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PubMed2006

Topology and static response of interaction networks in molecular biology.

We introduce a mathematical framework describing static response of networks occurring in molecular biology. This formalism has many similarities with the Laplace-Kirchhoff equations for electrical networks. We introduce the concept of graph boundary and we show how the response of the biological networks to external perturbations can be related to the Dirichlet or Neumann problems for the corresponding equations on the interaction graph. Solutions to these two problems are given in terms of path moduli (measuring path rigidity with respect to the propagation of interaction along the graph). Path moduli are related to loop products in the interaction graph via generalized Mason-Coates formulae. We apply our results to two specific biological examples: the lactose operon and the genetic regulation of lipogenesis. Our applications show consistency with experimental results and in the case of lipogenesis check some hypothesis on the behaviour of hepatic fatty acids on fasting.

Animals↗
PubMed2006

Modelling protein-protein interaction networks via a stickiness index.

What type of connectivity structure are we seeing in protein-protein interaction networks? A number of random graph models have been mooted. After fitting model parameters to real data, the models can be judged by their success in reproducing key network properties. Here, we propose a very simple random graph model that inserts a connection according to the degree, or 'stickiness', of the two proteins involved. This model can be regarded as a testable distillation of more sophisticated versions that attempt to account for the presence of interaction surfaces or binding domains. By computing a range of network similarity measures, including relative graphlet frequency distance, we find that our model outperforms other random graph classes. In particular, we show that given the underlying degree information, fitting a stickiness model produces better results than simply choosing a degree-matching graph uniformly at random. Therefore, the results lend support to the basic modelling methodology.

Adhesiveness↗
PubMed2002

Pseudofractal scale-free web.

We find that scale-free random networks are excellently modeled by simple deterministic graphs. Our graph has a discrete degree distribution (degree is the number of connections of a vertex), which is characterized by a power law with exponent gamma=1+ln 3/ln 2. Properties of this compact structure are surprisingly close to those of growing random scale-free networks with gamma in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For a large network (ln N>>1) the distribution tends to a Gaussian of width approximately sqrt[ln N] centered at (-)l approximately ln N. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2+gamma.

Journal Article↗
PubMed2004

Extremal optimization at the phase transition of the three-coloring problem.

We investigate the phase transition in vertex coloring on random graphs, using the extremal optimization heuristic. Three-coloring is among the hardest combinatorial optimization problems and is equivalent to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the "backbone," an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size n up to 512. For these graphs, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about O (n(3.5)) update steps. Finite size scaling yields a critical mean degree value alpha(c) =4.703 (28). Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.

Journal Article↗
PubMed2004

Perturbing general uncorrelated networks.

This paper is a direct continuation of an earlier work, where we studied Erdös-Rényi random graphs perturbed by an interaction Hamiltonian favoring the formation of short cycles. Here, we generalize these results. We keep the same interaction Hamiltonian but let it act on general graphs with uncorrelated nodes and an arbitrary given degree distribution. It is shown that the results obtained for Erdös-Rényi graphs are generic, at the qualitative level. However, scale-free graphs are an exception to this general rule and exhibit a singular behavior, studied thoroughly in this paper, both analytically and numerically.

Journal Article↗
PubMed2002

Explicitly solvable cases of one-dimensional quantum chaos.

We identify a set of quantum graphs with unique and precisely defined spectral properties called regular quantum graphs. Although chaotic in their classical limit with positive topological entropy, regular quantum graphs are explicitly solvable. The proof is constructive: we present exact, convergent periodic orbit expansions for individual energy levels, thus obtaining an analytical solution for the spectrum of regular quantum graphs that is complete, explicit, and exact.

Journal Article↗
PubMed2005

Accuracy and scaling phenomena in Internet mapping.

It was recently argued that sampling a network by traversing it with paths from a small number of sources, as with traceroutes on the Internet, creates a fundamental bias in observed topological features like the degree distribution. We examine this bias analytically and experimentally. For Erdo s-Re nyi random graphs with mean degree c, we show analytically that such sampling gives an observed degree distribution P(k) approximately k(-1) for k less, similarc, despite the underlying distribution being Poissonian. For graphs whose degree distributions have power-law tails P(k) approximately k(-alpha), sampling can significantly underestimate alpha when the graph has a large excess (i.e., many more edges than vertices). We find that in order to accurately estimate alpha, one must use a number of sources which grows linearly in the mean degree of the underlying graph. Finally, we comment on the accuracy of the published values of alpha for the Internet.

Journal Article↗
PubMed2006

Polynomial growth in branching processes with diverging reproductive number.

We study the spreading dynamics on graphs with a power law degree distribution pk approximately k-gamma, with 2<gamma<3, as an example of a branching process with a diverging reproductive number. We provide evidence that the divergence of the second moment of the degree distribution carries as a consequence a qualitative change in the growth pattern, deviating from the standard exponential growth. First, the population growth is extensive, meaning that the average number of vertices reached by the spreading process becomes of the order of the graph size in a time scale that vanishes in the large graph size limit. Second, the temporal evolution is governed by a polynomial growth, with a degree determined by the characteristic distance between vertices in the graph. These results open a path to further investigation on the dynamics on networks.

Animals↗
PubMedDate not supplied

Efficient parameterized algorithms for biopolymer structure-sequence alignment.

Computational alignment of a biopolymer sequence (e.g., an RNA or a protein) to a structure is an effective approach to predict and search for the structure of new sequences. To identify the structure of remote homologs, the structure-sequence alignment has to consider not only sequence similarity, but also spatially conserved conformations caused by residue interactions and, consequently, is computationally intractable. It is difficult to cope with the inefficiency without compromising alignment accuracy, especially for structure search in genomes or large databases. This paper introduces a novel method and a parameterized algorithm for structure-sequence alignment. Both the structure and the sequence are represented as graphs, where, in general, the graph for a biopolymer structure has a naturally small tree width. The algorithm constructs an optimal alignment by finding in the sequence graph the maximum valued subgraph isomorphic to the structure graph. It has the computational time complexity O[k(t)N(2)] for the structure of N residues and its tree decomposition of width t. Parameter k, small in nature, is determined by a statistical cutoff for the correspondence between the structure and the sequence. This paper demonstrates a successful application of the algorithm to RNA structure search used for noncoding RNA identification. An application to protein threading is also discussed.

Algorithms↗
PubMed2006

A generative sketch model for human hair analysis and synthesis.

In this paper, we present a generative sketch model for human hair analysis and synthesis. We treat hair images as 2D piecewise smooth vector (flow) fields and, thus, our representation is view-based in contrast to the physically-based 3D hair models in graphics. The generative model has three levels. The bottom level is the high-frequency band of the hair image. The middle level is a piecewise smooth vector field for the hair orientation, gradient strength, and growth directions. The top level is an attribute sketch graph for representing the discontinuities in the vector field. A sketch graph typically has a number of sketch curves which are divided into 11 types of directed primitives. Each primitive is a small window (say 5 x 7 pixels) where the orientations and growth directions are defined in parametric forms, for example, hair boundaries, occluding lines between hair strands, dividing lines on top of the hair, etc. In addition to the three level representation, we model the shading effects, i.e., the low-frequency band of the hair image, by a linear superposition of some Gaussian image bases and we encode the hair color by a color map. The inference algorithm is divided into two stages: 1) We compute the undirected orientation field and sketch graph from an input image and 2) we compute the hair growth direction forthe sketch curves and the orientation field using a Swendsen-Wang cut algorithm. Both steps maximize a joint Bayesian posterior probability. The generative model provides a straightforward way for synthesizing realistic hair images and stylistic drawings (rendering) from a sketch graph and a few Gaussian bases. The latter can be either inferred from a real hair image or input (edited) manually using a simple sketching interface. We test our algorithm on a large data set of hair images with diverse hair styles. Analysis, synthesis, and rendering results are reported in the experiments.

Algorithms↗
PubMed2006

Robust point matching for nonrigid shapes by preserving local neighborhood structures.

In previous work on point matching, a set of points is often treated as an instance of a joint distribution to exploit global relationships in the point set. For nonrigid shapes, however, the local relationship among neighboring points is stronger and more stable than the global one. In this paper, we introduce the lotion of a neighborhood structure for the general point matching problem. We formulate point matching as an optimization problem to preserve local neighborhood structures during matching. Our approach has a simple graph matching interpretation, where each point is a node in the graph, and two nodes are connected by an edge if they are neighbors. The optimal match between two graphs is the one that maximizes the number of matched edges. Existing techniques are leveraged to search for an optimal solution with the shape context distance used to initialize the graph matching, followed by relaxation labeling updates for refinement. Extensive experiments show the robustness of our approach under deformation, noise in point locations, outliers, occlusion, and rotation. It outperforms the shape context and TPS-RPM algorithms on most scenarios.

Algorithms↗
PubMed2004

A simplifier for propositional formulas with many binary clauses.

Deciding whether a propositional formula in conjunctive normal form is satisfiable (SAT) is an NP-complete problem. The problem becomes linear when the formula contains binary clauses only. Interestingly, the reduction to SAT of a number of well-known and important problems--such as classical AI planning and automatic test pattern generation for circuits--yields formulas containing many binary clauses. In this paper we introduce and experiment with 2-SIMPLIFY, a formula simplifier targeted at such problems. 2-SIMPLIFY constructs the transitive closure of the implication graph corresponding to the binary clauses in the formula and uses this graph to deduce new unit literals. The deduced literals are used to simplify the formula and update the graph, and so on, until stabilization. Finally, we use the graph to construct an equivalent, simpler set of binary clauses. Experimental evaluation of this simplifier on a number of bench-mark formulas produced by encoding AI planning problems prove 2-SIMPLIFY to be a useful tool in many circumstances.

Journal Article↗
PubMedDate not supplied

Time-varying contour topology.

The contour tree has been used to compute the topology of isosurfaces, generate a minimal seed set for accelerated isosurface extraction, and provide a user interface to segment individual contour components in a scalar field. In this paper, we extend the benefits of the contour tree to time-varying data visualization. We define temporal correspondence of contour components and describe an algorithm to compute the correspondence information in time-dependent contour trees. A graph representing the topology changes of time-varying isosurfaces is constructed in real-time for any selected isovalue using the precomputed correspondence information. Quantitative properties, such as surface area and volume of contour components, are computed and labeled on the graph. This topology change graph helps users to detect significant topological and geometric changes in time-varying isosurfaces. The graph is also used as an interactive user interface to segment, track, and visualize the evolution of any selected contour components over time.

Algorithms↗
PubMedDate not supplied

Smashing peacocks further: drawing quasi-trees from biconnected components.

Quasi-trees, namely graphs with tree-like structure, appear in many application domains, including bioinformatics and computer networks. Our new SPF approach exploits the structure of these graphs with a two-level approach to drawing, where the graph is decomposed into a tree of biconnected components. The low-level biconnected components are drawn with a force-directed approach that uses a spanning tree skeleton as a starting point for the layout. The higher-level structure of the graph is a true tree with meta-nodes of variable size that contain each biconnected component. That tree is drawn with a new area-aware variant of a tree drawing algorithm that handles high-degree nodes gracefully, at the cost of allowing edge-node overlaps. SPF performs an order of magnitude faster than the best previous approaches, while producing drawings of commensurate or improved quality.

Journal Article↗
PubMedDate not supplied

On the selection of stopping-power and mass energy-absorption coefficient ratios for high-energy x-ray dosimetry.

A method for the selection of average stopping-power (L/rho)medair and energy-absorption coefficient (mu en/rho)medair ratios has been developed. The quality of the x-ray beam is characterized by the ratio of ionization chamber readings at depths of 20 and 10 cm in water (TMR)2010. For convenience, a relationship is established between experimental (TMR)2010 and the nominal accelerating potential (MV) of the accelerator. Experimental (TMR)2010 are related to (L/rho)medair and (mu en/rho)medair in a three-step process. First, using experimental and theoretical spectra in the range 60Co to 45 MV, (TMR)2010 were calculated for primary and first-scatter photons, and a graph of experimental versus calculated (TMR)2010 for these same spectra was constructed. Second, (L/rho)medair and (mu en/rho)medair were calculated for a large number of primary spectra [for most of which experimental (TMR)2010 were not available] and a graph constructed that related these quantities and (TMR)2010 calculated as above for this group of spectra. Third, using the graphs from the preceding steps, graphs relating the calculated (L/rho)medair and (mu en/rho)medair with experimental (TMR)2010 were constructed. Data are presented for water, polystyrene, acrylic, graphite, A-150, C-552, Bakelite, and nylon for beams with nominal accelerating potentials in the range 2-45 MV.

Humans↗
PubMed1999

Prediction of survival for preterm births by weight and gestational age: retrospective population based study.

OBJECTIVE: To produce current data on survival of preterm infants. DESIGN: Retrospective population based study. SETTING: Trent health region. SUBJECTS: All European and Asian live births, stillbirths, and late fetal losses from 22 to 32 weeks' gestation, excluding those with major congenital malformations, in women resident in the Trent health region between 1 January 1994 and 31 December 1997. MAIN OUTCOME MEASURES: Birth weight and gestational age specific survival for both European and Asian infants (a) known to be alive at the onset of labour, and (b) admitted for neonatal care. RESULTS: 738 deaths occurred in 3760 infants born between 22 and 32 weeks' gestation during the study period, giving an overall survival rate of 80.4%. The survival rate for the 3489 (92.8%) infants admitted for neonatal care was 86.6%. For European infants known to be alive at the onset of labour, significant variations in gestation specific survival by birth weight emerged from 24 weeks' gestation: survival ranged from 9% (95% confidence interval 7% to 13%) for infants of birth weight 250-499 g to 21% (16% to 28%) for those of 1000-1249 g. At 27 weeks' gestation, survival ranged from 55% (49% to 61%) for infants of birth weight 500-749 g (below the 10th centile) to 80% (76% to 85%) for those of 1250-1499 g. Infants who were large for dates (>/=27 weeks' gestation) had a slightly reduced, but not significant, predicted survival. Similar survival rates were observed for Asian infants. The odds ratio for the survival of infants from a multiple birth compared with singleton infants was 1.4 (1.1 to 1.8). Survival graphs for infants admitted for neonatal care are presented by sex. CONCLUSION: Easy to use birth weight and gestational age specific predicted survival graphs for preterm infants facilitate decision making for clinicians and parents. It is important that these graphs are representative, are produced for a geographically defined population, and are not biased towards the outcomes of particular centres. Such graphs, produced in two stages, allow for the changing pattern of survival of infants from the start of the intrapartum period to immediately after admission for neonatal care.

Asia↗
PubMed1982

Isolated cat trabeculae in a simulated feline heart and arterial system. Contractile basis of cardiac pump function.

Isolated cat trabeculae were studied under conditions resembling those present for the muscle fibers in the wall of the left ventricle. To obtain such a situation experimental animals, perfusion fluid, temperature, stimulation frequency, peak stress values, contraction sequence, length, and force control were chosen with respect to that criterion. Results were compared with those described for the intact feline heart in previous studies. Special emphasis was placed on determinants of the pump function graph, i.e., the relationship between mean ventricular pressure and output. It was found that peak isometric stress values measured in the trabeculae were about twice as high as those existing on average at the base of the intact left ventricle in the circumferential direction. However, the duration of the mechanical activity, as measured in iso(volu)metric contractions, was in the isolated trabeculae (206 msec) significantly less (P less than 0.01) than found in intact right (292 msec) or intact left ventricle (344 msec). Furthermore the (maximum) output of the intact left ventricle at end-diastolic pressure could not be accounted for in a simple manner by the maximum amount of shortening found in isolated trabeculae. The points of the pump function graph obtained by varying the input impedance of the loading arterial system over a wide range of compliance and resistance values in the steady state deviated only little from the graph obtained from a series of constant pressure levels applied in a beat-to-beat fashion. Therefore, the insensitivity of the pump function graph to the nature of the arterial load is found in the intact heart as well as in isolated cardiac muscle.

Animals↗
PubMed1985

Optimal power generation by the left ventricle. A study in the anesthetized open thorax cat.

We studied the interaction of the left ventricle and the systemic arterial bed in the open thorax cat. In the steady state, the ventricle can be characterized by the pump function graph (i.e., the relationship between mean left ventricular pressure and mean outflow). From this pump function graph, the apparent source resistance of the heart is found. Apparent source resistance is defined as the ratio of the difference between maximal and actual mean left ventricular pressure, and mean outflow. The arterial system can be characterized by the ratio of mean aortic pressure and mean flow (peripheral resistance). The pressure and flow at which the heart operates is defined as the working point. We have investigated whether the ventricle in the intact cat is working optimally, i.e., that it cannot increase work output further at the end-diastolic volume, contractile state, and prevailing heart rate. This condition is considered as "matching" of ventricle and load. It could be shown that optimal power is transferred when the ratio of peripheral and apparent source resistance equals twice the ratio of mean aortic and mean left ventricular pressure (the matching principle). In four cats, we observed that mean aortic and mean left ventricular pressures are proportionally related. Mean external power (the time integral of the product of pressure and flow divided by cycle length) and steady power (the product of mean pressure and mean flow) were found to be proportional as well. These proportionalities allow for the calculation of peripheral resistance and mean external power from the pump function graph. Pump function graphs were determined in three groups: control (n = 9), atrial pacing (n = 8), and halothane (n = 5). We compared the ratio of peripheral and source resistance at the working point and at the point of optimal work output (expressed in steady ventricular power). It could be shown that, in all investigated groups, the power optimum and the working point coincide. It was concluded that circulatory control in the intact anesthetized cat keeps the ventricle at optimal work output under the conditions studied.

Animals↗