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Register a new userThe topological filtration of $gamma$-structures
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In this paper we study $gamma$-structures filtered by topological genus. $gamma$-structures are a class of RNA pseudoknot structures that plays a key role in the context of polynomial time folding of RNA pseudoknot structures. A $gamma$-structure is composed by specific building blocks, that have topological genus less than or equal to $gamma$, where composition means concatenation and nesting of such blocks. Our main results are the derivation of a new bivariate generating function for $gamma$-structures via symbolic methods, the singularity analysis of the solutions and a central limit theorem for the distribution of topological genus in $gamma$-structures of given length. In our derivation specific bivariate polynomials play a central role. Their coefficients count particular motifs of fixed topological genus and they are of relevance in the context of genus recursion and novel folding algorithms.
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A topological RNA structure is derived from a diagram and its shape is obtained by collapsing the stacks of the structure into single arcs and by removing any arcs of length one. Shapes contain key topological, information and for fixed topological genus there exist only finitely many such shapes. We shall express topological RNA structures as unicellular maps, i.e. graphs together with a cyclic ordering of their half-edges. In this paper we prove a bijection of shapes of topological RNA structures. We furthermore derive a linear time algorithm generating shapes of fixed topological genus. We derive explicit expressions for the coefficients of the generating polynomial of these shapes and the generating function of RNA structures of genus $g$. Furthermore we outline how shapes can be used in order to extract essential information of RNA structure databases.
In this paper we study properties of topological RNA structures, i.e.~RNA contact structures with cross-serial interactions that are filtered by their topological genus. RNA secondary structures within this framework are topological structures having genus zero. We derive a new bivariate generating function whose singular expansion allows us to analyze the distributions of arcs, stacks, hairpin- , interior- and multi-loops. We then extend this analysis to H-type pseudoknots, kissing hairpins as well as $3$-knots and compute their respective expectation values. Finally we discuss our results and put them into context with data obtained by uniform sampling structures of fixed genus.
In this paper, we develop a quantitative comparison method for two arbitrary protein structures. This method uses a root-mean-square deviation (RMSD) characterization and employs a series expansion of the proteins shape function in terms of the Wigner-D functions to define a new criterion, which is called a similarity value. We further demonstrate that the expansion coefficients for the shape function obtained with the help of the Wigner-D functions correspond to structure factors. Our method addresses the common problem of comparing two proteins with different numbers of atoms. We illustrate it with a worked example.
Currently, there is no effective antiviral drugs nor vaccine for coronavirus disease 2019 (COVID-19) caused by acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Due to its high conservativeness and low similarity with human genes, SARS-CoV-2 main protease (M$^{text{pro}}$) is one of the most favorable drug targets. However, the current understanding of the molecular mechanism of M$^{text{pro}}$ inhibition is limited by the lack of reliable binding affinity ranking and prediction of existing structures of M$^{text{pro}}$-inhibitor complexes. This work integrates mathematics and deep learning (MathDL) to provide a reliable ranking of the binding affinities of 92 SARS-CoV-2 M$^{text{pro}}$ inhibitor structures. We reveal that Gly143 residue in M$^{text{pro}}$ is the most attractive site to form hydrogen bonds, followed by Cys145, Glu166, and His163. We also identify 45 targeted covalent bonding inhibitors. Validation on the PDBbind v2016 core set benchmark shows the MathDL has achieved the top performance with Pearsons correlation coefficient ($R_p$) being 0.858. Most importantly, MathDL is validated on a carefully curated SARS-CoV-2 inhibitor dataset with the averaged $R_p$ as high as 0.751, which endows the reliability of the present binding affinity prediction. The present binding affinity ranking, interaction analysis, and fragment decomposition offer a foundation for future drug discovery efforts.
We show that the Friedlander-Mazur conjecture holds for a sequence of products of projective varieties such as the product of a smooth projective curve and a smooth projective surface, the product of two smooth projective surfaces, the product of arbitrary number of smooth projective curves. Moreover, we show that the Friedlander-Mazur conjecture is stable under a surjective map. As applications, we show that the Friedlander-Mazur conjecture holds for the Jacobian variety of smooth projective curves, uniruled threefolds and unirational varieties up to certain range.
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