Spatio-temporal dynamics in glycolysis

Thomas Mair, Christian Warnke and Stefan C. Mu� ller

Otto-von-Guericke- Magdeburg, Institute of Experimental Physics,Universita� tGroup of Biophysics, 2, 39106 Magdeburg, Germany.Universita� tsplatzE-mail : stefan.mueller=physik.uni-magdeburg.de

Received 10th May 2001First published as an Advance Article on the web 12th November 2001

During the glycolytic degradation of sugar in a thin layer of yeast extract, travelling wavesof NADH and protons can be generated that carry a state of high enzymatic activitythrough the system. The controlled initiation of such waves with an activator of theenzyme phosphofructokinase (PFK) and the inÑuence of various salts and co-factors onthe propagation dynamics are investigated. Furthermore a Ðrst study of the dispersion ofwaves is presented. The experimental characterisation of this in vitro system contributes tounravelling the possible role of glycolysis for biological information processing. In thiscontext, the provision of chemically available energy in the absence of compartmentationby glycolysis is of primary importance.

1 IntroductionRhythmicity is a common phenomenon of life, as we know from daily experience (e.g. our heartbeat or breathing). It occurs on all levels of biological organisation and on a wide range of time-scales. It is well known that cells and organs respond quite often to perturbations in their environ-ment by rhythmic changes of cellular activity. Such a response requires the exchange ofinformation between the cell and the environment and subsequent information processing withinthe cell. From this behaviour it has been concluded, that oscillatory reactions can have importantimpact on biological information processing, the oscillatory reaction being a measure of timeand/or signal strength.1h4

Temporal oscillations in glycolysis were one of the Ðrst type of metabolic rhythms to have beenintensively studied since the early sixties.5h7 The degradation of sugar to pyruvate via glycolysisrepresents the primary pathway for the generation of energy in all living cells, except for a fewbacteria. Moreover, the glycolytic intermediates are precursors for the synthesis of cellular com-ponents, such as amino acids or lipids. Due to this multitude of functions, glycolysis is connectedto many di†erent pathways which branch from or Ñow into glycolysis, thus representing acomplex metabolic network. Accordingly, it plays a central role for the regulation and co-ordination of the cellular metabolism. The question has been raised whether the long knownglycolytic oscillations have a particular biological function. Experimental results indicate an inter-action between glycolytic oscillations and control of insulin secretion in pancreatic b-cells.8 Fur-thermore, recent experiments on the activation and propagation of neutrophil cells have cast newlight on the possible role of glycolytic dynamic spatial patterns. It was found that the activation ofthese cells gives rise to the generation of traveling waves of nicotinamide adenine dinucleotidereduced form (NADH) and protons, which result from oscillatory glycolysis.9 The propagationdirection of activated neutrophil cells coincides with that of the intracellular waves, indicating a

DOI: 10.1039/b104106c Faraday Discuss., 2001, 120, 249È259 249

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functional coupling between cell movement and wave propagation. Similar patterns have beenobtained with a yeast extract, where such travelling waves were observed for the Ðrst time.10,11Theoretical analyses in the early seventies, however, already predicted the formation of glycolyticwaves.12 Interestingly, the wave velocity in the two experimental systems was practically the samedespite their di†erent origin, indicating the existence of common properties of such patterns.

The detection of spatio-temporal dynamics in glycolysis opens a new and innovative way inwhich oscillatory reactions of the metabolism can be involved in biological information pro-cessing. It is suggested, that the propagation dynamics, e.g. wave velocity and direction, can trans-late the metabolic state of the cell into signals that induce co-ordinated cellular responses.

Most of our current knowledge about the temporal dynamics of glycolysis comes from experi-ments with yeast, both from cells and organelle-free extract. We have chosen the glycolytic degra-dation of sugar in a yeast extract as a model system for our investigations, in order to unravel thepossible function of glycolytic waves for biological information processing. This in vitro system hasthe advantage of easy experimental handling. Moreover, the complexity of the metabolic networkcan be gradually changed by addition or removal of speciÐc cellular components, thus supportinga detailed investigation of mutual interactions between the spatio-temporal patterns and particu-lar metabolic pathways. This should give a basis for further investigations on the biological signiÐ-cance of reactionÈdi†usion waves.

2 Materials and methodsPreparation of yeast extract

Yeast extract was prepared from aerobically grown yeast Saccharomyces carlsbergensis (ATCC9080) as described in ref. 6, except that the phosphate bu†er was replaced by a 3-morpholino-propansulfonic acid (MOPS) bu†er, 25 mM pH 6.5, containing 50 mM KCl.

Data acquisition

Temporal oscillations of NADH were recorded by means of its absorption at 340 nm with aspectrophotometer (Shimadzu UV-2501PC). The oscillations were initiated by the addition oftrehalose and phosphate. Traveling waves of NADH were monitored with a home made 2D-spectrophotometer.13 BrieÑy, the yeast extract (500 ll) was mixed with 100 mM trehalose, 50 mM

pH 6.5 and 50 mM KCl (all Ðnal concentrations, Ðnal volume 550 ll) and transferred intoKPO4 ,a sealed reaction chamber. This reaction chamber was a petri dish made from quartz glass(diameter 35 mm covered with a cover glass, leaving a small air gap of 2 mm above the reactivelayer (thickness 1 mm). The cover glass was sealed with grease. This chamber was placed into thelight beam of the 2D-spectrophotometer (j \ 340 nm) and the spatial absorption changes ofNADH were recorded with a UV-sensitive camera (Hamatsu C1000-13). The resulting movieswere stored on a video tape (Sony video recorder EVT-801 CE).

Data evaluation

In order to analyse the propagation dynamics of the waves, the movies were digitised with apersonal computer, equipped with a frame grabber (Matrox pulsar) using the image processingprogram WinPic (home-written). The digitised frames were further processed for improvement ofimage quality (background subtraction, smoothing) and for extraction of data sets (e.g. timeÈspaceplots) using image processing programs that were written with the interactive data language (IDL)software package.

3 Results and discussionE†ect of salts and co-factors on temporal dynamics

Preparation of an excitable medium from biological sources always includes the possibility ofobtaining media of varying quality, since living systems are hard to maintain in a deÐned state.Therefore it is difficult to work with standardised material. As an example, the sugar-inducedglycolytic oscillations for two di†erent yeast extract preparations are shown in Fig. 1A and B. One

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Fig. 1 Restoration of glycolytic oscillations by magnesium addition. Two di†erent yeast extract preparationswere tested for trehalose-induced oscillations in NADH: P05 (A) and P02 (B, C). P05 displays normal NADHoscillations whereas P02 has strongly reduced oscillatory activity (B). After addition of 2 mM magnesium toP02 (C), the amplitude is doubled and the onset of oscillations occurs earlier.

preparation (P05) exhibits signiÐcantly earlier onset of oscillations and larger amplitudes than theother preparation (P02).

In order to obtain improved standardisation of di†erent preparations one has to control thebasic reactions. It is well known that the allosteric enzyme PFK plays an important role in thegeneration of glycolytic oscillations via positive and negative feed-back regulation.14 The adeninenucleotides serve as important e†ectors of this enzyme, with adenosin triphosphate (ATP) beingan inhibitor (and substrate) and adenosin diphosphate (ADP) being an activator (and product) ofthe PFK. The enzyme needs several ions, e.g. magnesium and potassium, for optimal activity. Thefact, that P02 has less oscillatory activity than P05 indicates that the enzyme PFK is impaired inP02. Therefore, we tried to restore this activity by means of magnesium addition. In fact, the onsetof glycolytic oscillations occurs earlier and the amplitude is doubled when 2 mM magnesium isincluded in P02 (Fig. 1C). This e†ect is dose-dependent as can be seen from the results shown inFig. 2, the optimum being around 5 mM.

Oscillatory control in glycolysis is subjected to complex interactions due to the multitude ofregulatory functions of PFK. There are two feed-back loops which can directly or indirectly inter-

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Fig. 2 Dose dependent e†ect of magnesium on amplitude and period of glycolytic oscillations. The yeastextract P02 was supplemented with trehalose, phosphate, KCl and varying amounts of magnesium. TheNADH oscillations were monitored by absorption at 340 nm and their amplitudes (circles) and periods(squares) were evaluated at the di†erent magnesium concentrations.

act with PFK activity, namely the adenine- and NAD/NADH cycles. The adenine nucleotide cyclecouples the upper and lower part of glycolysis via the dephosphorylation and phosphorylation ofATP and ADP by means of the enzymes phosphofructokinase and pyruvate kinase, respectively.The NAD/NADH cycle is coupled to the phosphorylation of ADP, and hence the adenine nucleo-tide cycle, via the enzymes glyceraldehyde phosphate dehydrogenase (GAPDH)/phosphoglyceratekinase (PGK). Some preparations of yeast extract had a quite low concentration of NADH, asestimated from their absorbance at 340 nm. Addition of extra NAD to these preparations drasti-cally reduced the time required for the onset of glycolytic oscillations and also increased theamplitude, indicating the impact of the NAD/NADH cycle for the dynamics of glycolytic oscil-lations.

These data demonstrate, that the preparation of a yeast extract as an excitable system requirescareful control of those experimental parameters that can a†ect the allosteric regulation of PFK.Improved standardisation may be obtained by careful adjustment of magnesium, as well asadenine nucleotides and NAD in the yeast extract.

Spontaneous generation of traveling waves

Periodic reactions are not restricted to biological systems, but they also occur in chemical andphysical systems and most of our current knowledge about the basic mechanisms of oscillatoryreactions comes from experimental and theoretical investigations of such systems. It could beshown that the coupling of an autocatalytic reaction (i.e. an oscillatory one) with transport canlead to the formation of highly ordered spatio-temporal patterns.15h17 Depending on the systemproperties, the patterns can be either dynamic (e.g. travelling waves) or stationary (e.g. Turingpatterns, cf. ref. 18 and 19). Traveling waves occur as the result of a local perturbation of theautocatalytic reaction, which propagates by means of di†usive coupling through the medium,leading to a circular or spiral shaped reactionÈdi†usion wave (for overview see ref. 20). Theprocess of wave generation and propagation has similarities with neuronal excitability, e.g. all-or-none reaction or threshold value, and that is why these waves are also called excitation waves.

The dynamics of glycolysis can be governed by autocatalytic reactions and hence pattern forma-tion is possible. When there is reactionÈdi†usion coupling, addition of sugar spontaneously trans-

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forms the yeast extract into an excitable state, after an initial induction period.10 Under theseconditions, the oscillations are not homogeneous in space, but are characterised by the formationof travelling waves of glycolytic intermediates, composed of NADH or protons. As known fromother excitable media, the shape of the waves can be either circular (Fig. 3AÈD) or spiral shaped(Fig. 4). When looking at the intensity proÐle of NADH-waves, some overshoot in the wave backcan be observed (Fig. 3E).

It is frequently seen, that the waves become narrower during the initial stage of wave formation(compare Fig. 3C and D). This wave sharpening is the result of di†erent velocities of the wavefront and back. In contrast to the wave front, the velocity of the wave back decreases with increas-ing radius until it coincides with the constant velocity of the wave front, which amounts to 5 lms~1 (Fig. 5).

Controlled initiation of waves

The excitable nature of the yeast extract can be demonstrated by controlled initiation of waves.For this, the autocatalytic reaction, i.e. the PFK activity, has to be perturbed, since this reaction ispart of the basic mechanism for excitability. One possible means to perform an excitation is toproduce a local concentration increase of an activator of the autocatalytic reaction. Since fructose-2,6-bisphosphate (F-2,6- is a strong activator of PFK, we have applied this sugar phosphate viaP2)small glass capillaries by means of an injector.

Fig. 3 Spontaneous formation of circular NADH waves. The spatial distribution of NADH in a yeast extractwas recorded with a 2D-spectrophotometer (see Materials and methods). After an initial induction time, circu-lar NADH waves propagated through the extract (AÈD). Time intervals were 57 s (A, B) 150 s (B, C) and 58 s(C, D). Wave velocity was 5 lm s~1. Scale bar : 1 mm. Note that the shape of the wave becomes narrow withtime. The intensity proÐle of the wave shows some overshoot in the wave back (E).

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Fig. 4 Spiral-shaped NADH waves. Spontaneous break-up of circular shaped NADH waves (compare withFig. 3) resulted in the formation of two open wave ends which subsequently generated a rotating double armedspiral (the image is contrast enhanced).

In particular, we have focused on the initiation of waves in the induction phase. During thisphase of about one hour, only global oscillations of glycolysis, i.e. spatial homogeneous ones, canbe observed. This induction phase is followed by an excitable phase which is characterised by thespontaneous formation of travelling waves. The question arises, whether travelling waves can formalso in the induction phase. In the given experiment, the global homogeneous NADH oscillationshad a period of about 12 min. In fact, a local injection of F-2,6- during a global NADHP2minimum leads to the generation of a travelling NADH wave (Fig. 6AÈC). However, the wave can

Fig. 5 Velocity of the wave back displays decreasing curvature dependence. During the initial stage of waveformation (see Fig. 3) the velocity of the wave back decreased with decreasing curvature until it matched theconstant velocity of the wave front (5 lm s~1).

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Fig. 6 Controlled initiation of NADH waves during the induction phase. The strong activator of phos-phofructokinase, fructose-2,6-bisphosphate (23 nl of 0.5 mM dissolved in 1 mM KOH), was injected into theyeast extract at an oscillatory minimum of NADH during the induction phase (where no spontaneous waveformation occurs). The arrow in A marks the tip of the capillary used for the injection. AÈC show the gener-ation of a travelling NADH wave. DÈE show the annihilation of the wave by a global NADH oscillation,leaving a small area of the yeast extract out of phase (F). Time intervals are 58 s (A, B), 63 s (B, C), 150 s (C, D),69 s (D, E) and 57 s (E, F).

propagate only partially through the extract. After about 4.5 min the overall NADH concentra-tion increases due to the global NADH oscillations and thereby stops the propagation of the wave(Fig. 6DÈE). Since the back of an excitation wave has high inhibitor concentrations, the globaloscillations cannot propagate across the back of the wave, leaving a small area that is out of phasewith respect to the other parts of the yeast extract (Fig. 6F).

Similar patterns have been observed, when the yeast extract is supplied with puriÐed ATPase.11The activity of this enzyme leads to a permanent break down of ATP into ADP, i.e. to a per-manent increase in the activator concentration and decrease in the inhibitor concentration. In this

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case, the induction phase is followed by a phase that is still oscillatory but concomitantly showsthe spontaneous formation of excitation waves. Obviously, the ATP/ADP ratio is an importantdeterminant for the transition of the yeast extract from an oscillatory to an excitable state,whereby a high ATP concentration favours the transition to excitability.

The fact that a controlled initiation of NADH waves is possible at a NADH minimum duringthe phase of global oscillations agrees with this point of view, because the phase relations betweenNADH and ATP show a shift of 180¡,21 i.e. the NADH minimum corresponds to a maximumconcentration of ATP. However, it cannot be excluded that other glycolytic intermediates can alsoinÑuence this transition.

Dispersion of glycolytic wave velocity

Many of the autocatalytic reactions exert their rhythmic dynamics by feed-back regulation of theactivator/inhibitor type. Since the autocatalytic reaction is the basic process for pattern formation,the proÐle of an excitation wave is determined by the underlying mechanisms of this reaction.Three di†erent zones can be distinguished : An excitable zone in front of the wave with lowconcentrations of both e†ectors, an excited zone in the leading edge of the wave with high activa-tor concentrations and a refractory zone in the back of the wave with high inhibitor concentra-tions. This proÐle markedly inÑuences the propagation dynamics of excitation waves.

It is often observed, that NADH waves are frequently generated at some “hot spots Ï in the yeastextract, which act as excitatory areas. As a result, we could observe the permanent generation ofwaves starting from a common centre. Fig. 7 shows the timeÈspace plot of such an experiment.The diagonal dark lines in this plot correspond to propagating waves. It is clearly seen that eachwave propagates with a constant velocity, as is expected for reactionÈdi†usion waves. However,this constant velocity di†ers from wave to wave. The data in Table 1 show that these di†erences invelocity correlate with di†erent periods between these waves.

It is found that the wave velocity increases with the period between these waves. This relation-ship is representative for many excitable media and reÑects a common form of the dispersionrelation.20,22,23 A mechanistic explanation for this is based on the existence of the refractory zonein the back of the waves. The medium in the back of the Ðrst wave is refractory and recovers onlyslowly to the full excitable state. When a subsequent wave follows too early, it will propagate intothe refractory medium and consequently slows down.

Fig. 7 TimeÈspace plot of spontaneously generated wave trains with low frequency. The intensity proÐle of ahorizontal line along the centre of a hot spot that permanently generated circular waves is plotted vs. time.The diagonal dark lines in the resulting timeÈspace plot represent propagating NADH waves. (cf. legend toFig. 8). The slope of these lines is a measure of wave velocity. At the right edge of the Ðrst diagonal line,mutual annihilation of colliding waves is visible.

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Table 1 Dispersion of glycolytic wave veloc-ity. Wave trains of spontaneous generatedNADH waves were analysed for their propa-gation velocity and their wavelength

Period/min Velocity/lm s~1

1. Wave n.d. 3.82. Wave 21.9 3.34. Wave 23.3 3.33. Wave 33.6 4.0

Due to the concentration proÐle of the inhibitor (ATP in the case of PFK), there is an absoluterefractory zone which is characterised by high inhibitor concentrations. The autocatalytic reactionis completely inhibited and consequently the generation or propagation of waves within thisabsolute refractory zone is not possible. This implies that the dispersion relation has a lower limitfor the period as well as the velocity. The data in Table 1 do not indicate such a lower limit,probably because the waves were formed spontaneously. In order to test the lower limit, we haveinitiated the waves by injection of F-2,6- into the yeast extract during the excitable phase (Fig.P28A), thus controlling the period between the waves. The corresponding timeÈspace plot (Fig. 8B)shows that a propagating wave can be initiated when the period is reduced to 12 min. Furtherreduction down to 10 min still allows the formation of a wave, but this wave propagates only ashort distance through the extract and then exhibits self-annihilation. The same is true for shorterperiods of 8 to 6 min. From these experimental results we can estimate the absolute refractoryperiod to be around 12 min. It should be noted, that the wave velocity in this experiment waslarger (period 12 min, velocity 4.2 lm s~1) than for the experiment with controlled initiation of

Fig. 8 Initiation of NADH waves at various time intervals. Controlled initiation of waves during the excit-able state of the yeast extract was performed by fructose-2,6-bisphosphate injection (cf. legend to Fig. 6). Asnapshot of three subsequently initiated waves is shown in A. The arrow head marks the position of the tip ofthe injection capillary. The horizontal line along the centre of the waves marks the image line that was used forthe construction of a timeÈspace plot. The intensity proÐle of this line from all images of the movie (i.e. frompropagating waves) was plotted as a function of time (B). The diagonal dark lines in the timeÈspace plotrepresent the propagating waves. Note, that the velocity of the waves across the Ðrst 0.5 mm seems to beaccelerated. This is only an apparent acceleration which is due to some initial streaming phenomena caused bythe injection procedure.

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waves (see Table 1). This may be due to the fact that 2 di†erent preparations of yeast extracts wereused for these experiments (P02 for Table 1, P05 for Fig. 8).

Further conÐrmation for these results is obtained from experiments where a somewhat largerparticle was present in the yeast extract. This particle acted as a strong excitatory signal andinduced the permanent formation of waves with periods between 9 and 7 min. Only the Ðrst wavecould propagate through the extract whereas all other waves displayed self-annihilation (Fig. 9).

Possible functions of glycolytic patterns

Glycolytic waves are of exceptional scientiÐc interest with respect to other known biological wavephenomena for two reasons. Firstly, the generation of glycolytic patterns does not require com-partmentation. They spontaneously form in a simple enzyme solution. Other known biologicalreactionÈdi†usion waves require compartmentation for structure formation, as for examplecalcium stores or amoeboid cells.24h26 Secondly, glycolysis transforms chemically inert energy(sugar) into chemically available energy (ATP). Other known self-organised patterns in biologycan only persist via the consumption of chemically available energy. In this sense, glycolytic wavesprovide energy instead of consuming it and therefore they are possibly involved in the genesis ofbiological structures, the formation of which requires spatial order and biological energy. Suchproperties might also have been of primary importance for the generation of a Ðrst primitive cellfrom an organic solution during the early days of evolution. Of course, the ancestor of the glyco-lytic pathway was surely constructed in a simpler way than the one we know from todayÏs eukary-otic or prokaryotic organisms. However, this does not exclude that similar behaviour andproperties were active in a proposed simple ancestor of glycolysis. Theoretical analyses of evolu-tionary optimisation indicate, that the overall concept of glycolysis can be realised with only a fewproteins.27

Investigations of self-organisation in physical, chemical and biological systems greatly increasedour knowledge about their fundamental properties. One main outcome of these studies is, that thespatio-temporal dynamics of reactionÈdi†usion waves are a universal property of dissipativesystems. The presented results about the spatio-temporal dynamics of glycolysis give furtherexperimental evidence for this view.

Very recently, investigations about NADH waves in neutrophil cells gave indications about onepossible involvement of glycolytic waves for information processing. It was found, that polarisedneutrophil cells generate travelling NADH waves and that the propagation direction of thesewaves coincides with the direction of cell migration. Upon receptor activation, the NADH waveschange their propagation dynamics and so also do the cells.28 Moreover, extracellular spatialgradients of N-formyl-methionyl-leucyl-phenylalanine induced reorientation of the waves. Theseresults indicate that the extracellular signals are translated into ordered metabolic patterns, whichin turn control cell migration.

Fig. 9 TimeÈspace plot of spontaneously generated NADH waves with high frequency. In this experiment, asmall particle acted as a permanent source for excitation. This particle is visible as a dark vertical line in thetimeÈspace plot. Only the Ðrst wave can propagate for a larger distance. All other subsequent waves stoppropagation shortly behind the wave centre. For construction of the timeÈspace plot see legend to Fig. 8.

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In general, the central meaning of glycolysis for the cellular metabolism opens a wide Ðeld ofpossible functions for glycolytic waves. The main goal behind this idea is that a marked part of themanifold metabolic interactions of a cell are translated into ordered glycolytic patterns which inturn participate in the control and co-ordination of cellular behaviour. The experimental modelsystem yeast extract is of particular interest, because there it is possible to investigate the meta-bolic interactions on a subcellular level. This allows detailed studies of the basic mechanisms,which are hard to perform with living cells.

References1 M. J. Berridge, P. H. Cobbold and K. S. R. Cuthbertson, Philos. T rans. R. Soc. L ondon, 1988, 320, 325.2 P. De Koninck and H. Schulman, Science, 1998, 279, 227.3 Y. Tang and H. G. Othmer, Proc. Natl. Acad. Sci. USA, 1995, 92, 7869.4 G. Hajnoczky, L. D. Robb-Gaspars, M. B. Seitz and A. P. Thomas, Cell, 1995, 82, 415.5 A. Gosh and B. Chance, Biochem. Biophys Res. Commun., 1964, 16, 174.6 B. Hess and A. Boiteux, Hoppe-SeylerÏs Z. Physiol. Chem., 1968, 349, 1567.7 A. Betz and B. Chance, Arch. Biochem. Biophys., 1965, 109, 585.8 B. E. Corkey, K. Tornheim, J. T. Deeney, M. C. Glennon, J. C. Parker, F. M. Matschinsky, N. B. Ruder-

mann and M. Prentki, J. Biol. Chem., 1988, 263, 4254.9 H. R. Petty, R. G. Worth and A. L. Kindzelskii, Phys. Rev. L ett., 2000, 84, 2754.

10 T. Mair and S. C. J. Biol. Chem., 1996, 271, 627.Mu� ller,11 S. C. T. Mair and O. Steinbock, Biophys. Chem., 1998, 72, 37.Mu� ller,12 A. Goldbeter, Proc. Natl. Acad. Sci. USA, 1973, 70, 3255.13 S. C. T. Plesser and B. Hess, Anal. Biochem., 1985, 146, 125.Mu� ller,14 A. Goldbeter and R. Lefever, Biophys. J., 1972, 12, 1302.15 A. N. Zaikin and A. M. Zhabotinsky, Nature, 1970, 225, 535.16 A. T. Winfree, Science, 1972, 175, 634.17 S. C. Th. Plesser and B. Hess, Science, 1985, 230, 661.Mu� ller,18 A. M. Turing, Philos. T rans. R. Soc. L ondon Ser. B, 1952, 237, 37.19 V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Phys. Rev. L ett., 1990, 64, 2953.20 J. P. Keener and J. J. Tyson, Physica D, 1986, 32, 327.21 B. Hess, A. Boiteux and J. Adv. Enzym. Regul., 1968, 7, 149.Kru� ger,22 J. M. Flesselles, A. Belmonte and V. J. Gaspar, J. Chem. Soc., Faraday T rans., 1988, 94, 851.23 M. Wussling and T. Mair, in L ecture Notes in Physics : T ransport and Structure, ed. S. C. J. ParisiMu� ller,

and W. Zimmermann, Springer Verlag, Berlin, 1999, p. 151.24 M. J. Berridge, Nature, 1993, 361, 315.25 J. D. Lechleiter and D. E. Clapham, Cell, 1992, 69, 283.26 G. Gerisch, Naturwissenschaften, 1971, 58, 430.27 E. T. G. Waddell, R. Heinrich and F. Montero, Eur. J. Biochem., 1997, 244, 527.Mele� ndez-Hevia,28 H. R. Petty and A. L. Kindzelskii, Proc. Natl. Acad. Sci. USA, 2001, 98, 3145.

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