Asymptotic behaviour of a model of hierarchically structured population dynamics

 A hierarchically structured population model with a dependence of the vital rates on a function of the population density (environment) is considered. The existence, uniqueness and the asymptotic behaviour of the solutions is obtained transforming the original non-local PDE of the model into a local one. Under natural conditions, the global asymptotical stability of a nontrivial equilibrium is proved. Finally, if the environment is a function of the biomass distribution, the existence of a positive total biomass equilibrium without a nontrivial population equilibrium is shown. Received 16 February 1996; received in revised form 16 September 1996     

On periodic cohort solutions of a size-structured population model

 We consider a size-structured population model with discontinuous reproduction and feedback through the environmental variable ‘substrate’. The model admits solutions with finitely many cohorts and in that case the problem is described by a system of ODEs involving a bifurcation parameter β. Existence of nontrivial periodic n-cohort solutions is investigated. Moreover, we discuss the question whether n cohorts (n≧2) with small size differences will tend to a periodic one-cohort solution as t→∞. Received 16 March 1995; received in revised form 7 January 1997     

Approximating the long-term behaviour of a model for parasitic infection

In a companion paper two stochastic models, useful for the initial behaviour of a parasitic infection, were introduced. Now we analyse the long term behaviour. First a law of large numbers is proved which allows us to analyse the deterministic analogues of the stochastic models. The behaviour of the deterministic models is analogous to the stochastic models in that again three basic reproduction ratios are necessary to fully describe the information needed to separate growth from extinction. The existence of stationary solutions is shown in the deterministic models, which can be used as a justification for simulation of quasi-equilibria in the stochastic models. Host-mortality is included in all models. The proofs involve martingale and coupling methods.     

A dynamic model for the p53 stress response networks under ion radiation

Summary. P53 controls the cell cycle arrest and cell apoptosis through interaction with the downstream genes and their signal pathways. To stimulate the investigation into the complicated responses of p53 under the circumstance of ion radiation (IR) in the cellular level, a dynamic model for the p53 stress response networks is proposed. The model can be successfully used to simulate the dynamic processes of generating the double-strand breaks (DSBs) and their repairing, ataxia telangiectasia mutated (ATM) activation, as well as the oscillations occurring in the p53-MDM2 feedback loop.     

A mathematical model of the stress induced during avascular tumour growth

In this paper a mathematical model is developed to describe the effect of nonuniform growth on the mechanical stress experienced by cells within an avascular tumour. The constitutive law combines the stress-strain relation of linear elasticity with a growth term that is derived by analogy with thermal expansion. To accommodate the continuous nature of the growth process, the law relates the rate of change of the stress tensor to the rate of change of the strain (rather than relating the stress to the strain directly). By studying three model problems which differ in detail, certain characteristic features are identified. First, cells near the tumour boundary, where nutrient levels and cell proliferation rates are high, are under compression. By contrast, cells towards the centre of the tumour, where nutrient levels are low and cell death dominant, are under tension. The implications of these results and possible model developments are also discussed. Received: 15 November 1999 / Published online: 5 May 2000     

A generalized transport model for biased cell migration in an anisotropic environment

 A generalized transport model is derived for cell migration in an anisotropic environment and is applied to the specific cases of biased cell migration in a gradient of a stimulus (taxis; e.g., chemotaxis or haptotaxis) or along an axis of anisotropy (e.g., contact guidance). The model accounts for spatial or directional dependence of cell speed and cell turning behavior to predict a constitutive cell flux equation with drift velocity and diffusivity tensor (termed random motility tensor) that are explicit functions of the parameters of the underlying random walk model. This model provides the connection between cell locomotion and the resulting persistent random walk behavior to the observed cell migration on longer time scales, thus it provides a framework for interpreting cell migration data in terms of underlying motility mechanisms. Received: 8 April 1999     

Asymptotics of the total net direct pharmacological effect for large drug doses

The direct pharmacological effect E is described by the E _max model relating E to the drug plasma concentration C _p . The area under the effect vs. time curve (AUC _E ) is used as the measurement of the total net pharmacological effect. The drug plasma concentrations are solutions of compartmental systems of ordinary differential equations with the input terminated after a finite time and controlled in a proportional manner by a single dose-like parameter. The asymptotics of the time derivative of C _p for large doses are derived and used as conditions which have to be satisfied by functions for which the asymptotics of the integral defining AUC _E are derived. The AUC _E is proportional to the time T _C>EC50 for which the drug concentration stays above the threshold level EC ₅₀. The threshold EC ₅₀ denotes the drug plasma concentration which elicits 50% of the maximum effect. The parameter T _C>EC50 is proportional to the logarithm of drug dose for large doses and its asymptotics is calculated up to the order o(1) as dose increases to infinity. The results are applied to basic pharmacokinetic systems. Received: 7 December 1999 / Revised version: 23 May 2000 / Published online: 23 October 2000     

Periodic solutions in a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor

We obtain necessary and sufficient conditions on the existence of a unique positive equilibrium point and a set of sufficient conditions on the existence of periodic solutions for a 3-dimensional system which arises from a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor. Our results improve the corresponding results obtained by Hsu, Luo, and Waltman [1]. Received: 20 November 1997 / Revised version: 12 February 1999 / Published online: 20 December 2000     

A dynamical model for plant cell wall architecture formation

We discuss a dynamical mathematical model to explain cell wall architecture in plant cells. The highly regular textures observed in cell walls reflect the spatial organisation of the cellulose microfibrils (CMFs), the most important structural component of cell walls. Based on a geometrical theory proposed earlier [A. M. C. Emons, Plant, Cell and Environment 17, 3–14 (1994)], the present model describes the space-time evolution of the density of the so-called rosettes, the CMF synthesizing complexes. The motion of these rosettes in the plasma membrane is assumed to be governed by an optimal packing constraint on the CMFs plus adherent matrix material, that couples the direction of motion, and hence the orientation of the CMF being deposited, to the local density of rosettes. The rosettes are created inside the cell in the endoplasmatic reticulum and reach the cell-membrane via vesicles derived from Golgi-bodies. After being inserted into the plasma membrane they are assumed to be operative for a fixed, finite lifetime. The plasma membrane domains within which rosettes are activated are themselves also supposed to be mobile. We propose a feedback mechanism that precludes the density of rosettes to rise beyond a maximum dictated by the geometry of the cell. The above ingredients lead to a quasi-linear first order PDE for the rosette-density. Using the method of characteristics this equation can be cast into a set of first order ODEs, one of which is retarded. We discuss the analytic solutions of the model that give rise to helicoidal, crossed polylamellate, helical, axial and random textures, since all cell walls are composed of (or combinations of) these textures. Received: 10 July 1999 / Revised version: 7 June 2000 / Published online: 16 February 2001     

Linear stability criteria for population models with periodically perturbed delays

We examine some simple population models that incorporate a time delay which is not a constant but is instead a known periodic function of time. We examine what effect this periodic variation has on the linear stability of the equilibrium states of scalar population models and of a simple predator prey system. The case when the delay differs from a constant by a small amplitude periodic perturbation can be treated analytically by using two-timing methods. Of particular interest is the case when the system is initially marginally stable. The introduction of variation in the delay can then have either a stabilising effect or a destabilizing one, depending on the frequency of the periodic perturbation. The case when the periodic perturbation has large amplitude is studied numerically. If the fluctuation is large enough the effect can be stabilising.     

Subchronic-intermittent caffeine amplifies the motor effects of amphetamine in rats

Summary. Caffeine, the most widely consumed psychostimulant drug, acutely stimulates motor behaviour and enhances dopamine agonists actions whilst chronically it induces tolerance to either caffeine- or dopamine agonist-induced motor activating effects. The present study examined whether subchronic caffeine administration (15 mg/kg, on alternate days for 14 days) induces enduring modifications in caffeine- and amphetamine-mediated motor activity. To this end, motor activation and rotational behaviour stimulated by either caffeine or D-amphetamine (0.5, 2 mg/kg), given 3 days after the last caffeine administration, were evaluated in neurologically intact and unilaterally 6-hydroxydopamine-lesioned rats respectively. Subchronic caffeine resulted in an increase in caffeine-induced motor and turning behaviour. Furthermore, caffeine pretreatment potentiated the motor effects of amphetamine in both intact and 6-hydroxydopamine-lesioned rats. These results suggest that subchronic caffeine treatment results in an enhancement of its motor stimulant effects, rather than in tolerance, and induces neuroadaptive facilitatory changes in dopamine transmission.     

Asynchronous exponential growth for a two-phase size-structured population model and comparison with the corresponding one-phase model

In this paper we study a two-phase size-structured population model with distributed delay in the birth process. This model distinguishes individuals by ‘active’ or ‘resting’ status. The individuals in the two life-stages have different growth rates. Only individuals in the ‘active’ stage give birth to the individuals in the ‘active’ stage or the ‘resting’ stage. The size of all the newborns is 0. By using the method of semigroups, we obtain that the model is globally well-posed and its solution possesses the property of asynchronous exponential growth. Moreover, we give a comparison between this two-phase model with the corresponding one-phase model and show that the asymptotic behaviours of the sum of the densities of individuals in the ‘active’ stage and the ‘resting’ stage and the solution of the corresponding one-phase model are different.     

Synthesis, tandem MS- and NMR-based characterization, and quantification of the carbon 13-labeled advanced glycation endproduct, 6-N-carboxymethyllysine

Summary. 6-N-carboxymethyllysine (CML), generated by the glycation and/or oxidation of lysine residues, has been measured in biological materials and food products using techniques such as ELISA, HPLC with fluorescence detection and mass spectrometry methods. Only limited information has been reported regarding the preparation of standards labeled with either deuterium, ¹³C or ¹⁵N atoms to be used as internal standards. In the present paper, a synthesis of carbon-13 labeled CML is described using l,2-¹³C₂-glyoxylic acid and 2-N-acetyllysine as starting materials. The resulting labeled 2-N-acetyl-CML was purified by HPLC-UV as a dibutyl ester. After a deprotection step, the yield was evaluated to be 53% when the reaction was conducted 17 h at 37°C. CML was extensively studied by ¹H- and ¹³C-NMR and the fragments observed in the collision induced dissociation (CID) spectrum were also assigned. Finally, the standards of CML and carbon-13 labeled CML were accurately quantified based on ¹H-NMR and tandem MS using lysine as an internal reference.     

A simple SIS epidemic model with a backward bifurcation

It is shown that an SIS epidemic model with a non-constant contact rate may have multiple stable equilibria, a backward bifurcation and hysteresis. The consequences for disease control are discussed. The model is based on a Volterra integral equation and allows for a distributed infective period. The analysis includes both local and global stability of equilibria.     

Stability analysis of the partial selfing selection model

We undertake a detailed study of the one-locus two-allele partial selfing selection model. We show that a polymorphic equilibrium can exist only in the cases of overdominance and underdominance and only for a certain range of selfing rates. Furthermore, when it exists, we show that the polymorphic equilibrium is unique. The local stability of the polymorphic equilibrium is investigated and exact analytical conditions are presented. We also carry out an analysis of local stability of the fixation states and then conclude that only overdominance can maintain polymorphism in the population. When the linear local analysis is inconclusive, a quadratic analysis is performed. For some sets of selective values, we demonstrate global convergence. Finally, we compare and discuss results under the partial selfing model and the random mating model.     

A spatially discrete model for aggregating populations

 In this paper we introduce a spatially discrete model for aggregating populations described by a system of ODEs. We study the long time behavior of the solutions and we show that the model contains mechanisms by which individuals in the population aggregate at particular points in space. Received: 29 June 1996 / Revised version: 5 August 1997     

Alterations of taurine in the brain of chronic kainic acid epilepsy model

Summary. The aim of the study was to investigate the changes of taurine in the kainic acid (KA, 10 mg/kg, s.c.) chronic model of epilepsy, six months after KA application. The KA-rats used were divided into a group of animals showing weak behavioural response to KA (WDS, rare focal convulsion; rating scale <2 up to 3 h after KA injection) and a group of strong response to KA (WDS, seizures; rating >3 up to 3 h after KA injection). The brain regions investigated were caudate nucleus, substantia nigra, septum, hippocampus, amygdala/piriform cortex, and frontal, parietal, temporal and occipital cortices. KA-rats with rating <2 developed spontaneous WDS which occurred chronically and six months after KA injection increased taurine levels were found in the hippocampus (125.4% of control). KA-rats with rating >3 developed spontaneous recurrent seizures and six months after injection increased taurine levels were found in the caudate nucleus (162.5% of control) and hippocampus (126.6% of control), while reduced taurine levels were seen in the septum (78.2% of control). In summary, increased taurine levels in the hippocampus may involve processes for membrane stabilisation, thus favouring recovery after neuronal hyperactivity. The increased taurine levels in the caudate nucleus could be involved in the modulation of spontaneous recurrent seizure activity.     

A cytoskeletal basis for wood formation in angiosperm trees: the involvement of microfilaments

The cortical microfilament (MF) component of the cytoskeleton within axial elements of the secondary vascular system of the angiosperm tree, Aesculus hippocastanum L. (horse-chestnut) was studied using transmission electron microscopy of ultrathin sections and indirect immunofluorescence microscopy of actin in thick sections. As seen by electron microscopy, MF bundles have a net axial orientation within fusiform cambial cells and their secondary vascular derivatives (i.e. in the axial xylem and phloem parenchyma, xylem fibres, vessel and sieve elements, and companion cells). Immunofluorescence studies, however, reveal that this axial orientation can be more accurately described as a helix of extremely high pitch; it is a persistent feature of all axial secondary vascular elements during their development. Helical MF arrays are the only arrangement seen in secondary phloem cells. However, in addition to helices, other MF arrays are seen in secondary xylem cells. For example, fibres possess ellipses of MFs associated with simple-pit formation, and vessel elements possess circular arrays of MFs that associate with the developing inter-vessel bordered pits, ray–vessel contact pits, and with the perforation plate. Linear MF arrays are seen co-oriented with the developing tertiary wall-thickenings in vessel elements. The possible roles of MFs during the cytodifferentiation of secondary vascular cells is discussed, and compared with that of microtubules. Received: 7 June 1999 / Accepted: 23 December 1999     

A unified hypothesis of coeliac disease with implications for management of patients

Summary. This mini-review presents the research carried out within the context of two of the main hypotheses of the aetiology of coeliac disease. The enzymopathic hypothesis of the disease has been placed clearly as the underlying deficiency causing increased levels of toxic peptides, while the immunological hypothesis has been implicated in the pathogenesis of the disorder as the result of the action of undigested peptides in the small intestine. As a consequence, we are proposing a unified hypothesis of coeliac disease, which takes into account the actions of these undigested peptides through their direct cytotoxicity and their immunoactivity. At the same time, work aimed at defining some of these biologically active peptides, which could be said to be involved in the aetiopathogenesis of coeliac disease, will be reported. The review also focusses on the use of enzyme therapy for management of the disease, which when used in conjunction with the gluten-free diet, offers a safeguard against damage to the small intestine caused by small amounts of gluten.