Scaling and self-organized criticality in proteins I

Folding-unfolding asymmetry and a RetroFold computational algorithm

We treat protein folding as molecular self-assembly, while unfolding is viewed as disassembly. Fracture is typically a much faster process than self-assembly. Self-assembly is often an exponentially decaying process, since energy relaxes due to dissipation, while fracture is a constant-rate process as the driving force is opposed by damping. Protein folding takes two orders of magnitude longer than unfolding. We suggest a mathematical transformation of variables, which makes it possible to view self-assembly as time-reversed disassembly, thus folding can be studied as reversed unfolding. We investigate the molecular dynamics modelling of folding and unfolding of the short Trp-cage protein. Folding time constitutes about 800 ns, while unfolding (denaturation) takes only about 5.0 ns and, therefore, fewer computational resources are needed for its simulation. This RetroFold approach can be used for the design of a novel computation algorithm, which, while approximate, is less time-consuming than traditional folding algorithms.

Topological bio-scaling analysis as a universal measure of protein folding

Scaling relationships for polymeric molecules establish power law dependencies between the number of molecular segments and linear dimensions, such as the radius of gyration. They also establish spatial topological properties of the chains, such as their dimensionality. In the spatial domain, power exponents α = 1 (linear stretched molecule), α = 0.5 (the ideal chain) and α = 0.333 (compact globule) are significant. During folding, the molecule undergoes the transition from the one-dimensional linear to the three-dimensional globular state within a very short time. However, intermediate states with fractional dimensions can be stabilized by modifying the solubility (e.g. by changing the solution temperature). Topological properties, such as dimension, correlate with the interaction energy, and thus by tuning the solubility one can control molecular interaction. We investigate these correlations using the example of a well-studied short model of Trp-cage protein. The radius of gyration is used to estimate the fractal dimension of the chain at different stages of folding. It is expected that the same principle is applicable to much larger molecules and that topological (dimensional) characteristics can provide insights into molecular folding and interactions.

Relating SARS-CoV-2 variants using cellular automata imaging

We classify the main variants of the SARS-CoV-2 virus representing a given biological sequence coded as a symbolic digital sequence and by its evolution by a cellular automata with a properly chosen rule. The spike protein, common to all variants of the SARS-CoV-2 virus, is then by the picture of the cellular automaton evolution yielding a visible representation of important features of the protein. We use information theory Hamming distance between different stages of the evolution of the cellular automaton for seven variants relative to the original Wuhan/China virus. We show that our approach allows to classify and group variants with common ancestors and same mutations. Although being a simpler method, it can be used as an alternative for building phylogenetic trees.

Hydropathicity-based prediction of pain-causing NaV1.7 variants

Background

Mutation-induced variations in the functional architecture of the NaV1.7 channel protein are causally related to a broad spectrum of human pain disorders. Predicting in silico the phenotype of NaV1.7 variant is of major clinical importance; it can aid in reducing costs of in vitro pathophysiological characterization of NaV1.7 variants, as well as, in the design of drug agents for counteracting pain-disease symptoms.

Results

In this work, we utilize spatial complexity of hydropathic effects toward predicting which NaV1.7 variants cause pain (and which are neutral) based on the location of corresponding mutation sites within the NaV1.7 structure. For that, we analyze topological and scaling hydropathic characteristics of the atomic environment around NaV1.7’s pore and probe their spatial correlation with mutation sites. We show that pain-related mutation sites occupy structural locations in proximity to a hydrophobic patch lining the pore while clustering at a critical hydropathic-interactions distance from the selectivity filter (SF). Taken together, these observations can differentiate pain-related NaV1.7 variants from neutral ones, i.e., NaV1.7 variants not causing pain disease, with 80.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document}% sensitivity and 93.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document}% specificity [area under the receiver operating characteristics curve = 0.872].

Conclusions

Our findings suggest that maintaining hydrophobic NaV1.7 interior intact, as well as, a finely-tuned (dictated by hydropathic interactions) distance from the SF might be necessary molecular conditions for physiological NaV1.7 functioning. The main advantage for using the presented predictive scheme is its negligible computational cost, as well as, hydropathicity-based biophysical rationalization.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-021-04119-2.

Self-organized cell migration across scales - from single cell movement to tissue formation

Self-organization is a key feature of many biological and developmental processes, including cell migration. Although cell migration has traditionally been viewed as a biological response to extrinsic signals, advances within the past two decades have highlighted the importance of intrinsic self-organizing properties to direct cell migration on multiple scales. In this Review, we will explore self-organizing mechanisms that lay the foundation for both single and collective cell migration. Based on in vitro and in vivo examples, we will discuss theoretical concepts that underlie the persistent migration of single cells in the absence of directional guidance cues, and the formation of an autonomous cell collective that drives coordinated migration. Finally, we highlight the general implications of self-organizing principles guiding cell migration for biological and medical research.

Non-extensitivity and criticality of atomic hydropathicity around a voltage-gated sodium channel's pore: a modeling study

Voltage-gated sodium channels (NavChs) are pore-forming membrane proteins that regulate the transport of sodium ions through the cell membrane. Understanding the structure and function of NavChs is of major biophysical, as well as clinical, importance given their key role in cellular pathophysiology. In this work, we provide a computational framework for modeling system-size-dependent, i.e., cumulative, atomic properties around a NavCh's pore. We illustrate our methodologies on the bacterial NavAb channel captured in a closed-pore state where we demonstrate that the atomic environment around its pore exhibits a bi-phasic spatial organization dictated by the structural separation of the pore domains (PDs) from the voltage-sensing domains (VSDs). Accordingly, a mathematical model describing packing of atoms around NavAb's pore is constructed that allows-under certain conservation conditions-for a power-law approximation of the cumulative hydropathic dipole field effect acting along NavAb's pore. This verified the non-extensitivity hypothesis for the closed-pore NavAb channel and revealed a long-range hydropathic interactions law regulating atom-packing around the NavAb's selectivity filter. Our model predicts a PDs-VSDs coupling energy of [Formula: see text] kcal/mol corresponding to a global maximum of the atom-packing energy profile. Crucially, we demonstrate for the first time how critical phenomena can emerge in a single-channel structure as a consequence of the non-extensive character of its atomic porous environment.

Survival of Virus Particles in Water Droplets: Hydrophobic Forces and Landauer's Principle

Many small biological objects, such as viruses, survive in a water environment and cannot remain active in dry air without condensation of water vapor. From a physical point of view, these objects belong to the mesoscale, where small thermal fluctuations with the characteristic kinetic energy of k_BT (where k_B is the Boltzmann’s constant and T is the absolute temperature) play a significant role. The self-assembly of viruses, including protein folding and the formation of a protein capsid and lipid bilayer membrane, is controlled by hydrophobic forces (i.e., the repulsing forces between hydrophobic particles and regions of molecules) in a water environment. Hydrophobic forces are entropic, and they are driven by a system’s tendency to attain the maximum disordered state. On the other hand, in information systems, entropic forces are responsible for erasing information, if the energy barrier between two states of a switch is on the order of k_BT, which is referred to as Landauer’s principle. We treated hydrophobic interactions responsible for the self-assembly of viruses as an information-processing mechanism. We further showed a similarity of these submicron-scale processes with the self-assembly in colloidal crystals, droplet clusters, and liquid marbles.

Long-range correlation in protein dynamics: Confirmation by structural data and normal mode analysis

Proteins in cellular environments are highly susceptible. Local perturbations to any residue can be sensed by other spatially distal residues in the protein molecule, showing long-range correlations in the native dynamics of proteins. The long-range correlations of proteins contribute to many biological processes such as allostery, catalysis, and transportation. Revealing the structural origin of such long-range correlations is of great significance in understanding the design principle of biologically functional proteins. In this work, based on a large set of globular proteins determined by X-ray crystallography, by conducting normal mode analysis with the elastic network models, we demonstrate that such long-range correlations are encoded in the native topology of the proteins. To understand how native topology defines the structure and the dynamics of the proteins, we conduct scaling analysis on the size dependence of the slowest vibration mode, average path length, and modularity. Our results quantitatively describe how native proteins balance between order and disorder, showing both dense packing and fractal topology. It is suggested that the balance between stability and flexibility acts as an evolutionary constraint for proteins at different sizes. Overall, our result not only gives a new perspective bridging the protein structure and its dynamics but also reveals a universal principle in the evolution of proteins at all different sizes.

The long-range correlated fluctuations are closely related to many biological processes of the proteins, such as catalysis, ligand binding, biomolecular recognition, and transportation. In this paper, we elucidate the structural origin of the long-range correlation and describe how native contact topology defines the slow-mode dynamics of the native proteins. Our result suggests an evolutionary constraint for proteins at different sizes, which may shed light on solving many biophysical problems such as structure prediction, multi-scale molecular simulations, and the design of molecular machines. Moreover, in statistical physics, as the long-range correlations are notable signs of the critical point, unveiling the origin of such criticality can extend our understanding of the organizing principle of a large variety of complex systems.

A protein interaction free energy model based on amino acid residue contributions: Assessment of point mutation stability of T4 lysozyme

Here we present a model to estimate the interaction free energy contribution of each amino acid residue of a given protein. Protein interaction energy is described in terms of per-residue interaction factors, μ. Multibody interactions are implicitly captured in μ through the combination of amino acid terms (γ) guided by local conformation indices (σ). The model enables construction of an interaction factor heat map for a protein in a given fold, allows prima facie assessment of the degree of residue-residue interaction, and facilitates a qualitative and quantitative evaluation of protein association properties. The model was used to compute thermal stability of T4 bacteriophage lysozyme mutants across seven sites. Qualitative assessment of mutational effects provides a straightforward rationale regarding whether a particular site primarily perturbs native or non-native states, or both. The presented model was found to be in good agreement with experimental mutational data (R 2 = 0.73) and suggests an approach by which to convert structure space into energy space.

Why Ubiquitin Has Not Evolved

Ubiquitin, discovered less than 50 years ago, tags thousands of diseased proteins for destruction. It is small (only 76 amino acids), and is found unchanged in mammals, birds, fish, and even worms, indicating that ubiquitin is perfect. Key features of its functionality are identified here using critical point thermodynamic scaling theory. These include synchronized pivots and hinges, a stabilizing central pivot, and Fano interference between first- and second-order elements of correlated long-range (allosteric) globular surface shape transitions. Comparison with its closest relative, 76 amino acid Nedd8, shows that the latter lacks all these features. A cracked elastic network model is proposed for the common target shared by many diseased proteins.

Critical Fluctuations in the Native State of Proteins

Based on protein structural ensembles determined by nuclear magnetic resonance, we study the position fluctuations of residues by calculating distance-dependent correlations and conducting finite-size scaling analysis. The fluctuations exhibit high susceptibility and long-range correlations up to the protein sizes. The scaling relations between the correlations or susceptibility and protein sizes resemble those in other physical and biological systems near their critical points. These results indicate that, at the native states, motions of each residue are felt by every other one in the protein. We also find that proteins with larger susceptibility are more frequently observed in nature. Overall, our results suggest that the protein's native state is critical.

Poised for survival: criticality, natural selection, and excitation-transcription coupling

Neurologically-complex species utilize two intricately coupled information-processing systems to adapt to their social and natural environments. Action potentials (APs) facilitate rapid responses to the nearly continuous fluctuations in the animal's surroundings. By contrast, genetic encodings comprise a molecular memory of ancient adaptive responses expressed as cognitive, emotional, or behavioral phenotypes. The linking of the two systems via intracellular Ca(2+) networks which address transcription factors - e.g., cAMP response element-binding protein (CREB) - is an appropriate focus for the biology of human behavior. Computational modeling utilizing Boolean networks (BNs) is a suitable qualitative method, requiring no kinetic information, for eliciting the systems' architectural properties. In particular, BNs can reveal critical intracellular regimes of possible evolutionary significance. As a platform for future research, we propose that those networks sufficiently robust to attenuate damaging intracellular noise and deleterious mutations, yet sufficiently close to chaos to permit or amplify adaptive noise and favorable mutations, would be favored by natural selection. Critical regimes of this type would be, literally, "poised for survival", and would stabilize and promote the survival of their correlated cultural phenotypes.

From superhydrophobicity to icephobicity: forces and interaction analysis

The term “icephobicity” has emerged in the literature recently. An extensive discussion took place on whether the icephobicity is related to the superhydrophobicity, and the consensus is that there is no direct correlation. Besides the parallel between the icephobicity and superhydrophobicity for water/ice repellency, there are similarities on other levels including the hydrophobic effect/hydrophobic interactions, mechanisms of protein folding and ice crystal formation. In this paper, we report how ice adhesion is different from water using force balance analysis, and why superhydrophobic surfaces are not necessary icephobic. We also present experimental data on anti-icing of various surfaces and suggest a definition of icephobicity, which is broad enough to cover a variety of situations relevant to de-icing including low adhesion strength and delayed ice crystallization and bouncing.

Fractal nature of protein surface roughness: a note on quantification of change of surface roughness in active sites, before and after binding

Year 2010 marked the 25th year since we came to know that roughness of a protein surface has fractal symmetry. Ever since the publication of Lewis and Rees' paper, hundreds of works from a spectrum of perspectives have established that fractal dimension (FD) can be considered as a reliable marker that describes roughness of protein surface objectively. In this article, we introduce readers to the fundamentals of fractals and present categorical biophysical and geometrical reasons as to why FD-based constructs can describe protein surface roughness more accurately. We then review the commonality (and the lack of it) between numerous approaches that have attempted to investigate protein surface with fractal measures, before exploring the patterns in the results that they have produced. Apart from presenting the genealogy of approaches and results, we present an analysis that quantifies the difference in surface roughness in stretches of protein surface containing the active site, before and after binding to ligands, to underline the utility of FD-based measures further. It has been found that surface stretches containing the active site, in general, undergo a significant increment in its roughness after binding. After presenting the entire repertoire of FD-based surface roughness studies, we talk about two yet-unexplored problems where application of FD-based techniques can help in deciphering underlying patterns of surface interactions. Finally, we list the limitations of FD-based constructs and put down several precautions that one must take while working with them.

Scaling and self-organized criticality in proteins: Lysozyme c

Proteins appear to be the most dramatic natural example of self-organized criticality (SOC), a concept that explains many otherwise apparently unlikely phenomena. Protein functionality is often dominated by long-range hydro(phobic/philic) interactions, which both drive protein compaction and mediate protein-protein interactions. In contrast to previous reductionist short-range hydrophobicity scales, the holistic Moret-Zebende hydrophobicity scale [Phys. Rev. E 75, 011920 (2007)] represents a hydroanalytic tool that bioinformatically quantifies SOC in a way fully compatible with evolution. Hydroprofiling identifies chemical trends in the activities and substrate binding abilities of model enzymes and antibiotic animal lysozymes c , as well as defensins, which have been the subject of tens of thousands of experimental studies. The analysis is simple and easily performed and immediately yields insights not obtainable by traditional methods based on short-range real-space interactions, as described either by classical force fields used in molecular-dynamics simulations, or hydrophobicity scales based on transference energies from water to organic solvents or solvent-accessible areas.

Revisiting the myths of protein interior: studying proteins with mass-fractal hydrophobicity-fractal and polarizability-fractal dimensions

A robust marker to describe mass, hydrophobicity and polarizability distribution holds the key to deciphering structural and folding constraints within proteins. Since each of these distributions is inhomogeneous in nature, the construct should be sensitive in describing the patterns therein. We show, for the first time, that the hydrophobicity and polarizability distributions in protein interior follow fractal scaling. It is found that (barring ‘all-α’) all the major structural classes of proteins have an amount of unused hydrophobicity left in them. This amount of untapped hydrophobicity is observed to be greater in thermophilic proteins, than that in their (structurally aligned) mesophilic counterparts. ‘All-β’(thermophilic, mesophilic alike) proteins are found to have maximum amount of unused hydrophobicity, while ‘all-α’ proteins have been found to have minimum polarizability. A non-trivial dependency is observed between dielectric constant and hydrophobicity distributions within (α+β) and ‘all-α’ proteins, whereas absolutely no dependency is found between them in the ‘all-β’ class. This study proves that proteins are not as optimally packed as they are supposed to be. It is also proved that origin of α-helices are possibly not hydrophobic but electrostatic; whereas β-sheets are predominantly hydrophobic in nature. Significance of this study lies in protein engineering studies; because it quantifies the extent of packing that ensures protein functionality. It shows that myths regarding protein interior organization might obfuscate our knowledge of actual reality. However, if the later is studied with a robust marker of strong mathematical basis, unknown correlations can still be unearthed; which help us to understand the nature of hydrophobicity, causality behind protein folding, and the importance of anisotropic electrostatics in stabilizing a highly complex structure named ‘proteins’.

Self-similarity and protein compactness

The hydrophobic effect is the major factor that drives a protein toward collapse and folding. As a consequence of the folding process a hydrophobic core is shielded by the solvent-accessible surface area of the protein. We analyze the solvent-accessible surface area of 1825 nonhomolog protein chains deposited in the Brookhaven Protein Data Bank. This solvent-accessible surface area presents an intrinsic self-similarity behavior. The comparison between the accessible surface area as function of the number of amino acids and the accessible surface area as function of gyration radius supplies a measure of the scaling exponent close to the one observed by volume as function of radius of gyration or by mass-size exponent. The present finding indicates that the fractal analysis describes the protein compactness as an object packing between random spheres in percolation threshold and crumpled wires.

Scaling and self-organized criticality in proteins II

The complexity of proteins is substantially simplified by regarding them as archetypical examples of self-organized criticality (SOC). To test this idea and to elaborate it, this article applies the Moret-Zebende (MZ) SOC hydrophobicity scale to transport repeat proteins of the HEAT superfamily, importin beta, and transportin, as well as the export protein Cse1p, and their ubiquitous cargo manager Ran. The difference between the MZ scale and conventional hydrophobicity scales reflects long-range conformational forces that are central to protein functionality. These compete with long-range Coulomb forces associated with cationic and anionic side chains in a revealing way.