On the dynamics of controlled metabolic network and cellular behavior

The existence of elaborate control mechanisms for the various biochemical processes inside and within living cells is responsible for the coherent behaviour observed in its spatio-temporal organisation. Stability and sensitivity are both necessary properties of living systems and these are achieved through negetive and positive feedback loops as in other control systems. We have studied a three-step reaction scheme involving a negative and a positive feedback loop in the form of end-product inhibition and allosteric activation. The variety of behaviour exhibited by this system, under different conditions, includes steady state, simple limit cycle oscillations, complex oscillations and period bifurcations leading to random oscillations or chaos. The system also shows the existence of two distinct chaotic regimes under the variation of a single parameter. These results, in comparison with single biochemical control loops, show that new behaviours can be exhibited in a more complex network which are not seen in the single control loops. The results are discussed in the light of a diverse variety of cellular functions in normal and altered cells indicating the role of controlled metabolic network as the underlying basis for cellular behaviour.     

Modeling the interrelations between the calcium oscillations and ER membrane potential oscillations

A refined electrochemical model accounting for intracellular calcium oscillations and their interrelations with oscillations of the potential difference across the membrane of the endoplasmic reticulum (ER) or other intracellular calcium stores is established. The ATP dependent uptake of Ca2+ from the cytosol into the ER, the Ca2+ release from the ER through channels following a calcium-induced calcium release mechanism, and a potential-dependent Ca2+ leak flux out of the ER are included in the model and described by plausible rate laws. The binding of calcium to specific proteins such as calmodulin is taken into account. The quasi-electroneutrality condition allows us to express the transmembrane potential in terms of the concentrations of cytosolic calcium and free binding sites on proteins, which are the two independent variables of the model. We include monovalent ions in the model, because they make up a considerable portion in the balance of electroneutrality. As the permeability of the endoplasmic membrane for these ions is much higher than that for calcium ions, we assume the former to be in Nernst equilibrium. A stability analysis of the steady-state solutions (which are unique or multiple depending on parameter values) is carried out and the Hopf bifurcation leading from stable steady states to self-sustained oscillations is analysed with the help of appropriate mathematical techniques. The oscillations obtained by numerical integration exhibit the typical spike-like shape found in experiments and reasonable values of frequency and amplitude. The model describes the process of switching between stationary and pulsatile regimes as well as changes in oscillation frequency upon parameter changes. It turns out that calcium oscillations can arise without a permanent influx of calcium into the cell, when a calcium-buffering system such as calmodulin is included.     

A two variable delay model for the circadian rhythm of Neurospora crassa

A two variable model with delay in both the variables, is proposed for the circadian oscillations of protein concentrations in the fungal species Neurospora crassa. The dynamical variables chosen are the concentrations of FRQ and WC-1 proteins. Our model is a two variable simplification of the detailed model of Smolen et al. (J. Neurosci. 21 (2001) 6644) modeling circadian oscillations with interlocking positive and negative feedback loops, containing 23 variables. In our model, as in the case of Smolen's model, a sustained limit cycle oscillation takes place in both FRQ and WC-1 protein in continuous darkness, and WC-1 is anti-phase to FRQ protein, as observed in experiments. The model accounts for various characteristic features of circadian rhythms such as entrainment to light dark cycles, phase response curves and robustness to parameter variation and molecular fluctuations. Simulations are carried out to study the effect of periodic forcing of circadian oscillations by light-dark cycles. The periodic forcing resulted in a rich bifurcation diagram that includes quasiperiodicity and chaotic oscillations, depending on the magnitude of the periodic changes in the light controlled parameter. When positive feedback is eliminated, our model reduces to the generic one dimensional delay model of Lema et al. (J. Theor. Biol. 204 (2000) 565), delay model of the circadian pace maker with FRQ protein as the dynamical variable which represses its own production. This one-dimensional model also exhibits all characteristic features of circadian oscillations and gives rise to circadian oscillations which are reasonably robust to parameter variations and molecular noise.     

Delay model of circadian gene expression with two negative feedback loops

In this theoretical paper we propose a quantitative minimal model for circadian gene expression based on two negative feedback loops. We perform numerical simulations to analyse its dynamics and parameter sensitivities in free-running conditions, and verify the entrainability by a single periodic driver. We furthermore apply two simultaneously acting external drivers, leading to aperiodic oscillations in the case of a single-loop system. These can be turned into regular periodic oscillations by introduction of a second loop. Our studies confirm the increasing evidence that multiple feedback loops increase the robustness of regulatory systems, and stress the particular situation of systems that are close to transition from free-running oscillation to steady-state behaviour. We discuss possible molecular realisations of the featured feedback loops and suggest the application of complex patterns of external stimulation as a generally useful approach to assess the functionality of models of circadian systems.     

Population oscillations in a discrete model of neural networks of the brain

A discrete model of biological neural networks is used to find out how synchronized firing of neurons emerges in a randomly connected neural population. The objective is to understand the mechanisms underlying brain waves and to find and characterize conditions which support spontaneous switching from disordered to rhythmic population activity as in case of an epileptic seizure. The model is kept as simple as possible to achieve on one hand a fast performance of computer simulations of networks with up to 10,000 neurons and to keep on the other hand an overview of parameter dependences. Dynamics of the model can be classified into different regimes: random fluctuations, rhythmic oscillations and silence. When the ratio of the inhibitory/excitatory connectivity is raised the system crosses from the fluctuating regime through the rhythmic oscillating region to the silence regime. Close to the boundary between the fluctuating and the oscillating regimes the network shows spontaneous bursting of high amplitude rhythmic oscillations, which is characteristic of epileptiform behavior. The simulation results are in agreement with recent theories saying that focal epilepsy after injury of the brain could result from axonal sprouting of GABAergic neurons in the injured region.     

Dynamic Processes in Regulation and Some Implications for Biofeedback and Biobehavioral Interventions

Systems theory has long been used in psychology, biology, and sociology. This paper applies newer methods of control systems modeling for assessing system stability in health and disease. Control systems can be characterized as open or closed systems with feedback loops. Feedback produces oscillatory activity, and the complexity of naturally occurring oscillatory patterns reflects the multiplicity of feedback mechanisms, such that many mechanisms operate simultaneously to control the system. Unstable systems, often associated with poor health, are characterized by absence of oscillation, random noise, or a very simple pattern of oscillation. This modeling approach can be applied to a diverse range of phenomena, including cardiovascular and brain activity, mood and thermal regulation, and social system stability. External system stressors such as disease, psychological stress, injury, or interpersonal conflict may perturb a system, yet simultaneously stimulate oscillatory processes and exercise control mechanisms. Resonance can occur in systems with negative feedback loops, causing high-amplitude oscillations at a single frequency. Resonance effects can be used to strengthen modulatory oscillations, but may obscure other information and control mechanisms, and weaken system stability. Positive as well as negative feedback loops are important for system function and stability. Examples are presented of oscillatory processes in heart rate variability, and regulation of autonomic, thermal, pancreatic and central nervous system processes, as well as in social/organizational systems such as marriages and business organizations. Resonance in negative feedback loops can help stimulate oscillations and exercise control reflexes, but also can deprive the system of important information. Empirical hypotheses derived from this approach are presented, including that moderate stress may enhance health and functioning.     

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.     

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.     

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.     

Effect of positive feedback loops on the robustness of oscillations in the network of cyclin-dependent kinases driving the mammalian cell cycle

The transitions between the G(1) , S, G(2) and M phases of the mammalian cell cycle are driven by a network of cyclin-dependent kinases (Cdks), whose sequential activation is regulated by intertwined negative and positive feedback loops. We previously proposed a detailed computational model for the Cdk network, and showed that this network is capable of temporal self-organization in the form of sustained oscillations, which govern ordered progression through the successive phases of the cell cycle [Gérard and Goldbeter (2009) Proc Natl Acad Sci USA106, 21643-21648]. We subsequently proposed a skeleton model for the cell cycle that retains the core regulatory mechanisms of the detailed model [Gérard and Goldbeter (2011) Interface Focus1, 24-35]. Here we extend this skeleton model by incorporating Cdk regulation through phosphorylation/dephosphorylation and by including the positive feedback loops that underlie the dynamics of the G(1) /S and G(2) /M transitions via phosphatase Cdc25 and via phosphatase Cdc25 and kinase Wee1, respectively. We determine the effects of these positive feedback loops and ultrasensitivity in phosphorylation/dephosphorylation on the dynamics of the Cdk network. The multiplicity of positive feedback loops as well as the existence of ultrasensitivity promote the occurrence of bistability and increase the amplitude of the oscillations in the various cyclin/Cdk complexes. By resorting to stochastic simulations, we further show that the presence of multiple, redundant positive feedback loops in the G(2) /M transition of the cell cycle markedly enhances the robustness of the Cdk oscillations with respect to molecular noise.     

Self-sustained pH oscillations in a compartmentalized enzyme reactor system

This work represents our continued effort toward fulfilling the need to discover a model system for experimental investigations of temporal oscillations in an enzyme-membrane system. In this paper, the regions in the parameter space where self-sustained pH oscillations can be induced for a compartmentalized enzyme reactor system, which consists of a well-stirred reactor, a reservoir and a membrane containing no enzyme, were determined via numerical simulation with two proteolytic enzymes: papain (EC 3.4.22.2) and alpha-chymotrypsin (EC 3.4.21.1). The sizes of the regions were qualitatively compared with those associated with enzymic membrane system. As a result, we found that the possibility of experimentally observing self-sustained oscillations in the compartmentalized papain reactor system, as well as in the papain-membrane system, is high. However, self-sustained pH oscillations are less likely in the compartmentalized alpha-chymotrypsin reactor system than in the alpha-chymotrypsin-membrane system.     

Kelvin-Helmholtz instability on the magnetopause, magnetohydrodynamic waveguide in the outer magnetosphere, and Alfvén resonance deep in the magnetosphere

Oscillations of the “magnetosphere-solar wind” system are studied analytically in the framework of a plane-stratified model of the medium. The properties of oscillations are determined by three phenomena: Kelvin-Helmholtz instability on the tangential discontinuity (magnetopause) separating the magnetosphere and the solar wind, the presence of a waveguide for fast magnetosonic waves in the magnetosphere, and the Alfvén resonance—a sharp increase in the amplitude of oscillations having the properties of Alfvén waves—in the inner magnetosphere. The oscillations of the system form a discrete spectrum of eigenmodes. Analytical expressions are obtained for the frequency and growth rate of instability of each mode, as well as for the functions describing the spatial structure of these modes. All these characteristics of the eigenmodes are shown to depend on the velocity of the solar wind as a parameter. The dependences of the main mode characteristics (such as the instability thresholds, the points of the maximum and minimum growth rate, and the spatial distributions of the oscillation energy) on this parameter are determined for each eigenmode.     

Understanding cyclical thrombocytopenia: a mathematical modeling approach

Cyclical thrombocytopenia (CT) is a rare hematological disease characterized by periodic oscillations in the platelet count. Although first reported in 1936, the pathogenesis and an effective therapy remain to be identified. Since besides fluctuations in platelet levels the patients hematological profile have been consistently normal, a destabilization of a peripheral control mechanism might play an important role in the genesis of this disorder. In this paper, we investigate through computer simulations the mechanisms underlying the platelet oscillations observed in CT. First, we collected the data published in the last 40 years and quantified the significance of the platelet fluctuations using Lomb-Scargle periodograms. Our analysis reveals that the incidence of the statistically significant periodic data is equally distributed in men and women. The mathematical model proposed in this paper captures the essential features of hematopoiesis and successfully duplicates the characteristics of CT. With the same parameter changes, the model is able to fit the platelet counts and to qualitatively reproduce the TPO oscillations (when data is available). Our results indicate that a variation in the megakaryocyte maturity, a slower relative growth rate of megakaryocytes, as well as an increased random destruction of platelets are the critical elements generating the platelet oscillations in CT.     

Oscillations, shocks, and fine wave structures arising during the coalescence of two force-free current loops

Two-dimensional numerical simulations of the magnetic reconnection of two parallel force-free current loops are carried out using a high-resolution MHD code in which an artificial wind scheme is employed. Two typical cases (namely, co-helicity and counter-helicity reconnection) are investigated. The simulation results show that co-helicity reconnection involves only the reconnection of the poloidal component of the magnetic field, while counter-helicity reconnection involves the reconnection of both the poloidal and axial components of the magnetic field. Therefore, counter-helicity reconnection is much more complicated and violent as compared to co-helicity reconnection. In both cases, jetlike flows are generated. Counter-helicity reconnection is accompanied by oscillations of both the axial magnetic field and the axial component of the velocity. Due to these oscillations, quasi-steady models of a current sheet appear to be inapplicable, because the current sheet structure also changes. The complicated and unsteady structure of the current distribution shows that magnetic reconnection occurs not only in the central sheet between two loops in the earlier stage of the process, but also inside each loop in later stages. Rather complicated flows and waves with fine structures are also generated during reconnection. Some of the waves appear to be shock waves.     

Thermodynamics of three-way multibranch loops in RNA

RNA multibranch loops (junctions) are loops from which three or more helices exit. They are nearly ubiquitous in RNA secondary structures determined by comparative sequence analysis. In this study, systems in which two strands combine to form three-way junctions were used to measure the stabilities of RNA multibranch loops by UV optical melting and isothermal titration calorimetry (ITC). These data were used to calculate the free energy increment for initiation of a three-way junction on the basis of a nearest neighbor model for secondary structure stability. Imino proton NMR spectra were also measured for two systems and are consistent with the hypothesized helical structures. Incorporation of the experimental data into the mfold and RNA structure computer programs has contributed to an improvement in prediction of RNA secondary structure from sequence.     

Age structure effects in predator-prey interactions

A mathematical model of predator-prey interactions is proposed which incorporates both age structure in the predators and density dependence in the prey. The properties of the model are investigated by a linearized analysis, which enables the conditions for stability to be formulated. The analysis indicates that for a substantial domain of parameter space, a stable equilibrium is possible with the prey well below its carrying capacity. The effect of violating the stability conditions on the behaviour of the model was investigated by computer simulation. Two further types of behaviour are noted in which coexistence is possible. The first is a two point limit cycle in which young and old predators occur in alternate time periods. The second involves a limit cycle in which the component population trajectories lie on closed curves in phase space.