Birhythmicity in a model for the cyclic AMP signalling system of the slime mold Dictyostelium discoideum

Space-dependent intermittent feedback can control birhythmicity

Birhythmicity is evident in many nonlinear systems, which include physical and biological systems. In some living systems, birhythmicity is necessary for response to the varying environment while unnecessary in some physical systems as it limits their efficiency. Therefore, its control is an important area of research. This paper proposes a space-dependent intermittent control scheme capable of controlling birhythmicity in various dynamical systems. We apply the proposed control scheme in five nonlinear systems from diverse branches of natural science and demonstrate that the scheme is efficient enough to control the birhythmic oscillations in all the systems. We derive the analytical condition for controlling birhythmicity by applying harmonic decomposition and energy balance methods in a birhythmic van der Pol oscillator. Further, the efficacy of the control scheme is investigated through numerical and bifurcation analyses in a wide parameter space. Since the proposed control scheme is general and efficient, it may be employed to control birhythmicity in several dynamical systems.

Impulsive feedback control of birhythmicity: Theory and experiment

We study the dynamic control of birhythmicity under an impulsive feedback control scheme where the feedback is made ON for a certain rather small period of time and for the rest of the time, it is kept OFF. We show that, depending on the height and width of the feedback pulse, the system can be brought to any of the desired limit cycles of the original birhythmic oscillation. We derive a rigorous analytical condition of controlling birhythmicity using the harmonic decomposition and energy balance methods. The efficacy of the control scheme is investigated through numerical analysis in the parameter space. We demonstrate the robustness of the control scheme in a birhythmic electronic circuit where the presence of noise and parameter fluctuations are inevitable. Finally, we demonstrate the applicability of the control scheme in controlling birhythmicity in diverse engineering and biochemical systems and processes, such as an energy harvesting system, a glycolysis process, and a p53-mdm2 network.

Multi-synchronization and other patterns of multi-rhythmicity in oscillatory biological systems

While experimental and theoretical studies have established the prevalence of rhythmic behaviour at all levels of biological organization, less common is the coexistence between multiple oscillatory regimes (multi-rhythmicity), which has been predicted by a variety of models for biological oscillators. The phenomenon of multi-rhythmicity involves, most commonly, the coexistence between two (birhythmicity) or three (trirhythmicity) distinct regimes of self-sustained oscillations. Birhythmicity has been observed experimentally in a few chemical reactions and in biological examples pertaining to cardiac cell physiology, neurobiology, human voice patterns and ecology. The present study consists of two parts. We first review the mechanisms underlying multi-rhythmicity in models for biochemical and cellular oscillations in which the phenomenon was investigated over the years. In the second part, we focus on the coupling of the cell cycle and the circadian clock and show how an additional source of multi-rhythmicity arises from the bidirectional coupling of these two cellular oscillators. Upon bidirectional coupling, the two oscillatory networks generally synchronize in a unique manner characterized by a single, common period. In some conditions, however, the two oscillators may synchronize in two or three different ways characterized by distinct waveforms and periods. We refer to this type of multi-rhythmicity as ‘multi-synchronization’.

Network synchronization, stability and rhythmic processes in a diffusive mean-field coupled SEIR model

Connectivity and rates of movement have profound effect on the persistence and extinction of infectious diseases. The emerging disease spread rapidly, due to the movement of infectious persons to some other regions, which has been witnessed in case of novel coronavirus disease 2019 (COVID-19). So, the networks and the epidemiology of directly transmitted infectious diseases are fundamentally linked. Motivated by the recent empirical evidence on the dispersal of infected individuals among the patches, we present the epidemic model SEIR (Susceptible-Exposed-Infected-Recovered) in which the population is divided into patches which form a network and the patches are connected through mean-field diffusive coupling. The corresponding unstable epidemiology classes will be synchronized and achieve stable state when the patches are coupled. Apart from synchronization and stability, the coupled model enables a range of rhythmic processes such as birhythmicity and rhythmogenesis which have not been investigated in epidemiology. The stability of Disease Free Equilibrium (or Endemic Equilibrium) is attained through cessation of oscillation mechanism namely Oscillation Death (OD) and Amplitude Death (AD). Corresponding to identical and non-identical epidemiology classes of patches, the different steady states are obtained and its transition is taking place through Hopf and transcritical bifurcation.

Effect of filtered feedback on birhythmicity: Suppression of birhythmic oscillation

The birhythmic oscillation, generally known as birhythmicity, arises in a plethora of physical, chemical, and biological systems. In this paper we investigate the effect of filtered feedback on birhythmicity as both are relevant in many living and engineering systems. We show that the presence of a low-pass filter in the feedback path of a birhythmic system suppresses birhythmicity and supports monorhythmic oscillations depending on the filtering parameter. Using harmonic decomposition and energy balance methods we determine the conditions for which birhythmicity is removed. We carry out a detailed bifurcation analysis to unveil the mechanism behind the quenching of birhythmic oscillations. Finally, we demonstrate our theoretical findings in analog simulation with electronic circuit. This study may have practical applications in quenching birhythmicity in several biochemical and physical systems.

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.

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.

Nonequilibrium phase transition in a self-activated biological network

We present a lattice model for a two-dimensional network of self-activated biological structures with a diffusive activating agent. The model retains basic and simple properties shared by biological systems at various observation scales, so that the structures can consist of individuals, tissues, cells, or enzymes. Upon activation, a structure emits a new mobile activator and remains in a transient refractory state before it can be activated again. Varying the activation probability, the system undergoes a nonequilibrium second-order phase transition from an active state, where activators are present, to an absorbing, activator-free state, where each structure remains in the deactivated state. We study the phase transition using Monte Carlo simulations and evaluate the critical exponents. As they do not seem to correspond to known values, the results suggest the possibility of a separate universality class.

Noise-induced Hopf-bifurcation-type sequence and transition to chaos in the lorenz equations

We study the effects of noise on the Lorenz equations in the parameter regime admitting two stable fixed point solutions and a strange attractor. We show that noise annihilates the two stable fixed point attractors and evicts a Hopf-bifurcation-like sequence and transition to chaos. The noise-induced oscillatory motions have very well defined period and amplitude, and this phenomenon is similar to stochastic resonance, but without a weak periodic forcing. When the noise level exceeds certain threshold value but is not too strong, the noise-induced signals enable an objective computation of the largest positive Lyapunov exponent, which characterize the signals to be truly chaotic.

Annihilation of one of the coexisting attractors in a bistable system

A small change of one of the system parameters may not in general convert a bistable system to a monostable system. However, an external control in the form of a slow periodic parameter modulation can annihilate one of the coexisting states, and thus results in controlled monostability. The annihilation takes place because the state becomes chaotic via the period doubling route and the chaotic state undergoes boundary crisis within a small range of the control amplitude. These features are observed theoretically in two standard models, namely, Henon map and laser rate equations, and confirmed experimentally in a cavity loss modulated CO2 laser.

Oscillatory behavior of a simple kinetic model for proteolysis during cell invasion

Extracellular proteolysis during cell invasion is thought to be tightly organized, both temporally and spatially. This work presents a simple kinetic model that describes the interactions between extracellular matrix (ECM) proteins, proteinases, proteolytic fragments, and integrins. Nonmonotonous behavior arises from enzyme de novo synthesis consecutive to integrin binding to fragments or entire proteins. The model has been simulated using realistic values for kinetic constants and protein concentrations, with fibronectin as the ECM protein. The simulations show damped oscillations of integrin-complex concentrations, indicating alternation of maximal adhesion periods with maximal mobility periods. Comparisons with experimental data from the literature confirm the similarity between this system behavior and cell invasion. The influences on the system of cryptic functions of ECM proteins, proteinase inhibitors, and soluble antiadhesive peptides were examined. The first critical parameter for oscillation is the discrepancy between integrin affinity for intact ECM proteins and the respective proteolytic fragments, thus emphasizing the importance of cryptic functions of ECM proteins in cell invasion. Another critical parameter is the ratio between proteinase and the initial ECM protein concentration. These results suggest new insights into the organization of the ECM degradation during cell invasion.

Cyclic AMP oscillations in Dictyostelium discoideum: models and observations

Oscillations in intra- and extracellular cyclic AMP are believed to underlie aggregation and morphogenesis in Dictyostelium discoideum. Upon comparing mathematical models with observations we find that the models are, qualitatively speaking, quite successful. At the same time many features remain unexplained. A strong case can be made for cyclic AMP-independent oscillations whose basis remains to be explored.

Suppression of chaos and other dynamical transitions induced by intercellular coupling in a model for cyclic AMP signaling in Dictyostelium cells

The effect of intercellular coupling on the switching between periodic behavior and chaos is investigated in a model for cAMP oscillations in Dictyostelium cells. We first analyze the dynamic behavior of a homogeneous cell population which is governed by a three-variable differential system for which bifurcation diagrams are obtained as a function of two control parameters. We then consider the mixing of two populations behaving in a chaotic and periodic manner, respectively. Cells are coupled through the sharing of a common chemical intermediate, extracellular cAMP, which controls its production and release by the cells into the extracellular medium; the dynamics of the mixed suspension is governed by a five-variable differential system. When the two cell populations differ by the value of a single parameter which measures the activity of the enzyme that degrades extracellular cAMP, the bifurcation diagram established for the three-variable homogeneous population can be used to predict the dynamic behavior of the mixed suspension. The analysis shows that a small proportion of periodic cells can suppress chaos in the mixed suspension. Such a fragility of chaos originates from the relative smallness of the domain of aperiodic oscillations in parameter space. The bifurcation diagram is used to obtain the minimum fraction of periodic cells suppressing chaos. These results are related to the suppression of chaos by the small-amplitude periodic forcing of a strange attractor. Numerical simulations further show how the coupling of periodic cells with chaotic cells can produce chaos, bursting, simple periodic oscillations, or a stable steady state; the coupling between two populations at steady state can produce similar modes of dynamic behavior.

Suppression of chaos by periodic oscillations in a model for cyclic AMP signalling in Dictyostelium cells

We investigate how the introduction of cells oscillating periodically affects the behaviour of a suspension of Dictyostelium discoideum amoebae undergoing chaotic oscillations of cyclic AMP. The analysis of a model indicates that a tiny proportion of periodic cells suffices to transform chaos into periodic oscillations in such suspensions. A similar result is obtained by forcing the aperiodic oscillations by a small-amplitude, periodic input of cyclic AMP. The results provide an explanation for the observation of regular oscillations in suspensions of a putatively chaotic mutant of Dictyostelium discoideum. More generally, the results show how chaos in biological systems may disappear through the coupling with periodic oscillations.

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.

Cyclic AMP oscillations in suspensions of Dictyostelium discoideum

A model developed previously for signal relay and adaptation in the cellular slime mould Dictyostelium discoideum is shown to account for the observed oscillations of calcium and cyclic AMP in cellular suspensions. A qualitative argument is given which explains how the oscillations arise, and numerical computations show how characteristics such as the period and amplitude of the periodic solutions depend on parameters in the model. Several extensions of the basic model are investigated, including the effect of cell aggregation and the effect of time delays in the activation and adaptation processes. The dynamics of mixed cell populations in which only a small fraction of the cells are capable of autonomous oscillation are also studied.

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.