Classical Homology Research Articles

A Multi-Angle Approach to Predict Peptide-GPCR Complexes: The N/OFQ-NOP System as a Successful AlphaFold Application Case Study.

With nearly 700 structures solved and a growing number of customized structure prediction algorithms being developed at a fast pace, G protein-coupled receptors (GPCRs) are an optimal test case for validating new approaches for the prediction of receptor active state and ligand bioactive conformation complexes. In this study, we leveraged the availability of hundreds of peptide GPCRs in the active state and both classical homology and artificial intelligence (AI) based protein modeling combined with docking and AI-based peptide structure prediction approaches to predict the nociceptin/orphanin FQ-NOP receptor active state complex (N/OFQ-NOPa). The In Silico generated hypotheses were validated via the design, synthesis, and pharmacological characterization of novel linear N/OFQ(1-13)-NH2 analogues, leading to the discovery of a novel antagonist (3B; pKB = 6.63) bearing a single ring-constrained residue in place of the Gly2-Gly3 motif of the N/OFQ message sequence (FGGF). While the experimental validation was ongoing, the availability of the Cryo-EM structure of the predicted complex enabled us to unambiguously validate the generated hypotheses. To the best of our knowledge, this is the first example of a peptide-GPCR complex predicted with atomistic accuracy (full complex Cα RMSD < 1.0 Å) and of the N/OFQ message moiety being successfully modified with a rigid scaffold.

Dynamical obstructions to classification by (co)homology and other TSI-group invariants

In the spirit of Hjorth’s turbulence theory, we introduce “unbalancedness”: a new dynamical obstruction to classifying orbit equivalence relations by actions of Polish groups which admit a two-sided invariant metric (TSI). Since abelian groups are TSI, unbalancedness can be used for identifying which classification problems cannot be solved by classical homology and cohomology theories. In terms of applications, we show that Morita equivalence of continuous-trace C ∗ C^* -algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by actions of TSI groups. In the process, we show that the Wreath product of any two non-compact subgroups of S ∞ S_{\infty } admits an action whose orbit equivalence relation is generically ergodic against any action of a TSI group and we deduce that there is an orbit equivalence relation of a CLI group which is not classifiable by actions of TSI groups.

Comparisons between small ribosomal RNA and theoretical minimal RNA ring secondary structures confirm phylogenetic and structural accretion histories

Ribosomal RNAs are complex structures that presumably evolved by tRNA accretions. Statistical properties of tRNA secondary structures correlate with genetic code integration orders of their cognate amino acids. Ribosomal RNA secondary structures resemble those of tRNAs with recent cognates. Hence, rRNAs presumably evolved from ancestral tRNAs. Here, analyses compare secondary structure subcomponents of small ribosomal RNA subunits with secondary structures of theoretical minimal RNA rings, presumed proto-tRNAs. Two independent methods determined different accretion orders of rRNA structural subelements: (a) classical comparative homology and phylogenetic reconstruction, and (b) a structural hypothesis assuming an inverted onion ring growth where the three-dimensional ribosome’s core is most ancient and peripheral elements most recent. Comparisons between (a) and (b) accretions orders with RNA ring secondary structure scales show that recent rRNA subelements are: 1. more like RNA rings with recent cognates, indicating ongoing coevolution between tRNA and rRNA secondary structures; 2. less similar to theoretical minimal RNA rings with ancient cognates. Our method fits (a) and (b) in all examined organisms, more with (a) than (b). Results stress the need to integrate independent methods. Theoretical minimal RNA rings are potential evolutionary references for any sequence-based evolutionary analyses, independent of the focal data from that study.

Some remarks on hypergestural homology of spaces and its relation to classical homology

Classical homology of a topological space provides invariants of the space by means of triangulation or squaring made up from singular simplices (simplicial homology) or singular cubes (cubical homology) in the space. In much the same way, Mazzola's hypergestural homology intends to associate invariants to topological categories and, in particular, topological spaces by means of approximation with hypergestures playing the role of singular simplices and singular cubes. In this article, we locate Mazzola's hypergestural homology as a special kind of abstract cubical homology and propose two variations of Mazzola's construction, corresponding to simple geometric and physical interpretations of boundaries of hypergestures. Moreover, we discuss the relationship between hypergestural homology and classical cubical homology and prove that in many cases, one of our hypergestural homologies is invariant under homotopy equivalence of spaces, which is the main result of the article. Also, based on some examples, several structural improvements of hypergestural homology are suggested. However, one of these examples suggests that hypergestural homology could provide combinatorial information about a topological space beyond classical homology. Our computations are based on an explicit presentation of hypergestures, not included in previous works on gesture theory. This article has an Online Supplement, in which we expose some technical details, including the proof of the main result.

Homology of path complexes and hypergraphs

The path complex and its homology were defined in previous papers of the authors. The notion of a path complex is a natural discrete generalization of the notion of a simplicial complex. The theory of path complexes contains homotopy invariant homology theory of digraphs and (nondirected) graphs.In the paper we study the homology theory of path complexes. In particular, we describe functorial properties of paths complexes, introduce the notion of homotopy for path complexes and prove the homotopy invariance of path homology groups. We prove also several theorems that are similar to the results of classical homology theory of simplicial complexes. Then we apply this approach to construct homology theories on various categories of hypergraphs. We describe basic properties of these homology theories and relations between them. As a particular case, these results give new homology theories on the category of simplicial complexes.

Tropical Coordinates on the Space of Persistence Barcodes

The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e., a collection of intervals, to a finite metric space. Because of the nature of the invariant, barcodes are not well adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes, and these outputs can then be used as input to standard algorithms. It is the purpose of this paper to identify tropical coordinates on the space of barcodes and prove that they are stable with respect to the bottleneck distance and Wasserstein distances.

Leibniz and Hochschild homology

ABSTRACTWe construct and study the map from Leibniz homology HL∗(𝔥) of an abelian extension 𝔥 of a simple real Lie algebra 𝔤 to the Hochschild homology HH∗−1(U(𝔥)) of the universal envelopping algebra U(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.

Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation

This paper deals with left non-degenerate set-theoretic solutions to the Yang–Baxter equation (= LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution there is associated a shelf (i.e., a self-distributive structure) which captures its major properties. We consider two (co)homology theories for LND solutions, one of which was previously known, in a reduced form, for biracks only. An explicit isomorphism between these theories is described. For groups and racks we recover their classical (co)homology, whereas for cycle sets we get new constructions. For a certain type of LND solutions, including quandles and non-degenerate cycle sets, the (co)homologies split into the degenerate and the normalized parts. We express 2-cocycles of our theories in terms of group cohomology, and, in the case of cycle sets, establish connexions with extensions. This leads to a construction of cycle sets with interesting properties.

Synthesis, evaluation and structure-activity relationship of new 3-carboxamide coumarins as FXIIa inhibitors

Inhibitors of the coagulation factor XIIa (FXIIa) are attractive to detail the roles of this protease in hemostasis and thrombosis, to suppress artifact due to contact pathway activation in blood coagulation assays, and they are promising as antithrombotic therapy. The 3-carboxamide coumarins have been previously described as small-molecular-weight FXIIa inhibitors. In this study, we report a structure-activity relationship (SAR) study around this scaffold with the aim to discover new selective FXIIa inhibitors with an improved physico-chemical profile. To better understand these SAR, docking experiments were undertaken. For this purpose, we built an original hybrid model of FXIIa. This model has the advantage to gather the best features from the recently published crystal structure of FXIIa in its zymogen form and a more classical homology model. Results with the hybrid model are encouraging as they help understanding the activity and selectivity of our best compounds.

Insight into the Binding Mode of Agonists of the Nicotinic Acetylcholine Receptor from Calculated Electron Densities.

Insect nicotinic acetylcholine receptors (nAChRs) are among the most prominent and most economically important insecticide targets. Thus, an understanding of the modes of binding of respective agonists is important for the design of specific compounds with favorable vertebrate profiles. In the case of nAChRs, the lack of available high-resolution X-ray structures leaves theoretical considerations as the only viable option. Starting from classical homology and docking approaches, binding mode hypotheses are created for five agonists of the nAChR, covering insecticides in the main group 4 of the Insecticide Resistance Action Committee (IRAC) mode of action (MoA) classification, namely, neonicotinoids, nicotine, sulfoxaflor, and butenolides. To better understand these binding modes, the topologies of calculated electron densities of small-model systems are analyzed in the framework of the quantum theory of atoms in molecules. The theoretically obtained modes of binding are very much in line with the biology-driven IRAC MoA classification of the investigated ligands.

Existence of minimum molar volumes (maximum packing densities) in morphotropic series of mixed oxides and fluorides

Plots of the cubic root of the formula volume, (V/Z)1/3, vs. cation radius have been built for several morphotropic series of mixed oxides and fluorides. These plots are linear for isostructural compounds but exhibit discontinuities at morphotropic transitions. The classical homology rule ‘larger cation – higher its coordination number (CN) – denser the structure’ is only valid (with few exceptions) for substitutions of relatively small cations. Substitutions of large cations usually lead to opposite volume changes, regardless of increase or decrease in CNs. As a result, the densest structure types often appear with ions of somewhat intermediate size in a series. The discovery of the post-perovskite silicate phase, which greatly influenced mineralogy and geophysics, could be predicted four decades earlier, if proper attention had been paid to the small negative deviation of the layered CaIrO3 from the plot for CaMO3 perovskites. An analysis of the volume relations reveals some errors and leads to several predictions.

The weight filtration for real algebraic varieties II: classical homology

We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with \(\mathbb Z _2\) coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration of Borel–Moore homology.

On the Quantum Homology of Real Lagrangians in Fano Toric Manifolds

We study the Lagrangian quantum homology of real parts of Fano toric manifolds of minimal Chern number at least 2, using coefficients in a ring of Laurent polynomials over Z/2Z. We show that these Lagrangians are wide, in the sense that their quantum homology is isomorphic as a module to their classical homology tensored with this ring. Moreover, we show that the quantum homology is isomorphic as a ring to the quantum homology of the ambient symplectic manifold.

Structural chemistry of A 2 MX 4 compounds (X = O, F) with isolated tetrahedral anions: search for the densest structure types

The packing density of various structures is important not only for understanding and the prediction of high-pressure phase transitions, but also because of its reported correlation with thermodynamic stability. Plotting the cube root of formula volume against the cation radii (R) for nine morphotropic series with isolated tetrahedral anions, A(2)MO(4) (M = Si, Ge, S, Se, Cr, Mn, Mo, W) and A(2)BeF(4), permits the comparison of packing densities for 13 structure types (about 80 individual compounds and several solid solutions) stable at (or near) ambient temperature. The spinel type is the densest. The next densest types are those of K(2)MoO(4), Tl(2)CrO(4), β-Ca(2)SiO(4), β-K(2)SO(4), Ag(2)CrO(4) and Sr(2)GeO(4). In three series (M = Ge, Mo, W) the densest type comes with somewhat intermediate values of R, and not the largest, in contrast to the classical homology rule. Another contradiction with traditional views is that some of the densest phases have abnormally low overall binding energies. The correlation between packing density and coordination number (CN) is better when CN of A counts entire MX(4) groups rather than individual X atoms; many, but not all, A(2)MX(4) structures have binary A(2)M analogues (of course, A and M are not necessarily the same in these structure types). The most frequent arrangement of A around M is of the Ni(2)In type: a (distorted) pentacapped trigonal prism.

Protein Kinases: Docking and Homology Modeling Reliability

A database of about 700 high-resolution kinase structures was used to test the reliability of 17 docking procedures (using six docking software packages) by means of self- and cross-docking studies. The analysis of about 80 000 docking calculations suggests that the docking of an unknown ligand into a kinase has a probability of only 30-37% to be a correct ligand pose. However, based on the hypothesis that docking calculations are more reliable if the ligand to be docked is similar to the ligand present in the complex from which the target docking protein has been extracted, we propose an automated procedure that is able to improve the docking accuracy, suggest the best protein for docking studies, and assess the statistical reliability of docking calculations. The results were also transferred to the homology modeling field and led us to propose an alternative strategy based on ligand similarity for the development of kinase models whose experimental structure was not known. Our results suggest that in many cases this approach can give better results than the classical homology modeling procedure based exclusively on the sequence homology.

On noncontractible compacta with trivial homology and homotopy groups

We construct an example of a Peano continuum X X such that: (i) X X is a one-point compactification of a polyhedron; (ii) X X is weakly homotopy equivalent to a point (i.e. π n ( X ) \pi _n(X) is trivial for all n ≥ 0 n \geq 0 ); (iii) X X is noncontractible; and (iv) X X is homologically and cohomologically locally connected (i.e. X X is an HLC and c l c clc space). We also prove that all classical homology groups (singular, Čech, and Borel-Moore), all classical cohomology groups (singular and Čech), and all finite-dimensional Hawaiian groups of X X are trivial.

Simploidals sets: Definitions, operations and comparison with simplicial sets

The combinatorial structure of simploidal sets generalizes both simplicial complexes and cubical complexes. More precisely, cells of simploidal sets are cartesian product of simplices. This structure can be useful for geometric modeling (e.g. for handling hybrid meshes) or image analysis (e.g. for computing topological properties of parts of n -dimensional images). In this paper, definitions and basic constructions are detailed. The homology of simploidal sets is defined and it is shown to be equivalent to the classical homology. It is also shown that products of Bézier simplicial patches are well suited for the embedding of simploidal sets.

On the (non)-coincidence of Milnor–Thurston homology theory with singular homology theory

The paper investigates a homology theory based on the ideas of Milnor and Thurston that by considering measures on the set of all singular simplices one should get alternate possibilities for describing the cycles of classical homology theory. It suggests slight changes to Milnor’s and Thurston’s original definitions (giving differences for wild topological spaces only) which ensure that their homology theory is well-defined on all topological spaces. It further proves that Milnor-Thurston homology theory gives the same homology groups as the singular homology theory with real coefficients for all triangulable spaces. An example showing that the coincidence between these both homology theories does not hold for all topological spaces is also included.

Set-valued homology

We establish homology groups based on a class of set-valued mappings and prove they are isomorphic to the classical homology groups. Any set-valued mapping in this class induces homomorphisms between homology groups of two topological spaces. Set-valued homology can be used to study set-valued mappings between two spaces whose homology groups are already known. For instance, it follows a version of the Kakutani Fixed Point Theorem in the nonconvex and nonacyclic case. We can get some other properties of set-valued mappings. Besides, applying set-valued homology, we can get a result for ordinary single-valued mappings, which is analogous to the Vietoris Mapping Theorem.

Homology with Models and Tor

We consider the extension of the notion of a projective module to that of a projective functor relative to a model set (as in Dold, MacLane, Oberst, 1967). Then taking projective resolutions of functors, we consider the usual associated homology. We show that in some cases, including the classical simplicial homology of topological spaces, the model set can be replaced by a model set having only one element. We show that when the model set consists of a single element the homology modules can be interpreted as values of the Torsion functor. In the case of simplicial homology of topological spaces these Tors will be shown to be analogous to the Tors which occur in group homology.