Orthogonal Expansion of Real Polynomials, Location of Zeros, and an L2 Inequality

AbstractLet f(z)=a0φ0(z)+a1φ1(z)+…+anφn(z) be a polynomial of degree n, given as an orthogonal expansion with real coefficients. We study the location of the zeros of f relative to an interval and in terms of some of the coefficients. Our main theorem generalizes or refines results due to Turán and Specht. In particular, it includes a best possible criterion for the occurrence of real zeros. Our approach also allows us to establish a weighted L2 inequality giving a lower estimate for the product of two polynomials

Localness of energy cascade in hydrodynamic turbulence, II. Sharp spectral filter

We investigate the scale-locality of subgrid-scale (SGS) energy flux and inter-band energy transfers defined by the sharp spectral filter. We show by rigorous bounds, physical arguments and numerical simulations that the spectral SGS flux is dominated by local triadic interactions in an extended turbulent inertial-range. Inter-band energy transfers are also shown to be dominated by local triads if the spectral bands have constant width on a logarithmic scale. We disprove in particular an alternative picture of ``local transfer by nonlocal triads,'' with the advecting wavenumber mode at the energy peak. Although such triads have the largest transfer rates of all {\it individual} wavenumber triads, we show rigorously that, due to their restricted number, they make an asymptotically negligible contribution to energy flux and log-banded energy transfers at high wavenumbers in the inertial-range. We show that it is only the aggregate effect of a geometrically increasing number of local wavenumber triads which can sustain an energy cascade to small scales. Furthermore, non-local triads are argued to contribute even less to the space-average energy flux than is implied by our rigorous bounds, because of additional cancellations from scale-decorrelation effects. We can thus recover the -4/3 scaling of nonlocal contributions to spectral energy flux predicted by Kraichnan's ALHDIA and TFM closures. We support our results with numerical data from a $512^3$ pseudospectral simulation of isotropic turbulence with phase-shift dealiasing. We conclude that the sharp spectral filter has a firm theoretical basis for use in large-eddy simulation (LES) modeling of turbulent flows.Comment: 42 pages, 9 figure

Creation of a GPR18 homology model using conformational memories

G-protein coupled receptors (GPCRs) make up the largest family of eukaryotic membrane receptors, covering a broad range of cellular responses in the body. This wide range of activity makes them important pharmacological targets. In general, all Class A GPCRs share a common structure that consists of seven transmembrane alpha helices, connected by extracellular and intracellular loops, an extracellular N-terminus, and an intracellular C-terminus. These similarities can be used to construct a model of an unknown receptor, which can then be used to help guide further studies of this receptor and its pharmacology. The orphan GPCR GPR18 is a member of the Class A subfamily of GPCRs. GPR18 binds both lipid-like and small molecule ligands, such as NAGly and abnormal-cannabidiol (Abn-CBD), leading to belief that GPR18 may be the Abnormal Cannabinoid Receptor. The goal of this project was to construct a model of GPR18 in its inactive state and to explore the binding site of a key antagonist already identified for this receptor. A model of the GPR18 inactive (R) state was created using the μ-Opioid receptor (MOR) crystal structure as template (PDB: 4DKL). The Monte Carlo/simulated annealing method, Conformational Memories (CM) was used to study the accessible conformations of three GPR18 transmembrane helices (TMHs) with important sequence divergences from the MOR template: TMH3, TMH4, and TMH7. CM was also used to calculate the accessible conformations for TMH6, which allowed the choice of TMH6 conformers appropriate for the GPR18 R and R* models. Docking studies were guided by the hypothesis that a positively charged residue (either R2.60 or R5.42) may be the primary ligand interaction site in the GPR18 binding pocket. The binding pocket of the antagonist, cannabidiol (CBD) was explored in the inactive state GPR18 model using Glide, an automatic docking program in the Schrödinger modeling suite. These studies identified that both of these argenines are the primary interaction site for CBD. With the pocket determined, extracellular and intracellular loops were calculated using another Monte Carlo technique, Modeler. Once loops were attached, the N and C termini were modeled and added as well. Much like the S1PR1 receptor, and continuing the hypothesis that GPR18 used a lipid level access to the binding pocket, the N terminus displayed a small helical portion that lay atop the bundle, effectively blocking the extracellular side along with EC2. With the identification of key residues and a complete bundle, further mutation studies and dynamic simulations can be used to further refine and test these modeling results

Reconstruction of Bandlimited Functions from Unsigned Samples

We consider the recovery of real-valued bandlimited functions from the absolute values of their samples, possibly spaced nonuniformly. We show that such a reconstruction is always possible if the function is sampled at more than twice its Nyquist rate, and may not necessarily be possible if the samples are taken at less than twice the Nyquist rate. In the case of uniform samples, we also describe an FFT-based algorithm to perform the reconstruction. We prove that it converges exponentially rapidly in the number of samples used and examine its numerical behavior on some test cases

Multiplication and Composition in Weighted Modulation Spaces

We study the existence of the product of two weighted modulation spaces. For this purpose we discuss two different strategies. The more simple one allows transparent proofs in various situations. However, our second method allows a closer look onto associated norm inequalities under restrictions in the Fourier image. This will give us the opportunity to treat the boundedness of composition operators.Comment: 49 page

Identification of direct residue contacts in protein-protein interaction by message passing

Understanding the molecular determinants of specificity in protein-protein interaction is an outstanding challenge of postgenome biology. The availability of large protein databases generated from sequences of hundreds of bacterial genomes enables various statistical approaches to this problem. In this context covariance-based methods have been used to identify correlation between amino acid positions in interacting proteins. However, these methods have an important shortcoming, in that they cannot distinguish between directly and indirectly correlated residues. We developed a method that combines covariance analysis with global inference analysis, adopted from use in statistical physics. Applied to a set of >2,500 representatives of the bacterial two-component signal transduction system, the combination of covariance with global inference successfully and robustly identified residue pairs that are proximal in space without resorting to ad hoc tuning parameters, both for heterointeractions between sensor kinase (SK) and response regulator (RR) proteins and for homointeractions between RR proteins. The spectacular success of this approach illustrates the effectiveness of the global inference approach in identifying direct interaction based on sequence information alone. We expect this method to be applicable soon to interaction surfaces between proteins present in only 1 copy per genome as the number of sequenced genomes continues to expand. Use of this method could significantly increase the potential targets for therapeutic intervention, shed light on the mechanism of protein-protein interaction, and establish the foundation for the accurate prediction of interacting protein partners.Comment: Supplementary information available on http://www.pnas.org/content/106/1/67.abstrac

A quantum search for zeros of polynomials

A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical measurements of the observable output directly the numerical values of the zeros. Performing the measurements is the only computational resource involved