Model with asymptotically stable dynamics for Drosophila gap gene network

We consider a model of gap gene expression during the early development of Drosophila embryo. Parameter values in the model have been obtained by fitting to experimental patterns under an additional condition that the solution be asymptotically stable at large times. The patterns at the beginning of gastrulation in such solutions are very close to an actual attractor in the model. It is shown that such solutions are more robust to perturbations of concentrations and parameter values in the model.     

Spatial stability of traveling wave solutions of a nerve conduction equation.

A simplified FitzHugh-Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency wmax. The slower one is found to be unstable and the faster one to be stable, while that at wmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973. Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Effects of quarantine in six endemic models for infectious diseases

Thresholds, equilibria, and their stability are found for SIQS and SIQR epidemiology models with three forms of the incidence. For most of these models, the endemic equilibrium is asymptotically stable, but for the SIQR model with the quarantine-adjusted incidence, the endemic equilibrium is an unstable spiral for some parameter values and periodic solutions arise by Hopf bifurcation. The Hopf bifurcation surface and stable periodic solutions are found numerically.     

Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model

 Reaction-diffusion equations are ubiquitous as models of biological pattern formation. In a recent paper [4] we have shown that incorporation of domain growth in a reaction-diffusion model generates a sequence of quasi-steady patterns and can provide a mechanism for increased reliability of pattern selection. In this paper we analyse the model to examine the transitions between patterns in the sequence. Introducing a piecewise linear approximation we find closed form approximate solutions for steady-state patterns by exploiting a small parameter, the ratio of diffusivities, in a singular perturbation expansion. We consider the existence of these steady-state solutions as a parameter related to the domain length is varied and predict the point at which the solution ceases to exist, which we identify with the onset of transition between patterns for the sequence generated on the growing domain. Applying these results to the model in one spatial dimension we are able to predict the mechanism and timing of transitions between quasi-steady patterns in the sequence. We also highlight a novel sequence behaviour, mode-tripling, which is a consequence of a symmetry in the reaction term of the reaction-diffusion system. Received: 19 December 2000 / Revised version: 24 May 2001 / Published online: 7 December 2001     

Turing Patterns from Dynamics of Early HIV Infection

We have developed a mathematical model for in-host virus dynamics that includes spatial chemotaxis and diffusion across a two-dimensional surface representing the vaginal or rectal epithelium at primary HIV infection. A linear stability analysis of the steady state solutions identified conditions for Turing instability pattern formation. We have solved the model equations numerically using parameter values obtained from previous experimental results for HIV infections. Simulations of the model for this surface show hot spots of infection. Understanding this localization is an important step in the ability to correctly model early HIV infection. These spatial variations also have implications for the development and effectiveness of microbicides against HIV.     

Robustness,flexibility, and the role of lateral inhibition in the neurogenic network

BACKGROUND: Many gene networks used by developing organisms have been conserved over long periods of evolutionary time. Why is that? We showed previously that a model of the segment polarity network in Drosophila is robust to parameter variation and is likely to act as a semiautonomous patterning module. Is this true of other networks as well? RESULTS: We present a model of the core neurogenic network in Drosophila. Our model exhibits at least three related pattern-resolving behaviors that the real neurogenic network accomplishes during embryogenesis in Drosophila. Furthermore, we find that it exhibits these behaviors across a wide range of parameter values, with most of its parameters able to vary more than an order of magnitude while it still successfully forms our test patterns. With a single set of parameters, different initial conditions (prepatterns) can select between different behaviors in the network's repertoire. We introduce two new measures for quantifying network robustness that mimic recombination and allelic divergence and use these to reveal the shape of the domain in the parameter space in which the model functions. We show that lateral inhibition yields robustness to changes in prepatterns and suggest a reconciliation of two divergent sets of experimental results. Finally, we show that, for this model, robustness confers functional flexibility. CONCLUSIONS: The neurogenic network is robust to changes in parameter values, which gives it the flexibility to make new patterns. Our model also offers a possible resolution of a debate on the role of lateral inhibition in cell fate specification.     

Parametric Pattern Selection in a Reaction-Diffusion Model

We compare spot patterns generated by Turing mechanisms with those generated by replication cascades, in a model one-dimensional reaction-diffusion system. We determine the stability region of spot solutions in parameter space as a function of a natural control parameter (feed-rate) where degenerate patterns with different numbers of spots coexist for a fixed feed-rate. While it is possible to generate identical patterns via both mechanisms, we show that replication cascades lead to a wider choice of pattern profiles that can be selected through a tuning of the feed-rate, exploiting hysteresis and directionality effects of the different pattern pathways.     

Cycles in cannibalistic egg-larval interactions

A model of a cannibalistic larval-egg interaction such as occurs in Tribolium is developed which leads to a system of nonlinear Volterra integral equations. I determine the local stability properties of the unique equilibrium point of the model. A Hopf bifurcation analysis shows that the model always undergoes a subcritical bifurcation when stability is lost. Numerical solutions confirm the presence of multiple attractors over a range of parameter values. The form of the cycles observed in the numerical solutions is analogous to that observed in laboratory populations of Tribolium.     

Self-organization in biological systems with multiple cellular contacts

The self-organizing properties of an ensemble of interconnected units are studied by linear stability analyses. Small perturbations of a uniform steady-state may result in bifurcations to other solutions that exhibit spatial or temporal order. We show that increasing the number of connections that a unit makes with its neighbors changes the nature of these solutions and tends to destroy spatiotemporal patterns. If an unconnected system is orginally stable, the formation of multiple interconnections can never induce temporal periodicity but may, under certain circumstances, allow the emergence of stationary spatial patterns. We have verified the predictions of the linear stability analysis on a model system and comment on the implications of these results for multicellular ensembles.     

Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model

In this paper we consider a modified spatiotemporal ecological system originating from the temporal Holling-Tanner model, by incorporating diffusion terms. The original ODE system is studied for the stability of coexisting homogeneous steady-states. The modified PDE system is investigated in detail with both numerical and analytical approaches. Both the Turing and non-Turing patterns are examined for some fixed parametric values and some interesting results have been obtained for the prey and predator populations. Numerical simulation shows that either prey or predator population do not converge to any stationary state at any future time when parameter values are taken in the Turing-Hopf domain. Prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability. With help of numerical simulations we have shown that Turing-Hopf bifurcation leads to onset of spatio-temporal chaos when predator's diffusivity is much higher compared to prey population. Our investigation reveals the fact that Hopf-bifurcation is essential for the onset of spatiotemporal chaos.     

Analysis of a time-delayed mathematical model for tumour growth with an almost periodic supply of external nutrients

In this paper, the existence, uniqueness and exponential stability of almost periodic solutions for a mathematical model of tumour growth are studied. The establishment of the model is based on the reaction–diffusion dynamics and mass conservation law and is considered with a delay in the cell proliferation process. Using a fixed-point theorem in cones, the existence and uniqueness of almost periodic solutions for different parameter values of the model is proved. Moreover, by the Gronwall inequality, sufficient conditions are established for the exponential stability of the unique almost periodic solution. Results are illustrated by computer simulations.     

Variability of bursting patterns in a neuron model in the presence of noise

Spiking and bursting patterns of neurons are characterized by a high degree of variability. A single neuron can demonstrate endogenously various bursting patterns, changing in response to external disturbances due to synapses, or to intrinsic factors such as channel noise. We argue that in a model of the leech heart interneuron existing variations of bursting patterns are significantly enhanced by a small noise. In the absence of noise this model shows periodic bursting with fixed numbers of interspikes for most parameter values. As the parameter of activation kinetics of a slow potassium current is shifted to more hyperpolarized values of the membrane potential, the model undergoes a sequence of incremental spike adding transitions accumulating towards a periodic tonic spiking activity. Within a narrow parameter window around every spike adding transition, spike alteration of bursting is deterministically chaotic due to homoclinic bifurcations of a saddle periodic orbit. We have found that near these transitions the interneuron model becomes extremely sensitive to small random perturbations that cause a wide expansion and overlapping of the chaotic windows. The chaotic behavior is characterized by positive values of the largest Lyapunov exponent, and of the Shannon entropy of probability distribution of spike numbers per burst. The windows of chaotic dynamics resemble the Arnold tongues being plotted in the parameter plane, where the noise intensity serves as a second control parameter. We determine the critical noise intensities above which the interneuron model generates only irregular bursting within the overlapped windows.     

A non-linear analysis for spatial structure in a reaction-diffusion model

A non-linear stability analysis using a multi-scale perturbation procedure is carried out on the practical Thomas reaction-diffusion mechanism which exhibits bifurcation to non-uniform states. The analytical results compare favourably with the numerical solutions. The sequential patterns generated by this model by variations in a parameter related to the reaction-diffusion domain indicate its capacity to represent certain key morphogenetic features required in a recent model by Kauffman for pattern formation in theDrosophila embryo.     

Stability and parameter dependency analysis of a facilitation tectal column (FTC) model

Mathematical models and computer simulations have been widely used to study the spatio-temporal characteristics of the processing of information carried out by the central nervous system. When trying to show whether or not a neural model accounts for the phenomena under study, if the number of parameters whose values need to be calculated becomes large, then computer simulations alone become very inefficient to define such values. Here, we developed stability and parameter dependency analyses of the mathematical representation of a single facilitation tectal column (FTC) model, to show how by using techniques from non-linear systems theory we can define the ranges of parameter values under which the model would explain the required performance of the neural net model. The benefits of these analyses can be grouped in two parts: first, the advantage of using non-linear systems techniques to analyze, analytically, the dynamics of neural net models; and second, we get a deeper understanding of why the hypotheses embedded in the models yield the appropriate behaviors and what are the critical situations (parametric combinations) under which these behaviors are displayed.     

A numerical study of the Belousov-Zhabotinskii reaction using Galerkin finite element methods

The Belousov-Zhabotinskii reaction has been modelled by Field and Noyes [5] as a pair of nonlinear parabolic equations. Previous studies of these, both theoretical and numerical, have assumed wave solutions travelling with constant velocity leading to a simplification of the mathematical model in the form of a system of ordinary differential equations. In the present study a finite element Galerkin method is used directly on the original parabolic system for a range of parameter values.     

Xanthan gum production under several operational conditions: molecular structure and rheological properties*

Xanthan gum production under several operational conditions has been studied. Temperature, initial nitrogen concentration and oxygen mass transfer rate have been changed and average molecular weight, pyruvilation and acetylation degree of xanthan produced have been measured in order to know the influence of these variables on the synthesised xanthan molecular structure. Also, xanthan gum solution viscosity has been measured, and rheological properties of the solutions have been related to molecular structure and operational conditions. The Casson model has been employed to describe the rheological behaviour. The parameter values of the Casson model, tau(0) and K(c), have been obtained for each polysaccharide synthesised under different operational conditions. Both pyruvilation and acetylation degrees and average molecular weight of xanthan increase with fermentation time at any operating conditions. Xanthan molecules with the highest average molecular weight have been obtained at 25 degrees C. Nevertheless, at this temperature acetate and pyruvate radical concentration are lowest. Nitrogen concentration in broth does not show any clear influence over xanthan average molecular weight, although with high nitrogen source concentration xanthan with low pyruvilation degree is produced.     

Regulated transport as a mechanism for pattern generation: Capabilities for phyllotaxis and beyond

Large-scale pattern formation is a frequently occurring phenomenon in biological organisms, and several local interaction rules for generating such patterns have been suggested. A mechanism driven by feedback between the plant hormone auxin and its polarly localized transport mediator PINFORMED1 has been proposed as a model for phyllotactic patterns in plants. It has been shown to agree with current biological experiments at a molecular level as well as with respect to the resulting patterns. We present a thorough investigation of variants of models based on auxin-regulated polarized transport and use analytical and numerical tools to derive requirements for these models to drive spontaneous pattern formation. We find that auxin concentrations in neighboring cells can feed back either on exocytosis or endocytosis and still produce patterns. In agreement with mutant experiments, the active cellular efflux is shown to be more important for pattern capabilities as compared to active influx. We also find that the feedback must originate from neighboring cells rather than from neighboring walls and that intracellular competition for the transport mediator is required for patterning. The importance of model parameters is investigated, especially regarding robustness to perturbations of experimentally estimated parameter values. Finally, the regulated transport mechanism is shown to be able to generate Turing patterns of various types.     

Traveling Wave Solutions of a Nerve Conduction Equation

We consider a pair of differential equations whose solutions exhibit the qualitative properties of nerve conduction, yet which are simple enough to be solved exactly and explicitly. The equations are of the FitzHugh-Nagumo type, with a piecewise linear nonlinearity, and they contain two parameters. All the pulse and periodic solutions, and their propagation speeds, are found for these equations, and the stability of the solutions is analyzed. For certain parameter values, there are two different pulse-shaped waves with different propagation speeds. The slower pulse is shown to be unstable and the faster one to be stable, confirming conjectures which have been made before for other nerve conduction equations. Two periodic waves, representing trains of propagated impulses, are also found for each period greater than some minimum which depends on the parameters. The slower train is unstable and the faster one is usually stable, although in some cases both are unstable.