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.

Complexity of a peroxidase-oxidase reaction model.

The peroxidase-oxidase oscillating reaction was the first (bio)chemical reaction to show chaotic behaviour. The reaction is rich in bifurcation scenarios, from period-doubling to peak-adding mixed mode oscillations. Here, we study a state-of-the-art model of the peroxidase-oxidase reaction. Using the model, we report systematic numerical experiments exploring the impact of changing the enzyme concentration on the dynamics of the reaction. Specifically, we report high-resolution phase diagrams predicting and describing how the reaction unfolds over a quite extended range of enzyme concentrations. Surprisingly, such diagrams reveal that the enzyme concentration has a huge impact on the reaction evolution. The highly intricate dynamical behaviours predicted here are difficult to establish theoretically due to the total absence of an adequate framework to solve nonlinearly coupled differential equations. But such behaviours may be validated experimentally.

Patterns of cell thickness oscillations during directional migration of Physarum polycephalum.

The functional relationship between the velocity of cell locomotion and intracellular spatial patterns of thickness oscillations in the acellular slime mould Physarum polycephalum was studied. The freely migrating plasmodial cells of 300-800 µm length were tadpole-shaped and displayed thickness oscillations along their longitudinal (body) axis. Two distinct patterns of intracellular thickness oscillations were observed in dependence on the locomotive velocity. The first mode consisted of a single travelling wave that propagated from the rear to the front of the cell. This pattern occurred when the plasmodium migrated slowly. The second mode was a multinodal standing wave that was found during events of fast propagation. Transitions between these two types of cell thickness oscillation patterns took place in narrow propagation velocity intervals. We discuss the possible mechanism leading to these patterns, which are conjectured to modulate both the intracellular pressure and the velocity of free locomotion of the cell.

Convective dynamics of traveling autocatalytic fronts in a modulated gravity field.

When traveling in thin solution layers, autocatalytic chemical fronts may be deformed and accelerated by convective currents that develop because of density and surface tension gradients related to concentration and thermal gradients across the front. On earth, both buoyancy and Marangoni related flows can act in solution layers open to the air while only buoyancy effects operate in covered liquid layers. The respective effects of density and surface tension induced convective motions are analysed here by studying experimentally the propagation of autocatalytic fronts in uncovered and covered liquid layers during parabolic flights in which the gravity field is modulated periodically. We find that the velocity and deformation of the front are increased during hyper-gravity phases and reduced in the micro-gravity phase. The experimental results compare well with numerical simulations of the evolution of the concentration of the autocatalytic product coupled to the flow field dynamics described by Navier-Stokes equations.

Unraveling dispersion and buoyancy dynamics around radial A + B → C reaction fronts: microgravity experiments and numerical simulations.

Radial Reaction-Diffusion-Advection (RDA) fronts for A + B → C reactions find wide applications in many natural and technological processes. In liquid solutions, their dynamics can be perturbed by buoyancy-driven convection due to concentration gradients across the front. In this context, we conducted microgravity experiments aboard a sounding rocket, in order to disentangle dispersion and buoyancy effects in such fronts. We studied experimentally the dynamics due to the radial injection of A in B at a constant flow rate, in absence of gravity. We compared the obtained results with numerical simulations using either radial one- (1D) or two-dimensional (2D) models. We showed that gravitational acceleration significantly distorts the RDA dynamics on ground, even if the vertical dimension of the reactor and density gradients are small. We further quantified the importance of such buoyant phenomena. Finally, we showed that 1D numerical models with radial symmetry fail to predict the dynamics of RDA fronts in thicker geometries, while 2D radial models are necessary to accurately describe RDA dynamics where Taylor-Aris dispersion is significant.

Functional organization of the vascular network of Physarum polycephalum.

The plasmodium of the slime mould Physarum polycephalum forms a transportation network of veins, in which protoplasm is transported due to peristaltic pumping. This network forms a planar, weighted, undirected graph that, for the first time, can be extracted automatically from photographs or movies. Thus, data from real transportation networks have now become available for the investigation of network properties. We determine the local drag of the vein segments and use these data to calculate the transport efficiency. We unravel which veins form the backbone of the transportation network by using a centrality measure from graph theory. The principal vein segments lie on relatively ample cycles of veins, and the most important segments are those that belong simultaneously to two of these principal cycles. Each principal cycle contains a series of smaller cycles of veins of lower transport efficiency, thus reflecting the hierarchical and self-similar structure of the transportation network. Finally, we calculate accessibility maps that show how easily different nodes of the network may be reached from a given reference node.

Twists of opposite handedness on a scroll wave.

The dynamic interaction of scroll waves in the Belousov-Zhabotinsky reaction with a vertically orientated gradient of excitability is studied by optical tomography. This study focuses on scroll waves, whose filaments were oriented almost perpendicular to the gradient. Whereas scroll waves with filaments exactly perpendicular to the gradient remain unaffected, filaments with a component parallel to the gradient develop a twist. Scroll waves with U-shaped filaments exhibit twists starting from both of its ends, resulting in scroll waves whose filaments display a pair of twists of opposite handedness. These twists are separated by a nodal plane where the filament remains straight and untwisted. The experimental findings were reproduced by numerical simulations using the Oregonator model and a linear gradient of excitability almost perpendicular to the orientation of the filament.

Interaction of a pair of parallel scroll waves.

Interactions of pairs of scroll waves in three-dimensional excitable media were studied experimentally in the Belousov-Zhabotinsky reaction by optical tomography. The behavior of two scroll waves depended on the distance d between their filaments. When the interfilament distance was shorter than the wavelength λ of the scroll waves (but larger than the diameter of the spiral core), the filaments repelled each other. Once d ≈ λ, the two scroll waves synchronized, rotating around their filaments with both a common rotation frequency and a common pitch. The interfilament distance of synchronized scroll waves did not change. When fluctuations broke the symmetry of the rotation periods, the scroll with higher rotation frequency displaced the slower rotating one, until the latter was ousted or even annihilated. These behaviors were independent of the sense of rotation (co- or counter-rotating), the filament dynamics (rigidly rotating or meandering tip motion in two-dimensional media), and the presence or absence of a gradient of excitability parallel to the filaments.

Modeling the light- and redox-dependent interaction of PpsR/AppA in Rhodobacter sphaeroides.

Facultative photosynthetic bacteria switch their energy generation mechanism from respiration to photosynthesis depending on oxygen tension and light. Part of this transition is mediated by the aerobic transcriptional repressor PpsR. In Rhodobacter sphaeroides, the repressive action of PpsR is antagonized by the redox- and blue-light-sensitive flavoprotein AppA which results in a unique phenotype: the repression of photosynthesis genes at intermediate oxygen levels and high light intensity, which is believed to reduce the risk of photooxidative stress. To analyze the underlying mechanism we developed a simple mathematical model based on the AppA-dependent reduction of a disulfide bond in PpsR and the light-sensitive complex formation between the reduced forms of AppA and PpsR. A steady-state analysis shows that high light repression can indeed occur at intermediate oxygen levels if PpsR is reduced on a faster timescale than AppA and if the electron transfer from AppA to PpsR is effectively irreversible. The model further predicts that if AppA copy numbers exceed those of PpsR by at least a factor of two, the transition from aerobic to anaerobic growth mode can occur via a bistable regime. We provide necessary conditions for the emergence of bistability and discuss possible experimental verifications.

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.

Macroscopic dynamics as reporter of mesoscopic organization: the Belousov-Zhabotinsky reaction in aqueous layers of DPPC lamellar phases.

The propagation of traveling chemical waves in the excitable Belousov-Zhabotinsky (BZ) system when performed in the presence of 1,2-dipalmitoyl-sn-glycero-3-phosphatidyl choline (DPPC) bilayers responds sensitively to the phospholipid content. The characteristic features of wave propagation, such as spiral pitch, rotation period, and size of the spiral core region, show two regions of different behavior, one below and the other above a DPPC content of 12.5% (w/w) thus suggesting a transition in the organization of the lipid domains at a DPPC content of ∼12.5% (w/w). This transition is supported by small-angle X-ray scattering data, which show pronounced changes in the coherence lengths of the lyotropic smectic domains. Thus, the dynamics of the chemical system occurring at a macroscopic length scale reflects the organization of the water/lipid domains which extend over mesoscopic lengths. These findings indicate that in the BZ/DPPC system, there is an interaction between processes that occurs at length scales differing by as much as 3 orders of magnitude.

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.

Spatial control of the energy metabolism of yeast cells through electrolytic generation of oxygen.

The metabolic dynamics of yeast cells is controlled by electric pulses delivered through a spatially extended yeast cell/Au electrode interface. Concomitant with voltage pulses, oxygen is generated electrolytically at the electrode surface and delivered to the cells. The generation of oxygen was investigated in dependence of the applied voltage, width of the voltage pulses and temperature of the electrolytic solution. The local oxygen pulses at the electrodes lead to a transient activation of the aerobic energy metabolism of the yeast cells causing a perturbation in their energy balance. The effect of these local perturbations on the temporal dynamics of glycolysis in yeast cells is quantified in dependence of the energy state of cells.

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.

Streamless aggregation of Dictyostelium in the presence of isopropylidenadenosin.

Starving cells of Dictyostelium discoideum undergo a developmental cycle where cAMP is autocatalytically produced and relayed from cell to cell, resulting in the propagation of excitation waves over a spatially extended population. Later on the homogeneous cell layer transforms into a pattern of cell streams directed perpendicular to the cAMP waves. Here we chemically influence aggregation competent cells by isopropylidenadenosin (IPA), an adenosine derivative. It can be assumed, that IPA acts via specific adenosine binding sites localized in the cellular membrane. We find, however, that pattern formation and cellular aggregation under the influence of IPA differ considerably compared to experiments with adenosine. In particular, our observations point towards an inhibitory effect on adenylate cyclase (ACA), the key enzyme in the autocatalytic production process of cAMP inside the cell. Our results suggest the existence of a direct coupling (via intracellular affection) or indirect coupling (via inhibition of cAMP binding) of the specific adenosine receptors to the regulatory circuit that controls cyclic intra- and extracellular cAMP concentration.

Stability of scroll ring orientation in an advective field.

The stability of the orientation of scroll rings in the excitable Belousov-Zhabotinsky reaction under an applied electrical current was investigated in experiments and simulations. The parallel and antiparallel orientations of the scroll ring unit vector with respect to the current are two stationary states, the first one unstable, the latter stable. For any other orientation, the scroll rings were forced to rotate by the current. At the stable stationary orientation, the scroll rings may contract or expand under the same applied current depending on the radius of the scroll rings. In simulations, delicate adjustments caused a scroll ring to propagate with a constant radius in an advective field.

Reorientation of scroll rings in an advective field.

When scroll rings in the excitable Belousov-Zhabotinsky reaction are subjected to an applied electrical current, a reorientation of the scroll ring is induced which is accompanied by a linear drift towards the cathode. The findings can be explained using a modified theory of local filament dynamics under parameter gradients. Numerical simulations using the Oregonator model with an additional advective term accounting for the applied electric field reproduce the experimental results and provide insights into the deformation of the structure of the filament during the reorientation.

Metamaterial-based transmit and receive system for whole-body magnetic resonance imaging at ultra-high magnetic fields.

Magnetic resonance imaging (MRI) at ultra-high fields (UHF), such as 7 T, provides an enhanced signal-to-noise ratio and has led to unprecedented high-resolution anatomic images and brain activation maps. Although a variety of radio frequency (RF) coil architectures have been developed for imaging at UHF conditions, they usually are specialized for small volumes of interests (VoI). So far, whole-body coil resonators are not available for commercial UHF human whole-body MRI systems. The goal of the present study was the development and validation of a transmit and receive system for large VoIs that operates at a 7 T human whole-body MRI system. A Metamaterial Ring Antenna System (MRAS) consisting of several ring antennas was developed, since it allows for the imaging of extended VoIs. Furthermore, the MRAS not only requires lower intensities of the irradiated RF energy, but also provides a more confined and focused injection of excitation energy on selected body parts. The MRAS consisted of several antennas with 50 cm inner diameter, 10 cm width and 0.5 cm depth. The position of the rings was freely adjustable. Conformal resonant right-/left-handed metamaterial was used for each ring antenna with two quadrature feeding ports for RF power. The system was successfully implemented and demonstrated with both a silicone oil and a water-NaCl-isopropanol phantom as well as in vivo by acquiring whole-body images of a crab-eating macaque. The potential for future neuroimaging applications was demonstrated by the acquired high-resolution anatomic images of the macaque's head. Phantom and in vivo measurements of crab-eating macaques provided high-resolution images with large VoIs up to 40 cm in xy-direction and 45 cm in z-direction. The results of this work demonstrate the feasibility of the MRAS system for UHF MRI as proof of principle. The MRAS shows a substantial potential for MR imaging of larger volumes at 7 T UHF. This new technique may provide new diagnostic potential in spatially extended pathologies such as searching for spread-out tumor metastases or monitoring systemic inflammatory processes.

Dynamics of scroll waves in a cylinder jacket geometry.

The dynamics of scroll waves in a narrow cylinder jacket-shaped reactor is investigated experimentally by optical tomography. The fate of the scroll waves of excitation in the Belousov-Zhabotinsky reaction depends on the thickness of the cylinder jacket. While at sufficiently wide cylinder jackets vertically oriented scroll waves remain stable, the probability that the filament of a scroll hits a lateral wall increases as the cylinder jacket narrows. This may lead to the rupture of the initial filament and pinning of the filament ends at the lateral walls. Filaments that pin to opposite lateral walls shrink and reorient to a horizontal orientation; such a reorientation corresponds to a transition from an intramural to a transmural scroll wave. The kinetics of the reorientation and shrinkage of the scrolls were studied. Furthermore, we find that no new filaments were generated upon collision of excitation waves at the side of the cylinder jacket opposite to the scroll wave. Thus, under the studied conditions, we do not observe any new generation of filaments due to a phenomenon like reentry.

Origin of bursting oscillations in an enzyme model reaction system.

The transition from simple periodic to bursting behavior in a three-dimensional model system of the hemin-hydrogen-peroxide-sulfite pH oscillator is investigated. A two-parameter continuation in the flow rate and the hemin decay rate is performed to identify the region of complex dynamics. The bursting oscillations emerge subsequent to a cascade of period-doubling bifurcations and the formation of a chaotic attractor in parameter space where they are found to be organized in periodic-chaotic progressions. This suggests that the bursting oscillations are not associated with phase-locked states on a two-torus. The bursting behavior is classified by a bifurcation analysis using the intrinsic slow-fast structure of the dynamics. In particular, we find a slowly varying quasispecies (i.e., a linear combination of two species) which acts as an "internal" or quasistatic bifurcation parameter for the remaining two-dimensional subsystem. A systematic two-parameter continuation in the internal parameter and one of the external bifurcation parameters reveals a transition in the bursting mechanism from sub-Hopf/fold-cycle to fold/sub-Hopf type. In addition, the slow-fast analysis provides an explanation for the origin of quasiperiodic behavior in the hemin system, even though the underlying mechanism might be of more general importance.