Traveling NADH and proton waves during oscillatory glycolysis in vitro

Noise-induced formation of heterogeneous patterns in the Turing stability zones of diffusion systems

We study a phenomenon of stochastic generation of waveform patterns for reaction-diffusion systems in the Turing stability zone where the homogeneous equilibrium is a single attractor. In this analysis, we use a distributed variant of the Selkov glycolytic model with diffusion and random forcing. It is shown that in the Turing stability zone, random disturbances can induce a diversity of metastable spatial patterns with different waveforms. We carry out the parametric analysis of statistical characteristics of evolution of these patterns, and reveal the dominant patterns in the stochastic flow of mixed spatial structures.

Turing Instabilities and Rotating Spiral Waves in Glycolytic Processes

We study single-frequency oscillations and pattern formation in the glycolytic process modeled by a reduction in the well-known Sel'kov's equations (Sel'kov in Eur J Biochem 4:79, 1968), which describe, in the whole cell, the phosphofructokinase enzyme reaction. By using averaging theory, we establish the existence conditions for limit cycles and their limiting average radius in the kinetic reaction equations. We analytically establish conditions on the model parameters for the appearance of unstable nonlinear modes seeding the formation of two-dimensional patterns in the form of classical spots and stripes. We also establish the existence of a Hopf bifurcation, which characterizes the reaction dynamics, producing glycolytic rotating spiral waves. We numerically establish parameter regions for the existence of these spiral waves and address their linear stability. We show that as the model tends toward a suppression of the relative source rate, the spiral wave solution loses stability. All our findings are validated by full numerical simulations of the model equations. Finally, we discuss in vitro evidence of spatiotemporal activity patterns found in glycolytic experiments, and propose plausible biological implications of our model results.

Synchronisation of glycolytic activity in yeast cells

Glycolysis is the central metabolic pathway of almost every cell and organism. Under appropriate conditions, glycolytic oscillations may occur in individual cells as well as in entire cell populations or tissues. In many biological systems, glycolytic oscillations drive coherent oscillations of other metabolites, for instance in cardiomyocytes near anorexia, or in pancreas where they lead to a pulsatile release of insulin. Oscillations at the population or tissue level require the cells to synchronize their metabolism. We review the progress achieved in studying a model organism for glycolytic oscillations, namely yeast. Oscillations may occur on the level of individual cells as well as on the level of the cell population. In yeast, the cell-to-cell interaction is realized by diffusion-mediated intercellular communication via a messenger molecule. The present mini-review focuses on the synchronisation of glycolytic oscillations in yeast. Synchronisation is a quorum-sensing phenomenon because the collective oscillatory behaviour of a yeast cell population ceases when the cell density falls below a threshold. We review the question, under which conditions individual cells in a sparse population continue or cease to oscillate. Furthermore, we provide an overview of the pathway leading to the onset of synchronized oscillations. We also address the effects of spatial inhomogeneities (e.g., the formation of spatial clusters) on the collective dynamics, and also review the emergence of travelling waves of glycolytic activity. Finally, we briefly review the approaches used in numerical modelling of synchronized cell populations.

Nonequilibrium thermodynamics of glycolytic traveling wave: Benjamin-Feir instability

Evolution of the nonequilibrium thermodynamic entities corresponding to dynamics of the Hopf instabilities and traveling waves at a nonequilibrium steady state of a spatially extended glycolysis model is assessed here by implementing an analytically tractable scheme incorporating a complex Ginzburg-Landau equation (CGLE). In the presence of self and cross diffusion, a more general amplitude equation exploiting the multiscale Krylov-Bogoliubov averaging method serves as an essential tool to reveal the various dynamical instability criteria, especially Benjamin-Feir (BF) instability, to estimate the corresponding nonlinear dispersion relation of the traveling wave pattern. The critical control parameter, wave-number selection criteria, and magnitude of the complex amplitude for traveling waves are modified by self- and cross-diffusion coefficients within the oscillatory regime, and their variabilities are exhibited against the amplitude equation. Unlike the traveling waves, a low-amplitude broad region appears for the Hopf instability in the concentration dynamics as the system phase passes through minima during its variation with the control parameter. The total entropy production rate of the uniform Hopf oscillation and glycolysis wave not only qualitatively reflects the global dynamics of concentrations of intermediate species but almost quantitatively. Despite the crucial role of diffusion in generating and shaping the traveling waves, the diffusive part of the entropy production rate has a negligible contribution to the system's total entropy production rate. The Hopf instability shows a more complex and colossal change in the energy profile of the open nonlinear system than in the traveling waves. A detailed analysis of BF instability shows a contrary nature of the semigrand Gibbs free energy with discrete and continuous wave numbers for the traveling wave. We hope the Hopf and traveling wave pattern around the BF instability in terms of energetics and dissipation will open up new applications of such dynamical phenomena.

Partial synchronisation of glycolytic oscillations in yeast cell populations

The transition between synchronized and asynchronous behaviour of immobilized yeast cells of the strain Saccharomyces carlsbergensis was investigated by monitoring the autofluorescence of the coenzyme NADH. In populations of intermediate cell densities the individual cells remained oscillatory, whereas on the level of the cell population both a partially synchronized and an asynchronous state were accessible for experimental studies. In the partially synchronized state, the mean oscillatory frequency was larger than that of the cells in the asynchronous state. This suggests that synchronisation occurred due to entrainment by the cells that oscillated more rapidly. This is typical for synchronisation due to phase advancement. Furthermore, the synchronisation of the frequency of the glycolytic oscillations preceded the synchronisation of their phases. However, the cells did not synchronize completely, as the distribution of the oscillatory frequencies only narrowed but did not collapse to a unique frequency. Cells belonging to spatially denser clusters showed a slightly enhanced local synchronisation during the episode of partial synchronisation. Neither the clusters nor a transition from partially synchronized glycolytic oscillations to travelling glycolytic waves did substantially affect the degree of partial synchronisation. Chimera states, i.e., the coexistence of a synchronized and an asynchronous part of the population, could not be found.

Spontaneous center formation in Dictyostelium discoideum

Dictyostelium discoideum (D.d.) is a widely studied amoeba due to its capabilities of development, survival, and self-organization. During aggregation it produces and relays a chemical signal (cAMP) which shows spirals and target centers. Nevertheless, the natural emergence of these structures is still not well understood. We present a mechanism for creation of centers and target waves of cAMP in D.d. by adding cell inhomogeneity to a well known reaction-diffusion model of cAMP waves and we characterize its properties. We show how stable activity centers appear spontaneously in areas of higher cell density with the oscillation frequency of these centers depending on their density. The cAMP waves have the characteristic dispersion relation of trigger waves and a velocity which increases with cell density. Chemotactically competent cells react to these waves and create aggregation streams even with very simple movement rules. Finally we argue in favor of the existence of bounded phosphodiesterase to maintain the wave properties once small cell clusters appear.

Influence of cadmium on oxidative stress and NADH oscillations in mussel mitochondria

Biological organisms evolved to take advantage of recurring environmental factors which enabled them to assimilate and process metabolic energy for survival. Mitochondria display non-linear oscillations in NADH levels (i.e. wave behavior) that result from the balance between NADH production (aerobic glycolysis) and oxidation for ATP synthesis. The purpose of this study was to examine the effects of cadmium (Cd) on mitochondrial NADH oscillations in quagga mussels Dreissena bugensis exposed to 50 and 100 μg/L CdCl2 for 7 days at 15 °C. Metallothionein (MT) levels, thioredoxin reductase (TrxR) activity and NADH oxidation rate were also determined, as were oscillations in NADH and the formation of dissipative structures (turbidity), in isolated mitochondria suspensions. The results show that exposure to Cd readily induced MT levels at both concentrations tested and that TrxR and NADH oxidase activity was induced at 100 μg/L Cd only. In control mussels, NADH levels oscillated in mitochondria suspensions with a natural period of 2 to 2.5 min for up to 40 min. Exposure to Cd increased the complexity of the frequency profile of NADH oscillations and reduced the amplitudes of the natural signal with a period of 2 to 2.5 min. The formation of dissipative structures decreased in response to a Cd concentration of 100 μg/L but increased at a level of 50 μg/L. The amplitudes at the natural frequency were significantly correlated with NADH oxidase activity (r = -0.91) and with the formation of dissipative structures (r = -0.59). We conclude that Cd could alter the natural frequency in oscillations of NADH in mitochondria, thereby contributing to an increase in NADH oxidation rate and disruption of the spatial organization of mitochondria in suspension. In conclusion, changes in the wave behavior of NADH in mitochondria are proposed as a novel biomarker of toxicity in aquatic organisms.

Collision of two action potentials in a single excitable cell

BACKGROUND

It is a common incident in nature, that two waves or pulses run into each other head-on. The outcome of such an event is of special interest, because it allows conclusions about the underlying physical nature of the pulses. The present experimental study dealt with the head-on meeting of two action potentials (AP) in a single excitable plant cell (Chara braunii internode).

METHODS

The membrane potential was monitored with multiple sensors along a single excitable cell. In control experiments, an AP was excited electrically at either end of the cell cylinder. Subsequently, stimuli were applied simultaneously at both ends of the cell in order to generate two APs that met each other head-on.

RESULTS

When two action potentials propagated into each other, the pulses did not penetrate but annihilated (N=26 experiments in n=10 cells).

CONCLUSIONS

APs in excitable plant cells did not penetrate upon meeting head-on. In the classical electrical model, this behavior is specifically attributed to relaxation of ion channel proteins. From an acoustic point of view, annihilation can be viewed as a result of nonlinear material properties (e.g. a phase change).

GENERAL SIGNIFICANCE

The present results suggest that APs in excitable animal and plant cells belong to a similar class of nonlinear phenomena. Intriguingly, other excitation waves in biology (intracellular waves, cortical spreading depression, etc.) also annihilate upon collision and are thus expected to follow the same underlying principles as the observed action potentials.

Modular organization of cardiac energy metabolism: energy conversion, transfer and feedback regulation

To meet high cellular demands, the energy metabolism of cardiac muscles is organized by precise and coordinated functioning of intracellular energetic units (ICEUs). ICEUs represent structural and functional modules integrating multiple fluxes at sites of ATP generation in mitochondria and ATP utilization by myofibrillar, sarcoplasmic reticulum and sarcolemma ion-pump ATPases. The role of ICEUs is to enhance the efficiency of vectorial intracellular energy transfer and fine tuning of oxidative ATP synthesis maintaining stable metabolite levels to adjust to intracellular energy needs through the dynamic system of compartmentalized phosphoryl transfer networks. One of the key elements in regulation of energy flux distribution and feedback communication is the selective permeability of mitochondrial outer membrane (MOM) which represents a bottleneck in adenine nucleotide and other energy metabolite transfer and microcompartmentalization. Based on the experimental and theoretical (mathematical modelling) arguments, we describe regulation of mitochondrial ATP synthesis within ICEUs allowing heart workload to be linearly correlated with oxygen consumption ensuring conditions of metabolic stability, signal communication and synchronization. Particular attention was paid to the structure-function relationship in the development of ICEU, and the role of mitochondria interaction with cytoskeletal proteins, like tubulin, in the regulation of MOM permeability in response to energy metabolic signals providing regulation of mitochondrial respiration. Emphasis was given to the importance of creatine metabolism for the cardiac energy homoeostasis.

Reversal of spiral waves in an oscillatory system caused by an inhomogeneity

Spatial heterogeneities are commonly found in realistic systems and play significant roles in dynamics of spiral waves. We here demonstrate a novel phenomenon that a localized inhomogeneity put around the spiral core could lead to the reversal of spiral waves in an oscillatory system, e.g., the complex Ginzburg-Landau equation. With the amplitude-phase representation, we analyze underling mechanism and conditions of the wave reversal in detail, which is found to agree with the numerical evidence.

pH wave-front propagation in the urea-urease reaction

The urease-catalyzed hydrolysis of urea displays feedback that results in a switch from acid (pH ~3) to base (pH ~9) after a controllable period of time (from 10 to >5000 s). Here we show that the spatially distributed reaction can support pH wave fronts propagating with a speed of the order of 0.1-1 mm min(-1). The experimental results were reproduced qualitatively in reaction-diffusion simulations including a Michaelis-Menten expression for the urease reaction with a bell-shaped rate-pH dependence. However, this model fails to predict that at lower enzyme concentrations, the unstirred reaction does not always support fronts when the well-stirred reaction still rapidly switches to high pH.

Metabolic synchronization by traveling waves in yeast cell layers

The coordination of cellular behavior is a prerequisite of functionality of tissues and organs. Generally, this coordination occurs by signal transduction, neuronal control, or exchange of messenger molecules. The extent to which metabolic processes are involved in intercellular communication is less understood. Here, we address this question in layers of resting yeast cells and report for the first time the observation of intercellular glycolytic waves. We use a combined experimental and theoretical approach and explain the radial velocity of the waves to arise from the substrate gradient due to local substrate addition. Our results show that metabolic processes introduce an additional level of local intercellular coordination.

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system

BACKGROUND

A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze.

RESULTS

Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC), can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations), but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations.

CONCLUSIONS

This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward rotating spiral waves in glycolysis can be explained in terms of an MWC, but not with a Hill mechanism for the activation of the allosteric enzyme phosphofructokinase. Our results also highlight the importance of enzyme oligomerization for a possible experimental generation of Turing patterns in biological systems.

Inward rotating spiral waves in glycolysis

We report on the first observation of inward rotating spiral waves (antispirals) in a biochemical reaction-diffusion system. Experiments are performed with extracts from yeast cells in an open spatial reactor. By increasing the protein concentration of the extract we observe a transition from outward to inward propagating waves of glycolytic activity. Numerical simulations with an allosteric model for the phosphofructokinase can reproduce these inward propagating waves over a wide range of parameters if the octameric structure of yeast phosphofructokinase is taken into account.

Spatiotemporal dynamics of glycolytic waves provides new insights into the interactions between immobilized yeast cells and gels

The immobilization of cells or enzymes is a promising tool for the development of biosensors, yet the interactions between the fixative materials and the cells are not fully understood, especially with respect to their impact on both cell metabolism and cell-to-cell signaling. We show that the spatiotemporal dynamics of waves of metabolic synchronization of yeast cells provides a new criterion to distinguish the effect of different gels on the cellular metabolism, which otherwise could not be detected. Cells from the yeast Saccharomyces carlsbergensis were immobilized into agarose gel, silica gel (TMOS), or a mixture of TMOS and alginate. We compared these immobilized cells with respect to their ability to generate temporal, intracellular oscillations in glycolysis as well as propagating, extracellular synchronization waves. While the temporal dynamics, as measured by the period and the number of oscillatory cycles, was similar for all three immobilized cell populations, significant differences have been observed with respect to the shape of the waves, wave propagation direction and velocity in the three gel matrices used.

Adenylate kinase and AMP signaling networks: metabolic monitoring, signal communication and body energy sensing

Adenylate kinase and downstream AMP signaling is an integrated metabolic monitoring system which reads the cellular energy state in order to tune and report signals to metabolic sensors. A network of adenylate kinase isoforms (AK1-AK7) are distributed throughout intracellular compartments, interstitial space and body fluids to regulate energetic and metabolic signaling circuits, securing efficient cell energy economy, signal communication and stress response. The dynamics of adenylate kinase-catalyzed phosphotransfer regulates multiple intracellular and extracellular energy-dependent and nucleotide signaling processes, including excitation-contraction coupling, hormone secretion, cell and ciliary motility, nuclear transport, energetics of cell cycle, DNA synthesis and repair, and developmental programming. Metabolomic analyses indicate that cellular, interstitial and blood AMP levels are potential metabolic signals associated with vital functions including body energy sensing, sleep, hibernation and food intake. Either low or excess AMP signaling has been linked to human disease such as diabetes, obesity and hypertrophic cardiomyopathy. Recent studies indicate that derangements in adenylate kinase-mediated energetic signaling due to mutations in AK1, AK2 or AK7 isoforms are associated with hemolytic anemia, reticular dysgenesis and ciliary dyskinesia. Moreover, hormonal, food and antidiabetic drug actions are frequently coupled to alterations of cellular AMP levels and associated signaling. Thus, by monitoring energy state and generating and distributing AMP metabolic signals adenylate kinase represents a unique hub within the cellular homeostatic network.

Spatial desynchronization of glycolytic waves as revealed by Karhunen-Loeve analysis

The dynamics of glycolytic waves in a yeast extract have been investigated in an open spatial reactor. At low protein contents in the extract, we find a transition from inwardly moving target patterns at the beginning of the experiment to outwardly moving spiral- or circular-shaped waves at later stages. These two phases are separated by a transition phase of more complex spatiotemporal dynamics. We have analyzed the pattern dynamics in these three intervals at different spatial scales by means of a Karhunen-Loeve (KL) decomposition. During the initial phase of the experiment, the observed patterns are sufficiently described by the two dominant KL modes independently of the spatial scale. However, during the last stage of the experiment, at least 6 KL modes are needed to account for the observed patterns at spatial scales larger than 3 mm, while for smaller scales, 2 KL modes are still sufficient. This indicates that in the course of the experiment, the local glycolytic oscillators become desynchronized at spatial scales larger than 3 mm. Possible reasons for the desynchronization of the glycolytic waves are discussed.

Turing patterns inside cells

Concentration gradients inside cells are involved in key processes such as cell division and morphogenesis. Here we show that a model of the enzymatic step catalized by phosphofructokinase (PFK), a step which is responsible for the appearance of homogeneous oscillations in the glycolytic pathway, displays Turing patterns with an intrinsic length-scale that is smaller than a typical cell size. All the parameter values are fully consistent with classic experiments on glycolytic oscillations and equal diffusion coefficients are assumed for ATP and ADP. We identify the enzyme concentration and the glycolytic flux as the possible regulators of the pattern. To the best of our knowledge, this is the first closed example of Turing pattern formation in a model of a vital step of the cell metabolism, with a built-in mechanism for changing the diffusion length of the reactants, and with parameter values that are compatible with experiments. Turing patterns inside cells could provide a check-point that combines mechanical and biochemical information to trigger events during the cell division process.

Fronts and pulses in an enzymatic reaction catalyzed by glucose oxidase

Waves and patterns in living systems are often driven by biochemical reactions with enzymes as catalysts and regulators. We present a reaction-diffusion system catalyzed by the enzyme glucose oxidase that exhibits traveling wave patterns in a spatially extended medium. Fronts and pulses propagate as a result of the coupling between the enzyme-catalyzed autocatalytic production and diffusion of hydrogen ions. A mathematical model qualitatively explains the experimental observations.

Trypanosomes and mammalian sperm: one of a kind?

Flagellar-mediated motility is an indispensable function for cell types as evolutionarily distant as mammalian sperm and kinetoplastid parasites, a large group of flagellated protozoa that includes several important human pathogens. Despite the obvious importance of flagellar motility, little is known about the signalling processes that direct the frequency and wave shape of the flagellar beat, or those that provide the motile cell with the necessary environmental cues that enable it to aim its movement. Similarly, the energetics of the flagellar beat and the problem of a sufficient ATP supply along the entire length of the beating flagellum remain to be explored. Recent proteome projects studying the flagella of mammalian sperm and kinetoplastid parasites have provided important information and have indicated a surprising degree of similarities between the flagella of these two cell types.

Spiral turbulence developed through the formation of superimposed target waves in an oscillatory reaction-diffusion medium

An approach leading to the development of spiral turbulence is reported here in an oscillatory reaction-diffusion medium, which is through the spontaneous formation of targetlike waves near the core of a spiral wave. The newly formed target wave emerges with its own characteristic frequency and propagates on top of the original spiral wave, which eventually leads to the breakup of the spiral at a location far from the spiral center. The radius of the surviving spiral segment decreases rapidly with the bifurcation control parameter. Calculation of power spectra suggests that the meandering of the spiral tip is responsible for the onset of the superimposed target and the phase desynchronization of the superimposed target waves.

Electrical stimulation of the energy metabolism in yeast cells using a planar Ti-Au-electrode interface

We report on the influence of dielectric pulse injection on the energy metabolism of yeast cells with a planar interdigitated electrode interface. The energy metabolism was measured via NADH fluorescence. The application of dielectric pulses results in a distinct decrease of the fluorescence, indicating a response of the energy metabolism of the yeast cells. The reduction of the NADH signal significantly depends on the pulse parameters, i.e., amplitude and width. Furthermore, the interface is used to detect electrical changes in the cell-electrolyte system, arising from glucose-induced oscillations in yeast cells and yeast extract, by dielectric spectroscopy at 10 kHz. These dielectric investigations revealed a beta(1)-dispersion for the system electrolyte/yeast cells as well as for the system electrolyte/yeast extract. In agreement with control measurements we obtained a glycolytic period of 45 s for yeast cells and of 11 min for yeast extract.

Simple and complex spatiotemporal structures in a glycolytic allosteric enzyme model

Pattern formation in glycolysis is studied with a classical reaction-diffusion allosteric enzyme model. It is found that, similar to recent experimental reports in the yeast extracts, a small magnitude local perturbation can induce transient target waves in a two dimensional oscillatory medium. An above threshold stimulation generates target waves which eventually evolve into spatiotemporal chaos upon collisions with the boundary or other wave activities. Detailed simulation studies show that the studied simple glycolytic reaction-diffusion model can support three types of spatiotemporal behaviors which are independent of the boundary conditions: (1) a spatially uniform stable steady state, (2) periodic global oscillations and (3) spatiotemporal chaos.

Nucleation of spiral wave patterns at surface defects

The nucleation of spiral waves at a surface defect during catalytic CO oxidation on Pt(110) has been studied with a low energy electron microscope system. It is found that reaction fronts originate from a boundary layer between the defect and the surrounding Pt(110) area. The findings are corroborated by numerical simulations within a realistic reaction-diffusion model of the surface reaction.

Analysis of the oscillatory kinetics of glycolytic intermediates in a yeast extract by FT-IR spectroscopy

In the present work we demonstrate that FT-IR spectroscopy is a powerful tool for the time resolved and noninvasive measurement of multi-substrate/product interactions in complex metabolic networks as exemplified by the oscillating glycolysis in yeast extract. We found that many of the glycolytic intermediates can be identified with FT-IR spectroscopy. For this, we have constructed a spectral library of most of the glycolytic intermediates and obtained the kinetics of single components in spectra from glycolysing yeast extract by the use of mathematical fitting procedures. The results are in good agreement with the known phase relationships of oscillatory glycolysis. They provide the basis for future application of this method to investigate the energy metabolism of living cells.

Glycolytic oscillations and waves in an open spatial reactor: Impact of feedback regulation of phosphofructokinase

An open spatial reactor has been designed for the investigation of spatio-temporal dynamics of glycolysis. The reactor consists of a diffusive layer made of gel-fixed yeast extract which is in contact with a continuously stirred reservoir to supply this layer with substrates. The coupling between reaction and diffusion in the gel layer enables the formation of spatio-temporal patterns. Temporal oscillations of glycolysis are simply induced by feeding the yeast extract with sugar. Under properly chosen conditions, these oscillations sustain for more than 12 h. A necessary prerequisite for the generation of oscillations is that the ATP concentration in the feeding solution must be high enough to allow for negative feedback of phosphofructokinase. Otherwise, the interplay between ATP-consuming and ATP-producing reactions leads to an unfavorable low ATP/AMP ratio. The generation of travelling NADH-waves is observed in the diffusive layer, when feeding the yeast extract with substrates. Break-up of circular-shaped waves is repeatedly observed, resulting in the formation of rotating NADH-spirals.

Dynamics of excitation pulses with attractive interaction: kinematic analysis and chemical wave experiments

We present a theoretical analysis of stacking and destacking wave trains in excitable reaction-diffusion systems with anomalous velocity-wavelength dependence. For linearized dispersion relations, kinematic analysis yields an analytical function that rigorously describes front trajectories. The corresponding accelerations have exactly one extremum that slowly decays with increasing pulse number. For subsequent pulses these maxima occur with a lag time equal to the inverse slope of the linearized dispersion curve. These findings are reproduced in experiments with chemical waves in the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction but should be also applicable to step bunching on crystal surfaces and certain traffic phenomena.

Control of glycolytic oscillations by temperature

External control of oscillatory glycolysis in yeast extract has been performed by application of either homogeneous temperature oscillations or stationary, spatial temperature gradients. Entrainment of the glycolytic oscillations by the 1/2- and 1/3-harmonic, as well as the fundamental input frequency, could be observed. From the phase response curve to a single temperature pulse, a distinct sensitivity of NADH-oxidizing processes, compared with NAD-reducing processes, is visible. Determination of glycolytic intermediates shows that the feedback-regulated phosphofructokinase as well as the glyceraldehyde-3-phosphate dehydrogenase are the most temperature-sensitive steps of glycolysis. We also find strong concentration changes in ATP and AMP at varying temperatures and, accordingly, in the energy charge. Construction of a feedback loop for spatial control of temperature by means of a Peltier element allowed us to apply a temperature gradient to the yeast extract. With this setup it is possible to initiate traveling waves and to control the wave velocity.

Phosphotransfer networks and cellular energetics

Precise coupling of spatially separated intracellular ATP-producing and ATP-consuming processes is fundamental to the bioenergetics of living organisms, ensuring a fail-safe operation of the energetic system over a broad range of cellular functional activities. Here, we provide an overview of the role of spatially arranged enzymatic networks, catalyzed by creatine kinase, adenylate kinase, carbonic anhydrase and glycolytic enzymes, in efficient high-energy phosphoryl transfer and signal communication in the cell. Studies of transgenic creatine kinase and adenylate kinase deficient mice, along with pharmacological targeting of individual enzymes, have revealed the importance of near-equilibrium reactions in the dissipation of metabolite gradients and communication of energetic signals to distinct intracellular compartments, including the cell nucleus and membrane metabolic sensors. Enzymatic capacities, isoform distribution and the dynamics of net phosphoryl flux through the integrated phosphotransfer systems tightly correlate with cellular functions, indicating a critical role of such networks in efficient energy transfer and distribution, thereby securing the cellular economy and energetic homeostasis under stress.

Apparent role of traveling metabolic waves in oxidant release by living neutrophils

Cell metabolism self-organizes into two types of dissipative structures: chemical oscillations and traveling metabolic waves. In the present study we test the hypothesis that traveling NAD(P)H waves within neutrophils are associated spatially and temporally with the release of reactive oxygen metabolites (ROMs). Using high-speed optical microscopy and taking advantage of the autofluorescence of NAD(P)H, we have observed the propagation of NAD(P)H waves within cells. When NAD(P)H waves reach the lamellipodium of morphologically polarized neutrophils, a diffusing plume of superoxide is released as evidenced by the conversion of hydroethidine in the extracellular environment to ethidium bromide. Parallel results were obtained by using high-speed emission microspectrophotometry. These experiments indicate that the spatial and temporal properties of NAD(P)H waves are transformed into ROM pulses in the extracellular environment. Propagating NAD(P)H waves allow neutrophils to specifically deliver substrate to the lamellipodium at high concentrations, thus facilitating the local and periodic release of ROMs in the direction of cell movement and/or a target.

Periodic and Bursting pH Oscillations in an Enzyme Model Reaction

Oscillatory behaviour in the pH value was observed during the oxidation of sulfite by hydrogen peroxide mediated by hemin in a continuous-flow stirred tank reactor. The dynamics of this reaction was studied for a variety of flow rates of the reactants. As the flow rates increase, the oscillations evolve from relaxation oscillations to more complex shapes, displaying, among others, bursting behaviour. A reaction mechanism is proposed that involves the autocatalytic oxidation of HSO

Adenylate kinase phosphotransfer communicates cellular energetic signals to ATP-sensitive potassium channels

Transduction of energetic signals into membrane electrical events governs vital cellular functions, ranging from hormone secretion and cytoprotection to appetite control and hair growth. Central to the regulation of such diverse cellular processes are the metabolism sensing ATP-sensitive K+ (K(ATP)) channels. However, the mechanism that communicates metabolic signals and integrates cellular energetics with K(ATP) channel-dependent membrane excitability remains elusive. Here, we identify that the response of K(ATP) channels to metabolic challenge is regulated by adenylate kinase phosphotransfer. Adenylate kinase associates with the K(ATP) channel complex, anchoring cellular phosphotransfer networks and facilitating delivery of mitochondrial signals to the membrane environment. Deletion of the adenylate kinase gene compromised nucleotide exchange at the channel site and impeded communication between mitochondria and K(ATP) channels, rendering cellular metabolic sensing defective. Assigning a signal processing role to adenylate kinase identifies a phosphorelay mechanism essential for efficient coupling of cellular energetics with K(ATP) channels and associated functions.

Dissipative metabolic patterns respond during neutrophil transmembrane signaling

Self-organization is a common theme in biology. One mechanism of self-organization is the creation of chemical patterns by the diffusion of chemical reactants and their nonlinear interactions. We have recently observed sustained unidirectional traveling chemical redox [NAD(P)H - NAD(P)(+)] waves within living polarized neutrophils. The present study shows that an intracellular metabolic wave responds to formyl peptide receptor agonists, but not antagonists, by splitting into two waves traveling in opposite directions along a cell's long axis. Similar effects were noted with other neutrophil-activating substances. Moreover, when cells were exposed to an N-formyl-methionyl-leucyl-phenylalanine (FMLP) gradient whose source was perpendicular to the cell's long axis, cell metabolism was locally perturbed with reorientation of the pattern in a direction perpendicular to the initial cellular axis. Thus, extracellular activating signals and the signals' spatial cues are translated into distinct intracellular dissipative structures.

Image processing techniques applied to excitation waves in the chicken retina

Image processing techniques are described in detail that are used to gain information about the dynamics of wave propagation in excitable media. We focus on a phenomenon called spreading depression (SD) observed in the chicken retina, but the techniques described here concern a large variety of excitable systems. Despite the impressive progress both in SD research of the past 50 years and, during nearly the same period, in the theory of self-organization of wave patterns, there is still little mutual overlap. However, the increasing demands for understanding complex systems, like neuronal tissue, require such theoretical concepts. Arguments are given why the chicken retina is a nearly perfect experimental system for assessing and further developing these concepts.

Imaging sustained dissipative patterns in the metabolism of individual living cells

Theoretical studies have predicted spatiotemporal organization of cell metabolism. Using a rapidly gated CCD camera, we demonstrate for the first time sustained traveling waves of NAD(P)H autofluorescence and protons in individual morphologically polarized living cells. Chemical concentration fronts moved in the direction of cell orientation, thus correlating dissipative structures with cell shape.

Sarcoplasmic reticulum vesicles embedded in agarose gel exhibit propagating calcium waves

In different cell types, activation of signal transduction pathways leads to the generation of calcium oscillations and/or waves. Due to this important impact for cellular function, calcium waves are the subject of intensive investigations. To study interactions of cell organelles with no influence of the cell membrane, sarcoplasmic reticulum (SR) vesicles and well-coupled mitochondria were reconstituted. For the first time, we demonstrate the generation and propagation of calcium waves in a suspension of sarcoplasmic reticulum vesicles, embedded in an agarose gel. The propagation dynamics resemble those of calcium waves in living cells. Moreover, the addition of well-coupled mitochondria leads to more pronounced and significantly faster propagating waves, demonstrating the importance of the mitochondrial Ca(2+) transport. The experimental and simulation results indicate the resemblance of the in vitro system to an excitable medium.

Traveling waves in yeast extract and in cultures of Dictyostelium discoideum

Biological self-organization was investigated in a biochemical and a cellular system: yeast extract and cultures of the slime mold Dictyostelium discoideum. In both systems traveling reaction-diffusion waves occur in response to oscillatory reactions. Glycolytic degradation of sugar in a yeast extract leads to the spontaneous formation of NADH and proton waves. Manipulation of the adenine nucleotide pool by addition of purified plasma membrane ATPase favors the formation of both reaction-diffusion waves and phase waves. The results indicate that the energy charge has an important impact for the dynamics of glycolytic patterns. When affecting the lower part of glycolysis by pyruvate addition the frequency of wave generation was increased with concomitant formation of rotating NADH and proton spirals. During morphogenesis of the cellular system Dictyostelium discoideum, circular and spiral shaped aggregation patterns of motile amoeboid cells form in response to traveling cAMP waves. Velocity analysis of the cell movements reveals that the cAMP waves guide the cells towards the site of wave initiation along optimized trajectories. The minimization of aggregation paths is based on a mechanism exploiting general properties of excitation waves. The resulting aggregation territories are reminiscent of Voronoi diagrams.

Subcellular metabolic transients and mitochondrial redox waves in heart cells

Precise matching of energy supply with demand requires delicately balanced control of the enzymes involved in substrate metabolism. In response to a change in substrate supply, the nonlinear properties of metabolic control may induce complex dynamic behavior. Using confocal imaging of flavoprotein redox potential and mitochondrial membrane potential, we show that substrate deprivation leads to subcellular heterogeneity of mitochondrial energization in intact cells. The complex spatiotemporal patterns of redox and matrix potential included local metabolic transients, cell-wide coordinated redox transitions, and propagated metabolic waves both within and between coupled cells. Loss of metabolic synchrony during mild metabolic stress reveals that intra- and intercellular control of mitochondrial function involves diffusible cytoplasmic messengers.

Velocity-curvature relationship of colliding spherical calcium waves in rat cardiac myocytes

Colliding spherical calcium waves in enzymatically isolated rat cardiac myocytes develop new wavefronts propagating perpendicular to the original direction. When investigated by confocal laser scanning microscopy (CLSM), using the fluorescent Ca2+ indicator fluo-3 AM, "cusp"-like structures become visible that are favorably approximated by double parabolae. The time-dependent position of the vertices is used to determine propagation velocity and negative curvature of the wavefront in the region of collision. It is evident that negatively curved waves propagate faster than positively curved, single waves. Considering two perfectly equal expanding circular waves, we demonstrated that the collision of calcium waves is due to an autocatalytic process (calcium-induced calcium release), and not to a simple phenomenon of interference. Following the spatiotemporal organization in simpler chemical systems maintained under conditions far from the thermodynamic equilibrium (Belousov-Zhabotinskii reaction), the dependence of the normal velocity on the curvature of the spreading wavefront is given by a linear relation. The so-called velocity-curvature relationship makes clear that the velocity is enhanced by curvature toward the direction of forward propagation and decreased by curvature away from the direction of forward propagation (with an influence of the diffusion coefficient). Experimentally obtained velocity data of both negatively and positively curved calcium waves were approximated by orthogonal weighted regression. The negative slope of the straight line resulted in an effective diffusion coefficient of 1.2 x 10(-4) mm2/s. From the so-called critical radius, which must be exceeded to initiate a traveling calcium wave, a critical volume (with enhanced [Ca2+]i) of approximately 12 microm3 was calculated. This is almost identical to the volume that is occupied by a single calcium spark.