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Temporal order in living matters reflects the self-organizing nature of dynamical processes driven out of thermodynamic equilibrium. Because of functional reason, the period of a biochemical oscillation must be tuned to a specific value with precision; however, according to the thermodynamic uncertainty relation (TUR), the precision of oscillatory period is constrained by the thermodynamic cost of generating it. After reviewing the basics of chemical oscillations using Brusselator as a model system, we study the glycolytic oscillation generated by octameric phosphofructokinase (PFK), which is known to display a period of several minutes. By exploring the phase space of glycolytic oscillations, we find that the glycolytic oscillation under the cellular condition is realized in a cost effective manner. Specifically, over the biologically relevant range of parameter values of glycolysis and octameric PFK, the entropy production from the glycolytic oscillation is minimal when the oscillation period is (5 - 10) minutes. Further, the glycolytic oscillation is found at work near the phase boundary of limit cycles, suggesting that a moderate increase of glucose injection rate leads to the loss of oscillatory dynamics, which is reminiscent of the loss of pulsatile insulin release resulting from elevated blood glucose level., Comment: 13 pages, 8 figures

Glycolytic oscillation is one of the first observed and described nonlinear phenomena in living objects. Our recent paper points out the similarity of the temperature and outer electric field to influence this oscillation. The electric field is absorbed and changes the molecules. Similarly to the effect of heating, molecules have various structural, dynamical and chemical changes promoted by electric field. The changes sometimes happen without increasing the temperature. Temperature, as the average energy of the included particles, has various kinds of “waste” energy used to heat up the particles which do not participate in the desired changes. The inaccuracy of the effects of temperature growth in local molecular changes could be remarkably high and could be corrected by the well-applied electric field absorption.

Possibilities of synchronized oscillations in glycolysis mediated by various extracellular metabolites are investigated theoretically using two-dimensional reaction-diffusion systems, which originate from the existing seven-variable model. Our simulation results indicate the existence of alternative mediators such as ATP and 1,3-bisphosphoglycerate, in addition to already known acetaldehyde or pyruvate. Further, it is also suggested that the alternative intercellular communicator plays a more important role in the respect that these can synchronize oscillations instantaneously not only with difference phases but also with different periods. Relations between intercellular coupling and synchronization mechanisms are also analyzed and discussed by changing the values of parameters such as the diffusion coefficient and the cell density that can reflect in tercellular coupling strength.

The damped glycolytic oscillation phenomenon occurring in starved cells of the yeast Saccharomyces cerevisiae (NBRC 0565) was characterization for application to a toxicity bioassay. S. cerevisiae was grown under semi-anaerobic conditions. The transient oscillations were observed photometrically as the time course of the fluorescent intensity of reduced pyridine nucleotide resulting from instantaneous addition of glucose to a cell suspension. In this study, simple and reproducible conditions inducing damped oscillations were obtained by modifying a literature method. For estimation of the wave shapes of the damped oscillations we used six indexes. To investigate the total reproducibility as the averaged relative standard deviation (RSD(av)) for the six indexes obtained from the wave shapes, the damped oscillations were induced under the optimum conditions and the RSD(av) values were calculated as 14% in a buffer cell suspension (n = 62) and 22% in a water cell suspension (n = 78). Finally, the effects of glucose concentration on the six indexes were examined, and all the indexes changed when the glucose concentration was changed. Excellent correlations were obtained between the index of oscillation-state time and the concentration of glucose in a buffer cell suspension (r = 0.9985, 0.5-250 mmol L(-1), 10 points) and in a water cell suspension (r = 0.9989, 2.5 micromol L(-1)-250 mmol L(-1), 12 points), respectively.

An ideal toxicity assay should utilize multiple indexes obtained from transient changes of metabolic activities. Here, we demonstrate the possibility for a novel toxicity bioassay using the damped glycolytic oscillation phenomenon occurring in starved yeast cells. In a previous study, the phenomenon was characterized in detail. Under optimum conditions to induce the phenomenon, the wave shapes of the damped glycolytic oscillations were changed by the instantaneous addition of both glucose and chemicals and by changing the chemical concentration. We estimated the changes in the oscillation wave shapes as six indexes, i.e., the number of wave cycles, maximum amplitude, oscillation frequency, attenuation coefficient, initial peak height, and non-steady-state time. These index changes were obtained from several kinds of chemicals. The chemicals, especially those for acids (0.01-100 mM HCl and 0.01-50 mM citric acid), bases (0.001-50 mM KOH), heavy metal ions (1-1,000 mg L(-1); Cu(2+), Pb(2+), Cd(2+), Hg(2+)), respiratory inhibitors (3-500 mg L(-1) NaN(3)), dissolved oxygen removers (10-300 mg L(-1) NaSO(3)), surfactants (10-200 mg L(-1) benzalkonium chloride), and aldehyde (10-1,000 mg L(-1) acetaldehyde), showed characteristic patterns depending on each chemical and its concentration. These significant results demonstrate the possibilities of new methods for both toxicity qualification and quantification.

The cell generation or cell cycle time in a clonal population of identical unicellular organisms growing under steady-state conditions in a carbon (or energy)-limited continuous culture (chemostat) shows a broad distribution. This indicates that the rate of cell division cycle traverse shows considerable variability. The basis for this variable temporal organisation has been the subject of a great deal of speculation, and many models have been suggested. This process is highly temperature dependent, and like all chemical and biochemical reactions, its rate approximately doubles for every 10 °C rise in temperature over a certain range of growth temperatures (i.e. the Q10 ♎ 2). Clock-controlled biological processes on the other hand are temperature-compensated, so that Q10 ♎ 1; two well-established examples are the circadian (τ ♎ 24 h) and ultradian clocks (τ ♎ 40 min in Saccharomyces cerevisiae). Other biological processes proceeding in faster time domains often show reactive oscillatory dynamics. A well-studied example is glycolysis, although a function for glycolytic oscillations is not yet established. As all these examples depend on three or more variables and are often in coupled sets, departure from regular oscillatory behaviour into deterministic aperiodicity is to be expected. Chaos has been demonstrated in biochemical reactions (enzyme-catalysed), as well as chemical reactions. It has also been shown in a metabolic pathway (glycolysis), and in the complex system of the cell division cycle. Chaos may also arise from coupled oscillators. One possible mechanism for spatio-temporal coherence (order) arises in unicells as an output from a tuneable multi-oscillator (controlled chaos). Self-similarity of oscillatory properties across multiple time domains indicates temporal coherence described by an inverse power law proportional to 1/fβ and long-term temporal correlations. Clocks , rhythms , oscillations, deterministic chaos and scale-free coherence are fundamental hallmarks of life on every time scale.

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.

A bioelectrochemical pyruvate sensor is applied to the monitoring of the glycolytic oscillation. This method allows acetaldehyde (or NADH) perturbation experiments. A moderate extent of acetaldehyde addition causes the phase reset of the oscillation of pyruvate as well as NADH. The phenomenon is reasonably explained by our model, whereby the NADH consumption catalyzed by alcohol dehydrogenase (ADH) activates glyceraldehyde-3-phosphate dehydrogenase (GAPDH) transiently and then ADP is effectively phosphorylated by the succeeding 3-phosphoglycerate kinase (3-PKG) reaction, which decelerates the phosphofructokinase (PFK) reaction and then the glycolytic flux. The fast transmission of information from the increase in the NAD + –NADH ratio by acetaldehyde addition to the decrease in the ADP–ATP ratio was verified experimentally by a model enzyme system composed of ADH, GAPDH, and 3-PKG. The acetaldehyde-driven ADP removal in the ADH–GAPDH–3-PKG system is proposed to play an important role in the glycolytic oscillation. The significance of the NAD + –NADH ratio on the period of the glycolytic oscillation is also discussed.

According ot previous studies, we may expect that slow oscillations can occur in dynamics of a parameterized family of systems in which a biochemical oscillator is coupled to a stationary systems. We introduce some numerical results that confirm our expectation. The results suggest that it is possible for slow oscillations to occur in dynamics of a system in which an active oscillatory unit is coupled to a passive medium, and that it is possible for circadian oscillations to arise from fast glycolytic oscillations in such a coupled system.

非線形生命現象に着目した新しい毒性アッセイ法の構築を究極の目的として, 酵母解糖系で見られるNADH濃度の振動現象が化学物質添加時に示す挙動を調べた.振動している解糖系に種々の化学物質を添加して波形変化を観測したところ, 水銀, フッ化物イオン, アセトアルデヒドなどでは顕著な波形変化が見られた.また解糖系とは関係のない呼吸活性阻害剤などでは添加の影響は見られなかった.さらに複数の化学物質を同時に添加することで, 相加的あるいは相乗的な影響が見られた.また実用的にも, 調製後, 凍結乾燥して1カ月以上-20℃で保存しても, 同様の解糖系振動が観察されることがわかり, 試薬化も可能であると考えられた.明確な波形変化が見られた化学物質の数がまだ限られていることや, 検出感度が低いことなど問題点も多いものの, 新しい毒性検出法としての可能性が示された.

When nitrate was added to anaerobic resting cultures of Escherichia coli, two different profiles of NAD(P)H fluorescence were observed. E. coli is known to reduce nitrate to ammonia via nitrite as an anaerobic respiration mechanism. The profile showing single-stage response corresponded to situations where the nitrite formed from nitrate reduction was immediately converted to ammonia. The other profile showing two-stage response resulted from a much slower reduction of nitrite than nitrate. Nitrite thus accumulated during the first stage and was gradually reduced to ammonia when nitrate was depleted, i.e. in the second stage. An undamped oscillation of NAD(P)H fluorescence was also observed in the cultures showing the two-stage response. The oscillation was always detected during the second stage and seldom during either the first stage or the recovered anaerobic stage (after complete nitrite reduction). It never occurred in the cultures showing the single-stage response. The period of oscillation ranged from 1 to 5min. The possibility of the common glycolytic oscillation being responsible is low, as judged from the current knowledge of the nitrate/nitrite reductases of E. coli and the observations in this study. This is the first report on the occurrence of oscillatory NAD(P)H fluorescence in E. coli.

The three main existing models for glycolytic oscillations in yeast were re-examined to investigate how these oscillations are controlled. We implemented the operational definitions provided by metabolic control analysis to quantify the control properties of enzymes with regard to glycolytic oscillations. In all three models, the control of the frequency and that of the amplitudes of the metabolites were distributed among the enzymes. There was no obvious correlation between the control of the average flax and the control of the frequency. Most importantly, the so-called ‘oscillophore’ of the system, traditionally the enzyme primarily held responsible for the generation of the oscillation, was not the only controlling step. We conclude that just like steady-state flux control is not necessarily limited to a rate-limiting step, oscillations are not dictated by a single ‘oscillophore’.

The thermodynamics of nonequilibrium states in the reversible Selkov model is reported. The reaction velocity and entropy production as a function of reaction affinity are computed.

In Wierschem and Bertram model describing bursting modulated by slow glycolytic oscillation, different complex bursting patterns are produced by interaction of two slow variables with different time scales. Generation mechanisms of the complex bursting patterns with one or multiple short bursts and a long burst, are investigated by an extended fast/slow analysis, when a faster slow variable is considered as a bifurcation parameter of fast subsystem, while a slower slow variable only has an effect on bifurcation curves of the fast subsystem.

Autocatalysis is necessary and ubiquitous in both engineered and biological systems but can aggravate control performance and cause instability. We analyze the properties of autocatalysis in the universal and well studied glycolytic pathway. A simple two-state model incorporating ATP autocatalysis and inhibitory feedback control captures the essential dynamics, including limit cycle oscillations, observed experimentally. System performance is limited by the inherent autocatalytic stoichiometry and higher levels of autocatalysis exacerbate stability and performance. We show that glycolytic oscillations are not merely a "frozen accident" but a result of the intrinsic stability tradeoffs emerging from the autocatalytic mechanism. This model has pedagogical value as well as appearing to be the simplest and most complete illustration yet of Bode’s integral formula.

We have investigated the short-wave instability due to Hopf bifurcation in a reaction−diffusion model of glycolytic oscillations. Very low values of the ratio d of the diffusion coefficient of the inhibitor (ATP) and that of the activator (ADP) do help to create short waves, whereas high values of the ratio d and the complexing reaction of the activator ADP reduces drastically the wave-instability domain, generating much longer wavelengths.

Glycolysis is the major source of metabolic energy in almost all living cells. A key feature of the glycolytic oscillations is their critical control by substrate injection rate. We show that in the limit of weak noise of the fluctuating substrate injection rate a new instability arises in the dynamics leading to collapse and revival of glycolytic oscillation reminiscent of "bursting" of action potential in nerve cells. The dynamical system in this limit also exhibits an interesting mirror image symmetry between growth and decay of fluctuations of the reaction product.

Since the pioneering work of Rapkine (1931) the hypothesis that many rhythmic life processes may involve cyclic interconversion of dithiols to disulphides provides a central theme that unifies ultradian, cell division cycle and circadian rhythm research. We have shown that in an autonomously-oscillating continuous culture of Saccharomyces cerevisiae ultradian (τ = 40 min) cycles between high and low respiratory states are accompanied by redox cycling of nicotinamide nucleotide(s) and glutathione. This system may provide insights into regulation of rhythmic processes with longer periods.

The basic idea of the concept of dynamic organization developed in Chapter 2 is that function, understood as the spatio–temporal coherence of events, results from the intrinsic dynamics of the processes taking place in living systems. This intrinsic dynamics may be either autonomous or forced by environmental (extra–or intracellular) perturbations. Dynamic organization results from the evolution of the spatio-temporal organization of biological processes between successive levels of organization. The temporal evolution of a biological process into successive dynamic regimes is achieved through instabilities (bifurcations) which give rise to the emergence of collective behaviour of, for example, supramolecular structures, enzyme activity, cells and spatio-temporal coherence.

Fructose 2,6-bisphosphate is physiologically one of the most potent activators of yeast 6-phosphofructo-1-kinase. The glycolytic oscillation observed in cell-free cytoplasmic extracts of the yeast Saccharomyces cerevisiae responds to the addition of fructose 2,6-bisphosphate in micromolar concentrations by showing a pronounced decrease of both the amplitude and the period. The oscillations can be suppressed completely by 10 microM and above of this activator but recovers almost fully (95%) to the unperturbed state after 3 h. Fructose 2,6-bisphosphate shifts the phases of the oscillations by a maximal +/- 60 degrees. Oscillations in concentration of endogenous fructose 2,6-bisphosphate in the extract were also observed. Fructose 2,6-bisphosphate alters the dynamic properties of 6-phosphofructo-1-kinase which are vital for its role as the 'oscillophore'. However, the minute amount (approximately 0.3 microM) of endogenous fructose 2,6-bisphosphate and the phase relationship of its oscillations compared with other metabolites indicate that this activator is not an essential component of the oscillatory mechanism. Further support for this conclusion is the observation of sustained oscillations in both the extracts and a population of intact cells of a mutant strain (YFA) of S. cerevisiae with no detectable fructose 2,6-bisphosphate (less than 5 nM).

Allosteric regulation is associated with a number of periodic phenomena in biochemical systems. The cooperative nature of such regulatory interactions provides a source of nonlinearity that favors oscillatory behavior. We assess the role of cooperativity in the onset of biochemical oscillations by analyzing two specific examples. First, we consider a model for a product-activated allosteric enzyme which has previously been proposed to account for glycolytic oscillations. While enzyme cooperativity plays an important role in the occurrence of oscillations, we show that these may nevertheless occur in the absence of cooperativity when the reaction product is removed in a Michaelian rather than linear manner. The second model considered was recently proposed to account for signal-induced oscillations of intracellular calcium. This phenomenon originates from a nonlinear process of calcium-induced calcium release. Here also, the cooperative nature of that positive feedback favors the occurrence of oscillations but is not absolutely required for periodic behavior. Besides underlining the importance of cooperativity, the results highlight the role of diffuse nonlinearities distributed over several steps within a regulated system: even in the absence of cooperativity, such mild nonlinearities (e.g., of the Michaelian type) may combine to raise the overall degree of nonlinearity up to the level required for oscillations.

We have modeled the energy metabolism of the perfused rat heart in order to elucidate the interaction of physiological and biochemical control mechanisms. This model which includes glycolysis, the Krebs cycle, and related metabolism, contains 68 submodels of individual enzymes and transport mechanisms including both cytosolic and mitochondrial reactions. The method of model construction, which relies heavily on fitting observed in situ behavior to known algebraic rate laws for isolated enzymes, and its data requirements and necessary assumptions are described. Simulation of a CO-induced anoxic preparation is described in detail. Here glycolysis increases sharply, due to both increased glucose uptake and phosphorylase activation (there is rapid interconversion between a and b forms, both of which are active here); this causes a damped glycolytic oscillation originating with the glycogen-handling enzymes rather than phosphofructokinase. The behavior and physiological consequences of ATPase activity and of a lactate permease which exports lactate to the perfusate are discussed.

The regular appearance of large NADH oscillations in mixtures of equal amounts of two suspensions of S. carlsbergensis originally oscillating approximately 180 ° out of phase is highly suggestive of the existence of an extracellular synchronizer of the glycolytic oscillation in this yeast. Two types of synchronization are usually observed: (a) both the phase and frequency of the mixed population oscillation are synchronized with one of the parent populations and (b) the frequency only is synchronized, i.e., the phase of the oscillation in the mixture is not close to either of the two parent suspensions. The large number of sustained oscillations at higher pH as compared to pH 4.5 suggests the possibility that damping may be due to a desynchronization process which is less effective at higher pH. In the absence of aldehyde traps such as KCN, Girard's reagent P, semicarbazide, or a mixture of NADH and ADH, the oscillations in the mixed suspension are highly damped and the frequency changes continuously. This suggests that acetaldehyde, secreted by the cells, may have a role as a desynchronizer of the oscillations. Mg2+ ion has the property of increasing the frequency of NADH oscillations in S. carlsbergensis. The synchronization of metabolism between individual yeast cells, which is described here, may be important as an example of a phenomenon common to many different cell types.

1. 1.|With an injection technique, glycolytic oscillation can be induced in yeast extract with hexoses, glucose-6-phosphate, fructose-6-phosphate, but not fructose-1,6-diphosphate, with an average rate corresponding to a QNglucose of 70 (per intact cell). 2. 2.|The enzyme and metabolic concentration pattern of glycolysis of a yeast extract is presented. Enzyme molarities per cytosol in the order of 10−6 to 10−4 are found. The enzyme activity pattern for optimal and oscillating conditions reveals large amplitude oscillations (∼ 60-fold) of the activity changes of phosphofructokinase during the oscillation and a mean activity of all glycolytic enzymes in the range of 10–40% of the optimal activity. 3. 3.|Glycolytic self-excitation is induced by substrate addition and generated by the allosteric oscillophor phosphofructokinase. The pulsed generation of products by phosphofructokinase is propagated with a dependent phase shift by means of the coupling variable adenosine phosphate via the kinases, phosphoglycerate kinase and pyruvate kinase. The function of fructose-diphosphate during the oscillations is masked because oscillations only occur above the stresshold level of fructose-diphosphate being critical for the activation of pyruvate kinase. The function of ATP and ADP is amplified by the strong activation of phosphofructokinase by AMP. 4. 4.|Oscillations are a general property of metabolic systems and an implicit function of their feedback structure, which involves cross-coupling and self-coupling with opposite sign in a two variable structure and might produce kinetic instability involving more than one singularity of the trajectories in a phase plane. The study of oscillations reveals the dynamics of a pathway over a large range of states. The physiological significance of controlled oscillations can not yet be evaluated.

Some years ago a simple topological theorem enabled discovery of arrhythmic “singularities” in biological clocks, e.g., in the glycolytic oscillation of yeast cells (1). The context then was homogeneous chemical reaction, i.e., ordinary differential equations. It turns out now that the problem in spatially distributed context, i.e., the partial differential equation of reaction and diffusion, has the same topology. The singularity in this context appears to include the “ROTOR” or “REVERBERATOR”. A heuristic extension of this theorem suggests a simple experimental procedure which may elicit rotors in a volume of oscillating reactants. This procedure resembles conditions which elicit fibrillation in oscillating heart muscle; the same singularity may underlie “SUDDEN CARDIAC DEATH”.

We have proposed a comprehensive model for the glycolytic reaction mechanism and have shown the possibility of self-tuning to resonance, with consequent increase in efficiency of energy transduction by separating the model into two subsystems: one for the phosphofructokinase (PFKase; ATP:D-fructose-6-phosphate 1-phosphotransferase, EC 2.7.1.11) reaction and the other for the pyruvate kinase (PKase; ATP:pyruvate 2- O -phosphotransferase, EC 2.7.1.40) reaction. The purpose of this article is to present an alternative theoretical approach—one more directly applicable to experimental situations—for the detection of these effects without the need for any decomposition. The approach consists in studying the response of such systems to externally applied periodic perturbations inside the fundamental entrainment band by computation. We show, in agreement with previous results obtained for simpler reaction schemes, that large increases and decreases in dissipation may occur in a narrow range of the period of entrainment of the entire mechanism. If the period of the autonomous glycolytic oscillation past marginal stability approaches a value, T 0 lim , that is close to the period of relaxation oscillation of the PKase subsystem, T PKase , then tuning of the primary oscillophor, the PFKase system, has been effected by the PKase system; in such cases we predict, for larger values of the amplitude of perturbation, two peaks in the response spectrum: one near the period of the autonomous oscillation and the other at the period of the PKase system. For small amplitudes of perturbation, there is only one resonance peak near the period of the autonomous oscillation. If T 0 lim is not close to T PKase , then there is only one resonance peak for small and large amplitudes of external perturbations. Our computer results lead to a classification of the chemical species in the reaction mechanism into two categories. The first category has the following two properties. ( i ) When fundamental entrainment occurs, the amplitude of the response in all species in the reaction mechanism is largest when the oscillation in the perturbation of a chemical species is in phase with the oscillation of that variable in the mechanism. ( ii ) If the amplitude of the external periodic perturbation is monitored in time so as to keep that variable in the mechanism always constant, then there is no oscillatory response in the whole reaction pathway. The variables having these two properties have been found to be essential for the generation of self-sustained oscillations in our model. For the second category neither of these two properties holds.

Publisher Summary Glycolytic self-excitation is produced by the allosteric oscillophor phosphofructokinase and propagated with a dependent phase shift by means of the coupling variable adenosine phosphates via the kinases, phosphoglycerate kinase and pyruvate kinase. The basis of a general mechanism of glycolytic oscillation is the establishment of a metabolite and energy balance of the pathway. Titration experiments with the oscillating system, as well as with isolated enzyme preparations, prove that the adenosine phosphate control variable is operating via the kinases of the glycolytic pathway. Glycolysis is controlled at any time by the adenosine phosphate variable. The variable is a function of the velocities of all adenosine triphosphate (ATP)-generating and ATP-splitting reactions and of the adenosine phosphate pool. Any experimental change of the ATP-splitting activities or adenosine phosphate pool influences the phase angle between the oscillophor and secondary oscillations of the other glycolytic intermediates. This has been tested under a great variety of conditions in the laboratory.

Regulation of glycolysis during sustained oscillations has been studies by simulating in detail the experimentally observed behavior of a supernatant fraction from beef heart. This preparation exhibits cyclic oscillations in concentrations of all observed glycolytic intermediates and coenzymes. A model consisting of 57 simultaneous differential equations representing 101 chemical reactions was used to simulate the experimental results. Most of the enzyme sub-models composing this are quite similar to those used for a nonoscillating glycolytic simulation. The differential equations were solved numerically by digital computers. An alternative calculation method, which speeds the process of computation by a factor of 100, was used in the later stages of model construction. Phosphofructokinase, primarily under adenine nucleotide control resulting in pulsed-type variations of active concentrations, causes oscillation in this system, although some other enzymes exert a secondary control. The discrepancies found between experimental and computed behavior suggest presently unknown control mechanisms and areas for further experimentation.

The need of taking into account the change of compartment volume when developing chemical kinetics analysis inside the living cell is discussed. Literature models of a single enzymatic Michaelis-Menten process, glycolytic oscillations, and mitotic cyclin oscillations were tested with appropriate theoretical extension in the direction of volume modification allowance. Linear and exponential type of volume increase regimes were compared. Due to the above, in a growing cell damping of the amplitude, phase shift, and time pattern deformation of the metabolic rhythms considered were detected, depending on the volume change character. The performed computer simulations allow us to conclude that evolution of the cell volume can be an essential factor of the chemical kinetics in a growing cell. The phenomenon of additional metabolite oscillations caused by the periodic cell growth and division was theoretically predicted and mathematically described. Also, the hypothesis of the periodized state in the growing cell as the generalization of the steady-state was formulated.

Background A well known example of oscillatory phenomena is the transient oscillations of glycolytic intermediates in Saccharomyces cerevisiae, their regulation being predominantly investigated by mathematical modeling. To our knowledge there has not been a genetic approach to elucidate the regulatory role of the different enzymes of the glycolytic pathway. Results We report that the laboratory strain BY4743 could also be used to investigate this oscillatory phenomenon, which traditionally has been studied using S. cerevisiae X2180. This has enabled us to employ existing isogenic deletion mutants and dissect the roles of isoforms, or subunits of key glycolytic enzymes in glycolytic oscillations. We demonstrate that deletion of TDH3 but not TDH2 and TDH1 (encoding glyceraldehyde-3-phosphate dehydrogenase: GAPDH) abolishes NADH oscillations. While deletion of each of the hexokinase (HK) encoding genes (HXK1 and HXK2) leads to oscillations that are longer lasting with lower amplitude, the effect of HXK2 deletion on the duration of the oscillations is stronger than that of HXK1. Most importantly our results show that the presence of beta (Pfk2) but not that of alpha subunits (Pfk1) of the hetero-octameric enzyme phosphofructokinase (PFK) is necessary to achieve these oscillations. Furthermore, we report that the cAMP-mediated PKA pathway (via some of its components responsible for feedback down-regulation) modulates the activity of glycoytic enzymes thus affecting oscillations. Deletion of both PDE2 (encoding a high affinity cAMP-phosphodiesterase) and IRA2 (encoding a GTPase activating protein- Ras-GAP, responsible for inactivating Ras-GTP) abolished glycolytic oscillations. Conclusions The genetic approach to characterising the glycolytic oscillations in yeast has demonstrated differential roles of the two types of subunits of PFK, and the isoforms of GAPDH and HK. Furthermore, it has shown that PDE2 and IRA2, encoding components of the cAMP pathway responsible for negative feedback regulation of PKA, are required for glycolytic oscillations, suggesting an enticing link between these cAMP pathway components and the glycolysis pathway enzymes shown to have the greatest role in glycolytic oscillation. This study suggests that a systematic genetic approach combined with mathematical modelling can advance the study of oscillatory phenomena.