The inherent structure landscape of a protein

Complexity and phase transitions in citation networks: insights from artificial intelligence research

In this study, we analyze the changes over time in the complexity and structure of words used in article titles and the connections between articles in citation networks, focusing on the topic of artificial intelligence (AI) up to 2020. By measuring unpredictability in word usage and changes in the connections between articles, we gain insights into shifts in research focus and diversity of themes. Our investigation reveals correspondence between fluctuations in word complexity and changes in the structure of citation networks, highlighting links between thematic evolution and network dynamics. This approach not only enhances our understanding of scientific progress but also may help in anticipating emerging fields and fostering innovation, providing a quantitative lens for studying scientific domains beyond AI.

Zirconium Coordination Chemistry and Its Role in Optimizing Hydroxymate Chelation: Insights from Molecular Dynamics

In the past decade, there has been a growth in using Zirconium-89 (89Zr) as a radionuclide in nuclear medicine for cancer diagnostic imaging and drug discovery processes. Although one of the most popular chelators for 89Zr, desferrioxamine (DFO) is typically presented as a hexadentate ligand, our work suggests a different scenario. The coordination structure of the Zr4+-DFO complex has primarily been informed by DFT-based calculations, which typically ignore temperature and therefore entropic and dynamic solvent effects. In this work, free energy calculations using molecular dynamics simulations, where the conformational fluctuations of both the ligand and the solvent are explicitly included, are used to compare the binding of Zr4+ cations with two different chelators, DFO and 4HMS, the latter of which is an octadentate ligand that has been recently proposed as a better chelator due to the presence of four hydroxymate groups. We find that thermally induced disorder leads to an open hexadentate chelate structure of the Zr4+-DFO complex, leaving the Zr4+ metal exposed to the solvent. A stable coordination of Zr4+ with 4HMS, however, is formed by involving both hydroxamate groups and water molecules in a more closely packed structure.

Understanding temperature-modulated calorimetry through studies of a model system

Temperature modulated calorimetry is widely used but still raises some fundamental questions. In this paper we study a model system as a test sample to address some of them. The model has a nontrivial spectrum of relaxation times. We investigate temperature-modulated calorimetry at constant average temperature to precise the meaning of the frequency-dependent heat capacity, its relation with entropy production, and how such measurements can observe the aging of a glassy sample leading to a time-dependent heat capacity. The study of the Kovacs effect for an out-of-equilibrium system shows how temperature-modulated calorimetry could contribute to the understanding of this memory effect. Then we compare measurements of standard scanning calorimetry and temperature-modulated calorimetry and show how the two methods are complementary because they do not observe the same features. While it can probe the timescales of energy transfers in a system, even in the limit of low-frequency temperature-modulated calorimetry does not probe some relaxation phenomena which can be measured by scanning calorimetry, as suggested by experiments with glasses.

In-Depth Molecular Dynamics Study of All Possible Chondroitin Sulfate Disaccharides Reveals Key Insight into Structural Heterogeneity and Dynamism

GAGs exhibit a high level of conformational and configurational diversity, which remains untapped in terms of the recognition and modulation of proteins. Although GAGs are suggested to bind to more than 800 biologically important proteins, very few therapeutics have been designed or discovered so far. A key challenge is the inability to identify, understand and predict distinct topologies accessed by GAGs, which may help design novel protein-binding GAG sequences. Recent studies on chondroitin sulfate (CS), a key member of the GAG family, pinpointing its role in multiple biological functions led us to study the conformational dynamism of CS building blocks using molecular dynamics (MD). In the present study, we used the all-atom GLYCAM06 force field for the first time to explore the conformational space of all possible CS building blocks. Each of the 16 disaccharides was solvated in a TIP3P water box with an appropriate number of counter ions followed by equilibration and a production run. We analyzed the MD trajectories for torsional space, inter- and intra-molecular H-bonding, bridging water, conformational spread and energy landscapes. An in-house phi and psi probability density analysis showed that 1→3-linked sequences were more flexible than 1→4-linked sequences. More specifically, phi and psi regions for 1→4-linked sequences were held within a narrower range because of intra-molecular H-bonding between the GalNAc O5 atom and GlcA O3 atom, irrespective of sulfation pattern. In contrast, no such intra-molecular interaction arose for 1→3-linked sequences. Further, the stability of 1→4-linked sequences also arose from inter-molecular interactions involving bridged water molecules. The energy landscape for both classes of CS disaccharides demonstrated increased ruggedness as the level of sulfation increased. The results show that CS building blocks present distinct conformational dynamism that offers the high possibility of unique electrostatic surfaces for protein recognition. The fundamental results presented here will support the development of algorithms that help to design longer CS chains for protein recognition.

Energy landscapes of perfect and defective solids: from structure prediction to ion conduction

The energy landscape concept is increasingly valuable in understanding and unifying the structural, thermodynamic and dynamic properties of inorganic solids. We present a range of examples which include (i) structure prediction of new bulk phases including carbon nitrides, phosphorus carbides, LiMgF3 and low-density, ultra-flexible polymorphs of B2O3, (ii) prediction of graphene and related forms of ZnO, ZnS and other compounds which crystallise in the bulk with the wurtzite structure, (iii) solid solutions, (iv) understanding grossly non-stoichiometric oxides including the superionic phases of δ-Bi2O3 and BIMEVOX and the consequences for the mechanisms of ion transport in these fast ion conductors. In general, examination of the energy landscapes of disordered materials highlights the importance of local structural environments, rather than sole consideration of the average structure.

Memory effects in glasses: Insights into the thermodynamics of out-of-equilibrium systems revealed by a simple model of the Kovacs effect

Glasses are interesting materials because they allow us to explore the puzzling properties of out-of-equilibrium systems. One of them is the Kovacs effect in which a glass, brought to an out-of-equilibrium state in which all its thermodynamic variables are identical to those of an equilibrium state, nevertheless evolves, showing a hump in some global variable before the thermodynamic variables come back to their starting point. We show that a simple three-state system is sufficient to study this phenomenon using numerical integrations and exact analytical calculations. It also brings some light on the concept of fictive temperature, often used to extend standard thermodynamics to the out-of-equilibrium properties of glasses. We confirm that the concept of a unique fictive temperature is not valid, an show it can be extended to make a connection with the various relaxation processes in the system. The model also brings further insights on the thermodynamics of out-of-equilibrium systems. Moreover, we show that the three-state model is able to describe various effects observed in glasses such as the asymmetric relaxation to equilibrium discussed by Kovacs, or the reverse crossover measured on B_{2}O_{3}.

Inverse design of self-assembling colloidal crystals with omnidirectional photonic bandgaps

Open colloidal lattices possessing omnidirectional photonic bandgaps in the visible or near-visible regime are attractive optical materials the realization of which has remained elusive. We report the use of an inverse design strategy termed landscape engineering that rationally sculpts the free energy self-assembly landscape using evolutionary algorithms to discover anisotropic patchy colloids capable of spontaneously assembling pyrochlore and cubic diamond lattices possessing complete photonic bandgaps. We validate the designs in computer simulations to demonstrate the defect-free formation of these lattices via a two-stage hierarchical assembly mechanism. Our approach demonstrates a principled strategy for the inverse design of self-assembling colloids for the bottom-up fabrication of desired crystal lattices.

In silico Design of Laccase Thermostable Mutants From Lacc 6 of Pleurotus Ostreatus

Fungal laccase enzymes have a great biotechnological potential for bioremediation processes due to their ability to degrade compounds such as ρ-diphenol, aminophenols, polyphenols, polyamines, and aryldiamines. These enzymes have activity at different pH and temperature values, however, high temperatures can cause partial or total loss of enzymatic activity, so it is appropriate to do research to modify their secondary and/or tertiary structure to make them more resistant to extreme temperature conditions. In silico, a structure of the Lacc 6 enzyme of Pleurotus ostreatus was constructed using a laccase of Trametes versicolor as a template. From this structure, 16 mutants with possible resistance at high temperature due to ionic interactions, salt bridges and disulfide bonds were also obtained in silico. It was determined that 12 mutants called 4-DB, 3-DB, D233C-T310C, F468P, 3-SB, L132T, N79D, N372D, P203C, P203V, T147E, and W85F, presented the lowest thermodynamic energy. Based on the previous criterion and determining the least flexibility in the protein structures, three mutants (4-DB, 3-DB, and P203C) were selected, which may present high stability at high temperatures without affecting their active site. The obtained results allow the understanding of the molecular base that increase the structural stability of the enzyme Lacc 6 of Pleurotus ostreatus, achieving the in silico generation of mutants, which could have activity at high temperatures.

Nonnative Energetic Frustrations in Protein Folding at Residual Level: A Simulation Study of Homologous Immunoglobulin-like β-Sandwich Proteins

Nonnative interactions cause energetic frustrations in protein folding and were found to dominate key events in folding intermediates. However, systematically characterizing energetic frustrations that are caused by nonnative intra-residue interactions at residual resolution is still lacking. Recently, we studied the folding of a set of homologous all-α proteins and found that nonnative-contact-based energetic frustrations are highly correlated to topology of the protein native-contact network. Here, we studied the folding of nine homologous immunoglobulin-like (Ig-like) β-sandwich proteins, and examined nonnative-contact-based energetic frustrations Gō-like model. Our calculations showed that nonnative-interaction-based energetic frustrations in β-sandwich proteins are much more complicated than those in all-α proteins, and they exhibit highly heterogeneous effects on the folding of secondary structures. Further, the nonnative interactions introduced distinct correlations in the folding of different folding-patches of β-sandwich proteins. Taken together, a strong interplay might exist between nonnative-interaction energetic frustrations and the protein native-contact networks, which ensures that β-sandwich domains adopt a common folding mechanism.

Ergodicity breaking of iron displacement in heme proteins

We present a model of the dynamical transition of atomic displacements in proteins. Increased mean-square displacement at higher temperatures is caused by the softening of the force constant for atomic/molecular displacements by electrostatic and van der Waals forces from the protein-water thermal bath. Displacement softening passes through a nonergodic dynamical transition when the relaxation time of the force-force correlation function enters, with increasing temperature, the instrumental observation window. Two crossover temperatures are identified. The lower crossover, presently connected to the glass transition, is related to the dynamical unfreezing of rotations of water molecules within nanodomains polarized by charged surface residues of the protein. The higher crossover temperature, usually assigned to the dynamical transition, marks the onset of water translations. All crossovers are ergodicity breaking transitions depending on the corresponding observation windows. Allowing stretched exponential relaxation of the protein-water thermal bath significantly improves the theory-experiment agreement when applied to solid protein samples studied by Mössbauer spectroscopy.

Golf-course and funnel energy landscapes: Protein folding concepts in martensites

We use protein folding energy landscape concepts such as golf course and funnel to study re-equilibration in athermal martensites under systematic temperature quench Monte Carlo simulations. On quenching below a transition temperature, the seeded high-symmetry parent-phase austenite that converts to the low-symmetry product-phase martensite, through autocatalytic twinning or elastic photocopying, has both rapid conversions and incubation delays in the temperature-time-transformation phase diagram. We find the rapid (incubation delays) conversions at low (high) temperatures arises from the presence of large (small) size of golf-course edge that has the funnel inside for negative energy states. In the incubating state, the strain structure factor enters into the Brillouin-zone golf course through searches for finite transitional pathways which close off at the transition temperature with Vogel-Fulcher divergences that are insensitive to Hamiltonian energy scales and log-normal distributions, as signatures of dominant entropy barriers. The crossing of the entropy barrier is identified through energy occupancy distributions, Monte Carlo acceptance fractions, heat emission, and internal work.

Inherent structure versus geometric metric for state space discretization

Inherent structure (IS) and geometry-based clustering methods are commonly used for analyzing molecular dynamics trajectories. ISs are obtained by minimizing the sampled conformations into local minima on potential/effective energy surface. The conformations that are minimized into the same energy basin belong to one cluster. We investigate the influence of the applications of these two methods of trajectory decomposition on our understanding of the thermodynamics and kinetics of alanine tetrapeptide. We find that at the microcluster level, the IS approach and root-mean-square deviation (RMSD)-based clustering method give totally different results. Depending on the local features of energy landscape, the conformations with close RMSDs can be minimized into different minima, while the conformations with large RMSDs could be minimized into the same basin. However, the relaxation timescales calculated based on the transition matrices built from the microclusters are similar. The discrepancy at the microcluster level leads to different macroclusters. Although the dynamic models established through both clustering methods are validated approximately Markovian, the IS approach seems to give a meaningful state space discretization at the macrocluster level in terms of conformational features and kinetics.

Energetic frustrations in protein folding at residue resolution: a homologous simulation study of Im9 proteins

Energetic frustration is becoming an important topic for understanding the mechanisms of protein folding, which is a long-standing big biological problem usually investigated by the free energy landscape theory. Despite the significant advances in probing the effects of folding frustrations on the overall features of protein folding pathways and folding intermediates, detailed characterizations of folding frustrations at an atomic or residue level are still lacking. In addition, how and to what extent folding frustrations interact with protein topology in determining folding mechanisms remains unclear. In this paper, we tried to understand energetic frustrations in the context of protein topology structures or native-contact networks by comparing the energetic frustrations of five homologous Im9 alpha-helix proteins that share very similar topology structures but have a single hydrophilic-to-hydrophobic mutual mutation. The folding simulations were performed using a coarse-grained Gō-like model, while non-native hydrophobic interactions were introduced as energetic frustrations using a Lennard-Jones potential function. Energetic frustrations were then examined at residue level based on φ-value analyses of the transition state ensemble structures and mapped back to native-contact networks. Our calculations show that energetic frustrations have highly heterogeneous influences on the folding of the four helices of the examined structures depending on the local environment of the frustration centers. Also, the closer the introduced frustration is to the center of the native-contact network, the larger the changes in the protein folding. Our findings add a new dimension to the understanding of protein folding the topology determination in that energetic frustrations works closely with native-contact networks to affect the protein folding.

Characterization of the low-temperature properties of a simplified protein model

Prompted by results that showed that a simple protein model, the frustrated Gō model, appears to exhibit a transition reminiscent of the protein dynamical transition, we examine the validity of this model to describe the low-temperature properties of proteins. First, we examine equilibrium fluctuations. We calculate its incoherent neutron-scattering structure factor and show that it can be well described by a theory using the one-phonon approximation. By performing an inherent structure analysis, we assess the transitions among energy states at low temperatures. Then, we examine nonequilibrium fluctuations after a sudden cooling of the protein. We investigate the violation of the fluctuation-dissipation theorem in order to analyze the protein glass transition. We find that the effective temperature of the quenched protein deviates from the temperature of the thermostat, however it relaxes towards the actual temperature with an Arrhenius behavior as the waiting time increases. These results of the equilibrium and nonequilibrium studies converge to the conclusion that the apparent dynamical transition of this coarse-grained model cannot be attributed to a glassy behavior.

An energetic model for macromolecules unfolding in stretching experiments

We propose a simple approach, based on the minimization of the total (entropic plus unfolding) energy of a two-state system, to describe the unfolding of multi-domain macromolecules (proteins, silks, polysaccharides, nanopolymers). The model is fully analytical and enlightens the role of the different energetic components regulating the unfolding evolution. As an explicit example, we compare the analytical results with a titin atomic force microscopy stretch-induced unfolding experiment showing the ability of the model to quantitatively reproduce the experimental behaviour. In the thermodynamic limit, the sawtooth force-elongation unfolding curve degenerates to a constant force unfolding plateau.

Protein inherent structures by different minimization strategies

Network-based methods provide an accurate description of the free-energy landscape of peptides and proteins sampled by molecular dynamics simulations. To that end, it is necessary to group the individual snapshots in a meaningful way. The inherent structures (ISs) provide an appropriate discretization of the trajectory into microstates, avoiding problems that can arise in clustering algorithms that have been used previously. In this work, different minimization protocols to obtain the IS of a peptide are investigated on the basis of cut-based free-energy profiles. It is found that a computationally more efficient quasi-Newtonian algorithm provides quantitative agreement to the classical conjugate gradient method in terms of the population of the peptide substates and the energy barriers separating them. That is, despite the fact that the two algorithms can occasionally quench a given peptide snapshot in different potential energy minima, the overall properties of the system are not affected. As reported by others, atom permutations affect the calculation of the IS, requiring an improved implementation of current potential energy functions.

Protein dynamics investigated by inherent structure analysis

Molecular dynamics (MD) simulations provide essential information about the thermodynamics and dynamics of proteins. To construct the free-energy surface from equilibrium trajectories, it is necessary to group the individual snapshots in a meaningful way. The inherent structures (IS) are shown to provide an appropriate discretization of the trajectory and to avoid problems that can arise in clustering algorithms that have been employed previously. The IS-based approach is illustrated with a 30-ns room temperature "native" state MD simulation of a 10-residue peptide in a beta-hairpin conformation. The transitions between the IS are used to construct a configuration space network from which a one-dimensional free-energy profile is obtained with the mincut method. The results demonstrate that the IS approach is useful and that even for this simple system, there exists a nontrivial organization of the native state into several valleys separated by barriers as high as 3 kcal/mol. Further, by introducing a coarse-grained network, it is demonstrated that there are multiple pathways connecting the valleys. This scenario is hidden when the snapshots of the trajectory are used directly with rmsd clustering to compute the free-energy profile. Application of the IS approach to the native state of the PDZ2 signaling domain indicates its utility for the study of biologically relevant systems.

Critical examination of the inherent-structure-landscape analysis of two-state folding proteins

Recent studies attracted the attention on the inherent-structure-landscape (ISL) approach as a reduced description of proteins allowing to map their full thermodynamic properties. However, the analysis has been so far limited to a single topology of a two-state folding protein, and the simplifying assumptions of the method have not been examined. In this work, we construct the thermodynamics of four two-state folding proteins of different sizes and secondary structure by molecular dynamics (MD) simulations using the ISL method and critically examine possible limitations of the method. Our results show that the ISL approach correctly describes the thermodynamics function, such as the specific heat, on a qualitative level. Using both analytical and numerical methods, we show that some quantitative limitations cannot be overcome with enhanced sampling or the inclusion of harmonic corrections.

Computational protein design with side-chain conformational entropy

Recent advances in modeling protein structures at the atomic level have made it possible to tackle "de novo" computational protein design. Most procedures are based on combinatorial optimization using a scoring function that estimates the folding free energy of a protein sequence on a given main-chain structure. However, the computation of the conformational entropy in the folded state is generally an intractable problem, and its contribution to the free energy is not properly evaluated. In this article, we propose a new automated protein design methodology that incorporates such conformational entropy based on statistical mechanics principles. We define the free energy of a protein sequence by the corresponding partition function over rotamer states. The free energy is written in variational form in a pairwise approximation and minimized using the Belief Propagation algorithm. In this way, a free energy is associated to each amino acid sequence: we use this insight to rescore the results obtained with a standard minimization method, with the energy as the cost function. Then, we set up a design method that directly uses the free energy as a cost function in combination with a stochastic search in the sequence space. We validate the methods on the design of three superficial sites of a small SH3 domain, and then apply them to the complete redesign of 27 proteins. Our results indicate that accounting for entropic contribution in the score function affects the outcome in a highly nontrivial way, and might improve current computational design techniques based on protein stability.

Free-energy landscape of mechanically unfolded model proteins: extended Jarzinsky versus inherent structure reconstruction

The equilibrium free-energy landscape of off-lattice model heteropolymers as a function of an internal coordinate, namely the end-to-end distance, is reconstructed from out-of-equilibrium steered molecular dynamics data. This task is accomplished via two independent methods: By employing an extended version of the Jarzynski equality and the inherent structure formalism. A comparison of the free energies estimated with these two schemes with equilibrium results obtained via the umbrella sampling technique reveals a good quantitative agreement among all the approaches in a range of temperatures around the "folding transition" for the two examined sequences. In particular, for the sequence with good foldability properties, the mechanically induced structural transitions can be related to thermodynamical aspects of folding. Moreover, for the same sequence the knowledge of the landscape profile allows for a good estimation of the lifetimes of the native configuration for temperatures ranging from the folding to the collapse temperature. For the random sequence, mechanical and thermal unfolding appear to follow different paths along the landscape.

Geometry of the energy landscape and folding transition in a simple model of a protein

A geometric analysis of the global properties of the energy landscape of a minimalistic model of a polypeptide is presented, which is based on the relation between dynamical trajectories and geodesics of a suitable manifold, whose metric is completely determined by the potential energy. We consider different sequences, some with a definite proteinlike behavior, a unique native state and a folding transition, and others undergoing a hydrophobic collapse with no tendency to a unique native state. The global geometry of the energy landscape appears to contain relevant information on the behavior of the various sequences: in particular, the fluctuations of the curvature of the energy landscape, measured by means of numerical simulations, clearly mark the folding transition and allow the proteinlike sequences to be distinguished from the others.

Hidden structure in protein energy landscapes

Inherent structure theory is used to discover strong connections between simple characteristics of protein structure and the energy landscape of a Gō model. The potential energies and vibrational free energies of inherent structures are highly correlated, and both reflect simple measures of networks of native contacts. These connections have important consequences for models of protein dynamics and thermodynamics.

Reconstructing the free-energy landscape of a mechanically unfolded model protein

The equilibrium free-energy landscape of an off-lattice model protein as a function of an internal (reaction) coordinate is reconstructed from out-of-equilibrium mechanical unfolding manipulations. This task is accomplished via two independent methods: by employing an extended version of the Jarzynski equality (EJE) and the protein inherent structures (ISs). In a range of temperatures around the "folding transition" we find a good quantitative agreement between the free energies obtained via EJE and IS approaches. This indicates that the two methodologies are consistent and able to reproduce equilibrium properties of the examined system. Moreover, for the studied model the structural transitions induced by pulling can be related to thermodynamical aspects of folding.

Violation of the fluctuation-dissipation theorem in a protein system

We report the results of molecular dynamics simulations of the protein myosin carried out with an elastic network model. Quenching the system, we observe glassy behavior of a density correlation function and a density response function that are often investigated in structure glasses and spin glasses. In the equilibrium, the fluctuation-response relation, a representative relation of the fluctuation-dissipation theorem, holds that the ratio of the density correlation function to the density response function is equal to the temperature of the environment. We show that, in the quenched system that we study, this relation can be violated. In the case that this relation does not hold, this ratio can be regarded as an effective temperature. We find that this effective temperature of myosin is higher than the temperature of the environment. We discuss the relation between this effective temperature and energy transduction that occurs after ATP hydrolysis in the myosin molecule.

Conformational temperature characterizing the folding of a protein

The time sequences of the molecular dynamics simulation for the folding process of a protein is analyzed with the inherent structure landscape which focuses on the configurational dynamics of the system. Time-dependent energy and entropy for inherent structures are introduced, and from these quantities a conformational temperature is defined. The conformational temperature follows the time evolution of a slow relaxation process and reaches the bath temperature when the system is equilibrated. We show that the nonequilibrium system is described by two temperatures, one for fast vibration and the other for slow configurational relaxation, while the equilibrium system is described by one temperature. The proposed formalism is applicable widely for systems with many metastable states.