The Effect of Loss of Immunity on Noise-Induced Sustained Oscillations in Epidemics

The effect of loss of immunity on sustained population oscillations about an endemic equilibrium is studied via a multiple scales analysis of a SIRS model. The analysis captures the key elements supporting the nearly regular oscillations of the infected and susceptible populations, namely, the interaction of the deterministic and stochastic dynamics together with the separation of time scales of the damping and the period of these oscillations. The derivation of a nonlinear stochastic amplitude equation describing the envelope of the oscillations yields two criteria providing explicit parameter ranges where they can be observed. These conditions are similar to those found for other applications in the context of coherence resonance, in which noise drives nearly regular oscillations in a system that is quiescent without noise. In this context the criteria indicate how loss of immunity and other factors can lead to a significant increase in the parameter range for prevalence of the sustained oscillations, without any external driving forces. Comparison of the power spectral densities of the full model and the approximation confirms that the multiple scales analysis captures nonlinear features of the oscillations.     

Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila

Thanks to genetic and biochemical advances on the molecular mechanism of circadian rhythms in Drosophila, theoretical models closely related to experimental observations can be considered for the regulatory mechanism of the circadian clock in this organism. Modeling is based on the autoregulatory negative feedback exerted by a complex between PER and TIM proteins on the expression of per and tim genes. The model predicts the occurrence of sustained circadian oscillations in continuous darkness. When incorporating light-induced TIM degradation, the model accounts for damping of oscillations in constant light, entrainment of the rhythm by light-dark cycles of varying period or photoperiod, and phase shifting by light pulses. The model further provides a molecular dynamical explanation for the permanent or transient suppression of circadian rhythmicity triggered in a variety of organisms by a critical pulse of light. Finally, the model shows that to produce a robust rhythm the various clock genes must be expressed at the appropriate levels since sustained oscillations only occur in a precise range of parameter values. BioEssays 22:84-93, 2000.     

Oscillations in stomatal conductance of hybrid poplar leaves in the light and dark

Cycling of stomatal conductance in three hybrid poplar ( Populus sp.) cultivars was observed under a variety of conditions. Illumination of plants kept previously in the dark induced very large oscillations with a period of about 40 min and large oscillations with a shorter period (< 10 min) were superimposed on the longer cycles. During these oscillations, large changes in conductance could occur very rapidly (1.0 cm s⁻¹ in 3 min). Plants in constant light also displayed both long and short term cycles in conductance, but these were smaller in amplitude than those induced by sudden illumination. Stomatal oscillations were also observed in darkness and after darkening of previously illuminated plants. These oscillations had shorter (< 30 min) and less regular periods than those observed in the light. Such cycling in the dark is rare. Cycling of the two leaf surfaces was sometimes in synchrony in the light, and more so after a perturbation. Little synchrony between the two surfaces was observed in the dark. Stomatal movements of different leaves on a plant were usually relatively independent. Transient stomatal opening occurred following leaf excision in the light or dark, and often after sudden darkening of intact leaves. Also, stomata of intact leaves sometimes transiently closed following illumination.     

Damped oscillations of the giant reed Arundo donax (Poaceae)

The slender upright culms of the giant reed (Arundo donax L.) are often exposed to dynamic wind loads causing significant swaying. The giant reed has slightly tapered hollow stems (4-6 m high) with flat leaves and an extensive underground rhizomatous system with solid branches bearing adventitious roots. Quantitative analyses of videorecordings prove that A. donax responds to dynamic deflections of the stem with damped harmonic bending oscillations. The logarithmic decrement can be used to calculate the relative damping, as a measure of the plant's capacity to dissipate vibrational energy. Plants with leaves have a significantly higher damping compared to plants without leaves. A comparison of the relative damping of plants with and without leaves shows that this finding is only partly due to aerodynamic resistance of the leaves. Structural damping also contributes considerably to the overall damping of the foliate A. donax stem. By stepwise removal of the underground plant organs the influence of rhizome, roots, and soil on the vibrational behavior was determined. The data indicate that underground plant organs as well as leaf sheaths covering the nodes have no significant influence on damping.     

Control design for sustained oscillation in a two-gene regulatory network

Control strategies for gene regulatory networks have begun to be explored, both experimentally and theoretically, with implications for control of disease as well as for synthetic biology. Recent work has focussed on controls designed to achieve desired stationary states. Another useful objective, however, is the initiation of sustained oscillations in systems where oscillations are normally damped, or even not present. Alternatively, it may be desired to suppress (by damping) oscillations that naturally occur in an uncontrolled network. Here we address these questions in the context of piecewise-affine models of gene regulatory networks with affine controls that match the qualitative nature of the model. In the case of two genes with a single relevant protein concentration threshold per gene, we find that control of production terms (constant control) is effective in generating or suppressing sustained oscillations, while control of decay terms (linear control) is not effective. We derive an easily calculated condition to determine an effective constant control. As an example, we apply our analysis to a model of the carbon response network in Escherichia coli, reduced to the two genes that are essential in understanding its behavior.     

Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour

Circadian rhythms are endogenous oscillations that occur with a period close to 24 h in nearly all living organisms. These rhythms originate from the negative autoregulation of gene expression. Deterministic models based on such genetic regulatory processes account for the occurrence of circadian rhythms in constant environmental conditions (e.g., constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. When the numbers of protein and mRNA molecules involved in the oscillations are small, as may occur in cellular conditions, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering the stochastic version of a deterministic model previously proposed for circadian oscillations of the PER and TIM proteins and their mRNAs in Drosophila. The model is based on repression of the per and tim genes by a complex between the PER and TIM proteins. Numerical simulations of the stochastic version of the model are performed by means of the Gillespie method. The predictions of the stochastic approach compare well with those of the deterministic model with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity from a bifurcation point beyond which the system evolves to stable steady state. Stochastic simulations indicate that robust circadian oscillations can emerge at the cellular level even when the maximum numbers of mRNA and protein molecules involved in the oscillations are of the order of only a few tens or hundreds. The stochastic model also reproduces the evolution to a strange attractor in conditions where the deterministic PER-TIM model admits chaotic behaviour. The difference between periodic oscillations of the limit cycle type and aperiodic oscillations (i.e. chaos) persists in the presence of molecular noise, as shown by means of Poincaré sections. The progressive obliteration of periodicity observed as the number of molecules decreases can thus be distinguished from the aperiodicity originating from chaotic dynamics. As long as the numbers of molecules involved in the oscillations remain sufficiently large (of the order of a few tens or hundreds, or more), stochastic models therefore provide good agreement with the predictions of the deterministic model for circadian rhythms.     

Ventilatory support: a dynamical systems approach

Misunderstanding of the dynamical behavior of the ventilatory system, especially under assisted ventilation, may explain the problems encountered in ventilatory support monitoring. Proportional assist ventilation (PAV) that theoretically gives a breath by breath assistance presents instability with high levels of assistance. We have constructed a mathematical model of interactions between three objects: the central respiratory pattern generator modelled by a modified Van der Pol oscillator, the mechanical respiratory system which is the passive part of the system and a controlled ventilator that follows its own law. The dynamical study of our model shows the existence of two crucial behaviors, i.e. oscillations and damping, depending on only two parameters, namely the time constant of the mechanical respiratory system and a cumulative interaction index. The same result is observed in simulations of spontaneous breathing as well as of PAV. In this last case, increasing assistance leads first to an increase of the tidal volume (V_T), a further increase in assistance inducing a decrease in V_T, ending in damping of the whole system to an attractive fixed point. We conclude that instabilities observed in PAV may be explained by the different possible dynamical behaviors of the system rather than changes in mechanical characteristics of the respiratory system.     

Flash-induced oxygen evolution in photosynthesis: simple solution for the extended S-state model that includes misses, double-hits, inactivation, and backward-transitions

Flash-induced oxygen evolution in higher plants, algae, and cyanobacteria exhibits damped period-four oscillations. To explain such oscillations, Kok suggested a simple phenomenological S-state model, in which damping is due to empirical misses and double-hits. Here we developed an analytical solution for the extended Kok model that includes misses, double-hits, inactivation, and backward-transitions. The solution of the classic Kok model (with misses and double-hits only) can be obtained as a particular case of this solution. Simple equations describing the flash-number dependence of individual S-states and oxygen evolution in both cases are almost identical and, therefore, the classic Kok model does not have a significant advantage in its simplicity over the extended version considered in this article. Developed equations significantly simplify the fitting of experimental data via standard nonlinear regression analysis and make unnecessary the use of many previously developed methods for finding parameters of the model. The extended Kok model considered here can provide additional insight into the effect of dark relaxation between flashes and inactivation.     

Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays

In the glucose-insulin regulatory system, ultradian insulin secretory oscillations are observed to have a period of 50-150 min. After pioneering work traced back to the 1960s, several mathematical models have been proposed during the last decade to model these ultradian oscillations as well as the metabolic system producing them. These currently existing models still lack some of the key physiological aspects of the glucose-insulin system. Applying the mass conservation law, we introduce two explicit time delays and propose a more robust alternative model for better understanding the glucose-insulin endocrine metabolic regulatory system and the ultradian insulin secretory oscillations for the cases of continuous enteral nutrition and constant glucose infusion. We compare the simulation profiles obtained from this two time delay model with those from the other existing models. As a result, we notice many unique features of this two delay model. Based on our intensive simulations, we suspect that one of the possibly many causes of ultradian insulin secretion oscillations is the time delay of the insulin secretion stimulated by the elevated glucose concentration.     

Endogenous rhythms and chaos in crassulacean acid metabolism

Endogenous free-running regular circadian oscillations of net CO₂ exchange in the crassulacean-acidmetabolism (CAM) plant Kalanchoë daigremontiana Hamet et Perrier de la Bâthie under constant external conditions in continuous light have been shown to change to irregular non-predictable (chaotic) time behaviour as irradiance or temperature are raised above a critical level. A model of CAM has been constructed with pools of major metabolites of varying concentrations, flows of metabolites leading to exchange between pools, metabolite transformations determined by chemical reactions, and feedback regulations. The model is described by a system of coupled non-linear differential equations. It shows stable rhythmicity in normal dark-light cycles and in continuous light and, like the K. daigremontiana leaves in the experiments, a change to chaos as irradiance is increased. The maintenance of endogenous oscillations in the model is brought about by a hysteresis switch or beat oscillator between two stable oscillation modes. In CAM these stable modes are vacuolar malate accumulation and remobilization. The model shows that the physical nature of the beat oscillator in the leaves can be explained by the balance between active and passive transport at the tonoplast.Abbreviations CAM crassulacean acid metabolism - D dark period - DL 12:12 h dark-light rhythm - L light period - LL constant illumination - PPFD photosynthetic photon flux density - T^L leaf temperature It is a great pleasure to thank Dr. G.-H. Vieweg, (Roßdorf, FRG) for his long-lasting efforts to have the phytotron in Darmstadt erected and for his persistent involvement during the various phases of planning and building. This made the present experiments possible. Dr. D. Kramer is thanked for all the time he spends to maintain functioning of the facility. Dr. P. Keller and Ms. Erika Ball assisted with the gas-exchange technology and helped with the surveillance of the long-running experiments, and Ms. Erika Ball performed all the integrations. Ms. Doris Schäfer is thanked for drawing the gas-exchange curves for publication. We are also most grateful to Professor Chr. Giersch and Professor M. Kluge (both Institut für Botanik, Technische Hochschule Darmstadt, FRG) for valuable discussions.     

Oscillations in microtubule polymerization: the rate of GTP regeneration on tubulin controls the period.

The essential reactions involved in the oscillatory kinetics of microtubule polymerization have been investigated. The rate of GDP dissociation from tubulin decreased cooperatively upon increasing tubulin concentration above 20 microM, consistent with the formation of GDP oligomers whose dissociation is rate limiting in nucleotide exchange. The apparent rate constant for nucleotide exchange at high tubulin concentration was 0.02 s-1 at 37 degrees C, which is the exact value needed in previous theoretical simulations to obtain oscillations with the real period of 70-80 s. A glass filter assay separating microtubules from oligomers and tubulin allowed nucleotide bound to non-microtubular tubulin during the oscillations to be monitored. In agreement with nucleotide exchange data, tubulin-bound GDP was found to oscillate in antiphase with microtubules. By varying the concentration of an enzymatic GTP-regenerating system, we could demonstrate that the period of the oscillations is directly controlled by the rate at which GTP is regenerated on tubulin, and oscillations can be observed under conditions where the dissociation of oligomers is no longer rate limiting. The possible physiological significance of GTP-regenerating systems in establishing synchrony in a microtubule population is evoked. The present data confirm and extend the model that we previously proposed to account for the oscillations.     

Effects of synergistic signaling by phytochrome A and cryptochrome1 on circadian clock-regulated catalase expression.

Persistent oscillation in constant conditions is a defining characteristic of circadian rhythms. However, in plants transferred into extended dark conditions, circadian rhythms in mRNA abundance commonly damp in amplitude over two or three cycles to a steady state level of relatively constant, low mRNA abundance. In Arabidopsis, catalase CAT3 mRNA oscillations damp rapidly in extended dark conditions, but unlike catalase CAT2 and the chlorophyll a/b binding protein gene CAB, in which the circadian oscillations damp to low steady state mRNA abundance, CAT3 mRNA oscillations damp to high steady state levels of mRNA abundance. Mutational disruption of either phytochrome- or cryptochrome-mediated light perception prevents damping of the oscillations in CAT3 mRNA abundance and reveals strong circadian oscillations that persist for multiple cycles in extended dark conditions. Damping of CAT3 mRNA oscillations specifically requires phytochrome A but not phytochrome B and also requires the cryptochrome1 blue light receptor. Therefore, we conclude that synergistic signaling mediated through both phytochrome A and cryptochrome1 is required for damping of circadian CAT3 mRNA oscillations in extended dark conditions.     

Oscillations in plant membrane transport: model predictions, experimental validation, and physiological implications

Although oscillations in membrane-transport activity are ubiquitous in plants, the ionic mechanisms of ultradian oscillations in plant cells remain largely unknown, despite much phenomenological data. The physiological role of such oscillations is also the subject of much speculation. Over the last decade, much experimental evidence showing oscillations in net ion fluxes across the plasma membrane of plant cells has been accumulated using the non-invasive MIFE technique. In this study, a recently proposed feedback-controlled oscillatory model was used. The model adequately describes the observed ion flux oscillations within the minute range of periods and predicts: (i) strong dependence of the period of oscillations on the rate constants for the H+ pump; (ii) a substantial phase shift between oscillations in net H+ and K+ fluxes; (iii) cessation of oscillations when H+ pump activity is suppressed; (iv) the existence of some 'window' of external temperatures and ionic concentrations, where non-damped oscillations are observed: outside this range, even small changes in external parameters lead to progressive damping and aperiodic behaviour; (v) frequency encoding of environmental information by oscillatory patterns; and (vi) strong dependence of oscillatory characteristics on cell size. All these predictions were successfully confirmed by direct experimental observations, when net ion fluxes were measured from root and leaf tissues of various plant species, or from single cells. Because oscillatory behaviour is inherent in feedback control systems having phase shifts, it is argued from this model that suitable conditions will allow oscillations in any cell or tissue. The possible physiological role of such oscillations is discussed in the context of plant adaptive responses to salinity, temperature, osmotic, hypoxia, and pH stresses.     

Computer simulations of synchrony and oscillations evoked by two coherent inputs

Coherent oscillations have been reported in multiple cortical areas. This study examines the characteristics of output spikes through computer simulations when the neural network model receives periodic/aperiodic spatiotemporal spikes with modulated/constant populational activity from two pathways. Synchronous oscillations which have the same period as the input are observed in response to periodic input patterns regardless of populational activity. The results confirm that the output frequency of synchrony is essentially determined by the period of the repeated input patterns. On the other hand, weak periodic outputs are observed when aperiodic spikes are input with modulated populational activity. In this case, higher firing rates are necessary to input for higher frequency oscillations. The spike-timing-dependent plasticity suppresses the spikes which do not contribute to the synchrony for periodic inputs. This effect corresponds to the experimental reports that learning sharpens the synchrony in the motor cortex. These results suggest that spatiotemporal spike patterns should be entrained on modulated populational activity to transmit oscillatory information effectively in the convergent pathway.     

Oscillations of the internal CO(2) concentration in tobacco leaves transferred to low CO(2)

Measurement of the internal CO(2) concentration (Ci) in tobacco leaves using a fast-response CO(2) exchange system showed that in the light, switching from 350 microLL(-1) to a low CO(2) concentration of 36.5 microLL(-1) (promoting high photorespiration) resulted in the Ci oscillating near the value of CO(2) compensation point (Gamma*). The oscillations are highly irregular, the range of Ci varying by 2-4 microLL(-1) in substomatal cavities with a period of a few seconds. The statistical properties of the time series became stationary after a transient of approximately 100s following transfer to low CO(2). Attractor reconstruction shows that the observed oscillations are not chaotic but exhibit stochastic behavior. The period of oscillations is consistent with the duration of photorespiratory post-illumination burst (PIB). We suggest that the observed oscillations may be due to a similar mechanism to that which leads to PIB, and may play a role in switching mitochondrial operation between oxidation of the photorespiratory glycine and of the tricarboxylic acid cycle substrates.     

The phantom burster model for pancreatic beta-cells

Pancreatic beta-cells exhibit bursting oscillations with a wide range of periods. Whereas periods in isolated cells are generally either a few seconds or a few minutes, in intact islets of Langerhans they are intermediate (10-60 s). We develop a mathematical model for beta-cell electrical activity capable of generating this wide range of bursting oscillations. Unlike previous models, bursting is driven by the interaction of two slow processes, one with a relatively small time constant (1-5 s) and the other with a much larger time constant (1-2 min). Bursting on the intermediate time scale is generated without need for a slow process having an intermediate time constant, hence phantom bursting. The model suggests that isolated cells exhibiting a fast pattern may nonetheless possess slower processes that can be brought out by injecting suitable exogenous currents. Guided by this, we devise an experimental protocol using the dynamic clamp technique that reliably elicits islet-like, medium period oscillations from isolated cells. Finally, we show that strong electrical coupling between a fast burster and a slow burster can produce synchronized medium bursting, suggesting that islets may be composed of cells that are intrinsically either fast or slow, with few or none that are intrinsically medium.     

Computer simulation of damped oscillations during peroxidase-catalyzed oxidation of indole-3-acetic acid

Oscillation patterns in horseradish peroxidase (HRP)-catalyzed oxidation of indole-3-acetic acid (IAA) at neutral pH were studied using computer simulation. Under certain conditions, such as the presence of a reaction promoter and continuous intake of oxygen from the gaseous phase, the simulated system exhibits damped oscillations of the concentrations of oxygen in the aqueous phase, [O(2)](aq), and of all the reaction intermediates. The critical concentration of oxygen in aqueous phase, [O(2)](cr)(aq), was used to describe the nature of the oscillations. The critical concentration is the concentration at which the system abruptly changes its properties. If [O(2)](aq) is higher than [O(2)](cr)(aq) then the reaction develops as an avalanche, otherwise, the reaction stops. The nature of oscillations is accounted for by the interaction of two processes: the consumption/accumulation of oxygen and the accumulation/consumption of reaction intermediates. Oscillations are always damped. Neither HRP or umbelliferone (Umb) deactivation nor IAA consumption can account for the damping. The nature of the damping is determined by the termination reactions of free radical intermediates and ROOH. The three major parameters of oscillations: period of oscillations, initial amplitude of oscillations and the rate of damping were studied as functions of: (i) oxygen concentration in the gaseous phase, (ii) initial oxygen concentration in aqueous phase, (iii) the concentration of IAA and (iv) the initial concentration of HRP.     

Bifurcations in a mathematical model for circadian oscillations of clock genes

Circadian oscillations with a period of about 24h are observed in nearly all living organisms as conspicuous biological rhythms. In this paper, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER-TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami's method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light-dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles.     

Oscillations of plants' stems and their damping: theory and experimentation

Free oscillations of upright plants' stems, or in technical terms slender tapered rods with one end free, can be modelled by considering the equilibrium between bending moments and moments resulting from inertia. For stems with apical loads and negligible mass of the stem and for stems with finite mass but without top loading, analytical solutions of the differential equations with appropriate boundary conditions are available for a finite number of cases. For other cases approximations leading to an upper and a lower estimate of the frequency of oscillation omega can be derived. For the limiting case of omega = 0, the differential equations are identical with Greenhill's equations for the stability against Euler buckling of slender poles. To illustrate, the oscillation frequencies of 25 spruce trees (Picea sitchensis (Bong.) Carr.) were compared with those calculated on the basis of their morphology, their density and their static elasticity modulus. For Arundo donax L. and Cyperus alternifolius L. the observed oscillation frequency was used in turn to calculate the dynamic elasticity modulus, which was compared with that determined in three-point bending. Oscillation damping was observed for A. donax and C. alternifolius for plants' stems with and without leaves or inflorescence. In C. alternifolius the difference can be attributed to the aerodynamic resistance of the leaves, whereas in A. donax structural damping in addition plays a major role.     

A study of the regulation of photosynthesis by the analysis of the effect of kinetic parameters on the characteristics of the oscillatory regime and the rate of CO2 assimilation

Earlier at the biophysics department, the experimental data on the oscillations of delayed luminescence have been described with the help of a mathematical model. Here we studied the influence of the model parameters on the characteristics of the oscillatory regime. The frequencies and damping factors of the oscillations at different parameter values were calculated using the Lyapunov analysis. It was shown that, in addition to oscillations observed experimentally, other, rapidly damping oscillations may exist. The dependence of the CO2 assimilation rate on the model parameters was studied. It was shown that the intensity of the light absorbed by photosystems I and II may differently affect the assimilation of CO2.