Oscillatory isozymes as the simplest model for coupled biochemical oscillators

Periodicity, mixed-mode oscillations, and multiple timescales in a phosphoinositide-Rho GTPase network

While rhythmic contractile behavior is commonly observed at the cellular cortex, the primary focus has been on excitable or periodic events described by simple activator-delayed inhibitor mechanisms. We show that Rho GTPase activation in nocodazole-treated mitotic cells exhibits both simple oscillations and complex mixed-mode oscillations. Rho oscillations with a 20- to 30-s period are regulated by phosphatidylinositol (3,4,5)-trisphosphate (PIP3) via an activator-delayed inhibitor mechanism, while a slow reaction with period of minutes is regulated by phosphatidylinositol 4-kinase via an activator-substrate depletion mechanism. Conversion from simple to complex oscillations can be induced by modulating PIP3 metabolism or altering membrane contact site protein E-Syt1. PTEN depletion results in a period-doubling intermediate, which, like mixed-mode oscillations, is an intermediate state toward chaos. In sum, this system operates at the edge of chaos. Small changes in phosphoinositide metabolism can confer cells with the flexibility to rapidly enter ordered states with different periodicities.

Multi-rhythmicity generated by coupling two cellular rhythms

The cell cycle and the circadian clock represent two major cellular rhythms, which are coupled because the circadian clock governs the synthesis of several proteins of the network that drives the mammalian cell cycle. Analysis of a detailed model for these coupled cellular rhythms previously showed that the cell cycle can be entrained at the circadian period of 24 h, or at a period of 48 h, depending on the autonomous period of the cell cycle and on the coupling strength. We show by means of numerical simulations that multiple stable periodic regimes, i.e. multi-rhythmicity, may originate from the coupling of the two cellular rhythms. In these conditions, the cell cycle can evolve to any one of two (birhythmicity) or three stable periodic regimes (trirhythmicity). When applied at the right phase, transient perturbations of appropriate duration and magnitude can induce the switch between the different oscillatory states. Such switching is characterized by final state sensitivity, which originates from the complex structure of the attraction basins. By providing a novel instance of multi-rhythmicity in a realistic model for the coupling of two major cellular rhythms, the results throw light on the conditions in which multiple stable periodic regimes may coexist in biological systems.

Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator

We present an explicit solution based on the phase-amplitude approximation of the Fokker-Planck equation associated with the Langevin equation of the birhythmic modified van der Pol system. The solution enables us to derive probability distributions analytically as well as the activation energies associated with switching between the coexisting different attractors that characterize the birhythmic system. Comparing analytical and numerical results we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. Under the effect of noise, the two states that characterize the birhythmic system can merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when the noise intensity increases. The solution of the Fokker-Planck equation shows that in the birhythmic region, the two attractors are characterized by very different probabilities of finding the system in such a state. The probability becomes comparable only for a narrow range of the control parameters, thus the two limit cycles have properties in close analogy with the thermodynamic phases.

LONG-RANGE CORRELATIONS IN THE PHASE-SHIFTS OF NUMERICAL SIMULATIONS OF BIOCHEMICAL OSCILLATIONS AND IN EXPERIMENTAL CARDIAC RHYTHMS

In biochemical dynamical systems during each transition between periodical behaviors, all metabolic intermediaries of the system oscillate with the same frequency but with different phase-shifts. We have studied the behavior of phase-shift records obtained from random transitions between periodic solutions of a biochemical dynamical system. The phase-shift data were analyzed by means of Hurst's rescaled range method (introduced by Mandelbrot and Wallis). The results show the existence of persistent behavior: each value of the phase-shift depends not only on the recent transitions, but also on previous ones. In this paper, the different kind of periodic solutions were determined by different small values of the control parameter. It was assessed the significance of this results through extensive Monte Carlo simulations as well as quantifying the long-range correlations. We have also applied this type of analysis on cardiac rhythms, showing a clear persistent behavior. The relationship of the results with the cellular persistence phenomena conditioned by the past, widely evidenced in experimental observations, is discussed.

Global stability analysis of birhythmicity in a self-sustained oscillator

We analyze the global stability properties of birhythmicity in a self-sustained system with random excitations. The model is a multi-limit-cycle variation in the van der Pol oscillator introduced to analyze enzymatic substrate reactions in brain waves. We show that the two frequencies are strongly influenced by the nonlinear coefficients alpha and beta. With a random excitation, such as a Gaussian white noise, the attractor's global stability is measured by the mean escape time tau from one limit cycle. An effective activation energy barrier is obtained by the slope of the linear part of the variation in the escape time tau versus the inverse noise intensity 1/D. We find that the trapping barriers of the two frequencies can be very different, thus leaving the system on the same attractor for an overwhelming time. However, we also find that the system is nearly symmetric in a narrow range of the parameters.

Synchronization of two coupled self-excited systems with multi-limit cycles

We analyze the stability and optimization of the synchronization process between two coupled self-excited systems modeled by the multi-limit cycles van der Pol oscillators through the case of an enzymatic substrate reaction with ferroelectric behavior in brain waves model. The one-way and two-way couplings synchronization are considered. The stability boundaries and expressions of the synchronization time are obtained using the properties of the Hill equation. Numerical simulations validate and complement the results of analytical investigations.

A bifurcation analysis of two coupled calcium oscillators

In many cell types, asynchronous or synchronous oscillations in the concentration of intracellular free calcium occur in adjacent cells that are coupled by gap junctions. Such oscillations are believed to underlie oscillatory intercellular calcium waves in some cell types, and thus it is important to understand how they occur and are modified by intercellular coupling. Using a previous model of intracellular calcium oscillations in pancreatic acinar cells, this article explores the effects of coupling two cells with a simple linear diffusion term. Depending on the concentration of a signal molecule, inositol (1,4,5)-trisphosphate, coupling two identical cells by diffusion can give rise to synchronized in-phase oscillations, as well as different-amplitude in-phase oscillations and same-amplitude antiphase oscillations. Coupling two nonidentical cells leads to more complex behaviors such as cascades of period doubling and multiply periodic solutions. This study is a first step towards understanding the role and significance of the diffusion of calcium through gap junctions in the coordination of oscillatory calcium waves in a variety of cell types. (c) 2001 American Institute of Physics.

Kinetic asymmetry as a key source of functional diversity in biochemical networks

From the analysis of the dynamic properties of various symmetric and asymmetric kinetic schemes, the present report demonstrates that all kinetic schemes, which can be hypothetically divided into two equal halves about an axis of mirror symmetry, are endowed with structural metastability under mass-closed conditions. In mass-closed symmetric schemes, absolute symmetry in reaction conditions in two halves is essential for the occurrence of ordered dynamic behaviour. Even an infinitesimal deviation from the symmetry relations instantaneously drives such systems from limit-cycles to turbulence. Reaction schemes with no axes of symmetry may exhibit a large variety of complex, structurally stable temporal order for wide ranges of values of system parameters and variables. Kinetic asymmetry, therefore, may confer to biochemical networks the functional diversity as well as stability against environmental perturbations.

Transitions and new dynamical states induced by noise in a multiply regulated biochemical system

Noise-induced transitions between coexisting states, and the emergence of a new oscillatory state, are examined in a model for a multiply regulated biochemical system. For the undisturbed system, three oscillatory states, I, II, and III, coexist. It is found that noise above a critical amplitude can cause a transition between states III and II and between states III or II and state I, whereas a transition from state I to either states II or III is never observed. This indicates that the relative stability under noise perturbations is greatest for state I, and progressively less for states II and III. In addition to this transition behaviour, a purely noise-induced state is found. Under noise perturbations, the average concentration of metabolites may depend on both the time duration and amplitude of the superimposed noise. The implications of these results for understanding the in vivo behaviour of complex biochemical systems are discussed.

Complex dynamics of mass-closed coupled autocatalytic systems in response to minute asymmetric perturbations

The role of kinetic coupling in catering to a remote-control mechanism for the onset and regulation of self-organization phenomena in a multicompartmental biochemical system has been examined. Using two cyclic autocatalytic reaction networks operating in two chambers separated by a membrane and coupled through a common cofactor, it has been demonstrated that (i) in response to asymmetric perturbations, the coupled reaction networks exhibit a variety of temporal self-organization phenomena such as bistability, multiple periodicity, hard excitation and coexistence of aperiodic oscillation with limit cycle even in mass-closed conditions; (ii) without disturbing a network directly, its dynamic behaviour can be regulated by perturbing some other network kinetically coupled to it and (iii) the dynamics of two coupled networks can be made to flip-flop between oscillatory and steady-states simply by modulating the time of application of external perturbations. The extreme sensitivity of this model to minute asymmetric fluctuations in the environment can predict how the impact of local changes in physico-chemical conditions can be transmitted from one compartment to another through coupled biochemical pathways in a living cell.

Dynamic behavior in glycolytic oscillations with phase shifts

Practically all of the studies of glycolytic oscillations in homogeneous spatial mediums have been performed through the construction of systems of ordinary differential equations and the search for their solutions. In this kind of modelling, the system dynamic behavior is considered to depend only on the values adopted by the parameters related to the dependent variables. In the present work, the modeling of a biochemical system through a system of functional differential equations with delay allows us to analyse the consequences that the variations in the parametric values linked to the independent variable (time) have upon the integral solutions of the system. In our model, the delays correspond with phase shifts in the initial functions for two dependent variables. The results of our researches show that when a instability-generating multienzymatic mechanism suffers variations of the delay time in any of its variables, a wide range of different dynamic responses can be produced. Our work is presented as an enlargement on the dynamic study of biochemical oscillations in general and, particularly, the glycolytic oscillations, under the consideration of the existence of variations in the phase shifts during the oscillations of metabolites involved in the studied reactive processes.

Model for receptor-controlled cytosolic calcium oscillations and for external influences on the signal pathway

The external stimulation of many cells by a hormone, for example, often leads to an oscillating cytosolic calcium concentration. This periodic behavior is now designated the cytosolic calcium oscillator. A theoretical model is presented that describes this behavior on the basis of inositol(1,4,5)trisphosphate-induced calcium oscillations. In contrast to other models only a single positive feedback loop is taken into account to obtain oscillations. The model includes important innovations compared to other approaches. It includes the contribution of extracellular calcium and its modification after the stimulation of the cell. Furthermore, the signal pathway that leads to cytosolic calcium oscillations is described in more detail than in other models. This enables investigations on the influence of additional parameters like external electromagnetic fields on the signal transduction pathway. The model and the calculations are based on the theory of nonlinear self-sustained oscillators.

Mathematical modeling for the aging process: normal, abnormal and self-terminating phenomena in spatio-temporal organization

An elucidation of the aging process is attempted using a simple one-dimensional multicellular system, a prototype of living organisms. This model analysis has the advantage of making us investigate the two types of modes of their dynamical behavior: (i) the local modes of behavior of individual cells and (ii) the global modes of behavior of the total system. At first, each cell is assumed to have biochemically excitable kinetics for local modes, and then what kind of the global modes results with change in intercellular interaction is examined. With a simple interaction as possibly occurs in the early stage of differentiation, the model displays well-coordinated spatio-temporal patterns. This may be interpreted as a normal state. With a more complex interaction as possibly occurs in the late stage of growth, however, the model produces much more erratic patterns. This may refer to an abnormal state. Interestingly, these abnormal patterns can be transformed into normal patterns, when the activity of some parts of this model is turned 'off'; the system can survive at the sacrifice of its parts. This makes us imagine that programmed cell death plays an important role in development during morphogenesis. When individual cells become less sensitive to intercellular signals and still possess intrinsic excitability, then the normal patterns are developed for a short while before being replaced by rather irregular patterns. As time proceeds, however, all activity of them disappears. We call this a 'self-terminating' phenomenon, which may refer to aging. This strongly suggests that the loss of the total system function, leading to death, results from a global mode of system failure but not from a local mode of subsystem failure.

Bistabilities and annihilation phenomena in electrophysiological cardiac models

We have investigated the oscillatory behavior of cardiac cellular elements simulated by two electrophysiological models: the van Capelle and Durrer (VCD) model and the sinoatrial node cell model of Yanagihara, Noma, and Irisawa (YNI). The VCD model behavior was examined systematically by using continuation-bifurcation analysis. Bifurcation diagrams were constructed as a function of Qit1, an intrinsic parameter of the model, which sets both maximum diastolic potential and depolarization threshold of the cell. The existence of stable high amplitude oscillations was evidenced between two Hopf bifurcation points (HB). Near each HB, a zone of bistability was detected. Close to the HB that corresponded to high values of Qit1, a high amplitude periodic stable state coexisted with a stable steady state. Close to the other HB, in a narrow range of lower Qit1 values, a relatively high amplitude periodic stable state coexisted with a low amplitude periodic stable state. There was no stable steady state in the latter bistability zone. Through the use of phase-plane representations and the determination of separatrices between the different attractor basins, we could deduce the conditions of timing, polarity, and strength needed for a pulse perturbation to send the system from one state to another and vice versa. The YNI model was analyzed by numerical simulation, and the oscillatory behavior of the sinoatrial node cell was explored while applying a depolarizing bias current of various strengths. Results were similar to those obtained from the VCD model in that there were two bistability regions for two different ranges of applied bias current. Depending on current intensity, annihilation of pacemaker activity could be achieved in both zones. However, the coexistence of two oscillatory stable states was never observed in the YNI model. From the behavioral similarities of these different models, we can conclude that bistabilities and annihilation phenomena can be found in transitional zones between quiescence and rhythmic activity.

Multiple levels in the control of rhythms in enzyme synthesis and activity by circadian clocks: recent trends

It is now well established that circadian clocks can control rhythmicity at different stages of the sequence of events leading from gene activity to a functional enzyme molecule. Conceptual and experimental distinction of the effective control targets is emphasized, with particular attention to recent results obtained on circadian clock control of transcription and to data indicating that proteins can behave as conformational oscillators. Thus circadian rhythmicity both in gene expression and in the dynamics of the enzyme molecule would contribute to the temporal compartmentation of processes such as metabolic co-ordination and channeling necessary for the adaptive efficiency of physiological programmes.