Mapping the distribution of packing topologies within protein interiors shows predominant preference for specific packing motifs

<p>Abstract</p> <p>Background</p> <p>Mapping protein primary sequences to their three dimensional folds referred to as the 'second genetic code' remains an unsolved scientific problem. A crucial part of the problem concerns the geometrical specificity in side chain association leading to densely packed protein cores, a hallmark of correctly folded native structures. Thus, any model of packing within proteins should constitute an indispensable component of protein folding and design.</p> <p>Results</p> <p>In this study an attempt has been made to find, characterize and classify recurring patterns in the packing of side chain atoms within a protein which sustains its native fold. The interaction of side chain atoms within the protein core has been represented as a contact network based on the surface complementarity and overlap between associating side chain surfaces. Some network topologies definitely appear to be preferred and they have been termed 'packing motifs', analogous to super secondary structures in proteins. Study of the distribution of these motifs reveals the ubiquitous presence of typical smaller graphs, which appear to get linked or coalesce to give larger graphs, reminiscent of the nucleation-condensation model in protein folding. One such frequently occurring motif, also envisaged as the unit of clustering, the three residue clique was invariably found in regions of dense packing. Finally, topological measures based on surface contact networks appeared to be effective in discriminating sequences native to a specific fold amongst a set of decoys.</p> <p>Conclusions</p> <p>Out of innumerable topological possibilities, only a finite number of specific packing motifs are actually realized in proteins. This small number of motifs could serve as a basis set in the construction of larger networks. Of these, the triplet clique exhibits distinct preference both in terms of composition and geometry.</p

Packing 3-vertex paths in claw-free graphs and related topics

An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the domination number of G. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it has no induced subgraph isomorphic to a claw. Our results include the following. Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G. Then (a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2) if v(G) = 1 mod 3, then G - x has an L-factor, (a3) if v(G) = 2 mod 3, then G - {x,y} has an L-factor, (a4) if v(G) = 0 mod 3 and G is either cubic or 4-connected, then G - P has an L-factor, (a5) if G is cubic with v(G) > 5 and E is a set of three edges in G, then G - E has an L-factor if and only if the subgraph induced by E in G is not a claw and not a triangle, (a6) if v(G) = 1 mod 3, then G - {v,e} has an L-factor for every vertex v and every edge e in G, (a7) if v(G) = 1 mod 3, then there exist a 4-vertex path N and a claw Y in G such that G - N and G - Y have L-factors, and (a8) d(G) < v(G)/3 +1 and if in addition G is not a cycle and v(G) = 1 mod 3, then d(G) < v(G)/3. We explore the relations between packing problems of a graph and its line graph to obtain some results on different types of packings. We also discuss relations between L-packing and domination problems as well as between induced L-packings and the Hadwiger conjecture. Keywords: claw-free graph, cubic graph, vertex disjoint packing, edge disjoint packing, 3-vertex factor, 3-vertex packing, path-factor, induced packing, graph domination, graph minor, the Hadwiger conjecture.Comment: 29 page

TS2PACK: A Two-Level Tabu Search for the Three-dimensional Bin Packing Problem

Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing one

Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

A 3-path vertex cover in a graph is a vertex subset $C$ such that every path of three vertices contains at least one vertex from $C$. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most $k$. In this paper, we give a kernel of $5k$ vertices and an $O^*(1.7485^k)$-time and polynomial-space algorithm for this problem, both new results improve previous known bounds.Comment: in TAMC 2016, LNCS 9796, 201

Scheduling Storms and Streams in the Cloud

Motivated by emerging big streaming data processing paradigms (e.g., Twitter Storm, Streaming MapReduce), we investigate the problem of scheduling graphs over a large cluster of servers. Each graph is a job, where nodes represent compute tasks and edges indicate data-flows between these compute tasks. Jobs (graphs) arrive randomly over time, and upon completion, leave the system. When a job arrives, the scheduler needs to partition the graph and distribute it over the servers to satisfy load balancing and cost considerations. Specifically, neighboring compute tasks in the graph that are mapped to different servers incur load on the network; thus a mapping of the jobs among the servers incurs a cost that is proportional to the number of "broken edges". We propose a low complexity randomized scheduling algorithm that, without service preemptions, stabilizes the system with graph arrivals/departures; more importantly, it allows a smooth trade-off between minimizing average partitioning cost and average queue lengths. Interestingly, to avoid service preemptions, our approach does not rely on a Gibbs sampler; instead, we show that the corresponding limiting invariant measure has an interpretation stemming from a loss system.Comment: 14 page