Yeast glycolytic oscillations that are not controlled by a single oscillophore: a new definition of oscillophore strength

Biochemical oscillations, such as glycolytic oscillations, are often believed to be caused by a single so-called ‘oscillophore’. The main characteristics of yeast glycolytic oscillations, such as frequency and amplitude, are however controlled by several enzymes. In this paper, we develop a method to quantify to which extent any enzyme determines the occurrence of oscillations. Principles extrapolated from metabolic control analysis are applied to calculate the control exerted by individual enzymes on the real and imaginary parts of the eigenvalues of the Jacobian matrix. We propose that the control exerted by an enzyme on the real part of the smallest eigenvalue, in terms of absolute value, quantifies to which extent that enzyme contributes to the emergence of instability. Likewise the control exerted by an enzyme on the imaginary part of complex eigenvalues may serve to quantify the extent to which that enzyme contributes to the tendency of the system to oscillate. The method was applied both to a core model and to a realistic model of yeast glycolytic oscillations. Both the control over stability and the control over oscillatory tendency were distributed among several enzymes, of which glucose transport, pyruvate decarboxylase and ATP utilization were the most important. The distributions of control were different for stability and oscillatory tendency, showing that control of instability does not imply control of oscillatory tendency nor vice versa. The control coefficients summed up to 1, suggesting the existence of a new summation theorem. These results constitute proof that glycolytic oscillations in yeast are not caused by a single oscillophore and provide a new, subtle, definition for the oscillophore strength of an enzyme.     

A conceptual model of the dynamics of the Syamozero Lake community

We present a conceptual mathematical model of the dynamics of a spatially heterogeneous population system whose prototype is the fish community of Lake Syamozero. Analysis of the solutions of this model is used to demonstrate that interactions between the predator and prey populations in two neighboring biotopes (the pelagic and coastal zones) may result in either undamped oscillations or steady states of the population sizes. The model population densities are of the same order of magnitude as the values obtained in long-term observations of the Syamozero biota. It is also demonstrated that the transition to steady states may be accompanied by long-term (dozens or hundreds of years) damped oscillations of the prey and predator population densities. Under natural conditions, long transitional periods may prevent fish communities from reaching stationary modes.     

UNDAMPED OSCILLATIONS OF PYRIDINE NUCLEOTIDE AND OXYGEN TENSION IN CHEMOSTAT CULTURES OF KLEBSIELLA AEROGENES

Klebsiella aerogenes was grown in chemostat culture with the pH controlled to ±0.01 and temperature to ±0.1°C. The oxygen tension of the culture was regulated by changing the partial pressure of oxygen in the gas phase and recorded by means of an oxygen electrode. Reduced pyridine nucleotide was monitored continuously in the culture by means of direct fluorimetry. On applying an anaerobic shock to the culture, damped oscillations in pyridine nucleotide fluorescence were obtained. Further anaerobic shocks decreased the damping and eventually gave rise to undamped oscillations of a 2–3 min period which continued for several days. These oscillations were paralleled by oscillations of the same frequency in respiration rate. The amplitude of the oscillations in the respiration rate was equivalent to only 1% of the total steady-state respiration, whereas that of pyridine nucleotide oscillations was equivalent to 10% of the total aerobic/anaerobic fluorescence response. The oscillations ceased on interrupting the glucose feed but restarted on adding excess glucose to the culture. Addition of succinate also restarted the oscillations so that they appear not to be of glycolytic origin. The frequency of oscillations varied with growth rate and conditions. Oscillations of much lower frequency were obtained under limited-oxygen and anaerobic conditions than under fully aerobic conditions. Under glucose-limited conditions, fluctuations were found in adenosine triphosphate (ATP) content which were in phase with the pyridine nucleotide oscillations, but under nitrogen-limited growth conditions no such fluctuations in ATP were observed. The primary oscillating pathway could not be identified but the mechanism would appear to be quite different from that involved in oscillations observed in yeast cells. The synchronization of oscillations and observations of negative damping could be explained by a syntalysis effect.     

Mass-induced oscillations in the femur-tibia control system of stick insects

Attaching an inert mass to a freely moving tibia of an otherwise fixed stick insect Carausius morosus, induces undamped oscillations of the tibia. We describe the use of a rotational pendulum to observe these oscillations applying various amounts of inertia. The dependence of the frequency of these oscillations on the moment of inertia is similar to that of a purely mechanical system. The sequence of the oscillatory behavior can be separated into 3 distinct behavioural states. The transitions between some of these states could be elicited by external stimuli and partly showed characteristics of habituation and dishabituation. With a rotational pendulum on each middle leg, simultaneous oscillations of both legs were measured to investigate coupling effects between the neural control systems of the two legs. In some cases, significant coupling effects could be observed in phase and frequency. In many other cases, no coupling was found. The habituation and dishabituation effects were not transferred between the middle legs.     

Oscillatory Transpiration and Water Uptake of Avena Plants

Xylem vessels in the lower part of the leaf of young Avena plants have been exposed to deformation by application of an external pressure. In this way a resistance to the water flow at the deformation site has been achieved, inducing undamped oscillations in transpiration and water uptake, even after removal of the root system.     

A theory of membrane permeability: III. The effect of hydrostatic pressure

Using expressions derived in previous papers, the author investigates the behavior of a cell immersed in an infinite medium, under the influence of diffusion of a single solute and flow of water. The effect of hydrostatic pressure on the system is taken into account. It is found that, depending on the values of certain parameters, the cell can collapse, burst, reach a stationary stable state, or execute undamped oscillations; a cell must burst or collapse unless its volume is an increasing function of internal pressure, and it can execute stable oscillations only if its membrane acts as a “potential well” to the molecules of the solute.     

Physiological response ofCandida tropicalis grown on n-paraffin to mixing in a tubular closed loop fermenter

Summary A special tubular closed loop fermenter was used in order to simulate the particular mixing condition of a large scale recycle fermenter.Some mixing parameters of the system are characerized.During continuous cultivation ofCandida tropicalis on n-paraffin as a substrate the biomass yield with respect to carbon and oxygen increased, when a controlled oxygen limit was imposed on the culture.Mixing in the closed loop fermenter generates undamped short period oscillations in the respiration activity, in the dissolved oxygen tension and in the actual ATP content of the culture. These oscillations likely represent oscillations of allosteric feedback loops which manifest themselves by some synchronising action of the particular environmental transients in the closed loop fermenter.     

Nonlinear modulation of an extraordinary wave under the conditions of parametric decay

A self-consistent set of Hamilton equations describing nonlinear saturation of the amplitude of oscillations excited under the conditions of parametric decay of an elliptically polarized extraordinary wave in cold plasma is solved analytically and numerically. It is shown that the exponential increase in the amplitude of the secondary wave excited at the half-frequency of the primary wave changes into a reverse process in which energy is returned to the primary wave and nonlinear oscillations propagating across the external magnetic field are generated. The system of ??slow?? equations for the amplitudes, obtained by averaging the initial equations over the high-frequency period, is used to describe steady-state nonlinear oscillations in plasma.     

Stability and oscillations in coupled systems.

In this paper are given criteria for stability and oscillations in coupled systems. The criteria are calculated by the determination of eigenvalues of a coupling matrix of an input-output relation which can be described by a set of differential equations of 1st order.     

Electrokinetic membrane processes in relation to properties of excitable tissues. I. Experiments on oscillatory transport phenomena in artificial membranes

An artificial system is studied consisting of salt solutions of different concentrations separated by a porous, "charged" membrane, through which a constant electric current is passed. Experiments on such systems demonstrate rhythmic variations of the transmembrane potential and the membrane resistance, which are concomitant with an oscillatory streaming of water solution across the membrane. The repetitive oscillations can be of a damped or undamped type dependent on the "stimulating" current density. A qualitative discussion of the mechanism of the oscillations is given. It centers around the periodic resistance changes in the membrane, which result from a complicated interplay between the driving forces present. The importance of electro-osmotic effects is emphasized. A few comparisons relating to possible electrophysiological implications are presented. In the metastable state of this membrane oscillator, "make" and "break" responses can be triggered by electric as well as by mechanical (pressure) "stimuli."     

Oscillatory phenomena in a model of infectious diseases

Oscillations of the number of cases around an average endemic level are common in several infectious diseases. In this paper we study simple deterministic models, where the oscillations arise either solely from periodically varying contact rates or from the combined effect of large initial perturbation, small periodic variation of the contact rate, and the destabilizing nature of infectious and latent periods when described as time delays. The main results are: (a) For a model with a periodically varying contact rate and a recovery rate, a threshold amplitude of variation is found by numerical and analytic methods at which 2-year subharmonic resonance appears. (b) Approximate analytic relationships are derived for the amplitude and phase of the forced 1-year oscillations below this threshold and for the 2-year oscillations above it—in terms of the reproduction rate of the infection. (c) Similar calculations are performed when the recovery rate is replaced by a fixed infectious period represented by a pure time delay. The threshold amplitude of variation in the contact rate is found here to be smaller than in the recovery rate model. (d) A model with a fixed infectious period and a constant contact rate is considered. The nontrivial steady state is shown to be locally stable for the parameter range of interest. However, the ratio of the imaginary to real parts of the eigenvalues in the characteristic equation is increased as compared to the corresponding model with a recovery rate. (e) For the model with a fixed infectious period and a constant contact rate an approximation method indicates consistency in a certain range of contact rates with the existence of an unstable periodic solution about the locally stable steady state. The actual existence of such a solution is not verified. The interpretation is that the destabilizing effect of the introduction of a pure delay into the model becomes more significant as the distance in the variables space from the endemic steady state is increased. (f) For a fixed infectious period and very small subthreshold variation in the contact rate, two different types of solutions are found numerically: yearly small-amplitude oscillations about an endemic average and large-amplitude oscillations of a subharmonic period. The pattern seen depends on the initial conditions. For a sufficiently large initial deviation from the endemic level even very small seasonal variations lead to regular recurrent outbreaks of the disease. The effect of latent periods and of changing the form of the interaction are also considered.     

Functional states of an excitable membrane and their dependence on its parameter values

The property of an excitable membrane of a nerve cell to change the type of electrical activity has been examined with the change of the value of applied current (I). The dependence of this property on the values of the membrane parameters is determined. Two different functional states depending on the values of the membrane parameters are considered. For one of the states a change in the value of I is accompanied by a change in the type of activity (damped periodic oscillations jump to undamped periodic oscillations or vice versa). For the other state the type of activity remains phasic (damped periodic oscillations) for each value of I. For the mathematical model of a membrane we have considered the problem of obtaining the boundary, partitioning the parameter space into the regions to which these functional states correspond. We suggest a mathematical set of this problem and give its algorithm. These boundaries have been constructed for two different variable parameters of the model. A good agreement between the boundaries and the experimental values of sodium and potassium conductances for different excitable membranes has been obtained.     

Hidden pools,hidden modes,and visible repeated eigenvalues in compartmental models

In compartmental analysis, when time-series data are initially fitted by sums of exponential functions, it is usually assumed that the eigenvalues are real and distinct and the number of pools (compartments) equals the number of exponentials. However, repeated real or complex eigenvalues, although more difficult to detect, may be inherent in the data, and the number of pools may be larger than the number of exponentials. In order to describe compartmental models with such properties, the visible multiplicity of eigenvalues and concepts of hidden pools and visible and hidden modes are defined. It is then shown that if a model is state observable, each mode is visible (not hidden) in the zero-input response for some choice of initial state, but not conversely, and also that the visible multiplicity of eigenvalues is determined by the submodel of input-output connectable compartments. Compartmental models are analyzed, using their decomposition into what we define as strongly connected components. An upper bound is given for the visible multiplicity of eigenvalues in terms of the model's strongly connected components. For models with one input and one output, this bound is shown to be attained for what we call generalized trees.     

Intrinsic values of transformation systems representable by tree graphs

When transformation systems follow the microreversibility principle, the eigenvalues computed around the equilibrium are all real. The question is to know if this property remains satisfied even if the system is open or is submitted to radiations. Generally, the answer is negative. However, it is possible to define a relatively large class of systems which follow that rule. Tree graphs are involved, which are rigorously defined by giving application limits. From there it is deduced that a tree, however complex it may be cannot have the behaviour of a biological clock. On the other hand, such a property allows the restriction of the whole of classes of systems able to show free or damped oscillations.     

A conceptual model of the dynamics of the Syamozero lake community

We have developed a conceptual mathematical model of the dynamics of a spatially heterogeneous population system, the prototype of which is the Syamozero lake fish community. Based on the analysis of solutions of the model, we show that interrelations between prey and predator populations in two neighboring habitats (pelagic and offshore zones) can lead to both undamped oscillations and stationary values of the population size. The population density was found to be close to the values oblained in the course of long-term observations of the biota of the Syamozero lake. Besides, we showed that the transition to the stationary states can be accompanied by long-term (dozens and hundreds of years) damped oscillations of the prey and predator population size. In natural waters, long-term transition periods can prevent the attainment of stationary regimes of fish community functioning.     

Stability of discrete age-structured and aggregated delay-difference population models

Sufficiency conditions for local stability are derived for a class of density dependent Leslie matrix models. Four of the recruitment functions in common use in fisheries management are then considered. In two of these oscillating instability can never occur (Beverton and Holt and Cushing forms). In the other two (Deriso-Schnute and Shepherd forms) undamped oscillations are possible within the region of parameter space described here. An algorithm is developed for calculating necessary and sufficient local stability conditions for a simplified form of the general age-structured model. The complete spectrum of stability states (monotonic stability; monotonic instability; oscillating-stable; oscillating-unstable) and the bifurcation periods are given for selected examples of this model. The examples cover a large portion of the parameter space of interest in resource management. It is shown that in perfectly deterministic systems which are observed with error, oscillating instabilities may be missed, and such systems could be erroneously assumed to be stable.     

Oscillatory kinetics of the peroxidase-oxidase reaction in an open system. Experimental and theoretical studies.

1. The oscillations in the peroxidase (donor: hydrogen-peroxide oxidoreductase, EC 1.11.1.7)-catalyzed reaction between NADH and O2 are undamped when the reaction is carried out in a system open to both substrates and when 2,4-dichlorophenol and methylene blue are present in the solution. 2. The waveform of the oscillations changes when the concentration of peroxidase is varied. 3. The waveforms obtained experimentally can be simulated by a branched chain reaction model in which the branching is quadratic. 4. A correlation between the present knowledge of the reaction and the model can be made by combining well established and hypothetical reaction steps into a few reaction schemes. A selection among schemes however, is not possible at the present time. 5. Compound III participates in the reaction as an active intermediate. This is possible because dichlorophenol stimulates the break down of compound III.     

Acidic Deposition, Plant Pests, and the Fate of Forest Ecosystems

We present and analyze a nonlinear dynamical system modelling forest-pests interactions and the way they are affected by acidic deposition. The model includes mechanisms of carbon and nitrogen exchange between soil and vegetation, biomass decomposition and microbial mineralization, and defoliation by pest grazers, which are partially controlled by avian or mammalian predators. Acidic deposition is assumed to directly damage vegetation, to decrease soil pH, which in turn damages roots and inhibits microbial activity, and to predispose trees to increased pest attack. All the model parameters are set to realistic values except the inflow of protons to soil and the predation mortality inflicted to the pest which are allowed to vary inside reasonable ranges. A numerical bifurcation analysis with respect to these two parameters is carried out. Five functioning modes are uncovered: (i) pest-free equilibrium; (ii) pest persisting at endemic equilibrium; (iii) forest–pest permanent oscillations; (iv) bistable behavior with the system converging either to pest-free equilibrium or endemic pest presence in accordance with initial conditions; (v) bistable behavior with convergence to endemic pest presence or permanent oscillations depending on initial conditions. Catastrophic bifurcations between the different behavior modes are possible, provided the abundance of predators is not too small. Numerical simulation shows that increasing acidic load can lead the forest to collapse in a short time period without important warning signals.     

In-phase and antiphase self-oscillations in a model of two electrically coupled pacemakers

The dynamic behavior of a model of two electrically coupled oscillatory neurons was studied while the external polarizing current was varied. It was found that the system with weak coupling can demonstrate one of five stable oscillatory modes: (1) in-phase oscillations with zero phase shift; (2) antiphase oscillations with halfperiod phase shift; (3) oscillations with any fixed phase shift depending on the value of the external polarizing current; (4) both in-phase and antiphase oscillations for the same current value, where the oscillation type depends on the initial conditions; (5) both in-phase and quasiperiodic oscillations for the same current value. All of these modes were robust, and they persisted despite small variations of the oscillator parameters. We assume that similar regimes, for example antiphase oscillations, can be detected in neurophysiological experiments. Possible applications to central pattern generator models are discussed.     

Nonlinear extraordinary wave in dense plasma

Conditions for the propagation of a slow extraordinary wave in dense magnetized plasma are found. A solution to the set of relativistic hydrodynamic equations and Maxwell’s equations under the plasma resonance conditions, when the phase velocity of the nonlinear wave is equal to the speed of light, is obtained. The deviation of the wave frequency from the resonance frequency is accompanied by nonlinear longitudinal-transverse oscillations. It is shown that, in this case, the solution to the set of self-consistent equations obtained by averaging the initial equations over the period of high-frequency oscillations has the form of an envelope soliton. The possibility of excitation of a nonlinear wave in plasma by an external electromagnetic pulse is confirmed by numerical simulations.