Fractal symmetry of protein interior: what have we learned?

Abstract

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.

Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

“…Equations containing fractal operators are used to analyze the behavior of systems characterized by: (1) power-law nonlinearity (2) power-law long-range spatial correlations or long-term memory (3) fractal or multi-fractal properties [22]. During the last years, the number of applications of fractional dynamics in science and particularly in physics has been steadily growing and include models of fractional relaxation and oscillation phenomena, anomalous transport in fluids and plasma, wave propagation in complex media, viscoelastic materials, non-Markovian evolution of quantum fields, networks of fractional oscillators and so on [23][24][25][26][27][28][29].…”

“…In recent years fractal geometry analysis has attracted attention of biologist as a useful and desirable tool to characterize the configuration and structure of biological structures. The considered issues include: fractal geometry analysis of configuration, structure and diffusion of proteins [24][25][26][27][28][29], fractal analysis of the DNA sequence, walks and aggregation [30][31][32][33][34] and fractal model of light scattering in biological tissue and cells [35]. (…”

Concepts on the protein folding problem

“…Our remark gets supported when one measures the compactness of folded structures in terms of hydrophobicity, mass, or polarizability using fractal dimensions. For instance, the all β proteins were identified with a maximum amount of "unused hydrophobicity" compared to the all α proteins (Banerji & Ghosh, 2011). This may result in quite high remaining conformational entropy in the functional structure.…”

Identifying critical unrecognized sugar–protein interactions in GH10 xylanases from Geobacillus stearothermophilus using STD NMR

“…The structural scaffold of these enzymes, the (b/a) 8 (triosephosphate isomerase, TIM) barrel (Fig. 1A), is one of the most abundant and stable protein folds known [8,11]. Structural conservation and rigidity are demonstrated by the small RMSD (< 3 A) between crystalline and solution structures [12,13].…”