Observation of a chaotic multioscillatory metabolic attractor by real-time monitoring of a yeast continuous culture Marc R. Roussel 1,2 and David Lloyd 1 1 Microbiology Group, Cardiff School of Biosciences, Cardiff University, UK 2 Department of Chemistry and Biochemistry, University of Lethbridge, Canada Organisms carry out processes necessary for the main- tenance of life on many time scales [1]. Not all possible cellular processes are compatible, so either temporal or spatial separation of activity is required [2]. Temporal coordination is provided by biological clocks such as the circadian [2,3] and circahoralian (with periods, T, of $ 1 h) [4–9], both of which are known to function in a wide variety of organisms [10,11]. Other oscilla- tory phenomena observed in yeast cultures include cell-cycle-dependent oscillations [12–15], a collective behavior, and the well-known glycolytic oscillations [16–19]. There are other rhythms in eukaryotic cells which have not thus far been observed in continuous culture systems, such as mitochondrial ion transport [20–24] and calcium oscillations [25,26]. Mitochondrial oscillations have been observed in single yeast cells [27] although, to our knowledge, calcium oscillations have not. It is not clear if the former have any physiological role although calcium oscillations are now known to exercise a number of functions in metabolism [28], cell division [29–31], and differentiation and development [32–34]. The study of biological rhythms in continuous culture systems has important advantages over other techniques. First, oscillations can be studied under constant chemical and physical conditions, the rhythm itself notwithstanding. Second, long-term experiments can be undertaken, which is particularly important for slow rhythms, but also allows the very large amounts of data required by some mathematical analyses to be collected. Among possible continuous culture model organisms, the yeast Saccharomyces cerevisiae stands out due to its ability to synchronize its metabolic state across the population in a relatively short period, and Keywords biochemical oscillations; chaos; continuous culture; yeast Correspondence M. R. Roussel, Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada Fax: +1 403 329 2057 Tel: +1 403 329 2326 E-mail: roussel@uleth.ca Website: http://people.uleth.ca/$roussel (Received 7 November 2006, revised 6 December 2006, accepted 14 December 2006) doi:10.1111/j.1742-4658.2007.05651.x We monitored a continuous culture of the yeast Saccharomyces cerevisiae by membrane-inlet mass spectrometry. This technique allows very rapid simultaneous measurements (one point every 12 s) of several dissolved gases. During our experiment, the culture exhibited a multioscillatory mode in which the dissolved oxygen and carbon dioxide records displayed period- icities of 13 h, 36 min and 4 min. The 36- and 4-min modes were not vis- ible at all times, but returned at regular intervals during the 13-h cycle. The 4-min mode, which has not previously been described in continuous culture, can also be seen when the culture displays simpler oscillatory behavior. The data can be used to visualize a metabolic attractor of this system, i.e. the set of dissolved gas concentrations which are consistent with the multioscillatory state. Computation of the leading Lyapunov exponent reveals the dynamics on this attractor to be chaotic. Abbreviations DO, dissolved oxygen; IBI, interbeat interval; MIMS, membrane-inlet mass spectrometry; PSD, power spectral density. FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS 1011 often without the need for an initial kick to place the culture in a synchronous state [8,16–18,35–37]. Also, yeasts serve as useful experimental models for eukary- otic cell biology [38] so that knowledge gained through the study of these organisms often leads to advances in our general understanding of eukaryotes. We report here on the observation of chaotic oscilla- tions in a continuous culture of the budding yeast S. ce- revisiae which combined a slow, cell-cycle-dependent mode (T $ 13 h), the circahoralian mode (T $ 40 min), and a fast oscillatory mode with T $ 4 min, not previ- ously reported in continuous cultures. The latter rhythm may be a manifestation at the population level of mitochondrial oscillations [21–23]. Because of our observational technique (membrane-inlet mass spectr- ometry; MIMS), we were able to simultaneously meas- ure signals corresponding to several dissolved gases and thus to directly observe the metabolic attractor of this experimental system. Results Cell-cycle-dependent oscillations We monitored the state of a continuous culture of S. cerevisiae using MIMS, a technique that allows for the rapid simultaneous determination of several com- ponents in solution [39]. Culture conditions were iden- tical to those used to study the circahoralian clock in S. cerevisiae [6,7]. Figure 1 shows the partial pressures of O 2 and of CO 2 measured by MIMS relative to the smoothed partial pressure of argon-40 used as a con- trol, with the MIMS probe inserted directly into the culture. The complex oscillations seen in the figure started after a series of accidental disturbances, inclu- ding prolonged periods of starvation (hours to days) and, perhaps more importantly, loss of temperature control. Indeed, we have typically observed large- amplitude long-period oscillations overlaid with faster oscillatory modes after temperature shocks. Complex modes similar to this one, but with different phase relationships of the slow and faster components, can also be reached by pH jumps [40]. The most prominent feature of the traces in Fig. 1 is a large-amplitude oscillation with a period of 13.1 h (determined by Fourier analysis of the entire time ser- ies). We find substantial cycle-to-cycle variability in these oscillations, the cycle time varying from 11.7 to 15.5 h, with a mean of 13.6 ± 1.3 h (mean ± SD; n ¼ 8). The dilution rate in this experiment was D ¼ 0.0765 h )1 . Because dilution and division must, on average, be balanced, we can compute a mean doub- ling time from the dilution rate [41] of ln2 ⁄ D ¼ 9.06 h. The oscillatory period is thus significantly different from the mean doubling time. Long-period oscillations in yeast continuous cultures have been extensively studied [12–15]. These long-period oscillations are a collective phenomenon of the culture with a strong dependence on the dilution rate, and have therefore been described as a cell-cycle-dependent mode [35]. It is thought that the oscillatory mechanism involves par- tial cell-cycle synchronization [42,43]. Circahoralian bursts Bursts of the circahoralian mode are obvious in the O 2 trace and can also be seen on close inspection of the CO 2 data (Fig. 2). Three to six beats are clearly visible in each major cycle. The beats were spaced by 27 min at a minimum, ranging up to 52 min, with a mean of 36 ± 7min (n ¼ 30). The second burst (t ¼ 681–686 h, $ 60 h after the last disturbance to the culture) was much less regular than the others and probably repre- sents transient behavior, long transients being well known in yeast cultures [44]. It was thus excluded from further analysis. (The inclusion of the second burst does not sensibly affect the overall mean and standard deviation of the interbeat intervals [IBI], but it does have a significant effect on the statistics of the individ- ual IBI. Most of the first burst, which occurred from t ¼ 670 to 673 h, was acquired at a slower sampling rate and is also excluded from any of our analyses.) There was a marked tendency for the period to lengthen during a circahoralian burst. The last IBI in each burst, i.e. the time between the last two beats before the large peak in the O 2 signal, averaged Fig. 1. Relative MIMS signals of the m ⁄ z ¼ 32 and 44 components versus time. These mass components correspond, respectively, to O 2 and to CO 2 . Time is given in hours since the fermentor started continuous operation. A chaotic multioscillatory metabolic attractor M. R. Roussel and D. Lloyd 1012 FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS 42 ± 7 min (n ¼ 7), the penultimate IBI averaged 36 ± 4 min (n ¼ 7), whereas the mean of earlier IBIs reached a plateau of 30.9 ± 2.4 min (when observed; n ¼ 11). It is likely that the variation in the IBIs is due, at least in part, to the superposition of the cell- cycle-dependent and circahoralian oscillatory modes rather than to variability in the underlying circahora- lian clock. Indeed, the IBIs of a superposition of sine waves also vary according to the relative phases of the two waves. Fast oscillations There is at least one fast oscillatory component with a period of $ 4 min which, like the circahoralian rhythm, appears, disappears and reappears at regular intervals. Figure 3 shows how the power spectral den- sity (PSD) of the oxygen signal changes with time. The value of the PSD at frequency f is essentially the square of the magnitude of the corresponding Fourier coefficient [45]. In other words, it tells us how strong a particular periodic component is. We computed this figure using 1024-point windows of the normalized O 2 data. Because our sampling rate was Dt ¼ 12 s, each window covers $ 3.4 h, which is adequate to capture both the circahoralian mode and faster components, up to the Nyquist limit, T min ¼ 2Dt ¼ 24 s [45]. The recurrence at 13 h intervals of both the 40-min mode and of a rhythm with a period of $ 4 min is quite clear from the figure. This latter rhythm has not to our knowledge been previously reported in yeast continu- ous cultures. The 4-min mode appears in both the O 2 and CO 2 data (Fig. 2), although it is detectable at a different time and has a more complex waveform in the latter. Fourier analysis of windows of the data set where this oscillatory mode is particularly easily resolved in the normalized O 2 record from the mass analyzer, i.e. between the large excursions and the circahoralian bursts, gives a period of 3.58 ± 0.15 min (average from eight windows of the data set, each between 5 and 8.5 h long). In the O 2 record, the 4-min mode dis- appears when the oxygen level in the culture medium is high, at which time this mode is evident in the CO 2 data (Fig. 2). It continues in the latter time series until roughly the midway point between circahoralian bursts, at which point it gives way to large amplitude, apparently random fluctuations. The 4-min mode is, however, not evident in our recording at m ⁄ z ¼ 34 (Fig. 4), which is diagnostic for H 2 S. Although we analyze just one data set here, we have seen this 4-min rhythm repeatedly in this experimental system, often in combination with the circahoralian oscillations. We have even observed it in the off-gases when the MIMS probe was placed in the fermentor’s headspace. This rhythm is highly robust and reappears after inevitable disturbances to the fermentor during long-term operation (e.g. failures in the medium feed system). By contrast, it does disappear for a time after such disturbances, indicating that it is not a simple electrical or mechanical artifact. Moreover, although we emphasize the data from the MIMS measurements in this report, the 4-min oscillation is also visible in the recordings from the dissolved oxygen (DO) elec- trode (data not shown). Differences in the instrumental responses make the oscillation observable over a Fig. 2. One period of the oscillation shown in Fig. 1. The inset shows the individual data points for m ⁄ z ¼ 32 (O 2 ) for a 30-min span starting at 730 h. Fig. 3. Time evolution of the PSD of the normalized m ⁄ z ¼ 32 data. Equally spaced 1024-point (3.4 h) windows of the data set were Fourier transformed and normalized so that the area under each PSD versus f curve was the same. Colors represent the relative intensities of the frequency components of the signal in each of the time windows, except that all PSD values > 0.3 have been mapped to red to enhance the contrast. M. R. Roussel and D. Lloyd A chaotic multioscillatory metabolic attractor FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS 1013 greater part of the cycle with the mass analyzer than with the DO electrode, which no doubt explains in part why this 4-min rhythm has not previously been reported. Also, we have found no correlation between these oscillations, on the one hand, and pH, NaOH pump activation or heating cycles, on the other hand, the latter two having been recorded manually for a few hours to rule out such artifacts. In particular, the heater turns on every 1–2 min to maintain a tempera- ture of 30 °C, and the pH, and hence the alkali addi- tion rate, fluctuate much less regularly than the oscillations seen in the dissolved oxygen. We conclude that this is a real biochemical rhythm. Metabolic attractor Figure 5 shows the ‘metabolic attractor’, i.e. the set of biochemical states, as reflected in the concentrations of dissolved oxygen, carbon dioxide and hydrogen sulfide, through which this experimental system passes during the complex oscillations. We measured the capacity dimension of the attractor, one of several measures of fractal dimension [46], directly from the 3D data set. We found this dimension to be 2.09 ± 0.07 (95% con- fidence). This value very near two implies that the attractor is neither a simple cycle, which would give a dimension near one, nor does it fill 3D space the way a cycle fattened by a substantial level of noise would. To go further with our analysis, we need to recon- struct the attractor using a single time series because methods for dealing directly with multidimensional time series are not well developed. A standard time- delay embedding was constructed from the O 2 signal. The O 2 signal was chosen because it shows the three periodicities identified above most clearly. In the fol- lowing, we work with data which has been interpolated so that the points are separated by equal time intervals (see Experimental procedures for details). The points in the time series can thus be labeled by an index i. The analysis starts with a computation of the mutual information I(k) between points in the time ser- ies separated by k time intervals, i.e. between points i and i + k for all values of i. The mutual information I(k) measures the amount of information about point i + k we have if we know point i [46]. The mutual information curve is shown in Fig. 6. The oscillations in the mutual information are due to the strong 4-min mode of the time series: these have a mean period of 19.1 ± 0.9 points, or 3.82 ± 0.19 min. Construction of a time-delay embedding requires both a delay and an embedding dimension. The first minimum in the mutual information curve was used to set the delay [46] at 12 points (2.4 min). An embedding dimension of at least twice the capacity dimension is A B Fig. 4. Relative m ⁄ z ¼ 34 signal (H 2 S) versus time. Insets each show 1 h of data starting, respectively, at (A) 712 h a period of high H 2 S and (B) 730 h (low H 2 S). Fig. 5. Metabolic attractor. The values of the relative O 2 and CO 2 signals (m ⁄ z ¼ 32 and 44) are plotted on the axes, whereas the rel- ative H 2 S(m ⁄ z ¼ 34) signal is mapped onto the color scale. Circula- tion around the attractor is in the clockwise direction. Fig. 6. Mutual information I as a function of the dephasing k. A chaotic multioscillatory metabolic attractor M. R. Roussel and D. Lloyd 1014 FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS known to be sufficient to guarantee a good embedding [47], which suggests an embedding dimension of 5 or higher. However, this theoretical lower bound is often unnecessarily large. Moreover, our estimate of the capacity dimension based directly on the multidimen- sional time series is somewhat higher than estimates calculated from time-delay embeddings. These values depend weakly on the embedding dimension and delay, and cluster around 1.9. We therefore carried out the bulk of our analysis with an embedding dimension of 4, and spot-checked the results in five dimensions. We also checked our results with a delay of 25 points. Sta- tistics computed with all combinations of delays and embedding dimensions tried are in good agreement with each other. For our optimal delay of 12 points and an embed- ding dimension of 4, we found the capacity dimension to be 1.90 ± 0.08. Note that this value is in reasonable agreement with that found directly from the full 3D data set. Furthermore, the reconstructed attractor (Fig. 7) is qualitatively similar to the metabolic attrac- tor of Fig. 5. Both of these observations imply that the reconstruction has been successful, i.e. that most of the information carried by our 3D data set is preserved in the reconstructed attractor. A capacity dimension near 2 could either result from quasiperiodicity (multiple independent frequencies) or chaotic dynamics. To resolve this question, we calcula- ted the leading Lyapunov exponent, a measure of the mean rate of divergence of neighboring points on the attractor. A positive Lyapunov exponent indicates sen- sitive dependence on initial conditions, a hallmark of chaos [46]. By contrast, the leading Lyapunov expo- nent for quasiperiodic evolution would be 0. We found the leading Lyapunov exponent to have a value of 0.752 ± 0.004 h )1 (95% confidence). Because the lead- ing Lyapunov exponent is bounded well away from 0, our analysis implies that the O 2 time series is chaotic. By extension, we can conclude that the culture dynam- ics is chaotic in the regime studied in our experiment. Discussion The fact that the 4-min mode is continually visible during this mixed-mode oscillation, although not always in the same dissolved gas, strongly supports the hypothesis that we are observing an intrinsic cellular rhythm rather than a collective behavior due mainly to interactions between members of the population. As with all rhythms observed in bulk measurements, this one has to be synchronized across some portion of the population. Because the rhythm is robustly observed in this system and does not fade away with time, persist- ent chemical synchronization by a diffusible factor is indicated, rather than synchronization by a single initi- ating event such as the disturbances to the reactor which initiated the complex oscillations. What is the biochemical basis of the 4-min rhythm? The most attractive hypothesis is that we are observing a manifestation at the population level of mitochond- rial oscillations. Oscillations associated with ion-trans- port processes in mitochondria with periods of a few minutes have been observed in a variety of experimen- tal preparations [21–23], including studies with the same yeast strain as used in our experiments [27]. These oscillations are typically synchronized across a population of mitochondria [20,24], although complex spatial patterns can also be seen [23]. The ability of some strains of S. cerevisiae to spontaneously syn- chronize their metabolic state across a population to reveal the circahoralian [8,35,37] and glycolytic oscilla- tions [16–18,36] evidently creates conditions propitious to the synchronization of mitochondrial states, making the mitochondrial oscillations observable in the bulk measurements. The large-amplitude, apparently noisy fluctuations in the carbon dioxide data, which obscure the 4-min rhythm through part of the cycle, might also be worthy of investigation. Their amplitude far exceeds the noise level of the instrument. Moreover, the regu- larity with which they appear and disappear in the record again suggests coordinated action among the x i +2t x i +t Fig. 7. Reconstructed attractor of dimension 4 with a time delay of 12 points. Here, x is the normalized m ⁄ z ¼ 32 signal. The first three coordinates of the embedding are plotted as dots on the axes and the fourth coordinate is rendered using the color scale. The teal ‘shadows’ are projections of the attractor onto the (x i ,x i+s ) and (x i ,x i+2s ) coordinate planes, in this case with the points connected by lines. The projection onto the (x i+s ,x i+2s ) plane is identical to that onto the (x i ,x i+s ) plane. Note that the topology of the reconstructed attractor is very similar to that of the directly observed metabolic attractor (Fig. 5). M. R. Roussel and D. Lloyd A chaotic multioscillatory metabolic attractor FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS 1015 cells. It is possible that these fluctuations are due to other components whose mass spectra include frag- ments with m ⁄ z ¼ 44 such as volatile fatty acids [39]. Mixing several oscillators with periods of a few min- utes could conceivably produce fluctuations of this sort. However, it seems unlikely that we are observing a mixed signal in this case: The high concentration of carbon dioxide in the culture medium and its high per- meability in silicone mean that one or two other dis- solved species contributing to the signal at m ⁄ z ¼ 44 would have to display oscillations of very large ampli- tude in order to obscure the 4-min oscillation in CO 2 . Moreover, a very similar mode is seen at m ⁄ z ¼ 34 (H 2 S; Fig. 4). It thus seems likely that these fast fluctu- ations are due to the biochemical dynamics of the sys- tem, and not to an observational artifact. These fluctuations may in fact turn out to be a yet faster rhythm. Note also that these fluctuations may be responsible for the slightly lower estimate of the capa- city dimension from embeddings of the O 2 data versus the full 3D data set. Unfortunately, our instrument cannot collect data fast enough to decide these issues. The finding of chaotic dynamics in this system can be due to one of two possible factors. First, one par- ticular oscillator whose existence was revealed by this experiment could be chaotic of itself. The second and perhaps more likely possibility is that these oscillators (and perhaps others) interact. Biochemical pathways in a cell always interact so that the completely isola- ted functioning of any oscillator would at best be an approximate description. These data can thus be seen as supporting the view that multiple interacting oscil- lators are involved in and perhaps critical to cell function [11,48]. Such phenomena are certainly not confined to yeast cells. Consider for instance the opti- cal measurements of Visser et al. [49], which also revealed a complex evolution of the frequency spec- trum in suspensions of murine erythroleukemia cells. It seems likely that complex oscillations will ulti- mately be detected in most cell types when experi- ments of sufficient temporal resolution and duration are carried out. Rapid sampling technologies like MIMS now enable us to measure several variables from a single system simultaneously. In our analysis, we were, how- ever, mostly forced to treat each variable as a separ- ate time series due to the lack of suitable methods for analyzing the dynamical properties of multidimen- sional data sets. We would encourage our mathemat- ical colleagues to turn their attention to these problems. A richer understanding of data sets such as ours will no doubt emerge once such methods become available. Experimental procedures Strain and culture conditions A continuous culture of the yeast S. cerevisiae IFO 0233 was studied in an LH Engineering 500 series fermentor with a working volume of $ 800 mL. The fermentor was stirred at 900 rpm and aerated at 180 mL Æmin )1 , the aeration rate being controlled by a GEC-Elliott model 1100 air flow meter. The feed pump (Watson-Marlow 101 U) was calib- rated to deliver 1 mLÆmin )1 of a standard medium whose composition is described elsewhere [7]. The pH was main- tained at 3.4 by controlled addition of 2.5 m NaOH solu- tion. The temperature controller held the culture at 30 °C. Monitoring The state of the fermentor was monitored using oxygen and pH electrodes, as well as a mass analyzer (Hiden Ana- lytical, model HAL 301⁄ 3F) fitted with a membrane-inlet probe [50]. The probe is a closed stainless-steel tube with a small aperture drilled into its side wall near the closed end. This aperture was covered with silicone tubing (the mem- brane). The probe was inserted into the fermentor at a depth sufficient to ensure that it would be covered by the culture medium during operation. The mass analyzer was set to record partial pressures at m ⁄ z ¼ 32, 34, 40 and 44. Data analysis The m ⁄ z ¼ 40 signal (Ar) was smoothed by averaging a moving window of 300 points ($ 1 h of data). This smoothed signal, which we denote by  P 40 , was used to nor- malize the other signals from the mass analyzer in order to correct for long-term drift in the response of the instru- ment. The quantity P i =  P 40 , the relative signal of mass com- ponent i, is thus used in all further analyses. To determine the period of the large-amplitude oscilla- tion, we used a technique based on Poincare ´ sections [46]. We looked for pairs of points in the time series where the threshold P 32 =  P 40 ¼ 7 was crossed in the increasing direc- tion. The cycle time is then the time between crossings of this section. Noise sometimes caused the appearance of a cluster of repeated crossings of the section. In these cases, we averaged the crossing times in a cluster. Calculation of the cycle time based on the absolute maximum of each cycle is a little less accurate since the relative m ⁄ z ¼ 32 sig- nal versus time is relatively flat near the maximum (Fig. 2) but gives very similar results. To analyze the circahoralian periodicity, however, we simply used the absolute maximum of each beat to compute the IBI. The Hiden mass analyzer adapts its dwell times automat- ically in order to keep the error in measurements within acceptable limits. Accordingly, the points collected are not uniformly spaced in time. Prior to further analysis, we A chaotic multioscillatory metabolic attractor M. R. Roussel and D. Lloyd 1016 FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS therefore preprocessed the data, using linear interpolation to estimate the values of the normalized signal at equally spaced times covering the data window of interest. For Fourier analysis, we further applied a linear transformation which makes the two endpoints equal to each other and which sets the mean of the transformed time series to zero in order to reduce low-frequency artifacts [46]: For a time series with points x i , i ¼ 1,2, . , N, the transformed time series was computed by y i ¼ x i ) (A+Bi), where B ¼ (x N ) x 1 )/(N ) 1), and A ¼  x À BðN þ 1Þ=2,  x being the mean of the time series. The PSD was then computed from the fast Fourier transform of the y i in the usual way [45]. For Fig. 3, we defined a series of 1024-point equally spaced and overlapping windows of the normalized O 2 data (points 1–1024, 230–1253, 459–1482, , 35 351–36 374). The PSD was computed for each window and normalized to make the area under each of these curves identical. Some of the analysis relies on a time-delay embedding of the O 2 data, i.e. on analysis of data in the space (x i , x i+s , x i+2s , ,x i+(d)1)s ), where s is an appropriate delay and d is the embedding dimension [46]. The mutual information and leading Lyapunov exponent were calculated using standard algorithms [46]. The Lyapunov exponent was cal- culated as the slope of the longest linear section in the graph of the logarithmic separation as a function of time for nearest neighbors in the time-delay embedding. The determination of this linear segment was done by eye, but the results are not greatly sensitive to the choice of the win- dow chosen. The capacity dimension of the attractor was calculated using the algorithm of Liebovitch and Toth [51]. Acknowledgements We thank C. J. Roussel for technical assistance. MRR’s research is supported by the Natural Sciences and Engineering Research Council of Canada. References 1 Lloyd D & Gilbert D (1998) Temporal organisation of the cell division cycle in eukaryotic microbes. Symp Soc Gen Microbiol 56, 251–278. 2 Mitsui A, Kumazawa S, Takahashi A, Ikemoto H, Cao S & Arai T (1986) Strategy by which nitrogen-fixing unicellular cyanobacteria grow photoautotrophically. Nature 323, 720–722. 3 Dunlap JC, Loros JJ & DeCoursey PJ (2004) Chronobiol- ogy: Biological Timekeeping. Sinauer, Sunderland, MA. 4 Brodsky WYa (1975) Protein synthesis rhythm. J Theor Biol 55, 167–200. 5 Satroutdinov AD, Kuriyama H & Kobayashi H (1992) Oscillatory metabolism of Saccharomyces cerevi- siae in continuous culture. FEMS Microbiol Lett 98, 261–268. 6 Murray DB, Roller S, Kuriyama H & Lloyd D (2001) Clock control of ultradian respiratory oscillation found during yeast continuous culture. J Bacteriol 183, 7253– 7259. 7 Adams CA, Kuriyama H, Lloyd D & Murray DB (2003) The Gts1 protein stabilizes the autonomous oscil- lator in yeast. Yeast 20, 463–470. 8 Murray DB, Klevecz RR & Lloyd D (2003) Generation and maintenance of synchrony in Saccharomyces cerevi- siae continuous culture. Exp Cell Res 287, 10–15. 9 Klevecz RR, Bolen J, Forrest G & Murray DB (2004) A genomewide oscillation in transcription gates DNA replication and cell cycle. Proc Natl Acad Sci USA 101, 1200–1205. 10 Klevecz RR (1976) Quantized generation time in mam- malian cells as an expression of the cellular clock. Proc Natl Acad Sci USA 73, 4012–4016. 11 Lloyd D & Murray DB (2005) Ultradian metronome: timekeeper for orchestration of cellular coherence. Trends Biochem Sci 30, 373–377. 12 Ku ¨ enzi MT & Fiechter A (1969) Changes in carbohy- drate composition and trehalase-activity during the bud- ding cycle of Saccharomyces cerevisiae. Arch Mikrobiol 64, 396–407. 13 von Meyenburg HK (1973) Stable synchrony oscilla- tions in continuous cultures of Saccharomyces cerevisiae under glucose limitation. In Biological and Biochemical Oscillators (Chance B, Pye EK, Ghosh AK & Hess B, eds), pp. 411–417. Academic Press, New York, NY. 14 Parulekar SJ, Semones GB, Rolf MJ, Lievense JC & Lim HC (1986) Induction and elimination of oscillations in continuous cultures of Saccharomyces cerevisiae. Bio- techn Bioeng 28, 700–710. 15 Martegani E, Porro D, Ranzi BM & Alberghina L (1990) Involvement of a cell size control mechanism in the induction and maintenance of oscillations in contin- uous cultures of budding yeast. Biotechn Bioeng 36, 453–459. 16 Ghosh AK, Chance B & Pye EK (1971) Metabolic cou- pling and synchronization of NADH oscillations in yeast cell populations. Arch Biochem Biophys 145, 319–331. 17 Aon MA, Cortassa S, Westerhoff HV & Van Dam K (1992) Synchrony and mutual stimulation of yeast cells during fast glycolytic oscillations. J Gen Microbiol 138, 2219–2227. 18 Richard P, Bakker BM, Teusink B, Van Dam K & Westerhoff HV (1996) Acetaldehyde mediates the syn- chronization of sustained glycolytic oscillations in popu- lations of yeast cells. Eur J Biochem 235, 238–241. 19 Danø S, Sørensen PG & Hynne F (1999) Sustained oscillations in living cells. Nature 402, 320–322. 20 Gooch VD & Packer L (1971) Adenine nucleotide con- trol of heart mitochondrial oscillations. Biochim Biophys Acta 245, 17–20. M. R. Roussel and D. Lloyd A chaotic multioscillatory metabolic attractor FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS 1017 21 Williams MA, Stancliff RC, Packer L & Keith AD (1972) Relation of unsaturated fatty acid composition of rat liver mitochondria to oscillation period, spin label motion, permeability and oxidative phosphorylation. Biochim Biophys Acta 267, 444–456. 22 Evtodienko YV, Zinchenko VP, Holmuhamedov EL, Gylkhandanyan AV & Zhabotinsky AM (1980) The stoichiometry of ion fluxes during Sr 2+ -induced oscilla- tions in mitochondria. Biochim Biophys Acta 589, 157– 161. 23 Holmuhamedov EL (1986) Oscillating dissipative struc- tures in mitochondrial suspensions. Eur J Biochem 158, 543–546. 24 Aon MA, Cortassa S, Marba ´ n E & O’Rourke B (2003) Synchronized whole cell oscillations in mitochondrial metabolism triggered by a local release of reactive oxy- gen species in cardiac myocytes. J Biol Chem 278, 44735–44744. 25 Berridge MJ & Galione A (1988) Cytosolic calcium oscillators. FASEB J 2, 3074–3082. 26 Berridge MJ (1990) Calcium oscillations. J Biol Chem 265, 9583–9586. 27 Aon MA, Cortassa S, Lemar KM, Hayes AJ & Lloyd D (2007) Single and cell population respiratory oscillations in yeast: a 2-photon scanning laser microscopy study. FEBS Lett 581, 8–14. 28 Hajno ´ czky G, Robb-Gaspers LD, Seitz MB & Thomas AP (1995) Decoding of cytosolic calcium oscillations in the mitochondria. Cell 82, 415–424. 29 Whitaker M (1997) Calcium and mitosis. Prog Cell Cycle Res 3, 261–269. 30 Santella L (1998) The role of calcium in the cell cycle: facts and hypotheses. Biochem Biophys Res Commun 244, 317–324. 31 Suprynowicz FA, Groigno L, Whitaker M, Miller FJ, Sluder G, Sturrock J & Whalley T (2000) Activation of protein kinase C alters p34 cdc2 phosphorylation state and kinase activity in early sea urchin embryos by abol- ishing intracellular Ca 2+ transients. Biochem J 349, 489– 499 (2000). 32 Gu X & Spitzer NC (1995) Distinct aspects of neuronal differentiation encoded by frequency of spontaneous Ca 2+ transients. Nature 375, 784–787. 33 Dolmetsch RE, Xu K & Lewis RS (1998) Calcium oscil- lations increase the efficiency and specificity of gene expression. Nature 392, 933–936. 34 Feijo ´ JA, Sainhas J, Holdaway-Clarke T, Cordeiro MS, Kunkel JG & Hepler PK (2001) Cellular oscillations and the regulation of growth: pollen tube paradigm. Bioessays 23, 86–94. 35 Keulers M, Satroutdinov AD, Suzuki T & Kuriyama H (1996) Synchronization affector of autonomous short- period-sustained oscillation of Saccharomyces cerevisiae. Yeast 12, 673–682. 36 Sheppard JD & Dawson PSS (1999) Cell synchrony and periodic behaviour in yeast populations. Can J Chem Eng 77, 893–902. 37 Sohn H-Y, Murray DB & Kuriyama H (2000) Ultra- dian oscillation of Saccharomyces cerevisiae during aero- bic continuous culture: hydrogen sulphide mediates population synchrony. Yeast 16, 1185–1190. 38 Herskowitz I (1985) Yeast as the universal cell. Nature 316, 678–679. 39 Lloyd D, Boha ´ tka S & Szila ´ gyi J (1985) Quadrupole mass spectrometry in the monitoring and control of fermentations. Biosensors 1, 179–212. 40 Murray DB & Lloyd D (2006) A tuneable attractor underlies yeast respiratory dynamics. Biosystems, doi: 10.1016/j.biosystems.2006.09.032. 41 Cortassa S, Aon MA, Iglesias AA & Lloyd D (2002) An Introduction to Metabolic and Cellular Engineering. World Scientific, Singapore. 42 Chen C-I & McDonald KA (1990) Oscillatory beha- viour of Saccharomyces cerevisiae in continuous culture. II. Analysis of cell synchronization and metabolism. Biotechn Bioeng 36, 28–38. 43 Duboc Ph, Marison I & von Stockar U (1996) Physiol- ogy of Saccharomyces cerevisiae during cell cycle oscilla- tions. J Biotechn 51, 57–72. 44 Birol G, Zamamiri A-QM & Hjortsø MA (2000) Fre- quency analysis of autonomously oscillating yeast cul- tures. Process Biochem 35, 1085–1091. 45 Press WH, Flannery BP, Teukolsky SA & Vetterling WT (1989) Numerical Recipes. Cambridge University Press, Cambridge. 46 Sprott JC (2003) Chaos and Time-Series Analysis. Oxford University Press, Oxford. 47 Sauer T, Yorke JA & Casdagli M (1991) Embedology. J Stat Phys 65, 579–616. 48 Gilbert D & Lloyd D (2000) The living cell: a complex autodynamic multi-oscillator system? Cell Biol Int 24, 569–580. 49 Visser G, Reinten C, Coplan P, Gilbert DA & Ham- mond K (1990) Oscillations in cell morphology and redox state. Biophys Chem 37, 383–394. 50 Lloyd D, Thomas KL, Cowie G, Tammam JD & Wil- liams AG (2002) Direct interface of chemistry to micro- biological systems: membrane inlet mass spectrometry. J Microbiol Methods 48, 289–302. 51 Liebovitch LS & Toth T (1989) A fast algorithm to determine fractal dimensions by box counting. Phys Lett A 141, 386–390. A chaotic multioscillatory metabolic attractor M. R. Roussel and D. Lloyd 1018 FEBS Journal 274 (2007) 1011–1018 ª 2007 The Authors Journal compilation ª 2007 FEBS . Observation of a chaotic multioscillatory metabolic attractor by real-time monitoring of a yeast continuous culture Marc R. Roussel 1,2 and David. Mitsui A, Kumazawa S, Takahashi A, Ikemoto H, Cao S & Arai T (1986) Strategy by which nitrogen-fixing unicellular cyanobacteria grow photoautotrophically. Nature