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].. The
Trang 1attractor by real-time monitoring of a yeast continuous culture
Marc R Roussel1,2and David Lloyd1
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.
Trang 2often 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 O2 and of CO2 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;
n¼ 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 O2 trace and can also be seen on close inspection of the
CO2data (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 (n¼ 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
t¼ 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 O2 signal, averaged
Fig 1 Relative MIMS signals of the m ⁄ z ¼ 32 and 44 components
versus time These mass components correspond, respectively, to
O2and to CO2 Time is given in hours since the fermentor started
continuous operation.
Trang 342 ± 7 min (n¼ 7), the penultimate IBI averaged
36 ± 4 min (n¼ 7), whereas the mean of earlier IBIs
reached a plateau of 30.9 ± 2.4 min (when observed;
n¼ 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 O2
data Because our sampling rate was Dt¼ 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, Tmin¼ 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 O2 and CO2
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 O2 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 O2record, 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 CO2 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 H2S
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.
Trang 4greater 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 30C, 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 O2 signal The O2 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
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
H2S and (B) 730 h (low H2S).
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 H2S (m ⁄ z ¼ 34) signal is mapped onto the color scale
Circula-tion around the attractor is in the clockwise direcCircula-tion Fig 6 Mutual information I as a function of the dephasing k.
Trang 5known 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 O2 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 condioscilla-tions 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 (xi+s,xi+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).
Trang 6cells 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 CO2
Moreover, a very similar mode is seen at m⁄ z ¼ 34
(H2S; 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 O2 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 mLÆmin)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 30C
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 P40, 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 Pi=P40, 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 P32=P40¼ 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
Trang 7therefore 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 xi, i¼ 1,2, , N, the transformed time
series was computed by yi¼ xi) (A+Bi), where B ¼
(xN) x1)/(N) 1), and A ¼ x BðN þ 1Þ=2, x being the
mean of the time series The PSD was then computed from
the fast Fourier transform of the yiin the usual way [45]
For Fig 3, we defined a series of 1024-point equally
spaced and overlapping windows of the normalized O2data
(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 O2data, i.e on analysis of data in the space (xi, xi+s,
xi+2s, , xi+(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
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