María Sebastián

Generalization of Hermite functions by fractal interpolation

Fractal interpolation techniques provide good deterministic representations of complex phenomena. This paper approaches the Hermite interpolation using fractal procedures. This problem prescribes at each support abscissa not only the value of a function but also its first p derivatives. It is shown here that the proposed fractal interpolation function and its first p derivatives are good approximations of the corresponding derivatives of the original function. According to the theorems, the described method allows to interpolate, with arbitrary accuracy, a smooth function with derivatives prescribed on a set of points. The functions solving this problem generalize the Hermite osculatory polynomials.

Smooth fractal interpolation

Fractal methodology provides a general frame for the understanding of real-world phenomena. In particular, the classical methods of real-data interpolation can be generalized by means of fractal techniques. In this paper, we describe a procedure for the construction of smooth fractal functions, with the help of Hermite osculatory polynomials. As a consequence of the process, we generalize any smooth interpolant by means of a family of fractal functions. In particular, the elements of the class can be defined so that the smoothness of the original is preserved. Under some hypotheses, bounds of the interpolation error for function and derivatives are obtained. A set of interpolating mappings associated to a cubic spline is defined and the density of fractal cubic splines in is proven.

A relation between fractal dimension and Fourier transform-electroencephalographic study using spectral and fractal parameters

Abstract Fourier parameters display the spectral content of a signal and provide explicit representations in the frequency domain. In the case of electroencephalograms, the FFT algorithm was the only quantifying method used until the 1970s, but this analysis is not conclusive in most pathologies. Fractal techniques provide new tools for the processing of signals whose traces have a sophisticated geometric complexity. The fractal dimension is computed easily in our work by means of explicit formulae involving the time samples. In this paper we obtain, under some hypotheses, an exact relation between the Fourier Transform of a signal and its Fractal Dimension. In the last part of the paper, an electroencephalographic application involving fractal and spectral parameters is described.

Structural insights into the synthesis of FMN in prokaryotic organisms.

Riboflavin kinases (RFKs) catalyse the phosphorylation of riboflavin to produce FMN. In most bacteria this activity is catalysed by the C-terminal module of a bifunctional enzyme, FAD synthetase (FADS), which also catalyses the transformation of FMN into FAD through its N-terminal FMN adenylyltransferase (FMNAT) module. The RFK module of FADS is a homologue of eukaryotic monofunctional RFKs, while the FMNAT module lacks homologyto eukaryotic enzymes involved in FAD production. Previously, the crystal structure of Corynebacterium ammoniagenes FADS (CaFADS) was determined in its apo form. This structure predicted a dimer-of-trimers organization with the catalytic sites of two modules of neighbouring protomers approaching each other, leading to a hypothesis about the possibility of FMN channelling in the oligomeric protein. Here, two crystal structures of the individually expressed RFK module of CaFADS in complex with the products of the reaction, FMN and ADP, are presented. Structures are complemented with computational simulations, binding studies and kinetic characterization. Binding of ligands triggers dramatic structural changes in the RFK module, which affect large portions of the protein. Substrate inhibition and molecular-dynamics simulations allowed the conformational changes that take place along the RFK catalytic cycle to be established. The influence of these conformational changes in the FMNAT module is also discussed in the context of the full-length CaFADS protomer and the quaternary organization.

Quaternary Organization in a Bifunctional Prokaryotic Fad Synthetase: Involvement of an Arginine at its Adenylyltransferase Module on the Riboflavin Kinase Activity.

Prokaryotic FAD synthetases (FADSs) are bifunctional enzymes composed of two modules, the C-terminal module with ATP:riboflavin kinase (RFK) activity, and the N-terminus with ATP:FMN adenylyltransferase (FMNAT) activity. The FADS from Corynebacterium ammoniagenes, CaFADS, forms transient oligomers during catalysis. These oligomers are stabilized by several interactions between the RFK and FMNAT sites from neighboring protomers, which otherwise are separated in the monomeric enzyme. Among these inter-protomer interactions, the salt bridge between E268 at the RFK site and R66 at the FMNAT-module is particularly relevant, as E268 is the catalytic base of the kinase reaction. Here we have introduced point mutations at R66 to analyze the impact of the salt-bridge on ligand binding and catalysis. Interestingly, these mutations have only mild effects on ligand binding and kinetic properties of the FMNAT-module (where R66 is located), but considerably impair the RFK activity turnover. Substitutions of R66 also modulate the ratio between monomeric and oligomeric species and modify the quaternary arrangement observed by single-molecule methods. Therefore, our data further support the cross-talk between the RFK- and FMNAT-modules of neighboring protomers in the CaFADS enzyme, and establish the participation of R66 in the modulation of the geometry of the RFK active site during catalysis.

FITTING FRACTAL SURFACES ON NON-RECTANGULAR GRIDS

In a complex society, the visualization and interpretation of large amounts of data acquires an increasing importance. This can be done at once by means of two- or three-dimensional maps. To approach this problem, we undertake the construction of several variable fractal functions. In the first place, we introduce real fractal functions defined as perturbations of the classical ones. These basic mappings allow us to compute multidimensional approximations of experimental variables by means of linear combinations of products of fractal functions of Legendre type. The paper proposes a method of non-smooth representation for a large number of three-dimensional data on non-uniform grids. The procedures described are applied in the last part of the paper to the implementation of fitting maps for brain electrical activity.

The trimer interface in the quaternary structure of the bifunctional prokaryotic FAD synthetase from Corynebacterium ammoniagenes

Bifunctional FAD synthetases (FADSs) fold in two independent modules; The C-terminal riboflavin kinase (RFK) catalyzes the RFK activity, while the N-terminal FMN-adenylyltransferase (FMNAT) exhibits the FMNAT activity. The search for macromolecular interfaces in the Corynebacterium ammoniagenes FADS (CaFADS) crystal structure predicts a dimer of trimers organization. Within each trimer, a head-to-tail arrangement causes the RFK and FMNAT catalytic sites of the two neighboring protomers to approach, in agreement with active site residues of one module influencing the activity at the other. We analyze the relevance of the CaFADS head-to-tail macromolecular interfaces to stabilization of assemblies, catalysis and ligand binding. With this aim, we evaluate the effect of point mutations in loop L1c-FlapI, loop L6c, and helix α1c of the RFK module (positions K202, E203, F206, D298, V300, E301 and L304), regions at the macromolecular interface between two protomers within the trimer. Although none of the studied residues is critical in the formation and dissociation of assemblies, residues at L1c-FlapI and helix α1c particularly modulate quaternary architecture, as well as ligand binding and kinetic parameters involved with RFK and FMNAT activities. These data support the influence of transient oligomeric structures on substrate accommodation and catalysis at both CaFADS active sites.

Discovery of antimicrobial compounds targeting bacterial type FAD synthetases

Abstract The increase of bacterial strains resistant to most of the available antibiotics shows a need to explore novel antibacterial targets to discover antimicrobial drugs. Bifunctional bacterial FAD synthetases (FADSs) synthesise the flavin mononucleotide (FMN) and flavin adenine dinucleotide (FAD). These cofactors act in vital processes as part of flavoproteins, making FADS an essential enzyme. Bacterial FADSs are potential antibacterial targets because of differences to mammalian enzymes, particularly at the FAD producing site. We have optimised an activity-based high throughput screening assay targeting Corynebacterium ammoniagenes FADS (CaFADS) that identifies inhibitors of its different activities. We selected the three best high-performing inhibitors of the FMN:adenylyltransferase activity (FMNAT) and studied their inhibition mechanisms and binding properties. The specificity of the CaFADS hits was evaluated by studying also their effect on the Streptococcus pneumoniae FADS activities, envisaging differences that can be used to discover species-specific antibacterial drugs. The antimicrobial effect of these compounds was also evaluated on C. ammoniagenes, S. pneumoniae, and Mycobacterium tuberculosis cultures, finding hits with favourable antimicrobial properties. Graphical Abstract

Numerical integration of affine fractal functions

Abstract This paper studies a method for the numerical integration and representation of functions defined through their samples, when the original “signal” is not explicitly known, but it shows experimentally some kind of self-similarity. In particular, we propose a methodology based on fractal interpolation functions for the computation of the integral that generalize the compound trapezoidal rule. The convergence of the procedure is proved with the only hypothesis of continuity. The rate of convergence is specified in the case of original Holder-continuous functions, but not necessarily smooth.

Time domain indices and discrete power spectrum in electroencephalographic processing

The classical fast Fourier transform (FFT) methods to process experimental signals have the problem of providing multivariant measures with dimension and hence depend on the devices and measure units. These features make the comparisons between different teams and equipments rather difficult. The aim of the present study is to prove the exact mathematical relation between the Hjorth parameters and the spectral powers obtained by means of the quoted algorithm. The proof confirms that these quantifiers can also be used for the exploration of spectral properties of univariant recordings. The formulae proposed provide an easy and low-cost procedure to compute the descriptors if a standard FFT algorithm is performed on the signal. A numerical study of electroencephalographic recordings involving spectral and Hjorth parameters is described in the paper in order to quantify the electroencephalogram of children with problems of attention.