Living cells represent thermodynamically open systems, characterized by a nonequilibrium state. Their metabolism is often regulated by the action of enzymes with nonlinear, oscillatory reaction kinetics. A well known example is periodic oscillations in the degradation of sugar via glycolysis in resting yeast or heart cells (1, 2, 3) . Investigations of spatially extended chemical nonequilibrium systems have demonstrated that nonlinear reaction kinetics, coupled with a transport process such as molecular diffusion, lead to the formation of self-organized waves(4, 5) . Since similar thermodynamic principles apply for biological pathways, it has been suggested that wave patterns should occur in living cells, too(6, 7) . First evidence for intracellular waves came from measurements of the spatial distribution of calcium in inositol 1,4,5-trisphosphate-activated frog eggs and in heart cells(8, 9) . The observed calcium patterns and wave dynamics share great similarity with the patterns generated in chemical systems, suggesting that principles of self-organization hold for both systems.

For a detailed examination of the underlying mechanisms of intracellular self-organization one needs an in vitro system for experimental manipulations that are hard to perform with living cells. For this purpose, we chose glycolytic degradation of sugars in a yeast extract as a model system for such investigations. Under appropriate metabolic conditions glycolysis is characterized by oscillatory reaction kinetics (10, 11, 12) and fulfills all requirements for the generation of excitation waves: a nonequilibrium state and nonlinear reaction kinetics. Glycolytic degradation of sugars has been the subject of intense experimental and theoretical work regarding its metabolic control points and its nonlinear dynamic properties(13, 14, 15, 16, 17) . Although several attempts were made to detect spatiotemporal patterns associated with oscillatory glycolysis in a yeast extract(18, 19) , traveling excitation waves have yet not been shown. The aim of the present work is to demonstrate the generation of excitation waves during oscillatory glycolysis and to give first insights into the control of the pattern formation process.

MATERIALS AND METHODS

Cell extract was prepared from aerobically grown yeast according to (1) except that the yeast cells were ground with glass beads in a Braun-Melsungen homogenizer and the phosphate buffer was replaced by 25 mM MOPS, ()50 mM KCl, pH 6.5. For the detection of NADH waves 90 µl of yeast extract (40 mg of protein/ml) was mixed with 12.5 µl of 1 M trehalose, 6.3 µl of 1 M phosphate, pH 6.5, 4.3 µl of 3 M KCl, and 11.3 µl of twice distilled water. This mixture was then pipetted into a totally sealed reaction chamber and placed in the light beam of a two-dimensional spectrophotometer (= 340 nm; cf. (20) ). Spatially resolved absorption was monitored with a UV-sensitive camera (Hamamatsu C 1000), and the resulting movie was stored on a video recorder (SONY time lapse recorder EVT 801 CE). Image processing was performed with the Khoros program (version 1.05) on a SUN SPARC station. Proton waves were monitored with the fluorescent proton indicator fluorescein, using an inverted microscope (Zeiss IM 35). The excitation wavelength was set to 490 nm, and emission was recorded above 520 nm. For analysis of NADH concentration changes the gray levels of a selected image area of 40 40 pixels out of 512 512 pixels were summed up, and the arithmetic mean was plotted as a function of time.

RESULTS

In free open solution layers evaporative cooling easily leads to the generation of convective currents, which in turn can form stationary mosaic type patterns(21) . Such structures can mask the appearance of excitation patterns. To avoid disturbances by convection we used a totally sealed reaction chamber for our experiments. During the measurement the yeast extract was not stirred in order to enable the development of spatial inhomogeneities in the probe. When mixed with its glycolytic substrates trehalose and phosphate, the yeast extract exhibits an initial phase of oscillatory concentration changes of NADH, which lasted for about 60 min (Fig. 1e). We call this phase the ``induction period.'' No spatial inhomogeneities could be observed during this phase of glycolytic activity. The induction period is followed by a continuous increase in the average NADH concentration (corresponding to a decrease in transmitted light intensity). It is during this phase of NADH increase (at 60 min) that spontaneously generated circular NADH waves develop from the boundary with a frequency of 0.05 min and start to propagate through the probe (arrows in Fig. 1e). Since there are multiple foci of wave generation, several waves move toward each other. The collision of NADH fronts results in their mutual annihilation (Fig. 1, a and b). The wave velocity amounts to about 5 µm/s and is constant throughout the passage of one wave, thus agreeing well with predictions from model calculations(6, 7) . The velocity of the following waves gradually decreases for each subsequent one.

Figure 1: Propagation and annihilation of NADH and proton waves in a yeast extract. Yeast extract was mixed with 100 mM trehalose, 50 mM phosphate, pH 6.5, and 100 mM KCl (final concentrations). NADH waves (a and b) were detected by spatially resolved spectrophotometry as described under ``Materials and Methods.'' In a separate experiment, proton waves (c and d) were visualized with the fluorescent proton indicator fluorescein (100 µM) using an inverted microscope. a-d represent single snapshots of propagating waves. The time interval between images a and b is 384 s and between c and d is 240 s. Scale bar is 1 mm for a and b and 2 mm for c and d. Concentration changes of NADH during the experiment were monitored by gray level analysis (e). During the first 60 min, oscillations but no spatial inhomogeneities were observed (induction period). Thereafter, a state of spontaneous wave formation occurs. Arrows in e indicate the passage of NADH waves through the selected frame for gray level analysis.

During glycolytic sugar degradation there is a production of protons by ATP hydrolysis, oscillating with a phase shift of about 30° with respect to the NADH oscillations(11) . In addition, NADH itself is associated with a proton. Thus, it is probable that proton waves are generated in the system as well. In fact, when using the fluorescent proton indicator fluorescein, proton waves were detected (Fig. 1, c and d) with a shape and wave dynamics being similar to that of the NADH wave patterns (compare images a and b with c and d in Fig. 1).

Control experiments were performed to check whether the generation of waves is correlated to the oscillatory reaction of the glycolytic pathway. EDTA was added to the yeast extract to complex magnesium. In the absence of magnesium the glycolytic enzymes phosphofructokinase and pyruvate kinase are no longer active and glycolysis is stopped. Replacement of trehalose by glucose leads to an increased glycolytic flux through phosphofructokinase without oscillatory reaction(11) . As shown in Fig. 2in both experiments no oscillations occur and no NADH waves are generated.

Figure 2: Suppression of oscillations in a yeast extract when EDTA is present or trehalose is replaced by glucose. Experimental procedures and data analysis are the same as described for Fig. 1e. The composition of yeast extract is as in Fig. 1, except that 5 mM EDTA was added (a) or trehalose was replaced by 10 mM glucose (b).

It is now widely accepted that phosphofructokinase acts as the primary source of glycolytic oscillations in vitro. Due to a model of Sel'kov (17) autocatalysis is achieved by feedback control of phosphofructokinase by adenine nucleotides. Having this model in mind, we manipulated the adenine nucleotide pool by addition of AMP in order to influence phosphofructokinase regulation and thereby wave propagation dynamics.

We found that a stepwise increase of AMP addition from 0.25 to 1 mM did not lead to remarkable changes of the oscillatory frequency during the induction period nor of the wave velocity (Table 1). However, shape and dynamics of the wave patterns alter markedly. In the investigated concentration range of AMP, wave fronts break up spontaneously and open wave ends curl up to form rotating spirals (Fig. 3, a-d). The motion of the spiral tip proceeds along a loop-shaped trajectory (Fig. 3e) resembling patterns of spiral trajectories known from the chemical Belousov-Zhabotinski reaction with reduced excitability (so-called meandering(22) ). One loop was completed after about 25 min. Without added AMP spontaneous break up of circular fronts also occurs, but open wave ends fail to curl up.

Figure 3: Spontaneously generated spiral-shaped NADH waves by an overall increase in AMP concentration. Experimental procedures and composition of yeast extract are as described for Fig. 1, except that 0.25 mM AMP was added to the yeast extract concomitantly with trehalose, phosphate, and KCl. Spontaneous break up of wave fronts and subsequent spiral formation were observed after the induction period had passed (compare with Fig. 1). The time interval between images a and b is 90 s, between b and c is 105 s, and between c and d is 392 s. Bar corresponds to 1 mm. The trajectories of the left and right spiral tip are shown in e. X and Y are space coordinates. Arrows indicate direction of tip movement.

Controlled wave initiation could be carried out with the very potent activator of phosphofructokinase, fructose 2,6-bisphosphate (cf. (23) ). After the induction period has passed, a local injection of 0.5 mM fructose 2,6-bisphosphate (estimated final overall concentration, 5-20 µM) leads to generation and propagation of NADH waves (Fig. 4). When injected in the back of a wave, fructose 2,6-bisphosphate did not initiate a new NADH wave. To exclude artifacts of solvents, water or 2 mM KOH (solvent for fructose 2,6-bisphosphate) was injected instead of fructose 2,6-bisphosphate. No NADH waves could be initiated in these control experiments.

Figure 4: Initiation of NADH waves by locally applied fructose 2,6-bisphosphate. Experimental procedures and composition of yeast extract are as described for Fig. 1. After the induction period, wave initiation was performed between the passage of two subsequent spontaneously formed waves at the initiation site. A glass capillary tip (diameter, 5 µm) filled with fructose 2,6-bisphosphate (0.5 mM dissolved in 1 mM KOH) was inserted into the probe, and the sugar phosphate was injected by air pressure (indicated by an arrow). Timing of images is indicated. Scale bar corresponds to 1 mm.

DISCUSSION

The presented data give evidence that traveling excitation waves can be generated in a biological in vitro system. From chemical systems it is known that such waves are followed by a refractory zone, where inhibitor must be degraded before a new wave can pass through it (for review see (24) ). Besides their spontaneous formation (Fig. 1), NADH waves can be generated by injection of fructose 2,6-bisphosphate (Fig. 4), indicating the excitable character of the yeast extract. The failure to induce waves by injection of fructose 2,6-bisphosphate into the back of a NADH wave clearly demonstrates the existence of the refractory zone. This zone is responsible for the mutual annihilation of colliding waves as shown in Fig. 1. It has, besides the excitability of the system, an important influence in determining the formation of spiral-shaped waves from open wave ends. If the refractory zone is too large or the excitability too low, spirals do not develop(25) . Thus, the formation of rotating spirals, which is the main effect of AMP addition, can result either from a reduction of the refractory zone or an increase of excitability.

Phosphofructokinase as the primary source of glycolytic oscillations plays an important role for control of wave dynamics. Its importance is pointed out by the results of wave initiation with fructose 2,6-bisphosphate. The effect of AMP is most probably also related to phosphofructokinase activity. Either it stimulates phosphofructokinase activity and thereby leads to an increased breakdown of the enzyme's inhibitor (and substrate) ATP or the overall increase of adenine nucleotides by AMP addition reduces the sensitivity of the enzyme to ATP. The autocatalytic reaction of phosphofructokinase is a necessary prerequisite for wave generation, as shown by the control experiments with EDTA or glucose (Fig. 2). In view of this finding, the meaning of allosteric regulation of phosphofructokinase for oscillatory glycolysis should be extended to the control of spatial patterns.

Circular and spiral calcium waves in frog eggs, as an another example for intracellular self-organization, exhibit similar patterns. Despite their similar wave dynamics, there are significant differences in the mechanisms of wave propagation. Whereas calcium waves propagate via a calcium-induced calcium release mechanism, requiring the existence of cellular calcium stores(8, 26) , NADH and proton waves propagate by simple diffusion in an evenly distributed enzyme solution. Moreover, calcium waves develop without a pronounced time delay after the onset of the inositol trisphosphate signal transduction pathway, whereas glycolytic NADH and proton waves are generated after a prolonged induction period. We assume that accumulation of glycolytic intermediates and/or end products during the induction period is necessary for spontaneous NADH wave formation. It is likely that NADH and proton waves interact with organelles and thereby with calcium waves, either via electrochemical gradients and/or NADH oxidizing enzymes. Interactions between calcium signaling pathways and glycolysis have been already shown in rat pancreatic -cells(27, 28, 29) .

The occurrence of proton waves provides a way in which glycolysis-induced excitation patterns can interact with electric fields. Membrane potential changes, driven by oscillations of intracellular NADH, already could be observed in heart cells(30) . Yet, it is not clear whether the glycolytic proton waves originate from the NADH-associated proton or whether they are produced in the upper glycolytic pathway. In order to clarify the origin of the proton waves in glycolysis, simultaneous measurements of both compounds, NADH and protons, are necessary.

It is remarkable that both oscillatory behavior and generation of excitation waves manifest a change in the cellular state. This might serve as an indication for their role in cellular information processing(31, 32) . Since NADH and proton waves represent highly ordered structures of molecules that are involved in the cellular energy metabolism, they can act as spatially resolved signals of the energetic status of the cell.

FOOTNOTES

*

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore by hereby marked ``advertisement'' in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

§

To whom correspondence should be addressed. Fax: 49-231-1206389. thomas.mair@mpi-dortmund.mpg.de.

()

The abbreviation used is: MOPS, 4-morpholinepropanesulfonic acid.

ACKNOWLEDGEMENTS

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