We study several aspects of FitzHugh–Nagumo's (FH–N) equations without diffusion. Some global stability results as well as the boundedness of solutions are derived by using a suitably defined Lyapunov functional. We show the existence of both supercritical and subcritical Hopf bifurcations. We demonstrate that the number of all bifurcation diagrams is 8 but that the possible sequential occurrences of bifurcation events is much richer. We present a numerical study of an example exhibiting a series of various bifurcations, including subcritical Hopf bifurcations, homoclinic bifurcations and saddle-node bifurcations of equilibria and of periodic solutions. Finally, we study periodically forced FH–N equations. We prove that phase-locking occurs independently of the magnitude of the periodic forcing.

No abstract

BackgroundMost of the currently used methods for protein function prediction rely on sequence-based comparisons between a query protein and those for which a functional annotation is provided. A serious limitation of sequence similarity-based approaches for identifying residue conservation among proteins is the low confidence in assigning residue-residue correspondences among proteins when the level of sequence identity between the compared proteins is poor. Multiple sequence alignment methods are more satisfactory--still, they cannot provide reliable results at low levels of sequence identity. Our goal in the current work was to develop an algorithm that could help overcome these difficulties by facilitating the identification of structurally (and possibly functionally) relevant residue-residue correspondences between compared protein structures.ResultsHere we present StralSV (structure-alignment sequence variability), a new algorithm for detecting closely related structure fragments and quantifying residue frequency from tight local structure alignments. We apply StralSV in a study of the RNA-dependent RNA polymerase of poliovirus, and we demonstrate that the algorithm can be used to determine regions of the protein that are relatively unique, or that share structural similarity with proteins that would be considered distantly related. By quantifying residue frequencies among many residue-residue pairs extracted from local structural alignments, one can infer potential structural or functional importance of specific residues that are determined to be highly conserved or that deviate from a consensus. We further demonstrate that considerable detailed structural and phylogenetic information can be derived from StralSV analyses.ConclusionsStralSV is a new structure-based algorithm for identifying and aligning structure fragments that have similarity to a reference protein. StralSV analysis can be used to quantify residue-residue correspondences and identify residues that may be of particular structural or functional importance, as well as unusual or unexpected residues at a given sequence position. StralSV is provided as a web service at http://proteinmodel.org/AS2TS/STRALSV/.

A numerical method incorporating a combination of a difference scheme and several uniform and nonuniform quadrature rules is presented. The method is designed to solve size-structured population equations with linear growth rate and nonlinear fertility and mortality rates. A detailed analysis of the global discretization error is carried out. Examples with known exact solutions have been solved numerically using the proposed method. The computations show that the global error is of third order as predicted by the theory.

No abstract