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Dispersion analysis violin plots of the fractal dimension D of different biomolecular surfaces, including complete molecules (DNA, RNA, proteins, and ligands) and protein surface patches. Also shown are the liganded protein sites (LS), bound proteinprotein interfaces (PPI), cavities on the protein surface (Cavities), and the background, which was computed from randomly sampled protein surface patches (BG). The symbol ‘○’ indicates statistically insignificant differences between the distributions. We used a non-parametric test (Mann-Whitney-U). To compensate for α-errors, the p values of the 28 independent tests were Bonferroni-corrected. The null hypothesis could not be rejected for the Ligands-PPI, and LS-PPI pairs implying similarities in their distributions. The sample sizes were as follows: DNA (n=786); RNA (449); Proteins (604); Ligands (604); LS (604); PPI (466); Cavities (12,543); BG (11,953).
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Quantifying the properties of macromolecules is a prerequisite for understanding their roles in biochemical processes. One of the less‐explored geometric features of macromolecules is molecular surface irregularity, or ‘roughness’, which can be measured in terms of fractal dimension (D). In this study, we demonstrate that surface roughness correlat...
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Medicinal chemistry is facing new challenges in approaching precision medicine. Several powerful new tools or improvements of already used tools are now available to medicinal chemists to help in the process of drug discovery, from a hit molecule to a clinically used drug. Among the new tools, the possibility of considering folding intermediates or...
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... It has become a guiding idea in modern physical [35-37] and biological sciences [12,38,39]. Narrowing attention to the spatial symmetry at the molecular level, we can state that majority of biological molecules (such as amino acids and sugars) and macromolecules (including DNA, RNA, proteins, and lipids) exhibit the prevalent form of spatial symmetry in three domains of geometry: chirality, fractality, and topology [40][41][42][43][44]. We will be focused on the effects of molecular chirality represented by two classes of stereoisomers: chiral enantiomers (prevalent L-isoform) and achiral diastereomers (prevalent cis-stereoform) ( Figure 3). ...
In bilaterians organism, the signaling of pyramidal neurons (PyrNs) is linked to the relative (prevalent) molecular chirality and physiological, perceptual, cognitive, and psychological functions and dysfunctions, providing the coupling of the central nervous system downstream and upstream evolutionary and developmental processes. The most apparent and discriminating morphological specificity of PyrNs is the geometry of the cell body. However, the question "why/how PyrNs soma gains the shape of quasi-tetrahedral symmetry" has never been explicitly articulated. If the basic function of PyrNs is sensory space perception, supporting the orientation, and movement coordination, then the pyramidal shape of soma is the best evolutionary-selected geometry to perform sensory-motor coupling. In biology, the impact of chiral symmetry (handedness) is evident at all levels of biological organization, from the prevalent symmetry of biological molecules to the morphology and function of bilateral organisms. How the tetrahedral symmetry of biomolecules is linked to the morphology and functions of bilateral organisms remains a challenging question. Cell chirality represents an intermediate point connecting two poles of biochirality. In this holistic perspective, examining the PyrNs' morphology-circuitry-function link is crucial to understanding the complex interaction between genetic, epigenetic, and environmental factors in the evolution of the CNS. The predictive power of our hypothesis can be partially expressed by the statement that the most integral and reliable biomarker of the neurodegenerative (including aging) and mental (including all variants of psychiatric illness) disorders would be the detection of the hemispheric asymmetry of D-amino acids (D-AAs) level in pyramidal neurons (PyrNs).
... No correlation with circularity [149] Fractal geometry is essential for visualizing macromolecules and drug carriers' interaction and targets [148][149][150]. Todoroff et al.'s study shows that ligand binding is correlated with the roughness of the protein's surface, giving necessary background for future drug development [151]. Additionally, it can be applied to estimate the condition of specific tissues such as the cerebral cortex [151]. ...
... Todoroff et al.'s study shows that ligand binding is correlated with the roughness of the protein's surface, giving necessary background for future drug development [151]. Additionally, it can be applied to estimate the condition of specific tissues such as the cerebral cortex [151]. The box-counting algorithm is also used to find patterns on a dendritic model by analyzing 2D images based on the hypothesis that dendrites are of the same dimension [99]. ...
Even though the promising therapies against cancer are rapidly improved, the oncology patients population has seen exponential growth, placing cancer in 5th place among the ten deadliest diseases. Efficient drug delivery systems must overcome multiple barriers and maximize drug delivery to the target tumors, limiting the side effects simultaneously. Since the first observation of the quantum tunneling phenomenon, many multidisciplinary studies have offered quantum-inspired solutions to optimized tumor mapping and efficient nanodrug design. The property of a wave function to propagate through a potential barrier offer the capability of obtaining 3D surface profiles using imaging of individual atoms on the surface of a material. The application of quantum tunneling on a scanning tunneling microscope offers an exact surface roughness mapping of tumors and pharmaceutical particles. Critical elements to cancer nanotherapeutics apply the fractal theory and calculate the fractal dimension for efficient tumor surface imaging at the atomic level. This review study presents the latest biological approaches to cancer management based on fractal geometry.
... Furthermore, biological structures present complex fractal patterns similar to brain structures (Hofman, 1991;Varley et al., 2020). Therefore, depending on the image resolution and noise, FD may evaluate the condition of specific tissues such as the cerebral cortex (Blanton et al., 2001;Todoroff et al., 2014). In addition, the FD can influence the neurodegeneration analysis progression (Carstensen and Franchini, 1993;Jelinek and Fernandez, 1998;Pereira, 2010). ...
A few methods and tools are available for the quantitative measurement of the brain volume targeting mainly brain volume loss. However, several factors, such as the clinical conditions, the time of the day, the type of MRI machine, the brain volume artifacts, the pseudoatrophy, and the variations among the protocols, produce extreme variations leading to misdiagnosis of brain atrophy. While brain white matter loss is a characteristic lesion during neurodegeneration, the main objective of this study was to create a computational tool for high precision measuring structural brain changes using the fractal dimension (FD) definition. The validation of the BrainFD software is based on T1-weighted MRI images from the Open Access Series of Imaging Studies (OASIS)-3 brain database, where each participant has multiple MRI scan sessions. The software is based on the Python and JAVA programming languages with the main functionality of the FD calculation using the box-counting algorithm, for different subjects on the same brain regions, with high accuracy and resolution, offering the ability to compare brain data regions from different subjects and on multiple sessions, creating different imaging profiles based on the Clinical Dementia Rating (CDR) scores of the participants. Two experiments were executed. The first was a cross-sectional study where the data were separated into two CDR classes. In the second experiment, a model on multiple heterogeneous data was trained, and the FD calculation for each participant of the OASIS-3 database through multiple sessions was evaluated. The results suggest that the FD variation efficiently describes the structural complexity of the brain and the related cognitive decline. Additionally, the FD efficiently discriminates the two classes achieving 100% accuracy. It is shown that this classification outperforms the currently existing methods in terms of accuracy and the size of the dataset. Therefore, the FD calculation for identifying intracranial brain volume loss could be applied as a potential low-cost personalized imaging biomarker. Furthermore, the possibilities measuring different brain areas and subregions could give robust evidence of the slightest variations to imaging data obtained from repetitive measurements to Physicians and Radiologists.
... Any subtle but significant differences in structural characteristics such as symmetry of interaction between peptide dipoles, active chiral centers, charged amino acids and π-electron clouds and also the interaction between π-electron clouds and positively charged amino acids cannot be technically probed. Surface roughness measured as fractal dimensions is a local, dynamic and contact-dependent property that can play a crucial role in determining protein-protein interactions [30]. Therefore, we compared the extent of symmetry of surface roughness in binding sites, in the atomic organization of entire protein exterior and in the distribution of the above-said biophysical characteristics in these isoforms by fractal dimension (FD)-based analyses [31]. ...
Background:
Ras are small cellular GTPases which regulate diverse cellular processes. It has three isoforms: H-Ras, K-Ras, and N-Ras. Owing to the N-terminus (1-165 residues) sequence homology these isoforms were thought to be functionally redundant. However, only K-Ras-deficient mice but not H-Ras- and N-Ras-deficient mice show embryonic lethality. Similarly, mutations in a given Ras isoform are associated with a particular type of cancer. Moreover, we have previously reported that Ras isoforms perform unique functions in Leishmania major infection. Thus, Ras isoforms are implicated to have signaling and functional specificity but the mechanism remains to be elucidated.
Result:
Using CD40 as a model receptor, we showed that depending on the strength of signaling, specific Ras isoforms are activated. Weak CD40 signal activates N-Ras, whereas strong signal activates H-Ras and K-Ras. Additionally, we showed that suppression of N-Ras expression reduced CD40-induced extracellular signal-regulated kinase-1/2 (ERK-1/2) activation and Interleukin (IL)-10 production; whereas suppression of H-Ras or K-Ras reduced CD40-induced p38 mitogen-activated protein kinase (p38MAPK) activation and IL-12 production. Furthermore, we showed that Ras isoforms have activator (GEF) specificity as weak CD40 signal-activated N-Ras requires Sos-1/2 whereas strong CD40 signal-activated H-Ras/K-Ras requires Ras-GRP as the guanine-nucleotide exchange factor (GEF) inducing ERK-1/2- or p38MAPK-mediated IL-10 or IL-12 productions, respectively, in macrophages. Silencing of syk reduced CD40-induced N-Ras activation but silencing of lyn inhibited H-Ras and K-Ras activation. In CD40 signaling, Ras isoforms also showed effector specificity; while H-Ras and K-Ras showed specificity for phosphatidyl inositol-3 kinase activation at high dose of CD40 stimulation, N-Ras primarily associated with Raf-1 at low dose of CD40 stimulation. Moreover, fractal analysis showed that functional site surface roughness for H-Ras (SurfaceFD = 2.39) and K-Ras (SurfaceFD = 2.39) are similar but significantly different from N-Ras (SurfaceFD = 2.25).
Conclusion:
The activator and effector specificities of Ras isoforms in CD40 signaling indicated their differential involvement in CD40 pathway and in maintaining the reciprocity. Our observations reveal Ras-regulated signaling outcome and its potential for developing Ras isoform-targeted immunotherapy and prophylaxis.
... Prominent approaches in this group utilize the maximum possible overlap in the volume of pairs of molecules to compare and rank query molecules against a template compound. [9,10] Alignment-independent methods have been gaining in popularity, primarily owing to their speed. [11,12] These provide a straightforward description of the distance distribution between heavy atoms in a molecule and a set of fixed reference points, but fail to consider molecular surface curvature. ...
... To this end, we expand on the concepts elucidated in our earlier work in applying the concept of fractal dimensionality in fast, shape-based similaritybased virtual compound screening, and with minimal sampling of ligand conformational space. [9] This approach allows for the description of a molecule through shape analysis of its Connolly surface, [10] and for comparison of molecules through a simple distance measure. ...
Molecular shape and pharmacological function are interconnected. To capture shape, the fractal dimensionality concept was employed, providing a natural similarity measure for the virtual screening of de novo generated small molecules mimicking the structurally complex natural product (−)‐englerin A. Two of the top‐ranking designs were synthesized and tested for their ability to modulate transient receptor potential (TRP) cation channels which are cellular targets of (−)‐englerin A. Intracellular calcium assays and electrophysiological whole‐cell measurements of TRPC4 and TRPM8 channels revealed potent inhibitory effects of one of the computer‐generated compounds. Four derivatives of this identified hit compound had comparable effects on TRPC4 and TRPM8. The results of this study corroborate the use of fractal dimensionality as an innovative shape‐based molecular representation for molecular scaffold‐hopping.
... Similar fractal structure and function properties have also been suggested for proteoglycan networks e.g. in the cornea [95,96]. Furthermore, macromolecules in general have been proposed to have fractal properties with functional consequences, such as for molecular binding potential [97]. Likely, the extent of fractal scaling and degree of fractality would be relevant in many proteins and biological molecules, for specifying their function, localization, and interactions with other structures within the body. ...
Optimal levels of chaos and fractality are distinctly associated with physiological health and function in natural systems. Chaos is a type of nonlinear dynamics that tends to exhibit seemingly random structures, whereas fractality is a measure of the extent of organization underlying such structures. Growing bodies of work are demonstrating both the importance of chaotic dynamics for proper function of natural systems, as well as the suitability of fractal mathematics for characterizing these systems. Here, we review how measures of fractality that quantify the dose of chaos may reflect the state of health across various biological systems, including: brain, skeletal muscle, eyes and vision, lungs, kidneys, tumours, cell regulation, skin and wound repair, bone, vasculature, and the heart. We compare how reports of either too little or too much chaos and fractal complexity can be damaging to normal biological function, and suggest that aiming for the healthy dose of chaos may be an effective strategy for various biomedical applications. We also discuss rising examples of the implementation of fractal theory in designing novel materials, biomedical devices, diagnostics, and clinical therapies. Finally, we explain important mathematical concepts of fractals and chaos, such as fractal dimension, criticality, bifurcation, and iteration, and how they are related to biology. Overall, we promote the effectiveness of fractals in characterizing natural systems, and suggest moving towards using fractal frameworks as a basis for the research and development of better tools for the future of biomedical engineering.
... В таких работах объектами изучения обычно являются полимерные молекулы (пептиды и белки). При этом достаточно широкое распространение получили исследования структуры белков, связанные с расчётом фрактальных размерностей [7][8][9][10]. Однако анализу мономеров белковых молекул уделяется мало внимания. ...
A series of 20 proteinogenic amino acids was studied. Four types of fractal descriptors for 2 conformational states are calculated: α-helix and 1-strand β-sheet. Based on the analysis of the results obtained, it is established that when the conformational state of the amino acids (α-helix→β-sheet) changes, significant changes in the fractal descriptor Dtot, in the calculation of which all the atoms of the molecule are used, are not observed. However, the more specific descriptors Dval, Dvdw and Dunb, which reflect the aggregate of valence-coupled, van der Waals contact and unbound atoms, respectively, are more sensitive to the conformational transition. The increase Dval, Dvdw and the decrease Dunb values were established for a series of 7 amino acids.
... On the other hand, the concept of surface roughness expressed as FD was applied to address structure and function of G protein-coupled receptors. 19 In the protein-ligand interaction context, Todoroff et al 20 demonstrated that surface roughness correlates with ligand binding potential. ...
... Concerning that, it has been mainly used for identification of functional sites in proteins. 14,17,20 The use of FD for the study of protein-ligand interactions and the estimation of binding strength has been described in few works. 21,22 Our work focuses on this less explored aspect of FD. ...
The calculation of absolute binding affinities for protein-inhibitor complexes remains as one of the main challenges in computational structure-based ligand design. The present work explored the calculations of surface fractal dimension (as a measure of surface roughness) and the relationship with experimental binding free energies of Plasmepsin II complexes. Plasmepsin II is an attractive target for novel therapeutic compounds to treat malaria. However, the structural flexibility of this enzyme is a drawback when searching for specific inhibitors. Concerning that, we performed separate explicitly solvated molecular dynamics simulations using the available high-resolution crystal structures of different Plasmepsin II complexes. Molecular dynamics simulations allowed a better approximation to systems dynamics and, therefore, a more reliable estimation of surface roughness. This constitutes a novel approximation in order to obtain more realistic values of fractal dimension, because previous works considered only x-ray structures. Binding site fractal dimension was calculated considering the ensemble of structures generated at different simulation times. A linear relationship between binding site fractal dimension and experimental binding free energies of the complexes was observed within 20 ns. Previous studies of the subject did not uncover this relationship. Regression model, coined FD model, was built to estimate binding free energies from binding site fractal dimension values. Leave-one-out cross-validation showed that our model reproduced accurately the absolute binding free energies for our training set (R2 = 0.76; <|error|> =0.55 kcal/mol; SDerror = 0.19 kcal/mol). The fact that such a simple model may be applied raises some questions that are addressed in the article.
... The surface representation (b) has the same orientation as in (a). Temperature coloring is according to the computed "ligandability" [45,64]. A potential ligand interaction "hot spot" is predicted inside the active site (approximated by the dashed circle). ...
Recent studies have demonstrated that the bacterial chaperone and serine protease high temperature requirement A (HtrA) is closely associated with the establishment and progression of several infectious diseases. HtrA activity enhances bacterial survival under stress conditions, but also has direct effects on functions of the cell adhesion protein E-cadherin and extracellular matrix proteins, including fibronectin and proteoglycans. Although HtrA cannot be considered as a pathogenic factor per se, it exhibits favorable characteristics making HtrA a potentially attractive drug target to combat various bacterial infections.
... Similarly, multiparametric fractals have automorphism groups generated by multiple generators. Currently, fractals play a central role in numerous scientific concepts [46,[52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70], which shed light on the construction of many natural objects including complex physicochemical [56][57][58][59][60][61][62][63] and biological phenomena [64][65][66][67][68][69][70]. Here, recall also that semifractals allow (only) a limited scaling. ...
... Similarly, multiparametric fractals have automorphism groups generated by multiple generators. Currently, fractals play a central role in numerous scientific concepts [46,[52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70], which shed light on the construction of many natural objects including complex physicochemical [56][57][58][59][60][61][62][63] and biological phenomena [64][65][66][67][68][69][70]. Here, recall also that semifractals allow (only) a limited scaling. ...
Symmetry is one of the most fundamental concepts of all nature. We discuss the general theory of symmetry based on theory of semigroups. This includes and extends the theory of groups, which is by now broadly used as an algebraic tool for studying many types of symmetries observed by people. However, symmetry groups are only a particular case of more general, symmetry semigroups. The latter are in turn an instance of an even more general groupoids of symmetry, which may further be a topic of special consideration.
