Evolution towards oscillation or stability in a predator–prey system

We studied a prey–predator system in which both species evolve. We discuss here the conditions that result in coevolution towards a stable equilibrium or towards oscillations. First, we show that a stable equilibrium or population oscillations with small amplitude is likely to occur if the prey''s (host''s) defence is effective when compared with the predator''s (parasite''s) attacking ability at equilibrium, whereas large-amplitude oscillations are likely if the predator''s (parasite''s) attacking ability exceeds the prey''s (host''s) defensive ability. Second, a stable equilibrium is more likely if the prey''s defensive trait evolves faster than the predator''s attack trait, whereas population oscillations are likely if the predator''s trait evolves faster than that of the prey. Third, when the adaptation rates of both species are similar, the amplitude of the fluctuations in their abundances is small when the adaptation rate is either very slow or very fast, but at an intermediate rate of adaptation the fluctuations have a large amplitude. We also show the case in which the prey''s abundance and trait fluctuate greatly, while those of the predator remain almost unchanged. Our results predict that populations and traits in host–parasite systems are more likely than those in prey–predator systems to show large-amplitude oscillations.     

Stability of a quasi-flute mode under the Bernstein-Kadomtsev condition in a toroidal confinement system

It is shown that the growth rate of the MHD instability in toroidal configurations is slower in a situation in which the Bernstein-Kadomtsev condition is satisfied while the Mercier stability criterion is not. Under the Bernstein-Kadomtsev condition, Alfvénic Mercier modes are not excited, but quasi-flute acoustic Mercier modes develop instead. In confinement systems with closed magnetic field lines, the Bernstein-Kadomtsev condition ensures MHD stability; however, a small rotational transform produced by magnetic perturbations can give rise to a quasi-flute acoustic instability whose growth rate is proportional to the perturbation amplitude, in which case the fastest growing oscillations are those with the shortest wavelengths.     

Kelvin-Helmholtz instability for a bounded plasma flow in a longitudinal magnetic field

Kelvin-Helmholtz MHD instability in a plane three-layer plasma is investigated. A general dispersion relation for the case of arbitrarily orientated magnetic fields and flow velocities in the layers is derived, and its solutions for a bounded plasma flow in a longitudinal magnetic field are studied numerically. Analysis of Kelvin-Helmholtz instability for different ion acoustic velocities shows that perturbations with wavelengths on the order of or longer than the flow thickness can grow in an arbitrary direction even at a zero temperature. Oscillations excited at small angles with respect to the magnetic field exist in a limited range of wavenumbers even without allowance for the finite width of the transition region between the flow and the ambient plasma. It is shown that, in a low-temperature plasma, solutions resulting in kink-like deformations of the plasma flow grow at a higher rate than those resulting in quasi-symmetric (sausage-like) deformations. The transverse structure of oscillatory-damped eigenmodes in a low-temperature plasma is analyzed. The results obtained are used to explain mechanisms for the excitation of ultra-low-frequency long-wavelength oscillations propagating along the magnetic field in the plasma sheet boundary layer of the Earth’s magnetotail penetrated by fast plasma flows.     

Dynamics of a biochemical system with multiple oscillatory domains as a clue for multiple modes of neuronal oscillations

We analyze the behavior of a two-variable biochemical model in conditions where it admits multiple oscillatory domains in parameter space. The model represents an autocatalytic enzyme reaction with input of substrate both from a constant source and from non-linear recycling of product into substrate. This system was previously studied for birhythmicity, i.e. the coexistence between two stable periodic regimes (Moran and Goldbeter 1984), and for multithreshold excitability (Moran and Goldbeter 1985). When two distinct oscillatory domains obtain as a function of the substrate injection rate, the system is capable of exhibiting two markedly different modes of oscillations for slightly different values of this control parameter. Phase plane analysis shows how the multiplicity of oscillatory domains depends on the parameters that govern the underlying biochemical mechanism of product recycling. We analyze the response of the model to various kinds of transient perturbations and to periodic changes in the substrate input that bring the system through the two ranges of oscillatory behavior. The results provide a qualitative explanation for experimental observations (Jahnsen and Llinas 1984b) related to the occurrence of two different modes of oscillations in thalamic neurones.     

Experimental Assessment of the Sensitiveness of an Electrochemical Oscillator towards Chemical Perturbations

In this study we address the problem of the response of a (electro)chemical oscillator towards chemical perturbations of different magnitudes. The chemical perturbation was achieved by addition of distinct amounts of trifluoromethanesulfonate (TFMSA), a rather stable and non-specifically adsorbing anion, and the system under investigation was the methanol electro-oxidation reaction under both stationary and oscillatory regimes. Increasing the anion concentration resulted in a decrease in the reaction rates of methanol oxidation and a general decrease in the parameter window where oscillations occurred. Furthermore, the addition of TFMSA was found to decrease the induction period and the total duration of oscillations. The mechanism underlying these observations was derived mathematically and revealed that inhibition in the methanol oxidation through blockage of active sites was found to further accelerate the intrinsic non-stationarity of the unperturbed system. Altogether, the presented results are among the few concerning the experimental assessment of the sensitiveness of an oscillator towards chemical perturbations. The universal nature of the complex chemical oscillator investigated here might be used for reference when studying the dynamics of other less accessible perturbed networks of (bio)chemical reactions.     

Attractor analysis of asynchronous Boolean models of signal transduction networks

Prior work on the dynamics of Boolean networks, including analysis of the state space attractors and the basin of attraction of each attractor, has mainly focused on synchronous update of the nodes’ states. Although the simplicity of synchronous updating makes it very attractive, it fails to take into account the variety of time scales associated with different types of biological processes. Several different asynchronous update methods have been proposed to overcome this limitation, but there have not been any systematic comparisons of the dynamic behaviors displayed by the same system under different update methods. Here we fill this gap by combining theoretical analysis such as solution of scalar equations and Markov chain techniques, as well as numerical simulations to carry out a thorough comparative study on the dynamic behavior of a previously proposed Boolean model of a signal transduction network in plants. Prior evidence suggests that this network admits oscillations, but it is not known whether these oscillations are sustained. We perform an attractor analysis of this system using synchronous and three different asynchronous updating schemes both in the case of the unperturbed (wild-type) and perturbed (node-disrupted) systems. This analysis reveals that while the wild-type system possesses an update-independent fixed point, any oscillations eventually disappear unless strict constraints regarding the timing of certain processes and the initial state of the system are satisfied. Interestingly, in the case of disruption of a particular node all models lead to an extended attractor. Overall, our work provides a roadmap on how Boolean network modeling can be used as a predictive tool to uncover the dynamic patterns of a biological system under various internal and environmental perturbations.     

Multistability and dynamic transitions of intracellular Min protein patterns

Cells owe their internal organization to self‐organized protein patterns, which originate and adapt to growth and external stimuli via a process that is as complex as it is little understood. Here, we study the emergence, stability, and state transitions of multistable Min protein oscillation patterns in live Escherichia coli bacteria during growth up to defined large dimensions. De novo formation of patterns from homogenous starting conditions is observed and studied both experimentally and in simulations. A new theoretical approach is developed for probing pattern stability under perturbations. Quantitative experiments and simulations show that, once established, Min oscillations tolerate a large degree of intracellular heterogeneity, allowing distinctly different patterns to persist in different cells with the same geometry. Min patterns maintain their axes for hours in experiments, despite imperfections, expansion, and changes in cell shape during continuous cell growth. Transitions between multistable Min patterns are found to be rare events induced by strong intracellular perturbations. The instances of multistability studied here are the combined outcome of boundary growth and strongly nonlinear kinetics, which are characteristic of the reaction–diffusion patterns that pervade biology at many scales.     

Sensitivity and control analysis of periodically forced reaction networks using the Green's function method

A general sensitivity and control analysis of periodically forced reaction networks with respect to small perturbations in arbitrary network parameters is presented. A well-known property of sensitivity coefficients for periodic processes in dynamical systems is that the coefficients generally become unbounded as time tends to infinity. To circumvent this conceptual obstacle, a relative time or phase variable is introduced so that the periodic sensitivity coefficients can be calculated. By employing the Green's function method, the sensitivity coefficients can be defined using integral control operators that relate small perturbations in the network's parameters and forcing frequency to variations in the metabolite concentrations and reaction fluxes. The properties of such operators do not depend on a particular parameter perturbation and are described by the summation and connectivity relationships within a control-matrix operator equation. The aim of this paper is to derive such a general control-matrix operator equation for periodically forced reaction networks, including metabolic pathways. To illustrate the general method, the two limiting cases of high and low forcing frequency are considered. We also discuss a practically important case where enzyme activities and forcing frequency are modulated simultaneously. We demonstrate the developed framework by calculating the sensitivity and control coefficients for a simple two reaction pathway where enzyme activities enter reaction rates linearly and specifically.     

Stability of a stationary double layer

Electrostatic oscillations in a stationary beam double layer are studied in the hydrodynamic approximation. An analysis of the solutions to the linearized equations for perturbations shows that the double layer in an unbounded plasma may be subject only to convective instability, in which case the perturbations localized inside the region of the potential jump escape from the plasma without distorting the steady field structure. The double layer in a bounded plasma is investigated numerically by a particle-in-cell method. It is established that, as is the case in the Pierce system, the governing role in the stability of the double layer is played by the field of the charges induced on the surfaces of the conducting electrodes.     

Systems Pharmacology and Rational Polypharmacy: Nitric Oxide?Cyclic GMP Signaling Pathway as an Illustrative Example and Derivation of the General Case

Impaired nitric oxide (NO˙)-cyclic guanosine 3'', 5''-monophosphate (cGMP) signaling has been observed in many cardiovascular disorders, including heart failure and pulmonary arterial hypertension. There are several enzymatic determinants of cGMP levels in this pathway, including soluble guanylyl cyclase (sGC) itself, the NO˙-activated form of sGC, and phosphodiesterase(s) (PDE). Therapies for some of these disorders with PDE inhibitors have been successful at increasing cGMP levels in both cardiac and vascular tissues. However, at the systems level, it is not clear whether perturbation of PDE alone, under oxidative stress, is the best approach for increasing cGMP levels as compared with perturbation of other potential pathway targets, either alone or in combination. Here, we develop a model-based approach to perturbing this pathway, focusing on single reactions, pairs of reactions, or trios of reactions as targets, then monitoring the theoretical effects of these interventions on cGMP levels. Single perturbations of all reaction steps within this pathway demonstrated that three reaction steps, including the oxidation of sGC, NO˙ dissociation from sGC, and cGMP degradation by PDE, exerted a dominant influence on cGMP accumulation relative to other reaction steps. Furthermore, among all possible single, paired, and triple perturbations of this pathway, the combined perturbations of these three reaction steps had the greatest impact on cGMP accumulation. These computational findings were confirmed in cell-based experiments. We conclude that a combined perturbation of the oxidatively-impaired NO˙-cGMP signaling pathway is a better approach to the restoration of cGMP levels as compared with corresponding individual perturbations. This approach may also yield improved therapeutic responses in other complex pharmacologically amenable pathways.     

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.     

Distinguishing intrinsic limit cycles from forced oscillations in ecological time series

Ecological cycles are ubiquitous in nature and have triggered ecologists’ interests for decades. Deciding whether a cyclic ecological variable, such as population density, is part of an intrinsically emerging limit cycle or simply driven by a varying environment is still an unresolved issue, particularly when the only available information is in the form of a recorded time series. We investigate the possibility of discerning intrinsic limit cycles from oscillations forced by a cyclic environment based on a single time series. We argue that such a distinction is possible because of the fundamentally different effects that perturbations have on the focal system in these two cases. Using a set of generic mathematical models, we show that random perturbations leave characteristic signatures on the power spectrum and autocovariance that differ between limit cycles and forced oscillations. We quantify these differences through two summary variables and demonstrate their predictive power using numerical simulations. Our work demonstrates that random perturbations of ecological cycles can give valuable insight into the underlying deterministic dynamics.     

Effect of tubular inhomogeneities on filter properties of thick ascending limb of Henle's loop

We used a simple mathematical model of rat thick ascending limb (TAL) of the loop of Henle to predict the impact of spatially inhomogeneous NaCl permeability, spatially inhomogeneous NaCl active transport, and spatially inhomogeneous tubular radius on luminal NaCl concentration when sustained, sinusoidal perturbations were superimposed on steady-state TAL flow. A mathematical model previously devised by us that used homogeneous TAL transport and fixed TAL radius predicted that such perturbations result in TAL luminal fluid NaCl concentration profiles that are standing waves. That study also predicted that nodes in NaCl concentration occur at the end of the TAL when the tubular fluid transit time equals the period of a periodic perturbation, and that, for non-nodal periods, sinusoidal perturbations generate non-sinusoidal oscillations (and thus a series of harmonics) in NaCl concentration at the TAL end. In the present study we find that the inhomogeneities transform the standing waves and their associated nodes into approximate standing waves and approximate nodes. The impact of inhomogeneous NaCl permeability is small. However, for inhomogeneous active transport or inhomogeneous radius, the oscillations for non-nodal periods tend to be less sinusoidal and more distorted than in the homogeneous case and to thus have stronger harmonics. Both the homogeneous and non-homogeneous cases predict that the TAL, in its transduction of flow oscillations into concentration oscillations, acts as a low-pass filter, but the inhomogeneities result in a less effective filter that has accentuated non-linearities.     

Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?

A constant dilemma in theoretical ecology is knowing whether model predictions corrspond to real phenomena or whether they are artifacts of the modelling framework. The frequent absence of detailed ecological data against which models can be tested gives this issue particular importance. We address this question in the specific case of invasion in a predator-prey system with oscillatory population kinetics, in which both species exhibit local random movement. Given only these two basic qualitative features, we consider whether we can deduce any properties of the behaviour following invasion. To do this we study four different types of mathematical model, which have no formal relationship, but which all reflect our two qualitative ingredients. The models are: reaction-diffusion equations, coupled map lattices, deterministic cellular automata, and integrodifference equations. We present results of numerical simulations of the invasion of prey by predators for each model, and show that although there are certain differences, the main qualitative features of the behaviour behind invasion are the same for all the models. Specifically, there are either irregular spatiotemporal oscillations behind the invasion, or regular spatiotemporal oscillations with the form of a periodic travelling ''wake'', depending on parameter values. The observation of this behaviour in all types of model strongly suggests that it is a direct consequence of our basic qualitative assumptions, and as such is an ecological reality which will always occur behind invasion in actual oscillatory predator-prey systems.     

Controlled Ultrasound-Induced Blood-Brain Barrier Disruption Using Passive Acoustic Emissions Monitoring

The ability of ultrasonically-induced oscillations of circulating microbubbles to permeabilize vascular barriers such as the blood-brain barrier (BBB) holds great promise for noninvasive targeted drug delivery. A major issue has been a lack of control over the procedure to ensure both safe and effective treatment. Here, we evaluated the use of passively-recorded acoustic emissions as a means to achieve this control. An acoustic emissions monitoring system was constructed and integrated into a clinical transcranial MRI-guided focused ultrasound system. Recordings were analyzed using a spectroscopic method that isolates the acoustic emissions caused by the microbubbles during sonication. This analysis characterized and quantified harmonic oscillations that occur when the BBB is disrupted, and broadband emissions that occur when tissue damage occurs. After validating the system''s performance in pilot studies that explored a wide range of exposure levels, the measurements were used to control the ultrasound exposure level during transcranial sonications at 104 volumes over 22 weekly sessions in four macaques. We found that increasing the exposure level until a large harmonic emissions signal was observed was an effective means to ensure BBB disruption without broadband emissions. We had a success rate of 96% in inducing BBB disruption as measured by in contrast-enhanced MRI, and we detected broadband emissions in less than 0.2% of the applied bursts. The magnitude of the harmonic emissions signals was significantly (P<0.001) larger for sonications where BBB disruption was detected, and it correlated with BBB permeabilization as indicated by the magnitude of the MRI signal enhancement after MRI contrast administration (R² = 0.78). Overall, the results indicate that harmonic emissions can be a used to control focused ultrasound-induced BBB disruption. These results are promising for clinical translation of this technology.     

Oscillatory Transpiration of Avena Plants: Perturbation Experiments Provide Evidence for a Stable Point of Singularity

Oscillatory plant water regulation of young Avena plants was studied. The period of the oscillations was around 40 min. Pulse perturbations were given to plants showing oscillations in the transpiration rate. Perturbations consisted in temporary irradiance changes of the leaf or in water potential changes around the root. The effect of the pulse perturbations on the amplitude of the oscillations was recorded. The oscillations could be permanently halted after a perturbation of suitable magnitude given at a suitable phase of The oscillations. A subsequent perturbation could restart the oscillations again. By means of simulations it was shown that a feedback model For the transpiration oscillations could explain the experimental outcome if a non finearity of a special kind was incorporated. The circadian eclosion rhythm of Drosophila pseudoobscura and the petal rhythm of Kalanchoe blossfeldiana show many features in common with the experiments reported. Biophysically the present results indicate that the transpiration oscillations of Avena plants have a stable point of equilibrium or a stable point of singularity.     

Electromagnetic induced kinetic effects on charged substrates in localized enzyme systems.

An analytical expression for the rate efficiency factor of planar localized enzyme systems is derived. The derivation takes into account the isothermal kinetic effect under the externally imposed perturbation of combined electrostatic and high frequency time-varying fields. The contribution of each individual field to the enzyme reaction is examined through the basic mechanism in which charged substrates interact with the specific perturbing field. The interaction mechanisms for the electrostatic and for the time-varying fields are found to be different. This difference regulates the different manners in which enzymatic reaction rates are altered. Enzymatic reactions under electrostatic perturbation can be retarded or enhanced depending on the field polarization. At sufficiently high field intensities the reaction rate may approach zero or approach a maximum value equal to the turnover number of the enzyme. Time-varying field perturbations, on the other hand, always enhance the enzymatic reactions if bunching effects are negligible. At sufficiently high field intensities, the reaction may approach a value equal to that of the free enzyme system. Several typical numerical examples on pure eletrostatic field perturbations, pure time-varying field perturbations, and combined field perturbations are also presented.     

Oscillatory Transpiration and Water Uptake of Avena Plants I. Preliminary Observations

Oscillations in transpiration and water uptake of individual, young oat plants have been studied. The free-running period of these oscillations was about 30 minutes. Conditions were reached under which the oscillations were sustained for about two days. Short perturbations were given to the transpiration oscillations, the perturbations consisting of a short time increase or decrease in the illumination. The phase shifts of the oscillations as well as the amplitude effects caused by these perturbations were investigated. Simultaneous recordings of transpiration and water uptake of a single plant showed that these functions were oscillating in phase. Both oscillations disappeared if the leaf was excised.     

Modeling of Glucose-Induced cAMP Oscillations in Pancreatic β Cells: cAMP Rocks when Metabolism Rolls

Recent advances in imaging technology have revealed oscillations of cyclic adenosine monophosphate (cAMP) in insulin-secreting cells. These oscillations may be in phase with cytosolic calcium oscillations or out of phase. cAMP oscillations have previously been modeled as driven by oscillations in calcium, based on the known dependence of the enzymes that generate cAMP (adenylyl cyclase) and degrade it (phosphodiesterase). However, cAMP oscillations have also been reported to occur in the absence of calcium oscillations. Motivated by similarities between the properties of cAMP and metabolic oscillations in pancreatic β cells, we propose here that in addition to direct control by calcium, cAMP is controlled by metabolism. Specifically, we hypothesize that AMP inhibits adenylyl cyclase. We incorporate this hypothesis into the dual oscillator model for β cells, in which metabolic (glycolytic) oscillations cooperate with modulation of ion channels and metabolism by calcium. We show that the combination of oscillations in AMP and calcium in the dual oscillator model can account for the diverse oscillatory patterns that have been observed, as well as for experimental perturbations of those patterns. Predictions to further test the model are provided.     

Getting there and around: Host range oscillations during colonization of the Canary Islands by the parasitic nematode Spauligodon

Episodes of expansion and isolation in geographic range over space and time, during which parasites have the opportunity to expand their host range, are linked to the development of host–parasite mosaic assemblages and parasite diversification. In this study, we investigated whether island colonization events lead to host range oscillations in a taxon of host‐specific parasitic nematodes of the genus Spauligodon in the Canary Islands. We further investigated whether range oscillations also resulted in shifts in host breadth (i.e., specialization), as expected for parasites on islands. Parasite phylogeny and divergence time estimates were inferred from molecular data with Bayesian methods. Host divergence times were set as calibration priors after a priori evaluation with a global‐fit method of which individual host–parasite associations likely represent cospeciation links. Parasite colonization history was reconstructed, followed by an estimation of oscillation events and specificity level. The results indicate the presence of four Spauligodon clades in the Canary Islands, which originated from at least three different colonization events. We found evidence of host range oscillations to truly novel hosts, which in one case led to higher diversification. Contemporary host–parasite associations show strong host specificity, suggesting that changes in host breadth were limited to the shift period. Lineages with more frequent and wider taxonomic host range oscillations prior to the initial colonization event showed wider range oscillations during colonization and diversification within the archipelago. Our results suggest that a lineage's evolutionary past may be the best indicator of a parasite's potential for future range expansions.