Beyond static structures: protein dynamic conformations modeling in the post-AlphaFold era

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.

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...

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.

Ion channels are proteins that form pores of nanoscopic dimensions in cell membranes. As a consequence of advance in protein crystallography we now know the three-dimensional structures of a number of ion channels. However, X-ray diffraction techniques yield an essentially static (time- and space-averaged) structure of an ion channel, in an environment often somewhat distantly related to that which the protein experiences when in a cell membrane. Thus, additional techniques are required to fully understand the relationship between channel structure and function. Potassium (K) channels (Yellen, 2002) provide an opportunity to explore the relationship between membrane protein structure, dynamics, and function. Furthermore, K channels are of considerable physiological and biomedical interest. They regulate K + ion flux across cell membranes. K channel regulation is accomplished by a conformational change that allows the protein to switch between two alternative (closed vs. open) conformations, a process known as gating. Gating is an inherently dynamic process that cannot be fully characterized by static structures alone. The elucidation of the structures of several K + channels (Mackinnon, 2003;

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.

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.

Structural motions along a reaction pathway hold the secret about how a biological macromolecule functions. If each static structure were considered as a snapshot of the protein molecule in action, a large collection of structures would constitute a multidimensional conformational space of an enormous size. Here I present a joint analysis of hundreds of known structures of human hemoglobin in the Protein Data Bank. By applying singular value decomposition to distance matrices of these structures, I demonstrate that this large collection of structural snapshots, derived under a wide range of experimental conditions, arrange orderly along a reaction pathway. The structural motions along this extensive trajectory, including several helical transformations, arrive at a reverse engineered mechanism of the cooperative machinery (Ren, companion article), and shed light on pathological properties of the abnormal homotetrameric hemoglobins from α-thalassemia. This method of meta-analysis provides a general approach to structural dynamics based on static protein structures in this post genomics era.

Emergence of complex patterns induced by Dynamic Systems implemented in Dynamic Structures (DS) 2

The correlation between structural relaxation and the medium-range structures formed by icosahedral cluster packing in CuZr supercooled metallic glass-forming liquids was studied via molecular dynamics simulations. We find that compared to the amount of icosahedral clusters, the medium-range structures formed by icosahedral cluster packing play more important and controlling roles in the structural relaxation process of the supercooled liquids. The relaxation times of local structures depend exponentially on the connectivity of local structures. Despite the argument whether there is a specific connection between the static structure and dynamic property of glass formers in particle level, it does exist on larger length scale. Our results demonstrate that dynamical heterogeneity is indeed correlated with the medium-range static atomic structures in the supercooled glass-forming liquids.

This study investigates the relations between the shape of hydrologic responses and the dynamic transport properties of channel networks within the framework of random walks on fractal networks, focusing on the shape parameter of Nash model. To this end, we evaluate the static fractal structures and the dynamic transport properties of various channel networks and, then, validate Liu's conjecture (1992) for the shape of hydrologic responses. In the context of random walks on fractal networks, the fractal dimensions of channel networks can directly connect the static structure to the dynamic transport properties of channel networks through Horton's law of drainage composition. It is observed that the peak coordinates of hydrologic responses would have a more intimate relation to the connectivity of channel networks than the conductivity of those. The characteristic times of hydrologic responses also tend to be related to the connectivity of channel networks. Thereby, the shape of hydrologic responses would be expected directly affected by the fractal dimension of channel networks in terms of their static structure, while interpreted a combined result of the conductivity and the connectivity of channel networks in terms of their dynamic transport properties. So, the runoff hydrographs of a river basin could be considered shaped by the fractal dimensions of its channel networks following the linear hydrologic system theory.

Direct synthesis and NMR spectra of [(C 6H 5) 2PCH 2] 2CuB 5H 8: Comments on the solution structure of 2,3-μ-metallopentaboranes

The breakthrough of AlphaFold2 and the publication of AlphaFold DB represent a significant advance in the field of predicting static protein structures. However, AlphaFold2 models tend to represent a single static structure, and multiple-conformation prediction remains a challenge. In this work, we proposed a method named MultiSFold, which uses a distance-based multi-objective evolutionary algorithm to predict multiple conformations. To begin, multiple energy landscapes are constructed using different competing constraints generated by deep learning. Subsequently, an iterative modal exploration and exploitation strategy is designed to sample conformations, incorporating multi-objective optimization, geometric optimization and structural similarity clustering. Finally, the final population is generated using a loop-specific sampling strategy to adjust the spatial orientations. MultiSFold was evaluated against state-of-the-art methods using a benchmark set containing 80 protein targets, each characterized by two representative conformational states. Based on the proposed metric, MultiSFold achieves a remarkable success ratio of 56.25% in predicting multiple conformations, while AlphaFold2 only achieves 10.00%, which may indicate that conformational sampling combined with knowledge gained through deep learning has the potential to generate conformations spanning the range between different conformational states. In addition, MultiSFold was tested on 244 human proteins with low structural accuracy in AlphaFold DB to test whether it could further improve the accuracy of static structures. The experimental results demonstrate the performance of MultiSFold, with a TM-score better than that of AlphaFold2 by 2.97% and RoseTTAFold by 7.72%. The online server is at http://zhanglab-bioinf.com/MultiSFold .

We propose a semantic framework for dynamic systems, which, in a sense, extends the well-known algebraic approach for modelling static data structures to the dynamic case. The framework is based on a new mathematical structure, called a d-oid, consisting of a set of instant structures and a set of dynamic operations. An instant structure is a static structure, e.g. an algebra; a dynamic operation is a transformation of instant structures with an associated point to point map, which allows us to keep track of the transformations of single objects and thus is called a tracking map. By an appropriate notion of morphism, the d-oids over a dynamic signature constitute a category.It is shown that d-oids can model object systems and support an abstract notion of possibly unique object identity; moreover, for a d-oid satisfying an identity preserving condition, there exists an essentially equivalent d-oid where the elements of instant structures are just names.

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

Computational studies of organic systems are frequently limited to static pictures that closely align with textbook style presentations of reaction mechanisms and isomerization processes. Of course, in reality chemical systems are dynamic entities where a multitude of molecular conformations exists on incredibly complex potential energy surfaces (PES). Here, we borrow a computational technique originally conceived to be used in the context of biological simulations, together with empirical force fields, and apply it to organic chemical problems. Replica‐exchange molecular dynamics (REMD) permits thorough exploration of the PES. We combined REMD with density functional tight binding (DFTB), thereby establishing the level of accuracy necessary to analyze small molecular systems. Through the study of four prototypical problems: isomer identification, reaction mechanisms, temperature‐dependent rotational processes, and catalysis, we reveal new insights and chemistry that likely would be missed using static electronic structure computations. The REMD‐DFTB methodology at the heart of this study is powered by i‐PI, which efficiently handles the interface between the DFTB and REMD codes. © 2015 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.