Budding yeast cells (Saccharomyces cerevisiae) exhibit oscillatory dynamics in several cellular pathways, such as those involving the cell cycle, glucose metabolism, and respiration. Previous studies [1, 2] have observed metabolic cycles in yeast cultures in which most genes were expressed in a cyclic manner. These cycles are self-sustaining oscillatory patterns: once cells are synchronized, they continue to exhibit robust oscillations indefinitely [3, 4]. Microarray analysis revealed that cells exhibit cycles consisting of long reductive phases and short respiratory bursts, accompanied by corresponding changes in dissolved oxygen levels, and cell cycle events are restricted to the reductive phase [1, 2]. Interestingly, the periods of the yeast metabolic cycles vary in the two studies. Klevecz et al.  reports a period of ~40 mins, and Tu et al.  observed periods of ~300 mins. There is debate about the relationship between the cell division cycle and the yeast respiratory oscillations, and whether the short- and long-phase oscillations are fundamentally different, or whether the variation in phase length is a result of a difference in nutrient availability, cell spacing, or errors in data interpretation [5–7].
Analysis of the regulatory network of transcription factors involved in the genomewide oscillations may shed light on the underlying causes of the yeast metabolic cycle. A previous study suggests that the Cbf1-Met4-Met28-Met31-Met32 transcription regulatory complex and Gcn4p are important in regulating the short-period metabolic cycle, although it is not likely that there is a single pathway responsible for the observed oscillation. Rather, there are several coupled subsystems involved, with no hierarchical control . The underlying metabolite responsible for synchrony in cells seems to be hydrogen sulfide . There are no previous studies aimed specifically at determining the transcription factors regulating the long-period yeast metabolic cycle, although Lelandais et al.  proposes that the transcription factors Hap1, which is heme-activated and is known to function as an oxygen-sensor, and Hap4, a subunit of a heme-activated complex, may be important. Since each long-period cycle is characterized by the upregulation of several clusters of genes with different functions in the various phases, there is no doubt there are more key transcription factors regulating the timing of cellular events. The goal of this study is to reveal information about the network of transcription factors regulating the longer (~300 min) metabolic cycle.
The long-period yeast metabolic cycle consists of three phases: Ox (oxidative, respiratory), R/B (reductive, building) and R/C (reductive, charging). Each phase is associated with a characteristic change in dissolved oxygen levels in the yeast culture. During the Ox phase, oxygen levels drop drastically. In the R/B phase, oxygen levels increase, while in the R/C phase, the longest of the three, oxygen levels stay relatively constant. During the course of the experiment, the yeast culture is continuously infused with low levels of glucose, however glucose levels in the media are almost zero at all phases of the cycle; cells appear to adsorb and metabolize available glucose immediately. Analyses of microarray time series expression data revealed that ~57% of yeast genes exhibit periodic expression during the course of a metabolic cycle and cluster into one the three superclusters, corresponding to the three phases of the yeast metabolic cycle. Gene expression in different clusters peaks at different phases, and many common metabolites also oscillate, indicating that there is a clear temporal separation between various cellular events [2, 11].
In the oxidative phase, oxygen is rapidly consumed in a burst of respiration. Genes whose expression peaks during this phase are highly expressed during a very narrow window of the yeast metabolic cycle. Functional and metabolome analysis indicates that in the Ox phase, oxidative phosphorylation is using up previously accumulated acetyl-CoA while ATP is being rapidly produced. The oxidative cluster is enriched for genes involving amino acid synthesis and ribosomes, indicating that cells are preparing for cell division. Genes involved in sulfur metabolism and RNA metabolism also show increased expression. During the Ox phase, ATP is abundant, and this is what enables the assembly of translation machinery for the next phase: the reductive/building phase [2, 12].
In the R/B phase, 40-50% of cells enter the cell cycle during each cycle of the yeast metabolic cycle . Therefore, expression increases for genes involved in cell division. Examples of these are histone genes, spindle pole genes, and genes involved in DNA replication. Meanwhile, respiration is shut off, possibly to protect DNA from oxidative damage during cell division. Instead, yeast cells shift to glycolysis and fermentation. Oxygen consumption ceases, and mitochondria are rebuilt. Consequently, the R/B cluster is also enriched for genes involving mitochondrial biogenesis [2, 12].
Finally, in the R/C phase, cells become dependent on non-respiratory modes of metabolism, and acetyl-CoA accumulates, which is a precursor to the upcoming respiratory Ox phase. The R/C cluster is enriched for genes involving fatty acid oxidation, glycolysis, stress-associated response and protein degradation, and this also includes genes involved in peroxisomal function, vacuoles and ubiquination machinery. Little oxygen is being consumed, and dissolved oxygen levels continue to rise. Altogether, cycles in metabolism, respiration and mitochondrial function are all important components of the yeast metabolic cycle [2, 12].
Analysis of intracellular concentrations of metabolites shows that many metabolites show periodic oscillations during the yeast metabolic cycle, and some may be important in the establishment and regulation of cycles . NADP(H), sulfur and heme metabolic pathways may be especially important, and blocking the production of either of these metabolites prevents oscillations from appearing .
Time-series microarray data may be analyzed to determine the transcription factors that are most likely regulating the periodic genes. Other studies searched the promoters of periodic genes to find the most frequently occurring motifs and deduce the most significant transcription factors [8, 10]. Cokus et al.  developed an alternative method to reverse-engineer the regulatory network behind oscillating cellular systems. For each transcription factor, linear regression is used to calculate α-coefficients, which capture whether the genes a transcription factor binds to are differentially expressed or not, assuming that the effects of other transcription factors are held constant. They are essentially a measure of transcription factor activity, and when calculated for each time point, one can find the α-coefficients ("activities") over time for each transcription factor.
Transcription factor α-coefficient profiles can be further analyzed for periodicity, and treated as if they were time-series expression data. Cokus et al.  uses a Fourier-based periodogram method to find the most periodic transcription factors, under the assumption that the transcription factors exhibiting the most robust oscillations are the ones likely to be regulating the cycle involved. Fourier analysis works well for scoring α-coefficient profiles that resemble a sinusoidal curve. However, it would not be as effective for studying expression profiles from the yeast metabolic cycle, because some clusters of genes contain a sharp spike or two peaks per cycle  (Supporting Material). In this case, calculating the autocorrelation function would give a more accurate periodicity score. Since the length of each period is known, we expect to see a peak in a specific location, and the relative magnitude of the peak is a measure of periodicity.
In order to determine the connections between the transcription factors themselves, Cokus et al.  used the collection of time-dependent α-coefficients to compute a time-translation matrix. This can be used to determine α-coefficients for successive time points using matrix multiplication, and allows one to compute the asymptotic amplitudes and phases of the transcription factor α-coefficients, as well. Since each entry of the time-translation matrix can be interpreted as one transcription factor affecting the α-coefficient of another in the successive time point, this is a useful quantitative model of the dynamical properties of the system.
The amplitudes and phases of the α-coefficients may also be estimated directly from the α-coefficient profiles, instead of using the translation matrix. Lelandais et al.  developed a Bayesian decomposition based algorithm, EDPM, to decompose gene expression profiles into a sum of predefined model patterns: sine waves with equivalent periods but different phases. Thus, each time-series profile is represented as the sum of sine waves with various phases and amplitudes, and the magnitudes of the contributing model patterns are a unique "footprint" for each profile. From these, the best-fitting phases and amplitudes can be calculated for each gene or, for the purposes of this study, each transcription factor.
We analyzed time-series microarray data from a previous study  to identify the transcription factors regulating the yeast metabolic cycle. We used the methods of Cokus et al.  to calculate transcription factor α-coefficients using linear regression. We also calculated the time-translation matrix, with some modifications from the previous study's methods. Considering that oxygen is a major oscillating metabolite, and that Hap1, a likely candidate for metabolic cycle regulation, had been previously described to function as an oxygen sensor , we also included oxygen in the transition matrix, and hence, in the dynamical model. α-coefficient profiles were analyzed for periodicity using autocorrelation. Finally, the phase and amplitude of each transcription factor's α-coefficients were calculated using a simplified version of the EDPM algorithm . Essentially, for each transcription factor, we found the single sine wave that best fits its α-coefficient profile.
The advantage of the linear regression based method for estimating transcription factor activities is that calculations use existing high-throughput data to provide an elegant, purely computational solution for finding not only the most periodic and most robustly oscillating transcription factors, but also the network of relationships between them. The goal of this study is to identify the key transcription factors regulating the yeast metabolic cycle and to construct a dynamical model of the activities of these transcription factors. The hypothesis is that the most significant transcription factors encompass cellular functions corresponding to the known phases of the yeast metabolic cycle as previously defined in , .