In this paper we analyze the vibrational spectra of a large ensemble of non-homologous protein structures by means of a novel tool, that we coin the Hierarchical Network Model (HNM). Our coarse-grained scheme accounts for the intrinsic heterogeneity of force constants displayed by protein arrangements and also incorporates side-chain degrees of freedom.
Our analysis shows that vibrational entropy per unit residue correlates with the content of secondary structure. Furthermore, we assess the individual contribution to vibrational entropy of the novel features of our scheme as compared with the predictions of state-of-the-art network models. This analysis highlights the importance of properly accounting for the intrinsic hierarchy in force strengths typical of the different atomic bonds that build up and stabilize protein scaffolds.
Finally, we discuss possible implications of our findings in the context of protein aggregation phenomena.
An overview of theories related to vibrational energy relaxation (VER) in proteins is presented. VER of a selected mode in cytochrome c is studied using two theoretical approaches. One is the equilibrium simulation approach with quantum correction factors, and the other is the reduced model approach which describes the protein as an ensemble of normal modes interacting through nonlinear coupling elements. Both methods result in estimates of the VER time (sub ps) for a CD stretching mode in the protein at room temperature. The theoretical predictions are in accord with the experimental data of …
When the hydration shell of a protein is filled with at least 0.6 gram of water per gram of protein, a significant anti-correlation between the vibrational free energy and the potential energy of energy-minimized conformers is observed. This means that low potential energy, well-hydrated, protein conformers tend to be more rigid than high-energy ones. On the other hand, in the case of CASP target 624, when its hydration shell is filled, a significant average energy gap is observed between the crystal structure and the best conformers proposed during the prediction experiment, strongly suggesti…
Vibrational energy relaxation (VER) of a selected mode in cytochrome c (hemeprotein) in vacuum is studied using two theoretical approaches: One is the equilibrium simulation approach with quantum correction factors, and the other is the reduced model approach which describes the protein as an ensemble of normal modes coupled with nonlinear coupling elements. Both methods result in estimates of VER time (sub ps) for a CD stretching mode in the protein at room temperature, that are in accord with the experimental data of Romesberg's group. The applicability of the two methods is examined through…
Without invoking the Markov approximation, we derive formulas for vibrational energy relaxation (VER) and dephasing for an anharmonic system oscillator using a time-dependent perturbation theory. The system-bath Hamiltonian contains more than the third order coupling terms since we take a normal mode picture as a zeroth order approximation. When we invoke the Markov approximation, our theory reduces to the Maradudin-Fein formula which is used to describe VER properties of glass and proteins. When the system anharmonicity and the renormalization effect due to the environment vanishes, our formu…
Despite the recognized importance of the multi-scale spatio-temporal organization of proteins, most computational tools can only access a limited spectrum of time and spatial scales, thereby ignoring the effects on protein behavior of the intricate coupling between the different scales. Starting from a physico-chemical atomistic network of interactions that encodes the structure of the protein, we introduce a methodology based on multi-scale graph partitioning that can uncover partitions and levels of organization of proteins that span the whole range of scales, revealing biological features o…
Previous studies of the flexibilities of ancestral proteins suggests that proteins evolve their function by altering their native state ensemble. Here we propose a more direct method of visualizing this by measuring the changes in the vibrational density of states (VDOS) of proteins as they evolve. Through analysis of VDOS profiles of ancestral and extant proteins we observe that $β$-lactamase and thioredoxins evolve by altering their density of states in the terahertz region. Particularly, the shift in VDOS profiles between ancestral and extant proteins suggests that nature utilize dynamic al…
Recent experimental findings on a protein showed the diffusion of vibrational energy does not occur along the backbone interaction,as it might be expected, but prevalently on non-bonded contacts. These results are explained presenting a theoretical picture, supported by computational calculations, that accounts for these different behaviors in vibrational energy exchange process showing the collective motions on the backbone present a $superlocalized$ nature as their decay with the distance $r$ is $exp(-r^{d})$ with $d \sim 1.8$, whereas those associated to non-bonded contacts result simply lo…
It is shown that the density of modes of the vibrational spectrum of globular proteins is universal, i.e., regardless of the protein in question it closely follows one universal curve. The present study, including 135 proteins analyzed with a full atomic empirical potential (CHARMM22) and using the full complement of all atoms Cartesian degrees of freedom, goes far beyond previous claims of universality, confirming that universality holds even in the high-frequency range (300- 4000 1/cm), where peaks and turns in the density of states are faithfully reproduced from one protein to the next. We …
Following the ideas of Davydov's soliton theory, we study the bio-energy transport in protein molecules. By using a quantum Brownian motion model for a phonon dressed vibrational exciton, we calculate the time-dependence on the mean square distance, diffusion coefficient, and energy of the vibrational exciton. We find the time-dependence by solving the quantum Langevin equation and find oscillatory behaviors due to the super-diffusive non-ohmic dissipation. We find that the vibrational exciton gains an overall energy due to the coupling to the phonon bath; it also dissipates its energy to the …
We have extended our analytically derived PDB-NMA formulation, ATMAN [1], to include protein dimers using mixed internal and Cartesian coordinates. A test case on a 1.3Å resolution model of a small homodimer, ActVA-ORF6, consisting of two 112-residue subunits identically folded in a compact 50Å sphere, reproduces the distinct experimental Debye-Waller motility asymmetry for the two chains, demonstrating that structure sensitively selects vibrational signatures. The vibrational analysis of this PDB entry, together with biochemical and crystallographic data, demonstrates the cooperative nature o…
Using a time-dependent perturbation theory, vibrational energy relaxation (VER) of isotopically labeled amide I modes in cytochrome c solvated with water is investigated. Contributions to the VER are decomposed into two contributions from the protein and water. The VER pathways are visualized using radial and angular excitation functions for resonant normal modes. Key differences of VER among different amide I modes are demonstrated, leading to a detailed picture of the spatial anisotropy of the VER. The results support the experimental observation that amide I modes in proteins relax with sub…
Inherent structure theory is used to discover strong connections between simple characteristics of protein structure and the energy landscape of a Go 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.
Current all-atom potential based molecular dynamics (MD) allow the identification of a protein's functional motions on a wide-range of time-scales, up to few tens of ns. However, functional large scale motions of proteins may occur on a time-scale currently not accessible by all-atom potential based molecular dynamics. To avoid the massive computational effort required by this approach several simplified schemes have been introduced. One of the most satisfactory is the Gaussian Network approach based on the energy expansion in terms of the deviation of the protein backbone from its native conf…
We present a study on the near equilibrium dynamics of two small proteins in the family of truncated hemoglobins, developed under the framework of a Gaussian network approach. Effective beta carbon atoms are taken into account besides C-alphas for all residues but glycines in the coarse-graining procedure, without leading to an increase in the degrees of freedom (betaGaussian Model). Normalized covariance matrix and deformation along slowest modes with collective character are analyzed, pointing out anti-correlations between functionally relevant sites for the proteins under study. In particul…
Organization and maintenance of the chromosomal DNA in living cells strongly depends on the DNA interactions with a plethora of DNA-binding proteins. Single-molecule studies show that formation of nucleoprotein complexes on DNA by such proteins is frequently subject to force and torque constraints applied to the DNA. Although the existing experimental techniques allow to exert these type of mechanical constraints on individual DNA biopolymers, their exact effects in regulation of DNA-protein interactions are still not completely understood due to the lack of systematic theoretical methods able…
We present a model of the dynamical transition of atomic displacements in proteins. Increased mean-square displacement at higher temperatures is caused by softening of the vibrational force constant by electrostatic and van der Waals forces from the protein-water thermal bath. Vibrational 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 …
We used statistical thermodynamics of conformational fluctuations and the elements of algebraic graph theory together with data from 2000 protein crystal structures, and showed that folded native proteins with harmonic interactions exhibit distribution functions each of which appear to be universal across all proteins. The three universal distributions are: (i) the eigenvalue spectrum of the protein graph Laplacian, (ii) the B-factor distribution of residues, and (iii) the vibrational entropy difference per residue between the unfolded and the folded states. The three distributions, which look…
Genome-wide binding preferences of the key components of eukaryotic pre-initiation complex (PIC) have been recently measured with high resolution in Saccharomyces cerevisiae by Rhee and Pugh (Nature (2012) 483:295-301). Yet the rules determining the PIC binding specificity remain poorly understood. In this study we show that nonconsensus protein-DNA binding significantly influences PIC binding preferences. We estimate that such nonconsensus binding contribute statistically at least 2-3 kcal/mol (on average) of additional attractive free energy per protein, per core promoter region. The predict…
DeepAutoPIN: An automorphism orbits based deep neural network for characterizing the organizational diversity of protein interactomes across the tree of life
Centre for Computational Biology and Bioinformatics, Central University of Himahcal Pradesh（中央喜马偕尔大学计算生物学与生物信息学中心）
Proteins fold using a two-state or multi-state kinetic mechanisms, but up to now there isn't a first-principle model to explain this different behaviour. We exploit the network properties of protein structures by introducing novel observables to address the problem of classifying the different types of folding kinetics. These observables display a plain physical meaning, in terms of vibrational modes, possible configurations compatible with the native protein structure, and folding cooperativity. The relevance of these observables is supported by a classification performance up to 90%, even wi…
Neoplasms (NPs) and neurological diseases and disorders (NDDs) are amongst the major classes of diseases underlying deaths of a disproportionate number of people worldwide. To determine if there exist some distinctive features in the local wiring patterns of protein interactions emerging at the onset of a disease belonging to either of these two classes, we examined 112 and 175 protein interaction networks belonging to NPs and NDDs, respectively. Orbit usage profiles (OUPs) for each of these networks were enumerated by investigating the networks' local topology. 56 non-redundant OUPs (nrOUPs) …
The low-frequency collective vibrational modes in proteins as well as the protein-water interface have been suggested as dominant factors controlling the efficiency of biochemical reactions and biological energy transport. It is thus crucial to uncover the mystery of hydration structure and dynamics as well as their coupling to collective motions of proteins in aqueous solutions. Here we report dielectric properties of aqueous bovine serum albumin protein solutions as a model system using an extremely sensitive dielectric spectrometer with frequencies spanning from megahertz to terahertz. The …
Helices are a key folding motif in protein structure. The question which factors determine helix stability for a given polypeptide or protein is an ongoing challenge. Here we use van der Waals corrected density-functional theory to address a part of this question in a bottom-up approach. We show how intrinsic helical structure is stabilized with length and temperature for a series of experimentally well studied unsolvated alanine based polypeptides, Ac-Alan-LysH+. By exploring extensively the conformational space of these molecules, we find that helices emerge as the preferred structure in the…
Protein-protein binding enables orderly and lawful biological self-organization, and is therefore considered a miracle of nature. Protein-protein binding is steered by electrostatic forces, hydrogen bonding, van der Waals force, and hydrophobic interactions. Among these physical forces, only the hydrophobic interactions can be considered as long-range intermolecular attractions between proteins in intracellular and extracellular fluid. Low-entropy regions of hydration shells around proteins drive hydrophobic attraction among them that essentially coordinate protein-protein docking in rotationa…
Allosteric communication in proteins is a central and yet unsolved problem of structural biochemistry. Previous findings, from computational biology (Ota and Agard, 2005), have proposed that heat diffuses in a protein through cognate protein allosteric pathways. This work studied heat diffusion in the well-known PDZ-2 protein, and confirmed that this protein has two cognate allosteric pathways and that heat flows preferentially through these. Also, a new property was also observed for protein structures - heat diffuses asymmetrically through the structures. The underling structure of this asym…
Protein conformational transitions, which are essential for function, may be driven either by entropy or enthalpy when molecular systems comprising solute and solvent molecules are the focus. Revealing thermodynamic origin of a given molecular process is an important but difficult task, and general principles governing protein conformational distributions remain elusive. Here we demonstrate that when protein molecules are taken as thermodynamic systems and solvents being treated as the environment, conformational entropy is an excellent proxy for free energy and is sufficient to explain protei…
We investigate the presence of structural collective motions on a picosecond time scale for the heme protein, cytochrome c, as a function of oxidation and hydration, using terahertz (THz) time-domain spectroscopy and molecular dynamics simulations. The THz response dramatically increases with oxidation, with the largest increase for lowest hydrations and highest frequencies. For both oxidation states the THz response rapidly increases with hydration saturating above ~25% (g H2O/g protein). Quasi-harmonic vibrational modes and dipole-dipole correlation functions are calculated from molecular dy…
The tertiary structure of protein, as well as the local secondary structure organization are fully determined by the angles of the peptidic bound. The backbone dihedral angles not only determine the global fold of the protein, but also the details of the local chain organization. Although a wealth of structural information is available in different databases and numerous structural biology softwares have been developed, rapid conformational characterization remains challenging. We present here RamaDA, a program able to give a synthetic description of the conformation of a protein. The RamaDA p…
Membrane proteins deform the surrounding lipid bilayer, which can lead to membrane-mediated interactions between neighboring proteins. Using the mechanosensitive channel of large conductance (MscL) as a model system, we demonstrate how the observed differences in protein structure can affect membrane-mediated interactions and cooperativity among membrane proteins. We find that distinct oligomeric states of MscL lead to distinct gateway states for the clustering of MscL, and predict signatures of MscL structure and spatial organization in the cooperative gating of MscL. Our modeling approach es…
Although both RNA and proteins have densely packed native structures, chain organizations of these two biopolymers are fundamentally different. Motivated by the recent discoveries in chromatin folding that interphase chromosomes have territorial organization with signatures pointing to metastability, we analyzed the biomolecular structures deposited in the Protein Data Bank and found that the intrachain contact probabilities, $P(s)$ as a function of the arc length $s$, decay in power-law $\sim s^{-γ}$ over the intermediate range of $s$, $10\lesssim s\lesssim 110$. We found that the contact pro…
Starting from linear chains of amino acids, the spontaneous folding of proteins into their elaborate three-dimensional structures is one of the remarkable examples of biological self-organization. We investigated native state structures of 30 single-domain, two-state proteins, from complex networks perspective, to understand the role of topological parameters in proteins' folding kinetics, at two length scales-- as ``Protein Contact Networks (PCNs)'' and their corresponding ``Long-range Interaction Networks (LINs)'' constructed by ignoring the short-range interactions. Our results show that, b…
Argonne National Laboratory（阿贡国家实验室）；University of Chicago（芝加哥大学）；University of Pittsburgh（匹兹堡大学）
We outline recent developments in artificial intelligence (AI) and machine learning (ML) techniques for integrative structural biology of intrinsically disordered proteins (IDP) ensembles. IDPs challenge the traditional protein structure-function paradigm by adapting their conformations in response to specific binding partners leading them to mediate diverse, and often complex cellular functions such as biological signaling, self organization and compartmentalization. Obtaining mechanistic insights into their function can therefore be challenging for traditional structural determination techni…
We study vibrational thermodynamic stability of small-world oscillator networks, by relating the average mean-square displacement $S$ of oscillators to the eigenvalue spectrum of the Laplacian matrix of networks. We show that the cross-links suppress $S$ effectively and there exist two phases on the small-world networks: 1) an unstable phase: when $p\ll1/N$, $S\sim N$; 2) a stable phase: when $p\gg1/N$, $S\sim p^{-1}$, \emph{i.e.}, $S/N\sim E_{cr}^{-1}$. Here, $p$ is the parameter of small-world, $N$ is the number of oscillators, and $E_{cr}=pN$ is the number of cross-links. The results are ex…
We carry out a theoretical study of the vibrational and relaxation properties of naturally-occurring proteins with the purpose of characterizing both the folding and equilibrium thermodynamics. By means of a suitable model we provide a full characterization of the spectrum and eigenmodes of vibration at various temperatures by merely exploiting the knowledge of the protein native structure. It is shown that the rate at which perturbations decay at the folding transition correlates well with experimental folding rates. This validation is carried out on a list of about 30 two-state folders. Furt…
Proteins are key building blocks of virtually all life, providing the material foundation of spider silk, cells, and hair, but also offering other functions from enzymes to drugs, and pathogens like viruses. Based on a nanomechanical analysis of the structure and motions of atoms and molecules at multiple scales, we report sonified versions of the coronavirus spike protein of the pathogen of COVID-19, 2019-nCoV. The audio signal, created using a novel nanomechanical sonification method, features an overlay of the vibrational signatures of the protein's primary, secondary and higher-order struc…
In this paper we present a generative model for protein contact networks. The soundness of the proposed model is investigated by focusing primarily on mesoscopic properties elaborated from the spectra of the graph Laplacian. To complement the analysis, we study also classical topological descriptors, such as statistics of the shortest paths and the important feature of modularity. Our experiments show that the proposed model results in a considerable improvement with respect to two suitably chosen generative mechanisms, mimicking with better approximation real protein contact networks in terms…
In recent work we reported the vibrational spectrum of more than 100,000 known protein structures, and a self-consistent sonification method to render the spectrum in the audible range of frequencies (Extreme Mechanics Letters, 2019). Here we present a method to transform these molecular vibrations into materialized vibrations of thin water films using acoustic actuators, leading to complex patterns of surface waves, and using the resulting macroscopic images in further processing using deep convolutional neural networks. Specifically, the patterns of water surface waves for each protein struc…
