On Additive Combinatorics of Permutations of Z(n)

Let Z(n) denote the ring of integers modulo n. A permutation of Z(n) is a sequence of n distinct elements of Z(n). Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of Z(n), namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s (n) and t (n) respectively. The case when n is even is trivial in both the cases, with s (n) = 1 and t (n) = n!. For n odd, we prove (n phi(n))/2(k) <= s(n) <= n!.2(-)(n-1)/2/((n-1)/2)! and 2 (n-1)/2 . (n-1/2)! <= t (n) <= 2(k) . (n-1)!/phi(n), where k is the number of distinct prime divisors of n and phi is the Euler's totient function.

An optimal parallel algorithm for sparse matrix bandwidth reduction on a linear array

A simple and efficient algorithm for the bandwidth reduction of sparse symmetric matrices is proposed. It involves column-row permutations and is well-suited to map onto the linear array topology of the SIMD architectures. The efficiency of the algorithm is compared with the other existing algorithms. The interconnectivity and the memory requirement of the linear array are discussed and the complexity of its layout area is derived. The parallel version of the algorithm mapped onto the linear array is then introduced and is explained with the help of an example. The optimality of the parallel algorithm is proved by deriving the time complexities of the algorithm on a single processor and the linear array.

Distinct Disulfide Isomers of mu-Conotoxins KIIIA and KIIIB Block Voltage-Gated Sodium Channels

In the preparation of synthetic conotoxins containing multiple disulfide bonds, oxidative folding can produce numerous permutations of disulfide bond connectivities. Establishing the native disulfide connectivities thus presents a significant challenge when the venom-derived peptide is not available, as is increasingly the case when conotoxins are identified from cDNA sequences. Here, we investigate the disulfide connectivity of mu-conotoxin KIIIA, which was predicted originally to have a C1-C9,C2-C15,C4-C16] disulfide pattern based on homology with closely related mu-conotoxins. The two major isomers of synthetic mu-KIIIA formed during oxidative folding were purified and their disulfide connectivities mapped by direct mass spectrometric collision-induced dissociation fragmentation of the disulfide-bonded polypeptides. Our results show that the major oxidative folding product adopts a C1-C15,C2-C9,C4-C16] disulfide connectivity, while the minor product adopts a C1-C16,C2-C9,C4-C15] connectivity. Both of these peptides were potent blockers of Na(v)1.2 (K-d values of 5 and 230 nM, respectively). The solution structure for mu-KIIIA based on nuclear magnetic resonance data was recalculated with the C1-C15,C2-C9,C4-C16] disulfide pattern; its structure was very similar to the mu-KIIIA structure calculated with the incorrect C1-C9,C2-C15,C4-C16] disulfide pattern, with an alpha-helix spanning residues 7-12. In addition, the major folding isomers of mu-KIIIB, an N-terminally extended isoform of mu-KIIIA, identified from its cDNA sequence, were isolated. These folding products had the same disulfide connectivities as mu-KIIIA, and both blocked Na(v)1.2 (K-d values of 470 and 26 nM, respectively). Our results establish that the preferred disulfide pattern of synthetic mu-KIIIA and mu-KIIIB folded in vitro is 1-5/2-4/3-6 but that other disulfide isomers are also potent sodium channel blockers. These findings raise questions about the disulfide pattern(s) of mu-KIIIA in the venom of Conus kinoshitai; indeed, the presence of multiple disulfide isomers in the venom could provide a means of further expanding the snail's repertoire of active peptides.

CLAP: A web-server for automatic classification of proteins with special reference to multi-domain proteins

Background: The function of a protein can be deciphered with higher accuracy from its structure than from its amino acid sequence. Due to the huge gap in the available protein sequence and structural space, tools that can generate functionally homogeneous clusters using only the sequence information, hold great importance. For this, traditional alignment-based tools work well in most cases and clustering is performed on the basis of sequence similarity. But, in the case of multi-domain proteins, the alignment quality might be poor due to varied lengths of the proteins, domain shuffling or circular permutations. Multi-domain proteins are ubiquitous in nature, hence alignment-free tools, which overcome the shortcomings of alignment-based protein comparison methods, are required. Further, existing tools classify proteins using only domain-level information and hence miss out on the information encoded in the tethered regions or accessory domains. Our method, on the other hand, takes into account the full-length sequence of a protein, consolidating the complete sequence information to understand a given protein better. Results: Our web-server, CLAP (Classification of Proteins), is one such alignment-free software for automatic classification of protein sequences. It utilizes a pattern-matching algorithm that assigns local matching scores (LMS) to residues that are a part of the matched patterns between two sequences being compared. CLAP works on full-length sequences and does not require prior domain definitions. Pilot studies undertaken previously on protein kinases and immunoglobulins have shown that CLAP yields clusters, which have high functional and domain architectural similarity. Moreover, parsing at a statistically determined cut-off resulted in clusters that corroborated with the sub-family level classification of that particular domain family. Conclusions: CLAP is a useful protein-clustering tool, independent of domain assignment, domain order, sequence length and domain diversity. Our method can be used for any set of protein sequences, yielding functionally relevant clusters with high domain architectural homogeneity. The CLAP web server is freely available for academic use at http://nslab.mbu.iisc.ernet.in/clap/.

Genome-wide analysis correlates Ayurveda Prakriti

The practice of Ayurveda, the traditional medicine of India, is based on the concept of three major constitutional types (Vata, Pitta and Kapha) defined as ``Prakriti''. To the best of our knowledge, no study has convincingly correlated genomic variations with the classification of Prakriti. In the present study, we performed genome-wide SNP (single nucleotide polymorphism) analysis (Affymetrix, 6.0) of 262 well-classified male individuals (after screening 3416 subjects) belonging to three Prakritis. We found 52 SNPs (p <= 1 x 10(-5)) were significantly different between Prakritis, without any confounding effect of stratification, after 10(6) permutations. Principal component analysis (PCA) of these SNPs classified 262 individuals into their respective groups (Vata, Pitta and Kapha) irrespective of their ancestry, which represent its power in categorization. We further validated our finding with 297 Indian population samples with known ancestry. Subsequently, we found that PGM1 correlates with phenotype of Pitta as described in the ancient text of Caraka Samhita, suggesting that the phenotypic classification of India's traditional medicine has a genetic basis; and its Prakriti-based practice in vogue for many centuries resonates with personalized medicine.