Scaling behavior of some molecular shape descriptors of polymer chains and protein backbones

The Local Topological Free Energy of the SARS-CoV-2 Spike Protein

The novel coronavirus SARS-CoV-2 infects human cells using a mechanism that involves binding and structural rearrangement of its Spike protein. Understanding protein rearrangement and identifying specific amino acids where mutations affect protein rearrangement has attracted much attention for drug development. In this manuscript, we use a mathematical method to characterize the local topology/geometry of the SARS-CoV-2 Spike protein backbone. Our results show that local conformational changes in the FP, HR1, and CH domains are associated with global conformational changes in the RBD domain. The SARS-CoV-2 variants analyzed in this manuscript (alpha, beta, gamma, delta Mink, G614, N501) show differences in the local conformations of the FP, HR1, and CH domains as well. Finally, most mutations of concern are either in or in the vicinity of high local topological free energy conformations, suggesting that high local topological free energy conformations could be targets for mutations with significant impact of protein function. Namely, the residues 484, 570, 614, 796, and 969, which are present in variants of concern and are targeted as important in protein function, are predicted as such from our model.

Micromechanical model for isolated polymer-colloid clusters under tension

Binary polymer-colloid (PC) composites form the majority of biological load-bearing materials. Due to the abundance of the polymer and particles, and their simple aggregation process, PC clusters are used broadly by nature to create biomaterials with a variety of functions. However, our understanding of the mechanical features of the clusters and their load transfer mechanism is limited. Our main focus in this paper is the elastic behavior of close-packed PC clusters formed in the presence of polymer linkers. Therefore, a micromechanical model is proposed to predict the constitutive behavior of isolated polymer-colloid clusters under tension. The mechanical response of a cluster is considered to be governed by a backbone chain, which is the stress path that transfers most of the applied load. The developed model can reproduce the mean behavior of the clusters and is not dependent on their local geometry. The model utilizes four geometrical parameters for defining six shape descriptor functions which can affect the geometrical change of the clusters in the course of deformation. The predictions of the model are benchmarked against an extensive set of simulations by coarse-grained-Brownian dynamics, where clusters with different shapes and sizes were considered. The model exhibits good agreement with these simulations, which, besides its relative simplicity, makes the model an excellent add-on module for implementation into multiscale models of nanocomposites.

The impact of entropy on the spatial organization of synaptonemal complexes within the cell nucleus

We employ 4Pi-microscopy to study SC organization in mouse spermatocyte nuclei allowing for the three-dimensional reconstruction of the SC's backbone arrangement. Additionally, we model the SCs in the cell nucleus by confined, self-avoiding polymers, whose chain ends are attached to the envelope of the confining cavity and diffuse along it. This work helps to elucidate the role of entropy in shaping pachytene SC organization. The framework provided by the complex interplay between SC polymer rigidity, tethering and confinement is able to qualitatively explain features of SC organization, such as mean squared end-to-end distances, mean squared center-of-mass distances, or SC density distributions. However, it fails in correctly assessing SC entanglement within the nucleus. In fact, our analysis of the 4Pi-microscopy images reveals a higher ordering of SCs within the nuclear volume than what is expected by our numerical model. This suggests that while effects of entropy impact SC organization, the dedicated action of proteins or actin cables is required to fine-tune the spatial ordering of SCs within the cell nucleus.

Comparative modeling and protein-like features of hydrophobic-polar models on a two-dimensional lattice

Lattice models of proteins have been extensively used to study protein thermodynamics, folding dynamics, and evolution. Our study considers two different hydrophobic-polar (HP) models on the 2D square lattice: the purely HP model and a model where a compactness-favoring term is added. We exhaustively enumerate all the possible structures in our models and perform the study of their corresponding folds, HP arrangements in space and shapes. The two models considered differ greatly in their numbers of structures, folds, arrangements, and shapes. Despite their differences, both lattice models have distinctive protein-like features: (1) Shapes are compact in both models, especially when a compactness-favoring energy term is added. (2) The residue composition is independent of the chain length and is very close to 50% hydrophobic in both models, as we observe in real proteins. (3) Comparative modeling works well in both models, particularly in the more compact one. The fact that our models show protein-like features suggests that lattice models incorporate the fundamental physical principles of proteins. Our study supports the use of lattice models to study questions about proteins that require exactness and extensive calculations, such as protein design and evolution, which are often too complex and computationally demanding to be addressed with more detailed models.

Prediction of functionally important residues in globular proteins from unusual central distances of amino acids

BACKGROUND

Well-performing automated protein function recognition approaches usually comprise several complementary techniques. Beside constructing better consensus, their predictive power can be improved by either adding or refining independent modules that explore orthogonal features of proteins. In this work, we demonstrated how the exploration of global atomic distributions can be used to indicate functionally important residues.

RESULTS

Using a set of carefully selected globular proteins, we parametrized continuous probability density functions describing preferred central distances of individual protein atoms. Relative preferred burials were estimated using mixture models of radial density functions dependent on the amino acid composition of a protein under consideration. The unexpectedness of extraordinary locations of atoms was evaluated in the information-theoretic manner and used directly for the identification of key amino acids. In the validation study, we tested capabilities of a tool built upon our approach, called SurpResi, by searching for binding sites interacting with ligands. The tool indicated multiple candidate sites achieving success rates comparable to several geometric methods. We also showed that the unexpectedness is a property of regions involved in protein-protein interactions, and thus can be used for the ranking of protein docking predictions. The computational approach implemented in this work is freely available via a Web interface at http://www.bioinformatics.org/surpresi.

CONCLUSIONS

Probabilistic analysis of atomic central distances in globular proteins is capable of capturing distinct orientational preferences of amino acids as resulting from different sizes, charges and hydrophobic characters of their side chains. When idealized spatial preferences can be inferred from the sole amino acid composition of a protein, residues located in hydrophobically unfavorable environments can be easily detected. Such residues turn out to be often directly involved in binding ligands or interfacing with other proteins.

Fractal symmetry of protein interior: what have we learned?

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide-dipole units, charged amino acids, residues with the π-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to 'α/α toroid' (all-α class) and 'PLP-dependent transferase-like' domains (α/β class) are highly conducive. In contrast, the interiors of 'zinc finger design' ('designed proteins') and 'knottins' ('small proteins') were identified as folds with the least conducive electrostatic environments. The fold 'conotoxins' (peptides) could be unambiguously identified as one type with the least stability. The same analyses revealed that peptide-dipoles in the α/β class of proteins, in general, are more correlated to each other than are the peptide-dipoles in proteins belonging to the all-α class. Highly favorable electrostatic milieu in the interiors of TIM-barrel, α/β-hydrolase structures could explain their remarkably conserved (evolutionary) stability from a new light. Finally, we point out certain inherent limitations of fractal constructs before attempting to identify the areas and problems where the implementation of fractal dimension-based constructs can be of paramount help to unearth latent information on protein structural properties.

Size, shape, and flexibility of proteins and DNA

Size, shape, and flexibility are the important topological parameters which characterize the functional specificity and different types of interactions in proteins and DNA. The size of proteins and DNA, often measured by the radius of gyration (R(G)), are determined from the coordinates of their respective structures available in Protein Data Bank and Nucleic Acid Data Bank. The mean square radius of gyration obeys Flory's scaling law given by R(G) (2) approximately N(2nu) where N is the number of amino acid residues/nucleotides. The scaling exponent nu reflects the different characteristic features of nonglobular proteins, natively unstructured proteins, and DNA. The asymmetry in the shapes of proteins and DNA are investigated using the asphericity (Delta) parameter and the shape parameter (S), calculated from the eigenvalues of the moment of inertia tensor. The distributions of Delta and S show that most nonglobular proteins and DNA are aspherical and prolate (S>0). Natively unstructured proteins are comparatively spherically symmetrical having both prolate and oblate shapes. The flexibility of these molecules is characterized by the persistence length (l(p)). Persistence length for natively unstructured proteins is determined by fitting the distance distribution function P(r) to the wormlike chain (WLC) model in the limit of r>>R(G). For nonglobular proteins and DNA, l(p) may be computed from the Benoit-Doty approximation for unperturbed radius of gyration of the WLC. The flexibilities of the proteins and DNA increases with the chain length. This is due to an increase in the nonlocal interactions with the increase in N, needed to minimize the conformational fluctuations in the native state. The persistence length of these proteins has not yet been measured directly. Analysis of the two-body contacts for the proteins reveals that the nonglobular proteins are less densely packed compared to the natively unstructured proteins with least side-chain side chain contacts even though side-chain backbone contacts predominate in the two types of proteins.

Description of atomic burials in compact globular proteins by Fermi-Dirac probability distributions

We perform a statistical analysis of atomic distributions as a function of the distance R from the molecular geometrical center in a nonredundant set of compact globular proteins. The number of atoms increases quadratically for small R, indicating a constant average density inside the core, reaches a maximum at a size-dependent distance R(max), and falls rapidly for larger R. The empirical curves turn out to be consistent with the volume increase of spherical concentric solid shells and a Fermi-Dirac distribution in which the distance R plays the role of an effective atomic energy epsilon(R) = R. The effective chemical potential mu governing the distribution increases with the number of residues, reflecting the size of the protein globule, while the temperature parameter beta decreases. Interestingly, betamu is not as strongly dependent on protein size and appears to be tuned to maintain approximately half of the atoms in the high density interior and the other half in the exterior region of rapidly decreasing density. A normalized size-independent distribution was obtained for the atomic probability as a function of the reduced distance, r = R/R(g), where R(g) is the radius of gyration. The global normalized Fermi distribution, F(r), can be reasonably decomposed in Fermi-like subdistributions for different atomic types tau, F(tau)(r), with Sigma(tau)F(tau)(r) = F(r), which depend on two additional parameters mu(tau) and h(tau). The chemical potential mu(tau) affects a scaling prefactor and depends on the overall frequency of the corresponding atomic type, while the maximum position of the subdistribution is determined by h(tau), which appears in a type-dependent atomic effective energy, epsilon(tau)(r) = h(tau)r, and is strongly correlated to available hydrophobicity scales. Better adjustments are obtained when the effective energy is not assumed to be necessarily linear, or epsilon(tau)*(r) = h(tau)*r(alpha,), in which case a correlation with hydrophobicity scales is found for the product alpha(tau)h(tau)*. These results indicate that compact globular proteins are consistent with a thermodynamic system governed by hydrophobic-like energy functions, with reduced distances from the geometrical center, reflecting atomic burials, and provide a conceptual framework for the eventual prediction from sequence of a few parameters from which whole atomic probability distributions and potentials of mean force can be reconstructed.

Statistics of knots, geometry of conformations, and evolution of proteins

Like shoelaces, the backbones of proteins may get entangled and form knots. However, only a few knots in native proteins have been identified so far. To more quantitatively assess the rarity of knots in proteins, we make an explicit comparison between the knotting probabilities in native proteins and in random compact loops. We identify knots in proteins statistically, applying the mathematics of knot invariants to the loops obtained by complementing the protein backbone with an ensemble of random closures, and assigning a certain knot type to a given protein if and only if this knot dominates the closure statistics (which tells us that the knot is determined by the protein and not by a particular method of closure). We also examine the local fractal or geometrical properties of proteins via computational measurements of the end-to-end distance and the degree of interpenetration of its subchains. Although we did identify some rather complex knots, we show that native conformations of proteins have statistically fewer knots than random compact loops, and that the local geometrical properties, such as the crumpled character of the conformations at a certain range of scales, are consistent with the rarity of knots. From these, we may conclude that the known "protein universe" (set of native conformations) avoids knots. However, the precise reason for this is unknown--for instance, if knots were removed by evolution due to their unfavorable effect on protein folding or function or due to some other unidentified property of protein evolution.

Effect of confinement on coil-globule transition

The equilibrium thermodynamic properties of a linear polymer chain confined to a space between two impenetrable walls (lines) at a distance D under various solvent conditions have been studied using series analysis and exact enumeration technique. We have calculated the end-to-end distance of polymer chain, which shows a nonmonotonic behavior with inter wall separation D. The density distribution profile shows a maxima at a particular value of (D=)D*. Around this D*, our results show that the collapse transition occurs at higher temperature as compared to its bulk value of 2d and 3d. The variation of theta-temperature with D shows a re-entrance behavior. We also calculate the force of compression exerted by the walls (lines) on the polymer.

Geometrical complexity of conformations of ring polymers under topological constraints

One measure of geometrical complexity of a spatial curve is the average of the number of crossings appearing in its planar projection. The mean number of crossings averaged over some directions have been numerically evaluated for N-noded ring polymers with a fixed knot type. When N is large, the average crossing number of ring polymers under the topological constraint is smaller than that of no topological constraint. The decrease of the geometrical complexity is significant when the thickness of polymers is small. It is also suggested from the simulation that the relation between the average crossing number and the average size of ring polymers should depend on whether they are under a topological constraint or not.

Computing a New Family of Shape Descriptors for Protein Structures

The large-scale 3D structure of a protein can be represented by the polygonal curve through the carbon alpha atoms of the protein backbone. We introduce an algorithm for computing the average number of times that a given configuration of crossings on such polygonal curves is seen, the average being taken over all directions in space. Hereby, we introduce a new family of global geometric measures of protein structures, which we compare with the so-called generalized Gauss integrals.

Path-space ratio as a molecular shape descriptor of polymer conformation

Polymers at interfaces exhibit properties that cannot be completely captured by descriptors of mean molecular size. Recent work in the literature shows that a combined analysis of mean size and chain entanglement provides a more discriminating approach to understanding the onset of configurational transitions in these systems. Usually, chain entanglement is characterized by properties such as the mean overcrossing number or the chain's writhe; these are powerful properties but their evaluation can be computationally demanding. In this work, we introduce a geometrical descriptor of polymer shape, termed the path-space ratio zeta, aimed at quantifying essential features of chain complexity, but at a lower computational cost. The descriptor includes information on chain geometry and topology. The path-space ratio zeta is built by taking into account two key ideas: (a) a dimensionless measure of length along the backbone of the polymer, and (b) the behavior of topological "knot energies". Here, we compare zeta with other approaches to quantify polymer geometry and connectivity. Particular attention is devoted to the ability of these descriptors to discriminate and quantify conformational changes in grafted polymers under compression. We show that, for these types of applications, the path-space ratio presents a fast alternative to the mean overcrossing number.

Analytical estimation of scaling behavior for the entanglement complexity of a bond network

Geometrical entanglements in a polymer network can be characterized in terms of the mean number of projected bond-bond crossings, N. Here, we present an analytical method to study the dependence of N on the number of bonds in the network, n. Our approach shows the occurrence of power-law scaling, N - n(beta). The estimated upper bound to the exponent for maximally compact networks, Beta approximately 1.38, agrees well with the values observed in simulations of transient networks in liquids and in the folding features of native states of globular proteins.

Structural transitions in neutral and charged proteins in vacuo

In vacuo proteins provide a simple laboratory to explore the roles of sequence, temperature, charge state, and initial configuration in protein folding. Moreover, by the very absence of solvent, the study of anhydrous proteins in vacuo will also help us to understand specific environmental effects. From the experimental viewpoint, these systems are now beginning to be characterized at low resolution. Molecular dynamics (MD) simulations, in combination with tools for protein shape analysis, can complement experiments and provide further insights on the folding-unfolding transitions of these proteins. We review some aspects of this issue by using the results from a detailed MD study of hen egg-white lysozyme. For lysozyme ions, unfolding can be triggered by Coulombic repulsion. In neutral lysozyme, unfolding can be induced by centrifugal forces and also by weakening the monomer-monomer interaction. In both cases, the resulting unfolded transients can be used as initial configurations for relaxation dynamics. All trajectories are analyzed in terms of global molecular shape features of the backbone, including its anisometry and chain entanglement complexity. This strategy allows us to quantify separately the degree of polymer collapse and the evolution of large-scale folding features. Using these last two notions, we discuss some basic questions regarding the nature of the accessible paths associated with unfolding from, and refolding into, compact conformers.

Proteins in vacuo: denaturing and folding mechanisms studied with computer-simulated molecular dynamics

Mounting evidence from experiments suggests that the native fold in solution is metastable in dehydrated proteins. Results from a number of experiments that use mass spectrometry indicate also that folding-unfolding transitions take place in protein ions even in the absence of water. These observations on anhydrous proteins call for a re-evaluation of our understanding of the folding transition. In this context, computer-assisted simulations are an important complementary tool. Here, we provide an overview of recent progress on the simulation of proteins in vacuo. In particular, we discuss the response of proteins and protein ions to perturbations that trigger unfolding and re-folding transitions. By comparing the general patterns emerging from theory and experiment, we propose a series of new measurements that could help to validate, and improve, current simulation models.

Addendum to "Quantitative measure of folding in two-dimensional polymers"

Recently, we introduced a measure of folding complexity for two-dimensional polymers, N macro, the mean radial intersection number [Phys. Rev. E 59, 4209 (1999)]. In this addendum, we expand on three aspects of the previous work. First, we provide an analytical expression for N macro that is valid for two-dimensional networks. Second, we show that the power-law scaling N macro approximately equal to n(beta), with n the number of monomers, has a different critical exponent for random and self-avoiding walks. Finally, we find that the folding features in optimized projections of experimental three-dimensional (native) protein backbones fall between the latter limit models.

Characterization of fold diversity among proteins with the same number of amino acid residues

Chain entanglements provide a simple and global measure of folding in a macromolecule. The complexity of these entanglements can be expressed by the pattern of projected bond-bond crossings, or "overcrossings", associated with the molecular backbone. In this work, we use this approach to characterize quantitatively the range of tertiary folds observed in proteins with a given chain length. To discriminate among folding features, we use two shape descriptors derived from the probability distribution of overcrossings: the mean overcrossing number, N, and the most probable overcrossing number, N*. The values of N and N* relate to the content of secondary structure in a protein as well as its global three-dimensional organization. We propose a measure of folding diversity based on the properties of these descriptors. In addition, we discuss the application of our method to study how tertiary folds evolve during protein dynamics.