A linear programming approach for estimating the structure of a sparse linear genetic network from transcript profiling data

Background: A genetic network can be represented as a directed graph in which a node corresponds to a gene and a directed edge specifies the direction of influence of one gene on another. The reconstruction of such networks from transcript profiling data remains an important yet challenging endeavor. A transcript profile specifies the abundances of many genes in a biological sample of interest. Prevailing strategies for learning the structure of a genetic network from high-dimensional transcript profiling data assume sparsity and linearity. Many methods consider relatively small directed graphs, inferring graphs with up to a few hundred nodes. This work examines large undirected graphs representations of genetic networks, graphs with many thousands of nodes where an undirected edge between two nodes does not indicate the direction of influence, and the problem of estimating the structure of such a sparse linear genetic network (SLGN) from transcript profiling data. Results: The structure learning task is cast as a sparse linear regression problem which is then posed as a LASSO (l1-constrained fitting) problem and solved finally by formulating a Linear Program (LP). A bound on the Generalization Error of this approach is given in terms of the Leave-One-Out Error. The accuracy and utility of LP-SLGNs is assessed quantitatively and qualitatively using simulated and real data. The Dialogue for Reverse Engineering Assessments and Methods (DREAM) initiative provides gold standard data sets and evaluation metrics that enable and facilitate the comparison of algorithms for deducing the structure of networks. The structures of LP-SLGNs estimated from the INSILICO1, INSILICO2 and INSILICO3 simulated DREAM2 data sets are comparable to those proposed by the first and/or second ranked teams in the DREAM2 competition. The structures of LP-SLGNs estimated from two published Saccharomyces cerevisae cell cycle transcript profiling data sets capture known regulatory associations. In each S. cerevisiae LP-SLGN, the number of nodes with a particular degree follows an approximate power law suggesting that its degree distributions is similar to that observed in real-world networks. Inspection of these LP-SLGNs suggests biological hypotheses amenable to experimental verification. Conclusion: A statistically robust and computationally efficient LP-based method for estimating the topology of a large sparse undirected graph from high-dimensional data yields representations of genetic networks that are biologically plausible and useful abstractions of the structures of real genetic networks. Analysis of the statistical and topological properties of learned LP-SLGNs may have practical value; for example, genes with high random walk betweenness, a measure of the centrality of a node in a graph, are good candidates for intervention studies and hence integrated computational – experimental investigations designed to infer more realistic and sophisticated probabilistic directed graphical model representations of genetic networks. The LP-based solutions of the sparse linear regression problem described here may provide a method for learning the structure of transcription factor networks from transcript profiling and transcription factor binding motif data.

An upper bound for Cubicity in terms of Boxicity

An axis-parallel b-dimensional box is a Cartesian product R-1 x R-2 x ... x R-b where each R-i (for 1 <= i <= b) is a closed interval of the form [a(i), b(i)] on the real line. The boxicity of any graph G, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis-parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R-1 x R-2 x ... x R-b, where each R-i (for 1 <= i <= b) is a closed interval of the form [a(i), a(i) + 1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by cub(G)). In this paper we prove that cub(G) <= inverted right perpendicularlog(2) ninverted left perpendicular box(G), where n is the number of vertices in the graph. We also show that this upper bound is tight.Some immediate consequences of the above result are listed below: 1. Planar graphs have cubicity at most 3inverted right perpendicularlog(2) ninvereted left perpendicular.2. Outer planar graphs have cubicity at most 2inverted right perpendicularlog(2) ninverted left perpendicular.3. Any graph of treewidth tw has cubicity at most (tw + 2) inverted right perpendicularlog(2) ninverted left perpendicular. Thus, chordal graphs have cubicity at most (omega + 1) inverted right erpendicularlog(2) ninverted left perpendicular and circular arc graphs have cubicity at most (2 omega + 1)inverted right perpendicularlog(2) ninverted left perpendicular, where omega is the clique number.

Conductivity Scale In Disordered-Systems

A reanalysis of the correction to the Boltzmann conductivity due to maximally crossed graphs for degenerate bands explains why the conductivity scale in many-valley semiconductors is an order of magnitude higher than Mott's "minimum metallic conductivity." With the use of a reasonable assumption for the Boltzmann mean free path, the lowest-order perturbation theory is seen to give a remarkably good, semiquantitative, description of the conductivity variation in both uncompensated doped semiconductors and amorphous alloys.

Computerized synthesis of the structure of geared kinematic chains

The paper presents for the first time a fully computerized method for structural synthesis of geared kinematic chains which can be used to derive epicyclic gear drives. The method has been formulated on the basis of representing these chains by their graphs, the graphs being in turn represented algebraically by their vertex-vertex incidence matrices. It has thus been possible to make advantageous use of concepts and results from graph theory to develop a method amenable for implementation on a digital computer. The computerized method has been applied to the structural synthesis of single-freedom geared kinematic chains with up to four gear pairs, and the results obtained thereform are presented and discussed.

Analytical studies of gain optimization in CO2-N2 gasdynamic lasers employing two-dimensional wedge nozzles

An analytical method has been proposed to optimise the small-signaloptical gain of CO2-N2 gasdynamic lasers (gdl) employing two-dimensional (2D) wedge nozzles. Following our earlier work the equations governing the steady, inviscid, quasi-one-dimensional flow in the wedge nozzle of thegdl are reduced to a universal form so that their solutions depend on a single unifying parameter. These equations are solved numerically to obtain similar solutions for the various flow quantities, which variables are subsequently used to optimize the small-signal-gain. The corresponding optimum values like reservoir pressure and temperature and 2D nozzle area ratio also have been predicted and graphed for a wide range of laser gas compositions, with either H2O or He as the catalyst. A large number of graphs are presented which may be used to obtain the optimum values of small signal gain for a wide range of laser compositions without further computations.

Bending Of Circular Plate On Elastic-Foundation

The governing differential equation of linear, elastic, thin, circular plate of uniform thickness, subjected to uniformly distributed load and resting on Winkler-Pasternak type foundation is solved using ``Chebyshev Polynomials''. Analysis is carried out using Lenczos' technique, both for simply supported and clamped plates. Numerical results thus obtained by perturbing the differential equation for plates without foundation are compared and are found to be in good agreement with the available results. The effect of foundation on central deflection of the plate is shown in the form of graphs.

Cubicity, boxicity, and vertex cover

A k-dimensional box is the cartesian product R-1 x R-2 x ... x R-k where each R-i is a closed interval on the real line. The boxicity of a graph G,denoted as box(G), is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the cartesian product R-1 x R-2 x ... x R-k where each Ri is a closed interval on the real line of the form [a(i), a(i) + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G) <= t + inverted right perpendicularlog(n - t)inverted left perpendicular - 1 and box(G) <= left perpendiculart/2right perpendicular + 1, where t is the cardinality of a minimum vertex cover of G and n is the number of vertices of G. We also show the tightness of these upper bounds. F.S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph G, box(G) <= left perpendicularn/2right perpendicular and cub(G) <= inverted right perpendicular2n/3inverted left perpendicular, where n is the number of vertices of G, and these bounds are tight. We show that if G is a bipartite graph then box(G) <= inverted right perpendicularn/4inverted left perpendicular and this bound is tight. We also show that if G is a bipartite graph then cub(G) <= n/2 + inverted right perpendicularlog n inverted left perpendicular - 1. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to n/4. Interestingly, if boxicity is very close to n/2, then chromatic number also has to be very high. In particular, we show that if box(G) = n/2 - s, s >= 0, then chi (G) >= n/2s+2, where chi (G) is the chromatic number of G.

Intra and Inter-Molecular Communications Through Protein Structure Network

Communication within and across proteins is crucial for the biological functioning of proteins. Experiments such as mutational studies on proteins provide important information on the amino acids, which are crucial for their function. However, the protein structures are complex and it is unlikely that the entire responsibility of the function rests on only a few amino acids. A large fraction of the protein is expected to participate in its function at some level or other. Thus, it is relevant to consider the protein structures as a completely connected network and then deduce the properties, which are related to the global network features. In this direction, our laboratory has been engaged in representing the protein structure as a network of non-covalent connections and we have investigated a variety of problems in structural biology, such as the identification of functional and folding clusters, determinants of quaternary association and characterization of the network properties of protein structures. We have also addressed a few important issues related to protein dynamics, such as the process of oligomerization in multimers, mechanism on protein folding, and ligand induced communications (allosteric effect). In this review we highlight some of the investigations which we have carried out in the recent past. A review on protein structure graphs was presented earlier, in which the focus was on the graphs and graph spectral properties and their implementation in the study of protein structure graphs/networks (PSN). In this article, we briefly summarize the relevant parts of the methodology and the focus is on the advancement brought out in the understanding of protein structure-function relationships through structure networks. The investigations of structural/biological problems are divided into two parts, in which the first part deals with the analysis of PSNs based on static structures obtained from x-ray crystallography. The second part highlights the changes in the network, associated with biological functions, which are deduced from the network analysis on the structures obtained from molecular dynamics simulations.

In quest of a reliable and efficient computational test for detection of isomorphism in kinematic chains

The test based on comparison of the characteristic coefficients of the adjancency matrices of the corresponding graphs for detection of isomorphism in kinematic chains has been shown to fail in the case of two pairs of ten-link, simple-jointed chains, one pair corresponding to single-freedom chains and the other pair corresponding to three-freedom chains. An assessment of the merits and demerits of available methods for detection of isomorphism in graphs and kinematic chains is presented, keeping in view the suitability of the methods for use in computerized structural synthesis of kinematic chains. A new test based on the characteristic coefficients of the “degree” matrix of the corresponding graph is proposed for detection of isomorphism in kinematic chains. The new test is found to be successful in the case of a number of examples of graphs where the test based on characteristic coefficients of adjancency matrix fails. It has also been found to be successful in distinguishing the structures of all known simple-jointed kinematic chains in the categories of (a) single-freedom chains with up to 10 links, (b) two-freedom chains with up to 9 links and (c) three-freedom chains with up to 10 links.

A study of the effect of N2 reservoir temperature on a 16 μm CO2-N2 downstream-mixing gasdynamic laser

A study of the effect of N2 reservoir temperature on the small-signal gain in a downstream-mixing 16 μm CO2-N2 GDL is presented. It is shown that the small-signal gain decreases with the increase of N2 reservoir temperature. The conditions for reversing this trend are discussed and the results are presented in the form of graphs.

Synergistic Execution of Stream Programs on Multicores with Accelerators

The StreamIt programming model has been proposed to exploit parallelism in streaming applications oil general purpose multicore architectures. The StreamIt graphs describe task, data and pipeline parallelism which can be exploited on accelerators such as Graphics Processing Units (GPUs) or CellBE which support abundant parallelism in hardware. In this paper, we describe a novel method to orchestrate the execution of if StreamIt program oil a multicore platform equipped with an accelerator. The proposed approach identifies, using profiling, the relative benefits of executing a task oil the superscalar CPU cores and the accelerator. We formulate the problem of partitioning the work between the CPU cores and the GPU, taking into account the latencies for data transfers and the required buffer layout transformations associated with the partitioning, as all integrated Integer Linear Program (ILP) which can then be solved by an ILP solver. We also propose an efficient heuristic algorithm for the work-partitioning between the CPU and the GPU, which provides solutions which are within 9.05% of the optimal solution on an average across the benchmark Suite. The partitioned tasks are then software pipelined to execute oil the multiple CPU cores and the Streaming Multiprocessors (SMs) of the GPU. The software pipelining algorithm orchestrates the execution between CPU cores and the GPU by emitting the code for the CPU and the GPU, and the code for the required data transfers. Our experiments on a platform with 8 CPU cores and a GeForce 8800 GTS 512 GPU show a geometric mean speedup of 6.94X with it maximum of 51.96X over it single threaded CPU execution across the StreamIt benchmarks. This is a 18.9% improvement over it partitioning strategy that maps only the filters that cannot be executed oil the GPU - the filters with state that is persistent across firings - onto the CPU.

Recognition And Top-Down Generation Of Beta-Acyclic Database Schemes

Database schemes can be viewed as hypergraphs with individual relation schemes corresponding to the edges of a hypergraph. Under this setting, a new class of "acyclic" database schemes was recently introduced and was shown to have a claim to a number of desirable properties. However, unlike the case of ordinary undirected graphs, there are several unequivalent notions of acyclicity of hypergraphs. Of special interest among these are agr-, beta-, and gamma-, degrees of acyclicity, each characterizing an equivalence class of desirable properties for database schemes, represented as hypergraphs. In this paper, two complementary approaches to designing beta-acyclic database schemes have been presented. For the first part, a new notion called "independent cycle" is introduced. Based on this, a criterion for beta-acyclicity is developed and is shown equivalent to the existing definitions of beta-acyclicity. From this and the concept of the dual of a hypergraph, an efficient algorithm for testing beta-acyclicity is developed. As for the second part, a procedure is evolved for top-down generation of beta-acyclic schemes and its correctness is established. Finally, extensions and applications of ideas are described.

Robustness of the partitions in grouping multi-item inventories

A recent work obtained closed-form solutions to the.problem of optimally grouping a multi-item inventory into subgroups with a common order cycle per group, when the distribution by value of the inventory could be described by a Pareto function. This paper studies the sensitivity of the optimal subgroup boundaries so obtained. Closed-form expressions have been developed to find intervals for the subgroup boundaries for any given level of suboptimality. Graphs have been provided to aid the user in selecting a cost-effective level of aggregation and choosing appropriate subgroup boundaries for a whole range of inventory distributions. The results of sensitivity analyses demonstrate the availability of flexibility in the partition boundaries and the cost-effectiveness of any stock control system through three groups, and thus also provide a theoretical support to the intuitive ABC system of classifying the items.

DFL: A Data Flow Language

Many novel computer architectures like array and multiprocessors which achieve high performance through the use of concurrency exploit variations of the von Neumann model of computation. The effective utilization of the machines makes special demands on programmers and their programming languages, such as the structuring of data into vectors or the partitioning of programs into concurrent processes. In comparison, the data flow model of computation demands only that the principle of structured programming be followed. A data flow program, often represented as a data flow graph, is a program that expresses a computation by indicating the data dependencies among operators. A data flow computer is a machine designed to take advantage of concurrency in data flow graphs by executing data independent operations in parallel. In this paper, we discuss the design of a high level language (DFL: Data Flow Language) suitable for data flow computers. Some sample procedures in DFL are presented. The implementation aspects have not been discussed in detail since there are no new problems encountered. The language DFL embodies the concepts of functional programming, but in appearance closely resembles Pascal. The language is a better vehicle than the data flow graph for expressing a parallel algorithm. The compiler has been implemented on a DEC 1090 system in Pascal.

Quasi-three-dimensional elastic stresses in rotating disks

A method is presented to obtain stresses and displacements in rotating disks by taking into account the effect of out-of-plane restraint conditions at the hub. The stresses and displacements are obtained in a non-dimensional form, presented in the form of graphs and compared with the generalized plane stress solution.