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PubMed2001

A model for a network of phosphorylation-dephosphorylation cycles displaying the dynamics of dominoes and clocks.

We consider a model for a network of phosphorylation-dephosphorylation cycles coupled through forward and backward regulatory interactions, such that a protein phosphorylated in a given cycle activates the phosphorylation of a protein by a kinase in the next cycle as well as the dephosphorylation of a protein by a phosphatase in a preceding cycle. The network is cyclically organized in such a way that the protein phosphorylated in the last cycle activates the kinase in the first cycle. We study the dynamics of the network in the presence of both forward and backward coupling, in conditions where a threshold exists in each cycle in the amount of protein phosphorylated as a function of the ratio of kinase to phosphatase maximum rates. We show that this system can display sustained (limit-cycle) oscillations in which each cycle in the pathway is successively turned on and off, in a sequence resembling the fall of a series of dominoes. The model thus provides an example of a biochemical system displaying the dynamics of dominoes and clocks (Murray & Kirschner, 1989). It also shows that a continuum of clock waveforms exists of which the fall of dominoes represents a limit. When the cycles in the network are linked through only forward (positive) coupling, bistability is observed, while in the presence of only backward (negative) coupling, the system can display multistability or oscillations, depending on the number of cycles in the network. Inhibition or activation of any kinase or phosphatase in the network immediately stops the oscillations by bringing the system into a stable steady state; oscillations resume when the initial value of the kinase or phosphatase rate is restored. The progression of the system on the limit cycle can thus be temporarily halted as long as an inhibitor is present, much as when a domino is held in place. These results suggest that the eukaryotic cell cycle, governed by a network of phosphorylation-dephosphorylation reactions in which the negative control of cyclin-dependent kinases plays a prominent role, behaves as a limit-cycle oscillator impeded in the presence of inhibitors. We contrast the case where the sequence of domino-like transitions constitutes the clock with the case where the sequence of transitions is passively coupled to a biochemical oscillator operating as an independent clock.

Animals↗
PubMed2001

Increased coupling between subpopulations in a spatially structured environment can lead to population outbreaks.

Destruction and fragmentation of habitats is widely considered as a major threat to biological diversity. A theoretical framework aimed at understanding and predicting species responses to these destructive processes is still lacking, however. In this paper, the species dynamics in a spatially structured, two-habitat, patchy environment is considered subject to changes in individual migration intensity, i.e. coupling between the habitats. The subpopulation dynamics inside each habitat is assumed to be bistable but with different parameter values. By using space-discrete/continuous metapopulation dynamic models and computer simulations, we show that there can be two principally different regimes of metapopulation dynamics. With increasing intensity in the interplay between subpopulations, the total abundance can either gradually decrease or experience a sudden burst-like increase. This result is shown to be robust to the choice of mathematical models (discrete or continuous). Particularly, both the "self-excitation" and "self-inhibition" regimes of the metapopulation system are robust to variation in habitat size; however, when one of the habitats is much smaller than the other, the "self-excitation" regime can give way to the "self-inhibition" regime and vice versa.

Animals↗
PubMed2002

Modeling extracellular matrix degradation balance with proteinase/transglutaminase cycle.

Extracellular matrix mass balance is implied in many physiological and pathological events, such as metastasis dissemination. Widely studied, its destructive part is mainly catalysed by extracellular proteinases. Conversely, the properties of the constructive part are less obvious, cellular neo-synthesis being usually considered as its only element. In this paper, we introduce the action of transglutaminase in a mathematical model for extracellular matrix remodeling. This extracellular enzyme, catalysing intermolecular protein cross-linking, is considered here as a reverse proteinase as far as the extracellular matrix physical state is concerned. The model is based on a proteinase/transglutaminase cycle interconverting insoluble matrix and soluble proteolysis fragments, with regulation of cellular proteinase expression by the fragments. Under "closed" (batch) conditions, i.e. neglecting matrix influx and fragment efflux from the system, the model is bistable, with reversible hysteresis. Extracellular matrix proteins concentration abruptly switches from low to high levels when transglutaminase activity exceeds a threshold value. Proteinase concentration usually follows the reverse complementary kinetics, but can become apparently uncoupled from extracellular matrix concentration for some parameter values. When matrix production by the cells and fragment degradation are taken into account, the dynamics change to sustained oscillations because of the emergence of a stable limit cycle. Transitions out of and into oscillation areas are controlled by the model parameters. Biological interpretation indicates that these oscillations could represent the normal homeostatic situation, whereas the other exhibited dynamics can be related to pathologies such as tumor invasion or fibrosis. These results allow to discuss the insights that the model could contribute to the comprehension of these complex biological events.

Animals↗
PubMed2002

Neural correlates of visual-motion perception as object- or self-motion.

Both self-motion and objects moving in our visual field generate visual motion by displacing images on the retina. Resolving this ambiguity may seem effortless but large-field visual-motion stimuli can yield perceptual rivalry between the real percept of object-motion and the illusory percept of self-motion (vection). We used functional magnetic resonance imaging to record brain activity in human observers exposed to constant-velocity roll-motion. This stimulus induced responses in areas reaching from calcarine to parieto-occipital and to ventral and lateral temporo-occipital cortex and the anterior insula. During vection, early motion-sensitive visual areas and vestibular parieto-insular cortex deactivated, whereas higher-order parieto- and temporo-occipital areas known to respond to optical flow retained identical activity levels. Within this sustained response, these latter areas displayed transient activations in response to each perceptual switch as identified in event-related analyses. Our results thus show that these areas are responsive to the type of visual motion stimulus and highly sensitive to its perceptual bistability. The only region to be more active during perceived self-motion was in, or close to, the cerebellar nodulus. This activation may correspond to the gain increase of torsional optokinetic nystagmus during vection and/or to changes in sensory processing related to the rotational percept. In conclusion, we identified neural correlates of perceiving self-motion from vision alone, i.e., in the absence of confirmatory vestibular or proprioceptive input. These functional properties preserve the organism's ability to move accurately in its environment by relying on visual cues under conditions when the other spatial senses fail to provide such information.

Adult↗
PubMed1999

Modelling density-dependent resistance in insect-pathogen interactions.

We consider a mathematical model for a host-pathogen interaction where the host population is split into two categories: those susceptible to disease and those resistant to disease. Since the model was motivated by studies on insect populations, we consider a discrete-time model to reflect the discrete generations which are common among insect species. Whether an individual is born susceptible or resistant to disease depends on the local population levels at the start of each generation. In particular, we are interested in the case where the fraction of resistant individuals in the population increases as the total population increases. This may be seen as a positive feedback mechanism since disease is the only population control imposed upon the system. Moreover, it reflects recent experimental observations from noctuid moth-baculovirus interactions that pathogen resistance may increase with larval density. We find that the inclusion of a resistant class can stabilise unstable host-pathogen interactions but there is greatest regulation when the fraction born resistant is density independent. Nonetheless, inclusion of density dependence can still allow intrinsically unstable host-pathogen dynamics to be stabilised provided that this effect is sufficiently small. Moreover, inclusion of density-dependent resistance to disease allows the system to give rise to bistable dynamics in which the final outcome is dictated by the initial conditions for the model system. This has implications for the management of agricultural pests using biocontrol agents-in particular, it is suggested that the propensity for density-dependent resistance be determined prior to such a biocontrol attempt in order to be sure that this will result in the prevention of pest outbreaks, rather than their facilitation. Finally we consider how the cost of resistance to disease affects model outcomes and discover that when there is no cost to resistance, the model predicts stable periodic outbreaks of the insect population. The results are interpreted ecologically and future avenues for research to address the shortfalls in the present model system are discussed.

Animals↗
PubMed2001

Two-sex population dynamics in space: effects of gestation time on persistence.

Most single-species population models assume either that one sex dominates the growth dynamics (usually the female), or that the life cycles of the two sexes are identical; however, sexual differences in ontogenetic features can render this assumption invalid. Further, the interaction between sexes is necessarily nonlinear, and the dependence of dynamic behavior on sexual interactions can be complicated. Here we examine a two-sex population model, related to the well-known logistic model, with explicit sexual interactions. The model is bistable and, by the addition of diffusion, admits traveling wave solutions. Dominance of states via this spatial dynamic are examined. A simple condition for neutral dominance is obtained; sexual interactions inhibit the dominance of the nonzero population, making persistence more difficult.

Animals↗
PubMed1991

Patterns in the effects of infectious diseases on population growth.

An infectious disease may reduce or even stop the exponential growth of a population. We consider two very simple models for microparasitic and macroparasitic diseases, respectively, and study how the effect depends on a contact parameter kappa. The results are presented as bifurcation diagrams involving several threshold values of kappa. The precise form of the bifurcation diagram depends critically on a second parameter xi, measuring the influence of the disease on the fertility of the hosts. A striking outcome of the analysis is that for certain ranges of parameter values bistable behaviour occurs: either the population grows exponentially or it oscillates periodically with large amplitude.

Birth Rate↗
PubMed1992

Application of the metabolic control theory to the study of the dynamics of substrate cycles.

Substrate cycles are ubiquitous structures of the cellular metabolism (e.g. Krebs cycle, fatty acids beta-oxydation cycles, etc...). Moiety-conserved cycles (e.g. adenine nucleotides and NADH/NAD, etc...) are also important. The role played by such cycles in the metabolism and its regulation is not clearly understood so far. However, it was shown that these cycles can generate multistationarity (bistability), irreversible transitions, enhancement of sensitivity, temporal oscillations and chaotic motions (Hervagault & Canu, 1987; Hervagault & Cimino, 1989; Reich & Sel'kov, 1981; Ricard & Soulié, 1982). [formula: see text] Fig. 1: Scheme of the open binary substrate cycle under study. The substrate S is converted into P with a net rate v2. Substrate P is converted in turn into S with a net rate v3. Step v2 is inhibited by excess of the substrate, S. In addition, the cycle operates under open conditions, that is zero-order input of S at rates alpha 0(v1) and first order outputs of S and P at rates alpha S and alpha P(v4), respectively. The metabolic control theory (see also Fell, 1990), which shows how a metabolic network reacts to small perturbations in the vicinity of a steady state, and is formulated with the so-called "control coefficients", was applied to such a cycle in order to get a better knowledge on the importance of each step at the regulatory point of view. The behaviour of a binary substrate cycle (fig. 1) in which one of the enzymes may be subjected to inhibition by excess of its substrate (v2) was studied theoretically.(ABSTRACT TRUNCATED AT 250 WORDS)

Models, Biological↗
PubMed1992

Analytical modeling of the hysteresis phenomenon in guinea pig ventricular myocytes.

In the present study, we have demonstrated hysteresis phenomena in the excitability of single, enzymatically dissociated guinea pig ventricular myocytes. Membrane potentials were recorded with patch pipettes in the whole-cell current clamp configuration. Repetitive stimulation with depolarizing current pulses of constant cycle length and duration but varying strength led to predictable excitation (1:1) and non-excitation (1:0) patterns depending on current strength. In addition, transition between patterns depended on the direction of current intensity change and stable hysteresis loops were obtained in stimulus:response pattern vs. current intensity plots in 14 cells. Increase of pulse duration and decrease of stimulation rate contributed to a reduction in hysteresis loop areas. Changes in amplitude and shape of the subthreshold responses during the transitions from one stable pattern to the other, suggested that activity led to an increase in membrane resistance, particularly in the voltage domain between resting potential, and threshold. Therefore, we modelled the dynamic behaviour of the single cells as a function of diastolic membrane resistance, using previously published analytical solutions. Numerical iteration of the analytical model equations closely reproduced the experimental hysteresis loops in both qualitative and quantitative ways. In particular, the effect of stimulation frequency on the model was similar to the experimental findings. The overall study suggests that the excitability pattern of guinea pig ventricular myocytes accounts for hysteresis and bistabilities when current intensity is allowed to fluctuate around threshold levels.

Action Potentials↗
PubMed1995

Analysis of an autonomous phase model for neuronal parabolic bursting.

An understanding of the nonlinear dynamics of bursting is fundamental in unraveling structure-function relations in nerve and secretory tissue. Bursting is characterized by alternations between phases of rapid spiking and slowly varying potential. A simple phase model is developed to study endogenous parabolic bursting, a class of burst activity observed experimentally in excitable membrane. The phase model is motivated by Rinzel and Lee's dissection of a model for neuronal parabolic bursting (J. Math. Biol 25, 653-675 (1987)). Rapid spiking is represented canonically by a one-variable phase equation that is coupled bi-directionally to a two-variable slow system. The model is analyzed in the slow-variable phase plane, using quasi steady-state assumptions and formal averaging. We derive a reduced system to explore where the full model exhibits bursting, steady-states, continuous and modulated spiking. The relative speed of activation and inactivation of the slow variables strongly influences the burst pattern as well as other dynamics. We find conditions of the bistability of solutions between continuous spiking and bursting. Although the phase model is simple, we demonstrate that it captures many dynamical features of more complex biophysical models.

Animals↗
PubMed1992

The emergence of long-lasting transients of activity in simple neural networks.

The question was investigated whether long-lasting transients of activity, observed to occur in the intact cerebral cortex (EEG slow (delta) waves and 'K' complexes) as well as in isolated tissues cultured in vitro, can also emerge in a model network of excitatory and inhibitory cells. We show that such transients can indeed occur even if the cells do not have built-in slow kinetics. For certain parameter settings, the network is in a bistable state in which periods of increased activity (long-lasting transients) alternate with minimal activity. Transients are triggered by spontaneously firing cells ('noise'), which, rather than via a build-up of recurrent synaptic inhibition, also initiate their termination. During a transient, the network continually makes transitions from one equilibrium to another as a result of spontaneous firing until it is switched back to the quiescent state, i.e., after a variable period of time of noise-induced transitions the transient is terminated. If the network is small, activity can terminate even without inhibition. In large networks, inhibition keeps the network sensitive to spontaneously firing cells by holding it in the neighbourhood of a critical point between active and quiescent state.

Action Potentials↗
PubMed1986

Characteristics of reflex excitation in close synergist muscles evoked by muscle vibration.

The tonic vibration reflex evoked in the vibrated medial gastrocnemius muscle was compared with that induced simultaneously in an unvibrated close synergist, the lateral gastrocnemius muscle, in the unanesthetized decerebrated cat. The time course of the force rise and fall in the synergist muscle was found to be markedly different from that produced in the vibrated muscle. In addition the magnitude of force produced in the synergist was only weakly modulated with increasing vibration frequency, in that the synergist uniformly displayed much flatter force-frequency relations than the vibrated muscle, and even showed overt force saturation in several preparations. Comparable differences in reflex responses arose when the lateral gastrocnemius served as the unvibrated synergist. In a parallel series of experiments, low intensity electrical stimulation of the proximal end of the sectioned medial gastrocnemius muscle nerve at different frequencies weakly modulated the amount of reflex force induced in the lateral gastrocnemius muscle. High frequency electrical stimulation caused the synergist force to saturate in a similar way to that seen in the vibrated preparations. A variety of possible mechanisms are discussed, with particular emphasis on "bistable" properties of motoneurons and Ia excitatory interneuronal contributions.

Animals↗
PubMed1986

Periodic pulsatile stimulation of a nonlinear oscillator.

We consider pulsatile periodic stimulation of an integrate-and-fire oscillator, and investigate the possible phase-locked patterns between the intrinsic rhythm and the forcing system. These stimulations are varied according to their period and intensity, corresponding to controllable experimental parameters. Phase transition curves are derived and analyzed under functional iteration. Two different perturbative mechanisms are considered, leading to significant differences in possible behaviors. Bistability can be obtained in one case. The analysis is reduced to the investigation of two-parameter families of discontinuous maps of the circle into itself.

Mathematics↗
PubMed1985

Multiple stable recurrent outbreaks and predictability in seasonally forced nonlinear epidemic models.

A seasonally forced nonlinear SEIR epidemic model is used to simulate small and large amplitude periodic outbreaks. The model is shown to exhibit bistable behavior for a fixed set of parameters. Basins of attraction for each recurrent outbreak are computed, and it is shown that the basins of two coexisting stable outbreaks are intertwined in a complicated manner. The effect of such a basin structure is shown to result in an obstruction in predicting asymptotically the type of outbreak given an uncertainty in the initial population of susceptibles and infectives.

Epidemiology↗
PubMed1983

Scattering kinetics in a complex tryptophan hydroxylase preparation from rat brainstem raphe nuclei: statistical evidence that the lithium-induced sigmoid velocity function reflects two states of available catalytic potential.

Multiple measures on statistical patterns from a kinetic scattering paradigm for the activities of rat brainstem medial and dorsal raphé nuclear tryptophan hydroxylase preparations are consistent with the previous hypothesis (J. Neural Transmission 45: 1-15, 1979) that the lithium ion induces a state of kinetic bistability in the system. In this context, the lithium-induced sigmoid tetrahydrobiopterin-reaction velocity function is interpreted as a jump between two stable states of differing catalytic potential with a less kinetically accessible domain between them. These dynamics are qualitatively portrayed in three dimensions by Thom's classical cusp catastrophe.

Animals↗
PubMed1975

Five simultaneous steady states in a flipping two-compartment-system with optical antipodes.

The previously given systems-theoretic model for the synthesis of optical antipodes (AD and AL) in strongly asymmetric yield, which shows mono-bistable behaviour depending on the degree of "openess" of the chemical reaction system is reconsidered for two equal compartments (subscripts 1 and 2 on A) with coupling by diffusion. In this configuration three threshold values, j1, is less than j2 is less than j3, for the influx j of the common precursor substance appear. For j is less than j1 only one steady state (s.s.) with no optical activity (ADi = ALi, i-1;2) and equal distribution of the antipodes in both compartments (AD1 = AD2, AL1 = AL2) exists. For j is greater than j 1, this totally symmetric s.s. becomes unstable and a pair of s.s. with optical activity (AD1 is less than AL1, AD2 is less than AL2 or AD1 is greater than AL1, AD2 is greater than AL2) but no spatial asymmetry emerges (parallel flipping), i.e. both compartments be8have as a whole, showing a preponderance of either the D- or the L-form. For j is greater than j2 in addition two new s.s. are possible with antiparallel flipping (AD1 is less than AL1, AD2 is greater than AL2 or AD1 is greater than AL1, AD2 is less than AL2), i.e. in one compartment the D-form has the majority and in the other one the L-form, but these are stable only beyond a third threshold value j3. A third thinkable pair with no optical activity, but different sum concentrations in both cells, does not exist in this special circuitry, but can be obtained in a slightly changed arrangement. So for j is greater than j2, 5 different (4 stable, 1 unstable) s.s., exist for the same set of parameters, one of which is chosen by the system.

Catalysis↗
PubMed1996

A mathematical model for fibro-proliferative wound healing disorders.

The normal process of dermal wound healing fails in some cases, due to fibro-proliferative disorders such as keloid and hypertrophic scars. These types of abnormal healing may be regarded as pathologically excessive responses to wounding in terms of fibroblastic cell profiles and their inflammatory growth-factor mediators. Biologically, these conditions are poorly understood and current medical treatments are thus unreliable. In this paper, the authors apply an existing deterministic mathematical model for fibroplasia and wound contraction in adult mammalian dermis (Olsen et al., J. theor. Biol. 177, 113-128, 1995) to investigate key clinical problems concerning these healing disorders. A caricature model is proposed which retains the fundamental cellular and chemical components of the full model, in order to analyse the spatiotemporal dynamics of the initiation, progression, cessation and regression of fibro-contractive diseases in relation to normal healing. This model accounts for fibroblastic cell migration, proliferation and death and growth-factor diffusion, production by cells and tissue removal/decay. Explicit results are obtained in terms of the model processes and parameters. The rate of cellular production of the chemical is shown to be critical to the development of a stable pathological state. Further, cessation and/or regression of the disease depend on appropriate spatiotemporally varying forms for this production rate, which can be understood in terms of the bistability of the normal dermal and pathological steady states-a central property of the model, which is evident from stability and bifurcation analyses. The work predicts novel, biologically realistic and testable pathogenic and control mechanisms, the understanding of which will lead toward more effective strategies for clinical therapy of fibro-proliferative disorders.

Adult↗
PubMedDate not supplied

Liver system. IV. More on the liver micro-unit: physics and geometry of its action.

The geometry of liver functional micro-units is described in the light of the immunohistologic features of human hepatocytes stained with a retinol-binding protein antibody. The stained cells were distributed into two ordered sets which provided a provisional representation of the law regulating both the motion and the alternation of the rest-work phases (homopoiesis and homorhesis configurations, respectively) which reflect bistable liver micro-unit equilibrium (auto-isodiasostasis). Oscillations between the HP and HR states generate a relaxation cycle that can be assumed to be shared by all micro-units and to be regulated by feedback mechanisms. A model illustrating the compactness of the liver's global action as a result of the activity of each micro-unit is also proposed.

Biophysical Phenomena↗