“Coarse” stability and bifurcation analysis using time-steppers: A reaction-diffusion example

Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099–1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective (“coarse”) bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice–Boltzmann model.

Self-similar intermediate asymptotics for a degenerate parabolic filtration-absorption equation

The equation ∂tu = u∂xx2u − (c − 1)(∂xu)2 is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water-absorbing fissurized porous rock; therefore, we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.

RNA: a method to specifically inhibit PCR amplification of known members of a multigene family by degenerate primers

The polymerase chain reaction (PCR) is a versatile method to amplify specific DNA with oligonucleotide primers. By designing degenerate PCR primers based on amino acid sequences that are highly conserved among all known gene family members, new members of a multigene family can be identified. The inherent weakness of this approach is that the degenerate primers will amplify previously identified, in addition to new, family members. To specifically address this problem, we synthesized a specific RNA for each known family member so that it hybridized to one strand of the template, adjacent to the 3′-end of the primer, allowing the degenerate primer to bind yet preventing extension by DNA polymerase. To test our strategy, we used known members of the soluble, nitric oxide-sensitive guanylyl cyclase family as our templates and degenerate primers that discriminate this family from other guanylyl cyclases. We demonstrate that amplification of known members of this family is effectively and specifically inhibited by the corresponding RNAs, alone or in combination. This robust method can be adapted to any application where multiple PCR products are amplified, as long as the sequence of the desired and the undesired PCR product(s) is sufficiently distinct between the primers.

A gradient estimate to a degenerate parabolic equation with a singular absorption term: The global quenching phenomena

We prove global existence of nonnegative solutions to the one dimensional degenerate parabolic problems containing a singular term. We also show the global quenching phenomena for L1 initial datums. Moreover, the free boundary problem is considered in this paper.

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

In this article we present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many different phenomena in areas such as developmental and cancer biology, cell motility and material science. Often one is interested in identifying parameters which will lead to a particular pattern. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present various examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally we see that if two or more eigenvalues are in a permissible range then the inhomogeneous steady state can be a linear combination of the respective eigenfunctions. Finally we show an example which suggests that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.

Entanglement and bifurcations in Jahn-Teller models

We compare and contrast the entanglement in the ground state of two Jahn-Teller models. The Exbeta system models the coupling of a two-level electronic system, or qubit, to a single-oscillator mode, while the Exepsilon models the qubit coupled to two independent, degenerate oscillator modes. In the absence of a transverse magnetic field applied to the qubit, both systems exhibit a degenerate ground state. Whereas there always exists a completely separable ground state in the Exbeta system, the ground states of the Exepsilon model always exhibit entanglement. For the Exbeta case we aim to clarify results from previous work, alluding to a link between the ground-state entanglement characteristics and a bifurcation of a fixed point in the classical analog. In the Exepsilon case we make use of an ansatz for the ground state. We compare this ansatz to exact numerical calculations and use it to investigate how the entanglement is shared between the three system degrees of freedom.

A note on bifurcation from an interval

We study the global bifurcation of nonlinear Sturm-Liouville problems of the form -(pu')' + qu = lambda a(x)f(u), b(0)u(0) - c(0)u' (0) = 0, b(1)u(1) + c(1)u'(1) = 0 which are not linearizable in any neighborhood of the origin. (c) 2005 Published by Elsevier Ltd.

Asymptotic bifurcation results for quasilinear elliptic operators

We develop results for bifurcation from the principal eigenvalue for certain operators based on the p-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. The main result concerns an asymptotic estimate of the rate at which the solution branch departs from the eigenspace. The method can also be applied for nonpotential operators.

An exceptionally rich complex of Sanguinicolidae von Graff, 1907 (Platyhelminthes : Trematoda) from Siganidae, Labridae and Mullidae (Teleostei : Perciformes) from the Indo-West Pacific Region

We describe an unprecedented radiation of sanguinicolid blood flukes ( Digenea: Sanguinicolidae) from two species of Labridae (Choerodon venustus and C. cauteroma), seven species of Mullidae (Mulloidichthys vanicolensis, Parupeneus barberinoides, P. barberinus, P. bifasciatus, P. cyclostomus, P. indicus and P. multifasciatus) and ten species of Siganidae (Siganus argenteus, S. corallinus, S. doliatus, S. fuscescens, S. lineatus, S. margaritiferus, S. puellus, S. punctatus, S. virgatus and S. vulpinus) from sites off Australia and Palau. The flukes were morphologically similar in having the combination of a long thread-like body, tegumental spines in lateral transverse rows, a vestigial oral sucker bearing concentric rows of fine spines, an H-shaped intestine, a cirrussac, a notch level with the male genital pore, a lateral or post-ovarian uterus, a uterine chamber and separate genital pores. These species are divided into two genera on the basis of testis number. Sanguinicolids from Siganus fuscescens have a single large testis between the intestinal bifurcation and the ovary and are placed in Ankistromeces Nolan & Cribb, 2004. Species from the remaining nine species of Siganidae, Labridae and Mullidae are placed in Phthinomita n. g.; these species have two testes, the anterior testis being large and between the intestinal bifurcation and the ovary whereas the small posterior testis is at the posterior end of the body and appears rudimentary or degenerate and probably non-functional. The second internal transcribed spacer (ITS2) of ribosomal DNA ( rDNA) from 29 host/parasite/location combinations (h/p/l) was sequenced together with that of Ankistromeces mariae Nolan & Cribb, 2004 for comparison. From 135 samples we found 19 distinct genotypes which were interpreted as representing at least that many species. Replicate sequences were obtained for 25 of 30 h/p/l combinations ( including A. mariae); there was no intraspecific variation between replicates sequences for any of these. Interspecific variation ranged from 1 - 41 base differences (0.3 - 12.7% sequence divergence). The 19 putative species were difficult to recognise by morphological examination. We describe 13 new species; we do not describe (= name) six species characterised solely by molecular sequences and three putative species for which morphological data is available but for which molecular data is not. We have neither morphological nor molecular data for sanguinicolids harboured in five hosts species ( Siganus margaritiferus, S. puellus, Choerodon cauteroma, Parupeneus indicus and P. multifasciatus) in which we have seen infections. Where host species were infected in different localities they almost always harboured distinct species. Some host species ( for example, S. argenteus and S. lineatus from Lizard Island) harboured two or three species in a single geographical location. This suggests that, for parts of this system, parasite speciation has outstripped host speciation. Distance analysis of ITS2 showed species from each host family ( Siganidae, Mullidae and Labridae) did not form monophyletic clades to the exclusion of species from other host families. However, a host defined clade was formed by the sequences from sanguinicolids from S. fuscescens.

Degenerate dentition of the dugong (Dugong dugon), or why a grazer does not need teeth: morphology, occlusion and wear of mouthparts

The morphology and functional occlusion of the cheekteeth of 57 dugongs Dugong dugon of both sexes were examined using reflected light and scanning electron microscopy, radiography, hardness testing and skull manipulation. The functional morphology of the horny oral pads was also described. Mouthparts and body size allometry was examined for ontogenetic and gender-related trends. We found that the worn erupted cheekteeth of the dugong are simple flat pegs composed of soft degenerative dentine. During occlusion, the mandible moves in a mainly antero-lingual direction, with the possibility of mandibular retraction in some individuals. Anterior parts of the cheektooth row may become non-functional as a dugong ages. As a function of body size, dugong cheekteeth are extremely small compared with those of other mammalian herbivores, and with other hindgut fermenters in particular. The morphology, small size and occlusal variability of the cheekteeth suggest that there has not been strong selective pressure acting to maintain an effective dentition. In contrast, great development of the horny pads and associated skull parameters and their lower size variability suggest that the horny pads may have assumed the major role in food comminution.

Bifurcation analysis of a dynamic power system with SVC devices

Bifurcation analysis is a very useful tool for power system stability assessment. In this paper, detailed investigation of power system bifurcation behaviour is presented. One and two parameter bifurcation analysis are conducted on a 3-bus power system. We also examined the impact of FACTS devices on power system stability through Hopf bifurcation analysis by taking static Var compensator (SVC) as an example. A simplified first-order model of the SVC device is included in the 3-bus sample system. Real and reactive powers are used as bifurcation parameter in the analysis to compare the system oscillatory properties with and without SVC. The simulation results indicate that the linearized system model with SVC enlarge the voltage stability boundary by moving Hopf bifurcation point to higher level of loading conditions. The installation of SVC increases the dynamic stability range of the system, however complicates the Hopf bifurcation behavior of the system

Bifurcation contrasts between plane Poiseuille flow and plane Magnetohydrodynamic flow

The stability characteristics of an incompressible viscous pressure-driven flow of an electrically conducting fluid between two parallel boundaries in the presence of a transverse magnetic field are compared and contrasted with those of Plane Poiseuille flow (PPF). Assuming that the outer regions adjacent to the fluid layer are perfectly electrically insulating, the appropriate boundary conditions are applied. The eigenvalue problems are then solved numerically to obtain the critical Reynolds number Rec and the critical wave number ac in the limit of small Hartmann number (M) range to produce the curves of marginal stability. The non-linear two-dimensional travelling waves that bifurcate by way of a Hopf bifurcation from the neutral curves are approximated by a truncated Fourier series in the streamwise direction. Two and three dimensional secondary disturbances are applied to both the constant pressure and constant flux equilibrium solutions using Floquet theory as this is believed to be the generic mechanism of instability in shear flows. The change in shape of the undisturbed velocity profile caused by the magnetic field is found to be the dominant factor. Consequently the critical Reynolds number is found to increase rapidly with increasing M so the transverse magnetic field has a powerful stabilising effect on this type of flow.

ProxiMAX randomisation:a new technology for non-degenerate saturation mutagenesis of contiguous codons

Back in 2003, we published ‘MAX’ randomisation, a process of non-degenerate saturation mutagenesis using exactly 20 codons (one for each amino acid) or else any required subset of those 20 codons. ‘MAX’ randomisation saturates codons located in isolated positions within a protein, as might be required in enzyme engineering, or else on one face of an alpha-helix, as in zinc finger engineering. Since that time, we have been asked for an equivalent process that can saturate multiple, contiguous codons in a non-degenerate manner. We have now developed ‘ProxiMAX’ randomisation, which does just that: generating DNA cassettes for saturation mutagenesis without degeneracy or bias. Offering an alternative to trinucleotide phosphoramidite chemistry, ProxiMAX randomisation uses nothing more sophisticated than unmodified oligonucleotides and standard molecular biology reagents. Thus it requires no specialised chemistry, reagents nor equipment and simply relies on a process of saturation cycling comprising ligation, amplification and digestion for each cycle. The process can encode both unbiased representation of selected amino acids or else encode them in pre-defined ratios. Each saturated position can be defined independently of the others. We demonstrate accurate saturation of up to 11 contiguous codons. As such, ProxiMAX randomisation is particularly relevant to antibody engineering.

ProxiMAX Randomisation:a new technology for non-degenerate saturation mutagenesis of contiguous codons

ProxiMAX randomisation achieves saturation mutagenesis of contiguous codons without degeneracy or bias. Offering an alternative to trinucleotide phosphoramidite chemistry, it uses nothing more sophisticated than unmodified oligonucleotides and standard molecular biology reagents and as such, requires no specialised chemistry, reagents nor equipment. When particular residues are known to affect protein activity/specificity, their combinatorial replacement with all 20 amino acids, or a subset thereof, can provide a rapid route to generating proteins with desirable characteristics. Conventionally, saturation mutagenesis replaced key codons with degenerate ones. Although simple to perform, that procedure resulted in unnecessarily large libraries, termination codons and inherent uneven amino acid representation. ProxiMAX randomisation is an enzyme-based technique that can encode unbiased representation of all or selected amino acids or else can provide required codons in pre-defined ratios. Each saturated position can be defined independently of the others. ProxiMAX randomisation is achieved via saturation cycling: an iterative process comprising blunt end ligation, amplification and digestion with a Type IIS restriction enzyme. We demonstrate both unbiased saturation of a short 6-mer peptide and saturation of a hypervariable region of a scfv antibody fragment, where 11 contiguous codons are saturated with selected codons, in pre-defined ratios. As such, ProxiMAX randomisation is particularly relevant to antibody engineering. The development of ProxiMAX randomisation from concept to reality is described.