Mathematical topology and geometry-based classification of tauopathies.

Neurodegenerative diseases, like Alzheimer's, are associated with the presence of neurofibrillary lesions formed by tau protein filaments in the cerebral cortex. While it is known that different morphologies of tau filaments characterize different neurodegenerative diseases, there are few metrics of global and local structure complexity that enable to quantify their structural diversity rigorously. In this manuscript, we employ for the first time mathematical topology and geometry to classify neurodegenerative diseases by using cryo-electron microscopy structures of tau filaments that are available in the Protein Data Bank. By employing mathematical topology metrics (Gauss linking integral, writhe and second Vassiliev measure) we achieve a consistent, but more refined classification of tauopathies, than what was previously observed through visual inspection. Our results reveal a hierarchy of classification from global to local topology and geometry characteristics. In particular, we find that tauopathies can be classified with respect to the handedness of their global conformations and the handedness of the relative orientations of their repeats. Progressive supranuclear palsy is identified as an outlier, with a more complex structure than the rest, reflected by a small, but observable knotoid structure (a diagrammatic structure representing non-trivial topology). This topological characteristic can be attributed to a pattern in the beginning of the R3 repeat that is present in all tauopathies but at different extent. Moreover, by comparing single filament to paired filament structures within tauopathies we find a consistent change in the side-chain orientations with respect to the alpha carbon atoms at the area of interaction.

Following the entangled state of filaments.

California blackworms serve as a template for the topological design of active matter.

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.

The protein folding rate and the geometry and topology of the native state.

Proteins fold in 3-dimensional conformations which are important for their function. Characterizing the global conformation of proteins rigorously and separating secondary structure effects from topological effects is a challenge. New developments in applied knot theory allow to characterize the topological characteristics of proteins (knotted or not). By analyzing a small set of two-state and multi-state proteins with no knots or slipknots, our results show that 95.4% of the analyzed proteins have non-trivial topological characteristics, as reflected by the second Vassiliev measure, and that the logarithm of the experimental protein folding rate depends on both the local geometry and the topology of the protein's native state.

The local topological free energy of proteins.

Protein folding, the process by which proteins attain a 3-dimensional conformation necessary for their function, remains an important unsolved problem in biology. A major gap in our understanding is how local properties of proteins relate to their global properties. In this manuscript, we use the Writhe and Torsion to introduce a new local topological/geometrical free energy that can be associated to 4 consecutive amino acids along the protein backbone. By analyzing a culled protein dataset from the PDB, our results show that high local topological free energy conformations are independent of sequence and may be involved in the rate limiting step in protein folding. By analyzing a set of 2-state single domain proteins, we find that the total local topological free energy of these proteins correlates with the experimentally observed folding rates reported in Plaxco et al. (2000).

Oral health of overweight and obese children and adolescents: a comparative study with a multivariate analysis of risk indicators.

PURPOSE: The aim of this study was to investigate the oral health status of overweight and obese children compared to normal weight children and to explore the effect of risk indicators, including stress, on the oral health of the children. METHODS: For this cross-sectional study, 7-15-year-old children were enrolled: divided into a group of overweight and obese and a group of normal weight children. Parent-reported questionnaires were used to for data collection demographic data, children's medical and developmental history, brushing frequency and dietary habits. Participants' state and trait anxiety levels were evaluated using STAIC questionnaires while saliva cortisol was measured as a biomarker of stress. Dental Caries (DMFT/dmft), periodontal health (plaque index-PI and gingival index-GI), saliva flow rate and buffer capacity were recorded at the dental clinic by two calibrated examiners. Student's t-test, Pearson's r and Fisher's exact test were used for bivariate associations and backward stepwise multivariate regression analysis for each oral health outcome was performed (P ≤ 0.05). RESULTS: Overweight and obese children had significantly higher DMFT (3.92 ± 4.69 vs. 1.30 ± 2.49, P < 0.001), PI (0.9 ± 0.5 vs. 0.45 ± 0.52, P < 0,001) and GI (0.38 ± 0.39 vs. 0.18 ± 0.47, P < 0,001), salivary cortisol AUC (128.3 ± 55.5 vs. 99.4 ± 35.7, P = 0.004), lower saliva flow rate and decreased "high" saliva buffer capacity. There were no differences for the STAIC measures between the groups. Child's weight affected periodontal parameters but not dental caries, when controlling for other risk indicators. CONCLUSION: Oral status of overweight and obese children was worse compared to normal weight children and this did not correlate to stress variables.

Knot polynomials of open and closed curves.

In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

Theoretical prediction of an isotropic to nematic phase transition in bottlebrush homopolymer melts.

Bottlebrushes are an emerging class of polymers, characterized by a high density of side chains grafted to a linear backbone that offer promise in creating materials with unusual combinations of mechanical, chemical, and optoelectronic properties. Understanding the role of molecular architecture in the organization and assembly of bottlebrushes is of fundamental importance in polymer physics, but also enabling in applications. Here, we apply field-theoretic simulations to study the effect of grafting density, backbone length, and side-chain (SC) length on the structure and thermodynamics of bottlebrush homopolymer melts. Our results provide evidence for a phase transition from an isotropic to a nematic state with increasing grafting density and side-chain length. Variation in the backbone length is also observed to influence the location of the transition, primarily for short polymers just above the star to bottlebrush transition.

Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity.

We develop topological methods for characterizing the relationship between polymer chain entanglement and bulk viscoelastic responses. We introduce generalized Linking Number and Writhe characteristics that are applicable to open linear chains. We investigate the rheology of polymeric chains entangled into weaves with varying topologies and levels of chain density. To investigate viscoelastic responses, we perform non-equilibrium molecular simulations over a range of frequencies using sheared Lees⁻Edwards boundary conditions. We show how our topological characteristics can be used to capture key features of the polymer entanglements related to the viscoelastic responses. We find there is a linear relation over a significant range of frequencies between the mean absolute Writhe W r and the Loss Tangent tan ( δ ) . We also find an approximate inverse linear relationship between the mean absolute Periodic Linking Number L K P and the Loss Tangent tan ( δ ) . Our results show some of the ways topological methods can be used to characterize chain entanglements to better understand the origins of mechanical responses in polymeric materials.