Sustained oscillations in stochastic systems

Many non-linear deterministic models for interacting populations present damped oscillations towards the corresponding equilibrium values. However, simulations produced with related stochastic models usually present sustained oscillations which preserve the natural frequency of the damped oscillations of the deterministic model but showing non-vanishing amplitudes. The relation between the amplitude of the stochastic oscillations and the values of the equilibrium populations is not intuitive in general but scales with the square root of the populations when the ratio between different populations is kept fixed. In this work, we explain such phenomena for the case of a general epidemic model. We estimate the stochastic fluctuations of the populations around the equilibrium point in the epidemiological model showing their (approximated) relation with the mean values.     

Sustained simultaneous glycolytic and insulin oscillations in beta-cells

Chemical oscillation in glycolysis induced by glucose is an universal feature in all living cells. In beta-cells this is accompanied by sustained oscillations of concentration of insulin, which helps to keep the blood glucose level within optimum limits. Experiments in this regard had shown that the glycolytic and insulin oscillations are almost consistently in phase and their time periods are very close to each other at both high and low initial concentration of glucose. Experiments have also demonstrated the dynamical transition between the states of glycolytic oscillations indicating a saturation behaviour of glucose transporters at a higher glucose flow rate. We propose a phenomenological model to understand these simultaneous oscillations and how glycolysis provides a mechanism for pulsatory insulin secretion in the light of these basic experimental issues.     

Sustained oscillations in extended genetic oscillatory systems

Various dynamic cellular behaviors have been successfully modeled in terms of elementary circuitries showing particular characteristics such as negative feedback loops for sustained oscillations. Given, however, the increasing evidences indicating that cellular components do not function in isolation but form a complex interwoven network, it is still unclear to what extent the conclusions drawn from the elementary circuit analogy hold for systems that are highly interacting with surrounding environments. In this article, we consider a specific example of genetic oscillator systems, the so-called repressilator, as a starting point toward a systematic investigation into the dynamic consequences of the extension through interlocking of elementary biological circuits. From in silico analyses with both continuous and Boolean dynamics approaches to the four-node extension of the repressilator, we found that 1), the capability of sustained oscillation depends on the topology of extended systems; and 2), the stability of oscillation under the extension also depends on the coupling topology. We then deduce two empirical rules favoring the sustained oscillations, termed the coherent coupling and the homogeneous regulation. These simple rules will help us prioritize candidate patterns of network wiring, guiding both the experimental investigations for further physiological verification and the synthetic designs for bioengineering.     

Sustained oscillations via coherence resonance in SIR

Sustained oscillations in a stochastic SIR model are studied using a new multiple scale analysis. It captures the interaction of the deterministic and stochastic elements together with the separation of time scales inherent in the appearance of these dynamics. The nearly regular fluctuations in the infected and susceptible populations are described via an explicit construction of a stochastic amplitude equation. The agreement between the power spectral densities of the full model and the approximation verifies that coherence resonance is driving the behavior. The validity criteria for this asymptotic approximation give explicit expressions for the parameter ranges in which one expects to observe this phenomenon.     

Sustained glycolytic oscillations in individual isolated yeast cells

Yeast glycolytic oscillations have been studied since the 1950s in cell-free extracts and intact cells. For intact cells, sustained oscillations have so far only been observed at the population level, i.e. for synchronized cultures at high biomass concentrations. Using optical tweezers to position yeast cells in a microfluidic chamber, we were able to observe sustained oscillations in individual isolated cells. Using a detailed kinetic model for the cellular reactions, we simulated the heterogeneity in the response of the individual cells, assuming small differences in a single internal parameter. This is the first time that sustained limit-cycle oscillations have been demonstrated in isolated yeast cells. DATABASE: The mathematical model described here has been submitted to the JWS Online Cellular Systems Modelling Database and can be accessed at http://jjj.biochem.sun.ac.za/database/gustavsson/index.html free of charge.     

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.     

Expression and control of an operon from an intracellular symbiont which is homologous to the groE operon.

Members of the genus Buchnera are intracellular symbionts harbored by the aphid bacteriocyte which selectively synthesize symbionin, a homolog of the Escherichia coli GroEL protein, in vivo. Symbionin and SymS, a GroES homolog, are encoded in the symSL operon. Northern blotting and primer extension analyses revealed that the symSL operon invariably gives rise to a bicistronic mRNA under the control of a heat shock promoter, though the amount of the symSL mRNA in the isolated symbiont did not increase in response to heat shock. The sigma32 protein that recognizes the heat shock promoter in E. coli was scarcely detected in Buchnera cells even after heat shock. Although the functionally essential regions of the Buchnera sigma32 protein were well conserved, the Buchnera rpoH gene did not complement an E. coli delta rpoH mutant. On the one hand, the A-T evolutionary pressure imposed on the Buchnera genome may have not only decreased the activity of its sigma32 but also ruined the nucleotide sequences necessary for the expression of rpoH; on the other hand, it may have facilitated expression of the symSL operon without activation by sigma32.     

A note on facilitation and threshold phenomena

A neuron subjected to a Poisson shower of stimuli responds only ifh stimuli impinge upon it within the time interval ρ. It is shown that the derivative of the input-output curve cannot exceed unity.     

Calcium oscillations: phenomena, mechanisms and significance

Many hormones that mobilise intracellular calcium via inositol 1,4,5 trisphosphate induce oscillations in cytoplasmic free Ca. Two basic oscillatory patterns occur: quasi-sinusoidal oscillations and repetitive free Ca transients. The mechanisms responsible for generating these oscillations are not clear; calcium-induced calcium release, interplay between two intracellular calcium pools and repetitive generation of InsP3 are discussed. The significance of different oscillatory patterns induced by different agonists in the same cell is emphasised, and mechanisms by which the oscillators may retain-receptor specific information are proposed, such as negative feedback onto receptors or G-proteins by protein kinase C. Reasons why cells generate free Ca oscillations and possible consequences such as oscillations in downstream pathways are explored. The possibility that pathological conditions such as aluminium toxicity are exerted through distortion of oscillatory free Ca signalling is raised.     

Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model

Sustained and transient oscillations are frequently observed in clinical data for immune responses in viral infections such as human immunodeficiency virus, hepatitis B virus, and hepatitis C virus. To account for these oscillations, we incorporate the time lag needed for the expansion of immune cells into an immunosuppressive infection model. It is shown that the delayed antiviral immune response can induce sustained periodic oscillations, transient oscillations and even sustained aperiodic oscillations (chaos). Both local and global Hopf bifurcation theorems are applied to show the existence of periodic solutions, which are illustrated by bifurcation diagrams and numerical simulations. Two types of bistability are shown to be possible: (i) a stable equilibrium can coexist with another stable equilibrium, and (ii) a stable equilibrium can coexist with a stable periodic solution.     

Understanding shifts in wildfire regimes as emergent threshold phenomena

Ecosystems driven by wildfire regimes are characterized by fire size distributions resembling power laws. Existing models produce power laws, but their predicted exponents are too high and fail to capture the exponent's variation with geographic region. Here we present a minimal model of fire dynamics that describes fire spread as a stochastic birth-death process, analogous to stochastic population growth or disease spread and incorporating memory effects from previous fires. The model reproduces multiple regional patterns in fire regimes and allows us to classify different regions in terms of their proximity to a critical threshold. Transitions across this critical threshold imply abrupt and pronounced increases in average fire size. The model predicts that large regions in Canada are currently close to this transition and might be driven beyond the threshold in the future. We illustrate this point by analyzing the time series for large fires (>199 ha) from the Canadian Boreal Plains, found to have shifted from a subcritical regime to a critical regime in the recent past. By contrast to its predecessor, the model also suggests that a critical transition, and not self-organized criticality, underlies forest fire dynamics, with implications for other ecological systems exhibiting power-law-like patterns, in particular for their sensitivity to environmental change and control efforts.     

Mathematical models of threshold phenomena in the nerve membrane

The types of mathematical model which have been used to represent all-or-none behavior in the nerve membrane may be classified as follows: (1) thediscontinuous threshold phenomenon, in which differential equations with discontinuous functions provide both a discontinuity of response as a function of stimulus intensity at threshold and a finite maximum latency, (2) thesingular-point threshold phenomenon which exists in a phase space having analytic functions in its differential equations and having a singular point with one characteristic root positive and the rest with negative real parts, the latency being unbounded, and (3) thequasi threshold phenomenon, which has a finite maximum latency and continuous functions, but neither a true discontinuity in response nor an exact threshold. Several models of the nerve membrane in the literature are classified accordingly, and the applicability of the different types of threshold phenomena to the membrane is discussed, including an extension to a stochastic model.     

Sustained oscillations in a reconstituted enzyme system containing phosphofructokinase and fructose 1,6-bisphosphatase

In a reconstituted open and homogeneous enzyme system containing phosphofructokinase, fructose 1,6-bisphosphatase, pyruvate kinase, adenylate kinase, and glucose-6-phosphate isomerase sustained oscillations could experimentally be generated. The approach is based on a stirred flow-through reaction chamber. The periodic motions of the reactants are mainly caused by the antagonistic allosteric effects of the adenine nucleotides on the activities of the phosphofructokinase and fructose 1,6-bisphosphatase.     

Theta oscillations by synaptic excitation in a neocortical circuit model

Neocortical theta-band oscillatory activity is associated with cognitive tasks involving learning and memory. This oscillatory activity is proposed to originate from the synchronization of interconnected layer V intrinsic bursting (IB) neurons by recurrent excitation. To test this hypothesis, a sparsely connected spiking circuit model based on empirical data was simulated using Hodgkin-Huxley-type bursting neurons and use-dependent depressing synaptic connections. In response to a heterogeneous tonic current stimulus, the model generated coherent and robust oscillatory activity throughout the theta-band (4-12 Hz). These oscillations were not, however, self-sustaining without a driving current, and not dependent on N-methyl-D-aspartate receptor synaptic currents. At realistic connection strengths, synaptic depression was necessary to avoid instability and expanded the basin of attraction for theta oscillations by controlling the gain of recurrent excitation. These results support the hypothesis that IB neuron networks can generate robust and coherent theta-band oscillations in neocortex.     

Bistability explains threshold phenomena in protein aggregation both in vitro and in vivo

Neurodegenerative disease can originate from the misfolding and aggregation of proteins, such as Amyloid-beta, SOD1, or Huntingtin. Fortunately, all cells possess protein quality control machinery that sequesters misfolded proteins, either refolding or degrading them, before they can self-associate into proteotoxic oligomers and aggregates. This activity is largely performed by the stress response chaperones (i.e., Hsp70). However, the expression level of molecular chaperones varies widely among cell types. To understand the potential consequence of this variation, we studied the process of protein aggregation in the presence of molecular chaperones using mathematical modeling. We demonstrate that protein aggregation, in the presence of molecular chaperones, is a bistable process. Bistability in protein aggregation offers an explanation for threshold transitions to high aggregate concentration, which are observed both in vitro and in vivo. Additionally, we show that slight variations in chaperone concentration, due to natural fluctuations, have important consequences in a bistable system for the onset of protein aggregation. Therefore, our results offer a possible theoretical explanation for neuronal vulnerability observed in vivo and the onset of neurodegenerative phenotypes in neurons lacking an effective heat-shock response.     

Ketamine-induced oscillations in the motor circuit of the rat basal ganglia

Oscillatory activity can be widely recorded in the cortex and basal ganglia. This activity may play a role not only in the physiology of movement, perception and cognition, but also in the pathophysiology of psychiatric and neurological diseases like schizophrenia or Parkinson's disease. Ketamine administration has been shown to cause an increase in gamma activity in cortical and subcortical structures, and an increase in 150 Hz oscillations in the nucleus accumbens in healthy rats, together with hyperlocomotion.We recorded local field potentials from motor cortex, caudate-putamen (CPU), substantia nigra pars reticulata (SNr) and subthalamic nucleus (STN) in 20 awake rats before and after the administration of ketamine at three different subanesthetic doses (10, 25 and 50 mg/Kg), and saline as control condition. Motor behavior was semiautomatically quantified by custom-made software specifically developed for this setting.Ketamine induced coherent oscillations in low gamma (~ 50 Hz), high gamma (~ 80 Hz) and high frequency (HFO, ~ 150 Hz) bands, with different behavior in the four structures studied. While oscillatory activity at these three peaks was widespread across all structures, interactions showed a different pattern for each frequency band. Imaginary coherence at 150 Hz was maximum between motor cortex and the different basal ganglia nuclei, while low gamma coherence connected motor cortex with CPU and high gamma coherence was more constrained to the basal ganglia nuclei. Power at three bands correlated with the motor activity of the animal, but only coherence values in the HFO and high gamma range correlated with movement. Interactions in the low gamma band did not show a direct relationship to movement.These results suggest that the motor effects of ketamine administration may be primarily mediated by the induction of coherent widespread high-frequency activity in the motor circuit of the basal ganglia, together with a frequency-specific pattern of connectivity among the structures analyzed.     

Sustained oscillations of transmembrane Ca2+ fluxes in mitochondria and their possible biological significance

Sustained oscillations of transmembrane fluxes of Ca2+ and other ions in isolated mitochondria are described. The data are presented that the major cause of the oscillations is the Ca2+-induced Ca2+ efflux from the mitochondrial matrix and spontaneous opening/closing of the permeability transition pore in the inner mitochondrial membrane. Conditions favourable for the generation of oscillations are considered. The role of phospholipid peroxidation and hydrolysis in the generation of [Ca2+] oscillations is emphasized. Literature data concerning [Ca2+] changes in the mitochondrial matrix in intact cells and the data on the participation of mitochondria in intracellular Ca2+ oscillation and in the Ca2+ wave propagation are reviewed. The hypothesis that mitochondria are able to generate [Ca2+] oscillations in intact cells is put forward. It is assumed that Ca2+ oscillations can protect mitochondria of resting cells from osmotic shock and oxidative stress.     

The mdoA locus of Escherichia coli consists of an operon under osmotic control

In Escherichia coli, the 5 kb mdoA locus is involved in the osmotically controlled biosynthesis of periplasmic membrane-derived oligosaccharides (MDOs). The structure of this locus was analysed by in vitro cassette insertion, transposon mutagenesis, and gene-fusion analysis. A 'neo' cassette, derived from the neomycin phosphotransferase II region of transposon Tn5, was inserted into mdoA, borne by a multicopy plasmid. This plasmid was shown to complement two previously described mdoA mutations, depending on the orientation of the exogenous gene. Thus, the gene altered by these mutations could be expressed under the control of the exogenous promoter. Moreover, the 'neo' cassette inactivated another, uncharacterized, mdo gene, because when this insertion was transferred into the chromosome MDO synthesis was abolished. The existence of a second gene was confirmed by complementation analysis with a collection of Tn1000 insertions into mdoA. Two groups were defined, and the two genes are organized into an operon (mdoGH). This conclusion was reached because Tn1000 insertions in the first gene displayed a polar effect on the expression of the second gene. An active gene fusion was obtained on a multicopy plasmid between the beginning of mdoH and lacZ. The hybrid beta-galactosidase activity followed the same osmotically controlled response as that described for of MDO synthesis. This regulation was unaffected by the presence, or absence, of MDOs in the periplasm. Finally, the amount of mdoA-specific mRNAs, determined by dot blot hybridization, decreased when the osmolarity of the growth medium increased.