Natural modes of a coupled non-linear system

The natural modes of a non-linear system with two degrees of freedom are investigated. The system, which may contain either hard or soft springs, is shown to possess three modes of vibration one of which does not have any counterpart in the linear theory. The stability analysis indicates the existence of seven different modal stability patterns depending on the values of two parameters of non-linearity.

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.

On the solution of bivariate population balance equations for aggregation: X-discretization of space for expansion and contraction of computational domain

A new structured discretization of 2D space, named X-discretization, is proposed to solve bivariate population balance equations using the framework of minimal internal consistency of discretization of Chakraborty and Kumar [2007, A new framework for solution of multidimensional population balance equations. Chem. Eng. Sci. 62, 4112-4125] for breakup and aggregation of particles. The 2D space of particle constituents (internal attributes) is discretized into bins by using arbitrarily spaced constant composition radial lines and constant mass lines of slope -1. The quadrilaterals are triangulated by using straight lines pointing towards the mean composition line. The monotonicity of the new discretization makes is quite easy to implement, like a rectangular grid but with significantly reduced numerical dispersion. We use the new discretization of space to automate the expansion and contraction of the computational domain for the aggregation process, corresponding to the formation of larger particles and the disappearance of smaller particles by adding and removing the constant mass lines at the boundaries. The results show that the predictions of particle size distribution on fixed X-grid are in better agreement with the analytical solution than those obtained with the earlier techniques. The simulations carried out with expansion and/or contraction of the computational domain as population evolves show that the proposed strategy of evolving the computational domain with the aggregation process brings down the computational effort quite substantially; larger the extent of evolution, greater is the reduction in computational effort. (C) 2011 Elsevier Ltd. All rights reserved.

Solution nuclear magnetic resonance structure of the GATase subunit and structural Basis of the interaction between GATase and ATPPase subunits in a two-subunit-type GMPS from methanocaldococcus jannaschii

The solution structure of the monomeric glutamine amidotransferase (GATase) subunit of the Methanocaldococcus janaschii (Mj) guanosine monophosphate synthetase (GMPS) has been determined using high-resolution nuclear magnetic resonance methods. Gel filtration chromatography and N-15 backbone relaxation studies have shown that the Mj GATase subunit is present in solution as a 21 kDa (188-residue) monomer. The ensemble of 20 lowest-energy structures showed root-mean-square deviations of 0.35 +/- 0.06 angstrom for backbone atoms and 0.8 +/- 0.06 angstrom for all heavy atoms. Furthermore, 99.4% of the backbone dihedral angles are present in the allowed region of the Ramachandran map, indicating the stereochemical quality of the structure. The core of the tertiary structure of the GATase is composed of a seven-stranded mixed beta-sheet that is fenced by five alpha-helices. The Mj GATase is similar in structure to the Pyrococcus horikoshi (Ph) GATase subunit. Nuclear magnetic resonance (NMR) chemical shift perturbations and changes in line width were monitored to identify residues on GATase that were responsible for interaction with magnesium and the ATPPase subunit, respectively. These interaction studies showed that a common surface exists for the metal ion binding as well as for the protein-protein interaction. The dissociation constant for the GATase-Mg2+ interaction has been found to be similar to 1 mM, which implies that interaction is very weak and falls in the fast chemical exchange regime. The GATase-ATPPase interaction, on the other hand, falls in the intermediate chemical exchange regime on the NMR time scale. The implication of this interaction in terms of the regulation of the GATase activity of holo GMPS is discussed.

Protein sequence design on the basis of topology optimization techniques -Using continuous modeling of discrete amino acid types

The notion of optimization is inherent in protein design. A long linear chain of twenty types of amino acid residues are known to fold to a 3-D conformation that minimizes the combined inter-residue energy interactions. There are two distinct protein design problems, viz. predicting the folded structure from a given sequence of amino acid monomers (folding problem) and determining a sequence for a given folded structure (inverse folding problem). These two problems have much similarity to engineering structural analysis and structural optimization problems respectively. In the folding problem, a protein chain with a given sequence folds to a conformation, called a native state, which has a unique global minimum energy value when compared to all other unfolded conformations. This involves a search in the conformation space. This is somewhat akin to the principle of minimum potential energy that determines the deformed static equilibrium configuration of an elastic structure of given topology, shape, and size that is subjected to certain boundary conditions. In the inverse-folding problem, one has to design a sequence with some objectives (having a specific feature of the folded structure, docking with another protein, etc.) and constraints (sequence being fixed in some portion, a particular composition of amino acid types, etc.) while obtaining a sequence that would fold to the desired conformation satisfying the criteria of folding. This requires a search in the sequence space. This is similar to structural optimization in the design-variable space wherein a certain feature of structural response is optimized subject to some constraints while satisfying the governing static or dynamic equilibrium equations. Based on this similarity, in this work we apply the topology optimization methods to protein design, discuss modeling issues and present some initial results.

Engagement of CF3 Group in N-H center dot center dot center dot F-C Hydrogen Bond in the Solution State: NMR Spectroscopy and MD Simulation Studies

Unambiguous evidence for the engagement of CF3 group in N-H center dot center dot center dot F-C hydrogen bond in a low polarity solvent, the first observation of its kind, is reported. The presence of such weak molecular interactions in the solution state is convincingly established by one and two-dimensional H-1, F-19, and natural abundant N-15 NMR spectroscopic studies. The strong and direct evidence is derived by the observation of through-space couplings, such as, (1h)J(FH), (1h)J(FN), and (2h)J(FF), where the spin polarization is transmitted through hydrogen bond. In an interesting example of a molecule containing two CF3 groups getting simultaneously involved in hydrogen bond, where hydrogen bond mediated couplings are not reflected in the NMR spectrum, F-19-F-19 NOESY experiment yielded confirmatory evidence. Significant deviations in the strengths of (1)J(NH), variable temperature, and the solvent induced perturbations yielded additional support. The NMR results are corroborated by both DFT calculations and MD simulations, where the quantitative information on different ways of involvement of fluorine in two and three centered hydrogen bonds, their percentage of occurrences, and geometries have been obtained. The hydrogen bond interaction energies have also been calculated.

Some remarks on the symplectic pairing on the moduli space of representations of the fundamental group of surfaces

This work grew out of an attempt to understand a conjectural remark made by Professor Kyoji Saito to the author about a possible link between the Fox-calculus description of the symplectic structure on the moduli space of representations of the fundamental group of surfaces into a Lie group and pairs of mutually dual sets of generators of the fundamental group. In fact in his paper [3] , Prof. Kyoji Saito gives an explicit description of the system of dual generators of the fundamental group.

Structural basis for the function of anti-idiotypic antibody in immune memory

We had earlier proposed a hypothesis to explain the mechanism of perpetuation of immunological memory based on the operation of idiotypic network in the complete absence of antigen. Experimental evidences were provided for memory maintenance through anti-idiotypic antibody (Ab2) carrying the internal image of the antigen. In the present work, we describe a structural basis for such memory perpetuation by molecular modeling and structural analysis studies. A three-dimensional model of Ab2 was generated and the structure of the antigenic site on the hemagglutinin protein H of Rinderpest virus was modeled using the structural template of hemagglutinin protein of Measles virus. Our results show that a large portion of heavy chain containing the CDR regions of Ab2 resembles the domain of the hemagglutinin housing the epitope regions. The similarity demonstrates that an internal image of the H antigen is formed in Ab2, which provides a structural basis for functional mimicry demonstrated earlier. This work brings out the importance of the structural similarity between a domain of hemagglutinin protein to that of its corresponding Ab2. It provides evidence that Ab2 is indeed capable of functioning as surrogate antigen and provides support to earlier proposed relay hypothesis which has provided a mechanism for the maintenance of immunological memory.

Structural basis for the function of anti-idiotypic antibody in immune memory

We had earlier proposed a hypothesis to explain the mechanism of perpetuation of immunological memory based on the operation of idiotypic network in the complete absence of antigen. Experimental evidences were provided for memory maintenance through anti-idiotypic antibody (Ab(2)) carrying the internal image of the antigen. In the present work, we describe a structural basis for such memory perpetuation by molecular modeling and structural analysis studies. A three-dimensional model of Ab(2) was generated and the structure of the antigenic site on the hemagglutinin protein H of Rinderpest virus was modeled using the structural template of hemagglutinin protein of Measles virus. Our results show that a large portion of heavy chain containing the CDR regions of Ab(2) resembles the domain of the hemagglutinin housing the epitope regions. The similarity demonstrates that an internal image of the H antigen is formed in Ab(2), which provides a structural basis for functional mimicry demonstrated earlier. This work brings out the importance of the structural similarity between a domain of hemagglutinin protein to that of its corresponding Ab(2). It provides evidence that Ab(2) is indeed capable of functioning as surrogate antigen and provides support to earlier proposed relay hypothesis which has provided a mechanism for the maintenance of immunological memory.

Derivation of the solution of certain singular integral equations

It is shown that the a;P?lication of the Poincare-Bertrand fcm~ulaw hen made in a suitable manner produces the s~lutiano f certain singular integral equations very quickly, thc method of arriving at which, otherwise, is too complicaled. Two singular integral equations are considered. One of these quaiions is with a Cauchy-tyge kcrnel arid the other is an equalion which appears in the a a w guide theory and the theory of dishcations. Adifferent approach i? alw made here to solve the singular integralquation> of the waveguide theor? ind this i ~ v o l v eth~e use of the inversion formula of the Cauchy-type singular integral equahn and dudion to a system of TIilberl problems for two unknowns which can be dwupled wry easily to obi& tbe closed form solutim of the irilegral equatlou at band. The methods of the prescnt paper avoid all the complicaled approaches of solving the singular integral equaticn of the waveguide theory knowr todate.

Some direct numerical approaches to the solution of the singular nonlinear transonic integral equation

The nonlinear singular integral equation of transonic flow is examined, noting that standard numerical techniques are not applicable in solving it. The difficulties in approximating the integral term in this expression were solved by special methods mitigating the inaccuracies caused by standard approximations. It was shown how the infinite domain of integration can be reduced to a finite one; numerical results were plotted demonstrating that the methods proposed here improve accuracy and computational economy.

On the Solution of Certain Simultaneous Pairs of Dual Integral Equations

Mit einer direkten Methode, bei der der Erdelyi-Kober- und der modifizierte Hankel-Operator Anwendung finden, werden gewisse Systeme aus zwei bzw. drei Paaren dualer Integralgleichungen mit Bessel-Kernen in geschlossener Form gelöst. Für bestimmte Funktionenklassen und Ordnungen der Bessel-Funktionen ist die Vorgehensweise angebrachter und geeigneter als die bereits existierenden Methoden.

A unified approach to the solution of plane problems of Magneto-elasticity with special reference to a hole in a thin infinite conducting plate

Under certain specific assumption it has been observed that the basic equations of magneto-elasticity in the case of plane deformation lead to a biharmonic equation, as in the case of the classical plane theory of elasticity. The method of solving boundary value problems has been properly modified and a unified approach in solving such problems has been suggested with special reference to problems relating thin infinite plates with a hole. Closed form expressions have been obtained for the stresses due to a uniform magnetic field present in the plane of deformation of a thin infinite conducting plate with a circular hole, the plate being deformed by a tension acting parallel to the direction of the magnetic field.

Application of Laplace transform technique to the solution of certain third-order non-linear systems

A number of papers have appeared on the application of operational methods and in particular the Laplace transform to problems concerning non-linear systems of one kind or other. This, however, has met with only partial success in solving a class of non-linear problems as each approach has some limitations and drawbacks. In this study the approach of Baycura has been extended to certain third-order non-linear systems subjected to non-periodic excitations, as this approximate method combines the advantages of engineering accuracy with ease of application to such problems. Under non-periodic excitations the method provides a procedure for estimating quickly the maximum response amplitude, which is important from the point of view of a designer. Limitations of such a procedure are brought out and the method is illustrated by an example taken from a physical situation.

On a Special Co-cycle Basis of Graphs

In this paper we consider the problems of computing a minimum co-cycle basis and a minimum weakly fundamental co-cycle basis of a directed graph G. A co-cycle in G corresponds to a vertex partition (S,V ∖ S) and a { − 1,0,1} edge incidence vector is associated with each co-cycle. The vector space over ℚ generated by these vectors is the co-cycle space of G. Alternately, the co-cycle space is the orthogonal complement of the cycle space of G. The minimum co-cycle basis problem asks for a set of co-cycles that span the co-cycle space of G and whose sum of weights is minimum. Weakly fundamental co-cycle bases are a special class of co-cycle bases, these form a natural superclass of strictly fundamental co-cycle bases and it is known that computing a minimum weight strictly fundamental co-cycle basis is NP-hard. We show that the co-cycle basis corresponding to the cuts of a Gomory-Hu tree of the underlying undirected graph of G is a minimum co-cycle basis of G and it is also weakly fundamental.