Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.Comment: v.2, 39 pages. To appear in Invent. Mat

CMASA: an accurate algorithm for detecting local protein structural similarity and its application to enzyme catalytic site annotation

<p>Abstract</p> <p>Background</p> <p>The rapid development of structural genomics has resulted in many "unknown function" proteins being deposited in Protein Data Bank (PDB), thus, the functional prediction of these proteins has become a challenge for structural bioinformatics. Several sequence-based and structure-based methods have been developed to predict protein function, but these methods need to be improved further, such as, enhancing the accuracy, sensitivity, and the computational speed. Here, an accurate algorithm, the CMASA (Contact MAtrix based local Structural Alignment algorithm), has been developed to predict unknown functions of proteins based on the local protein structural similarity. This algorithm has been evaluated by building a test set including 164 enzyme families, and also been compared to other methods.</p> <p>Results</p> <p>The evaluation of CMASA shows that the CMASA is highly accurate (0.96), sensitive (0.86), and fast enough to be used in the large-scale functional annotation. Comparing to both sequence-based and global structure-based methods, not only the CMASA can find remote homologous proteins, but also can find the active site convergence. Comparing to other local structure comparison-based methods, the CMASA can obtain the better performance than both FFF (a method using geometry to predict protein function) and SPASM (a local structure alignment method); and the CMASA is more sensitive than PINTS and is more accurate than JESS (both are local structure alignment methods). The CMASA was applied to annotate the enzyme catalytic sites of the non-redundant PDB, and at least 166 putative catalytic sites have been suggested, these sites can not be observed by the Catalytic Site Atlas (CSA).</p> <p>Conclusions</p> <p>The CMASA is an accurate algorithm for detecting local protein structural similarity, and it holds several advantages in predicting enzyme active sites. The CMASA can be used in large-scale enzyme active site annotation. The CMASA can be available by the mail-based server (<url>http://159.226.149.45/other1/CMASA/CMASA.htm</url>).</p

Direct observation of particle interactions and clustering in charged granular streams

Biological and Soft Matter Physic