Model of calcium oscillations due to negative feedback in olfactory cilia

We present a mathematical model for calcium oscillations in the cilia of olfactory sensory neurons. The underlying mechanism is based on direct negative regulation of cyclic nucleotide-gated channels by calcium/calmodulin and does not require any autocatalysis such as calcium-induced calcium release. The model is in quantitative agreement with available experimental data, both with respect to oscillations and to fast adaptation. We give predictions for the ranges of parameters in which oscillations should be observable. Relevance of the model to calcium oscillations in other systems is discussed.     

Sustained oscillations in glycolysis: an experimental and theoretical study of chaotic and complex periodic behavior and of quenching of simple oscillations

We report sustained oscillations in glycolysis conducted in an open system (a continuous-flow, stirred tank reactor; CSTR) with inflow of yeast extract as well as glucose. Depending on the operating conditions, we observe simple or complex periodic oscillations or chaos. We report the response of the system to instantaneous additions of small amounts of several substrates as functions of the amount added and the phase of the addition. We simulate oscillations and perturbations by a kinetic model based on the mechanism of glycolysis in a CSTR. We find that the response to particular perturbations forms an efficient tool for elucidating the mechanism of biochemical oscillations.     

Analysis of chaotic oscillations induced in two coupled Wilson–Cowan models

Although it is known that two coupled Wilson–Cowan models with reciprocal connections induce aperiodic oscillations, little attention has been paid to the dynamical mechanism for such oscillations so far. In this study, we aim to elucidate the fundamental mechanism to induce the aperiodic oscillations in the coupled model. First, aperiodic oscillations observed are investigated for the case when the connections are unidirectional and when the input signal is a periodic oscillation. By the phase portrait analysis, we determine that the aperiodic oscillations are caused by periodically forced state transitions between a stable equilibrium and a stable limit cycle attractors around the saddle-node and saddle separatrix loop bifurcation points. It is revealed that the dynamical mechanism where the state crosses over the saddle-node and saddle separatrix loop bifurcations significantly contributes to the occurrence of chaotic oscillations forced by a periodic input. In addition, this mechanism can also give rise to chaotic oscillations in reciprocally connected Wilson–Cowan models. These results suggest that the dynamic attractor transition underlies chaotic behaviors in two coupled Wilson–Cowan oscillators.     

Simulation of an oscillatory reaction of alcohol dehydrogenase in an oil/water system

With numerical analysis of the enzyme-catalyzed reaction in the so called 'ordered bi-bi' mechanism and the model presented by Theorell and Chance (Acta Chem. Scand. 5 (1951) 1127) for alcohol dehydrogenase, oscillations of the concentrations of coenzyme and intermediate complex were obtained on gradual entry of substrate. Neither autocatalysis nor allostery was included in the reaction mechanism. The features of the reaction were that the oscillations did not occur in a limit cycle and the interval became longer with time. In addition, it was found that the oscillations with an upwardly directed peak occurred at very low concentrations of enzyme in the model of Theorell and Chance. The results were consistent with the experimentally determined data reported previously.     

A model of intracellular Ca2+ oscillations based on the activity of the intermediate-conductance Ca2+-activated K+ channels

Intracellular Ca2+ oscillations are observed in a large number of non-excitable cells. While most appear to reflect an intermittent Ca2+ release from intracellular stores, in some instances intracellular Ca2+ oscillations strongly depend on Ca2+ influx, and are coupled to oscillations of the membrane potential, suggesting that a plasma membrane-based mechanism may be involved. We have developed a theoretical model for the latter type of intracellular Ca2+ oscillations based on the Ca2+-dependent modulation of the intermediate-conductance, Ca2+-activated K+ (IKCa) channel. The functioning of this model relies on the Ca2+-dependent activation, and the much slower Ca2+-dependent rundown of this channel. We have shown that Ca2+-dependent activation of the IKCa channels, the consequent membrane hyperpolarization and the resulting increase in Ca2+ influx may confer the positive feedback mechanism required for the ascending phase of the oscillation. The much slower Ca2+-dependent rundown process will conversely halt this positive loop, and establish the descending phase of the intracellular Ca2+ oscillation. We found that this simple model gives rise to intracellular Ca2+ oscillations when using physiologically reasonable parameters, suggesting that IKCa channels could participate in the generation of intracellular Ca2+ oscillations.     

Delay model of the circadian pacemaker

We present a simple and realistic model of the circadian pacemaker that can be interpreted in molecular terms. The model, which consists of a single time-delay differential equation, simulates the expression of a generic clock protein that inhibits its own expression through a feedback mechanism. Despite its simplicity, this model fulfils most of the necessary characteristics of a realistic representation of natural circadian clocks: robust and stable oscillations with circadian free-running periods, typical phase response curves and entrainment to environmental zeitgebers. The present model reduces the molecular mechanism necessary to sustain stable oscillations to its bare bones, suggesting that the essential factor is the time-delayed negative feedback of the oscillating protein on its own expression.     

On the encoding and decoding of calcium signals in hepatocytes

Many different agonists use calcium as a second messenger. Despite intensive research in intracellular calcium signalling it is an unsolved riddle how the different types of information represented by the different agonists, is encoded using the universal carrier calcium. It is also still not clear how the information encoded is decoded again into the intracellular specific information at the site of enzymes and genes. After the discovery of calcium oscillations, one likely mechanism is that information is encoded in the frequency, amplitude and waveform of the oscillations. This hypothesis has received some experimental support. However, the mechanism of decoding of oscillatory signals is still not known. Here, we study a mechanistic model of calcium oscillations, which is able to reproduce both spiking and bursting calcium oscillations. We use the model to study the decoding of calcium signals on the basis of co-operativity of calcium binding to various proteins. We show that this co-operativity offers a simple way to decode different calcium dynamics into different enzyme activities.     

A Model of Oscillatory Protein Dynamics in Bacteria

Spatial oscillations of proteins in bacteria have recently attracted much attention. The cellular mechanism underlying these oscillations can be studied at molecular as well as at more macroscopic levels. We construct a minimal mathematical model with two proteins that is able to produce self-sustained regular pole-to-pole oscillations without having to take into account molecular details of the proteins and their interactions. The dynamics of the model is based solely on diffusion across the cell body and protein reactions at the poles, and is independent of stimuli coming from the environment. We solve the associated system of reaction-diffusion equations and perform a parameter scan to demonstrate robustness of the model for two possible sets of the reaction functions.     

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.     

Simulation of chemical oscillations in membranes

A computer simulation model has been developed to follow chemical oscillations in a membrane for immobilized enzyme systems. It is a discrete particle type model which follows the spatial and temporal fluctuations of the concentrations in a reaction involving two substrates. The parameters can be readily varied to allow dissipative structures to result from the sustained nonlinear reaction kinetics and to determine which parameters cause damping of the oscillations. The nature of the diffusion mechanism allows extension to more than one dimension.     

Delays in nutrient cycling and plant population oscillations

It is well known that delay-differential and delay-difference equations can produce plausible simulations of population oscillations, but many of these equations lack a specific mechanism responsible for the delay. We suggest that delays in release of nitrogen from decomposing litter, caused by microbial uptake, could produce oscillations in populations when the delay in the release of nitrogen is longer than the characteristic time scale of nitrogen uptake. We present a model which captures these dynamics. As the parameter controlling microbial uptake of nitrogen during litter decay increases, the model solutions bifurcate from fixed point equilibria, to periodic orbits (oscillations) which remain bounded for ecologically very long times, and finally to extinction of the plant population because of rapid increases in the amplitude of the oscillations. We suggest that such a mechanism may be especially important for annual plants which do not store nitrogen in perennial tissues to buffer delays. Natural populations of wild rice ( Zizania palustris ), an annual plant, oscillate with approximately four-year periods. Our model qualitatively mimics the period and shape of population oscillations in wild rice with parameter values in the range of those determined by experiments. The model therefore demonstrates a logical and experimentally plausible link between plant population dynamics and the ecosystem processes delaying the cycling of limiting nutrients.     

Modeling the mammalian circadian clock: sensitivity analysis and multiplicity of oscillatory mechanisms

We extend the study of a computational model recently proposed for the mammalian circadian clock (Proc. Natl Acad. Sci. USA 100 (2003) 7051). The model, based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, and Clock genes, can give rise to sustained circadian oscillations in conditions of continuous darkness. These limit cycle oscillations correspond to circadian rhythms autonomously generated by suprachiasmatic nuclei and by some peripheral tissues. By using different sets of parameter values producing circadian oscillations, we compare the effect of the various parameters and show that both the occurrence and the period of the oscillations are generally most sensitive to parameters related to synthesis or degradation of Bmal1 mRNA and BMAL1 protein. The mechanism of circadian oscillations relies on the formation of an inactive complex between PER and CRY and the activators CLOCK and BMAL1 that enhance Per and Cry expression. Bifurcation diagrams and computer simulations nevertheless indicate the possible existence of a second source of oscillatory behavior. Thus, sustained oscillations might arise from the sole negative autoregulation of Bmal1 expression. This second oscillatory mechanism may not be functional in physiological conditions, and its period need not necessarily be circadian. When incorporating the light-induced expression of the Per gene, the model accounts for entrainment of the oscillations by light-dark (LD) cycles. Long-term suppression of circadian oscillations by a single light pulse can occur in the model when a stable steady state coexists with a stable limit cycle. The phase of the oscillations upon entrainment in LD critically depends on the parameters that govern the level of CRY protein. Small changes in the parameters governing CRY levels can shift the peak in Per mRNA from the L to the D phase, or can prevent entrainment. The results are discussed in relation to physiological disorders of the sleep-wake cycle linked to perturbations of the human circadian clock, such as the familial advanced sleep phase syndrome or the non-24h sleep-wake syndrome.     

Biophysical basis for principles of universality of speech perception

There is increasing evidence that the synergistic interaction of the objectively accepted low-frequency rhythm and the subjectively accepted high-frequency sound is a source of speech "sense". The lack of one of these components or of their synergistic interaction results in the loss of "sense" of the acoustic signal being perceived. As for the universality of perception of the sound component of speech, there is a model of sound perception, which ensures this universality. This model describes the dynamics of audition using the synchronization model of oscillations in the Ruell-Takens scenario of transition to chaos. The oscillations are described by Hopf bifurcations known to be structurally stable, which just provides the universality of sound perception. However, up till now, nothing was known about the mechanism of rhythm. It was shown in our study that the mechanism of rhythm is described by the Feigenbaum scenario of transition to chaos. Upon transition from oscillations to chaos, this scenario incorporates a critical point near which the dynamics of the system is described in an universal way.     

Model of chromosome motility in Drosophila embryos: adaptation of a general mechanism for rapid mitosis

During mitosis, ensembles of dynamic MTs and motors exert forces that coordinate chromosome segregation. Typically, chromosomes align at the metaphase spindle equator where they oscillate along the pole-pole axis before disjoining and moving poleward during anaphase A, but spindles in different cell types display differences in MT dynamicity, in the amplitude of chromosome oscillations and in rates of chromatid-to-pole motion. Drosophila embryonic mitotic spindles, for example, display remarkably dynamic MTs, barely detectable metaphase chromosome oscillations, and a rapid rate of "flux-pacman-dependent" anaphase chromatid-to-pole motility. Here we develop a force-balance model that describes Drosophila embryo chromosome motility in terms of a balance of forces acting on kinetochores and kMTs that is generated by multiple polymer ratchets and mitotic motors coupled to tension-dependent kMT dynamics. The model shows that i), multiple MTs displaying high dynamic instability can drive steady and rapid chromosome motion; ii), chromosome motility during metaphase and anaphase A can be described by a single mechanism; iii), high kinetochore dynein activity is deployed to dampen metaphase oscillations, to augment the basic flux-pacman mechanism, and to drive rapid anaphase A; iv), modulation of the MT rescue frequency by the kinetochore-associated kinesin-13 depolymerase promotes metaphase chromosome oscillations; and v), this basic mechanism can be adapted to a broad range of spindles.     

On the role of enzyme cooperativity in metabolic oscillations: analysis of the Hill coefficient in a model for glycolytic periodicities.

The role of enzyme cooperativity in the mechanism of metabolic oscillations is analyzed in a concerted allosteric model for the phosphofructokinase reaction. This model of a dimer enzyme activated by the reaction product accounts quantitatively for glycolytic periodicities observed in yeast and muscle. The Hill coefficient characteristic of enzyme-substrate interactions is determined in the model, both at the steady state and in the course of sustained oscillations. Positive cooperativity is a prerequisite for periodic behavior. A necessary condition for oscillation in a dimer K system is a Hill coefficient larger than 1.6 at the unstable stationary state. The analysis suggests that positive as well as negative effectors of phosphofructokinase inhibit glycolytic oscillations by inducing a decrease in enzyme cooperativity. The results are discussed with respect to glycolytic and other metabolic periodicities.     

Synchronous versus asynchronous oscillations for antigenically varying Plasmodium falciparum with host immune response

We consider a deterministic intra-host model for Plasmodium falciparum (Pf) malaria infection, which accounts for antigenic variation between n clonal variants of PfEMP1 and the corresponding host immune response (IR). Specifically, the model separates the IR into two components, specific and cross-reactive, respectively, in order to demonstrate that the latter can be a mechanism for the sequential appearance of variants observed in actual Pf infections. We show that a strong variant-specific IR relative to the cross-reactive IR favours the asynchronous oscillations (sequential dominance) over the synchronous oscillations in a number of ways. The decay rate of asynchronous oscillations is smaller than that for the synchronous oscillations, allowing for the parasite to survive longer. With the introduction of a delay in the stimulation of the IR, we show that only a small delay is necessary to cause persistent asynchronous oscillations and that a strong variant-specific IR increases the amplitude of the asynchronous oscillations.     

Allosteric oscillatory enzymes: influence of the number of protomers on metabolic periodicities.

The study of a concerted allosteric model for an enzyme activated by the reaction product shows that this system can generate sustained metabolic oscillations regardless of the number of protomers constituting the enzyme. The analysis extends the results previously obtained in a dimeric model for the phosphofructokinase reaction which produces glycolytic periodicities. When the substrate and product concentrations evolve on comparable time scales, the amplitude of oscillations significantly drops as the number of enzyme subunits evolves from 2 to 8. The width of the domain of substrate injection rates which produce oscillations and the periodic variation in enzyme activity also depend on the number of protomers and on the time scale structure of the system. Theoretical predictions are compared with the experiments on glycolytic oscillations in yeast and muscle, and with the structural characteristics of phosphofructokinase. The results are also discussed in relation with the mechanism of cyclic AMP oscillations in the slime mold Dictyostelium discoideum.