTB, the so-called repulsive potential Vrep poses a major challenge. In this paper, we propose a machine-learning (ML) approach to fitting Vrep, using Gaussian Process Regression (GPR) to reconstruct Vrep with DFT-DFTB force residues as training data. The use of GPR circumvents the need for nonlinear or global parameter optimization, while at the same time offering arbitrary flexibility in terms of the functional form. We also show that the proposed method can be applied to multiple elements at once, by fitting repulsive potentials for organic molecules containing carbon, hydrogen, and oxygen. Overall, the new approach removes focus from the choice of functional form and parametrization procedure, in favor of a data-driven philosophy.

The Density-Functional Tight Binding (DFTB) method is a popular semiempirical approximation to Density Functional Theory (DFT). In many cases, DFTB can provide comparable accuracy to DFT at a fraction of the cost, enabling simulations on length- and time-scales that are unfeasible with first principles DFT. At the same time (and in contrast to empirical interatomic potentials and force-fields), DFTB still offers direct access to electronic properties such as the band-structure. These advantages come at the cost of introducing empirical parameters to the method, leading to a reduced transferability compared to true first-principle approaches. Consequently, it would be very useful if the parameter-sets could be routinely adjusted for a given project. While fairly robust and transferable parameterization workflows exist for the electronic structure part of DFTB, the so-called repulsive potential Vrep poses a major challenge. In this paper we propose a machine-learning (ML) approach to fitting Vrep, using Gaussian Process Regression (GPR). The use of GPR circumvents the need for non-linear or global parameter optimization, while at the same time offering arbitrary flexibility in terms of the functional form. We also show that the proposed method can be applied to multiple elements at once, by fitting repulsive potentials for organic molecules containing carbon, hydrogen and oxygen. Overall, the new approach removes focus from the choice of functional form and parameterization procedure, in favour of a data-driven philosophy.

β-Hairpin is an essential secondary structure unit of protein. Understanding the formation mechanism of the β hairpin and kinetic stability helps to gain insight into the formation of protein secondary structure as well as the mechanism of protein folding. Replica exchange molecular dynamics (REMD) approach is applied to investigate the folding mechanism of Trpzip4 β-hairpin. The REMD method is more efficient than conventional MD in searching for a representative set of the low energy minima in the existence of a large energy gap between the native state and any of the other possible state. The potential energy, backbone RMSD and solvent-accessible surface area (SASA) present descending trends in the process of REMD at 288 K. Methyl groups (most in threonines) show hydrophobic interactions with SASA decrements. The hydropho- bic interactions and intra hydrogen bond forming drive the folding and maintain the low energy structure. Two low energy structures, β-hairpin and helix-coil conformation, are sampled by REMD. In β-hairpin structure, the hydroxyl group (OH) of Thr49 at β turn is accessible to form hydrogen bonds with carboxyl group (COO - ) or C=O group of Asp46; while the hy- droxyl group (OH) of Thr51 is inclined to interact with water molecule with methyl orienting inwards. As a result, the strong interactions between Asp46 and Thr49 cause a bend at β turn. In helix-coil conformation, hydrophobic interactions play be- tween indole ring and methyl group. One of the zip-in paths of Trpzip4 indicates that, the forming of hydrogen bonds at β turn affects the whole folding process of β hairpin. The forming of hydrogen bonds at β turn takes place first, since it has advantages of distance and strong interactions in both side chains and backbone. Total solvent-accessible surface area de- creases significantly as it folds to the β hairpin structure. REMD method could sample large phase space for the low energy minima, and present scenarios of potential energy distribution on phase space with related variables as hydrophobic SASA, backbone RMSD, and hydrogen bonds for specific conformation. This work sheds some lights on the understanding of β hairpin folding. Keywords β-hairpin; folding; replica exchange molecular dynamics; hydrogen bond; hydrophobic interaction

Knowledge of the structure and dynamics of biomolecules is essential for elucidating the underlying mechanisms of biological processes. Given the stochastic nature of many biological processes, like protein unfolding, it is almost impossible that two independent simulations will generate the exact same sequence of events, which makes direct analysis of simulations difficult. Statistical models like Markov chains, transition networks, etc. help in shedding some light on the mechanistic nature of such processes by predicting long-time dynamics of these systems from short simulations. However, such methods fall short in analyzing trajectories with partial or no temporal information, for example, replica exchange molecular dynamics or Monte Carlo simulations. In this work, we propose a probabilistic algorithm, borrowing concepts from graph theory and machine learning, to extract reactive pathways from molecular trajectories in the absence of temporal data. A suitable vector representation was chosen to represent each frame in the macromolecular trajectory (as a series of interaction and conformational energies), and dimensionality reduction was performed using principal component analysis (PCA). The trajectory was then clustered using a density-based clustering algorithm, where each cluster represents a metastable state on the potential energy surface (PES) of the biomolecule under study. A graph was created with these clusters as nodes with the edges learned using an iterative expectation maximization algorithm. The most reactive path is conceived as the widest path along this graph. We have tested our method on RNA hairpin unfolding trajectory in aqueous urea solution. Our method makes the understanding of the mechanism of unfolding in the RNA hairpin molecule more tractable. As this method does not rely on temporal data, it can be used to analyze trajectories from Monte Carlo sampling techniques and replica exchange molecular dynamics (REMD).

Adsorption of peptides at the interface between a fluid and a solid occurs widely in both nature and applications. Knowing the dominant conformations of adsorbed peptides and the energy barriers between them is of interest for a variety of reasons. Molecular dynamics (MD) simulation is a widely used technique that can yield such understanding. However, the complexity of the energy landscapes of adsorbed peptides means that comprehensive exploration of the energy landscape by MD simulation is challenging. An alternative approach is energy landscape mapping (ELM), which involves the location of stationary points on the potential energy surface, and its analysis to determine, for example, the pathways and energy barriers between them. In the study reported here, a comparison is made between this technique and replica exchange molecular dynamics (REMD) for met-enkephalin adsorbed at the interface between graphite and the gas phase: the first ever direct comparison of these techniques for adsorbed peptides. Both methods yield the dominant adsorbed peptide conformations. Unlike REMD, however, ELM readily allows the identification of the connectivity and energy barriers between the favored conformations, transition paths, and structures between these conformations and the impact of entropy. It also permits the calculation of the constant volume heat capacity although the accuracy of this is limited by the sampling of high-energy minima. Overall, compared to REMD, ELM provides additional insights into the adsorbed peptide system provided sufficient care is taken to ensure that key parts of the landscape are adequately sampled.

Instead of static entropy we assert that the Kolmogorov complexity of a static structure such as a solid is the proper measure of disorder (or chaoticity). A static structure in a surrounding perfectly-random universe acts as an interfering entity which introduces local disruption in randomness. This is modeled by a selection rule R which selects a subsequence of the random input sequence that hits the structure. Through the inequality that relates stochasticity and chaoticity of random binary sequences we maintain that Lin’s notion of stability corresponds to the stability of the frequency of 1s in the selected subsequence. This explains why more complex static structures are less stable. Lin’s third law is represented as the inevitable change that static structure undergo towards conforming to the universe’s perfect randomness.

A new methodology for finding the low-energy structures of transition metal clusters is developed. A two-step strategy of successive density functional tight binding (DFTB) and density functional theory (DFT) investigations is employed. The cluster configuration space is impartially searched for candidate ground-state structures using a new single-parent genetic algorithm [I. Rata et al., Phys. Rev. Lett. 85, 546 (2000)] combined with DFTB. Separate searches are conducted for different total spin states. The ten lowest energy structures for each spin state in DFTB are optimized further at a first-principles level in DFT, yielding the optimal structures and optimal spin states for the clusters. The methodology is applied to investigate the structures of Fe4, Fe7, Fe10, and Fe19 clusters. Our results demonstrate the applicability of DFTB as an efficient tool in generating the possible candidates for the ground state and higher energy structures of iron clusters. Trends in the physical properties of iron clusters are also studied by approximating the structures of iron clusters in the size range n=2–26 by Lennard-Jones-type structures. We find that the magnetic moment of the clusters remains in the vicinity of 3μB/atom over this entire size range.

Prediction of three-dimensional structures and structural flexibilities of wild-type and mutant cytochrome P450 1A2 using molecular dynamics simulations

Complex molecular simulation methods are typically required to calculate the thermodynamic properties of biochemical systems. One example thereof is the thermodynamic profiling of (de)solvation of proteins, which is an essential driving force for protein-ligand and protein-protein binding. The thermodynamic state of water molecules depends on its enthalpic and entropic components; the latter is governed by dynamic properties of the molecule. Here, we developed, to the best of our knowledge, two novel machine learning methods based on deep neural networks that are able to generate the converged thermodynamic state of dynamic water molecules in the heterogeneous protein environment based solely on the information of the static protein structure. The applicability of our machine learning methods to predict the hydration information is demonstrated in two different studies, the qualitative analysis and quantitative prediction of structure-activity relationships, and the prediction of protein-ligand binding modes.