We propose a value for the Rosetta dataset dimensionality based on the average persistence calculations at each tree level made with ClutrFree using multiple Bayesian Decomposition simulations. The dimensionality has been inferred from the average persistence defined in the Methods section. As the number of basis vectors (i.e., patterns) k is increased, the curve shows a dramatic drop for k > 15 (see Figure 2). This drop is due to the reorganization of the groups of mutants constituting the basis vectors for k > 15 patterns, leading to an overly low average persistence.

The freedom to move between branches leads also to some loss of consistency in the annotations as one moves down a branch. This contrasts with the behavior of basis vectors obtained for less than 15 patterns where biological functions split logically as the number of patterns increases. We also observed a reduction in the number of genes related to each basis vector for 16 patterns in comparison to 15. Also, the standard deviation across samples of the obtained vectors is significantly higher for 16 patterns (1.7 × 10^-3) than for 15 patterns (7.9 × l0^-4) indicating that the Markov chain sampling is not as tightly constrained by the probability distribution. The behavior observed occurs because of the potential to overfit the data with 16 basis vectors allowing the algorithm to find multiple configurations to explain the variation in the data.

Bayesian Decomposition retrieves the two linked matrices: the P matrix (pattern matrix) groups mutants that share cellular functions, which can be deduced from the genes linked to each pattern contained in the A matrix. Each mutant (a column of the P matrix) can belong to multiple patterns, which models the fact that each mutant will have many cellular functions active. Each gene (a row of the A matrix) can be assigned to multiple patterns, reflecting the fact that evolution has led to genes being involved in multiple cellular processes. Interpretation of the results involves identifying cellular processes from the genes that are significantly expressed in a pattern (i.e., within a column of A).

We proceed by using the dimensionality estimate of 15 patterns and exploring for each pattern the genes associated with that pattern. These genes are interpreted using the MIPS ontology for yeast 51 in order to predict the cellular processes associated with a pattern. In addition, for patterns that can be linked to signaling pathways, we discuss the use of data on genes regulated by specific transcription factors and validate the results by analysis of specific key deletion mutants. For each pattern that shows enhancement of ontological terms we provide the terms, the enhancement (as defined in the methods section), and the p value for a hypergeometric test on the term.

We summarize the results in terms of patterns previously identified in other studies using this data set, then we present the new results isolating signatures for activity of the mating and filametation pathways.

Examination of pattern 1 shows expression of the overall common minimal processes necessary for survival, with 386 annotated genes associated with this pattern at a 3σ level. Measure of enhancement, e, of cellular functions, reveals two highly represented functional groups: 1) groups related to protein synthesis and 2) groups related to DNA synthesis. Group 1 includes genes enhanced in Protein Targeting, Sorting and Translocation (Term 14.04, e = 1.84, p = 0.0022), Protein Synthesis (Term 12, e = 1.55, p = 0.024), and Ribosome Biogenesis (Term 12.01, e = 1.60, p = 0.059). Group 2 includes DNA Processing (Term 10.01, e = 1.32, p = 0.097), DNA Recombination and DNA Repair (Term 10.01.05, e = 1.31, p = 0.16), and DNA Synthesis (Term 10.01.03, e = 1.58, p = 0.16). The p-values for the ontology terms remain high, due to the large number of genes associated with this pattern.

This pattern, which essentially includes genes necessary for viability, contains all the mutants of the dataset, although the Ssn6Δ mutant shows a lower level for this pattern than other mutants. As the Ssn6Δ mutant exhibits substantially greater overall expression than any other mutant (including the Tup1Δ mutant with the second highest level), this may reflect the high association of the Ssn6Δ mutant seen in almost all patterns, which will have some gene overlap with this pattern.

Pattern 5 contains 172 annotated genes. Highly enhanced ontologies include transport-related functions: Transported Compounds (Term 20.01, e = 2.02, p < 10^-4), C-compound and Carbohydrate Transport (Term 20.01.03, e = 2.60, p < 3 × 10^-4), and Cellular Transport, Transport Facilitation and Cellular Routes (Term 20, e = 1.62, p < 6 × 10^-4), in addition to other transport terms at p < 10^-3. The pattern contains the two deletion mutants, Ssn6Δ and Tup1Δ, and represents the strong response seen in the original study 53. Ssn6p and Tup1p form a system of transcriptional repression that appears to be highly conserved in eukaryotes 55. In yeast, the complex acts as a global transcriptional repressor over a large number of genes (more that 150), coordinating several cellular systems, including haploid specific genes, glucose repressible genes, and oxygen utilization genes 56. Turning off this repression leads to a large overall increase in gene expression (the overall expression in these two mutants is many fold higher than in other mutants).

Pattern 7 is related to the lack of cell wall functions (Cell Wall, Term 42.01, e = 0.0), as 28 cell wall genes (of 32 total) are absent from this pattern, while the other four genes have multiple annotations suggesting they have roles unrelated to Cell Wall function. Enhancement is present for Protein Modification (Term 14.07, e = 4.7, p < 10^-4) and Fermentation (Term 02.16, e = 4.5, p < 0.01). This pattern contains the mutants Gas1Δ and Fks1Δ, which impair cell-wall synthesis, as well as the mutant YER083cΔ, annotated as disrupting the cell wall in the original study 53. The pattern contains other mutants disrupting ergosterol biosynthesis as well, including Erg2Δ, She4Δ, as well as YER044cΔ. In addition, the pattern includes yeast treated with the drugs that are known to disrupt the cell wall, such as Tunicamycin and Glucosamine.

Pattern 11 is related to ribosomal function, with enhancements in terms for Ribosome Biogenesis (Term 12.01, e = 6.32, p < 4 × 10^-4) and Protein Synthesis (Term 12, e = 3,89, p < 6 × 10^-3). The pattern contains 8 mutants related to ribosomal proteins, Rpll2aΔ, Rpl27aΔ, Rpl34aΔ, Rpl6bΔ, Rp18aΔ, Rps24aΔ, Rps24aΔ (haploid), and Rps27bΔ, as well as some mutants with deleted ORFs of unknown function, YOR078wΔ, YMR269wΔ, and YHR034cΔ, proposed to be involved in ribosomal functions 53.

The two patterns that represent new insights into this data are 13 and 15, which appear related to two strongly coupled developmental pathways in yeast. Previous studies 4453 have identified the mating pathway transcriptional response, however this has included both the filamentation response and the mating response. It is difficult to separate these signatures, as the mating and filamentation pathways share many common elements in a MAPK cascade as shown in Figure 3575859.

Gene ontology (GO) was used to determine the biological function described by each pattern, with a term added specifically for transposable elements, as these are known to play a role during filamentation 6061. The terms that showed enhancement are summarized in Table 1. The patterns show strong overlap, since many genes are shared between the mating and filamentation responses. However, the filamentation ontology term is significantly higher only in pattern 15, which also shows a strong signature of transposable element genes. Meanwhile, the GO terms for meiosis and morphogenesis (such as for budding in S. cerevisiae) are significantly enhanced only in pattern 13. This allows association of pattern 13 with activation of the mating pathway, and pattern 15 with activation of the filamentation pathway.

In addition, we analyzed the 10 genes whose expression is most strongly linked to each pattern. These are shown in Table 2, which summarizes which genes are known to be regulated by the transcriptional activators related to the mating (Ste12p) and filamentation (Ste12p-Tec1p complex) pathways. The results show that the top 10 genes related to pattern 13 have 9 genes of known function, with 8 related to the mating response, of which five are known to be regulated by Ste12p. For pattern 15, 7 of the top 10 genes are known to be transposable element genes, with three other genes having unknown functions. This again links pattern 13 to mating and pattern 15 to filamentation.

In order to validate that the patterns were actually measuring activity of the mating and filamentation pathways, we explored deletion mutants related to these pathways 6162. The mating response in S. cerevisiae is mediated via a MAPK signaling cascade initiated by binding to the Ste2p or Ste3p membrane receptors (Figure 3). The signal is transduced through Ste11p, Ste7p, and Fus3p with Ste5p serving as a scaffolding protein. The signal activates the Ste12p transcription factor, leading to transcription of mating response genes. In addition, the signal is transduced to the MAPK cascade from the membrane receptor by a G protein complex or through the Ste20p protein. Pattern 13 shows near zero signal for the deletion mutants Ste11Δ, Ste7Δ Fus3Δ, Ste12Δ, Ste5Δ, and Ste2Δ, while showing signal for deletion mutants of Ste20Δ and Tec1Δ. This is exactly as expected, with the membrane receptor, all signaling proteins in the cascade, and the transcription factor necessary to generate the transcriptional response related to the mating signal (note that the Ste3Δ mutant is not in the data set). Ste20p is not necessary to raise the mating response, since the G-protein complex can trigger activation of Ste11p directly. For pattern 15, the response is very similar. The signal is near zero for the deletion mutants Ste2Δ, Ste11Δ, Ste7Δ, and Ste12Δ. The Fus3Δ mutant shows a signal for pattern 15, as appropriate, while the Fus3Δ, Kss1Δ double deletion mutant does not. In addition, the Tec1Δ mutant shows no signal for pattern 15, indicating that Tec1p is required for filamentation 61. Finally, the signal is greatly reduced for the Ste20Δ deletion mutant in pattern 15 relative to pattern 13, which agrees with previous work suggesting that the filamentation pathway is more dependent on Ste20p signaling than is the mating pathway 62.

The Rosetta Compendium comprises 300 conditions, including 276 deletion mutants, 11 tetracycline regulated genes, and 13 drug treatments, in S. cerevisiae growing in rich medium 53. The data were generated from a two color cDNA microarray hybridization assay 67, and transcriptional profiles were measured both with technical replication and biological replication (151 mutants). In parallel with the 300 experiments, 63 controls of wild-type S. cerevisiae were grown in identical conditions and compared against each other, permitting creation of a gene-specific error model. The data was downloaded from Rosetta Inpharmatics.

The data was filtered to retain only conditions characterized by at least a variation of 3 fold in a minimum of 2 genes. Then all genes that did not vary by 3-fold in at least 2 conditions were also removed, leading to a data matrix comprising 764 genes and 228 conditions.

The data used by Bayesian Decomposition included both the mean log ratio for each data point and the uncertainty in this measurement determined by the Rosetta error model. Since Bayesian Decomposition as applied here requires positivity 45, the log ratios were converted to ratios and the uncertainties propagated to uncertainties on the ratios. Although analysis of residuals suggested that seven dimensions fit the data 44, the analysis presented here suggests that this is due to overestimation of uncertainty in the data. Bayesian Decomposition is not highly sensitive to minor misestimations of noise however, so that this should not be a problem for this analysis.

Bayesian Decomposition (BD) has been applied to multiple types of data: in vivo spectroscopic data 68, medical imaging 69, microarray data from single cell organisms 4445, mammalian model organisms 47, humans 70, and on phylogenomic sequence data 71. A detailed description of the algorithm 46 and a review of applications 48 have been published.

Briefly, BD models the microarray data, comprising a matrix of estimates of the ratio of expression between the experimental condition and a control, as the result of the multiplication of two matrices describing behaviors across conditions (the P or pattern matrix) and the distribution of genes within those behaviors (the A or amplitude matrix). Naturally, the data, D, includes noise, so that the full relationship is defined by

D i j ≅ ∑ k = 1 K A i k P k j + ε i j       [ 1 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebardaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabgwKianaaqahabaGaemyqae0aaSbaaSqaaiabdMgaPjabdUgaRbqabaGccqWGqbaudaWgaaWcbaGaem4AaSMaemOAaOgabeaakiabgUcaRGGaciab=v7aLnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaaqaaiabdUgaRjabg2da9iabigdaXaqaaiabdUealbqdcqGHris5aOGaaCzcaiaaxMaacqGGBbWwcqaIXaqmcqGGDbqxaaa@4AD1@

where D_ij is the estimated ratio for gene i in mutant j, A_ik is the strength of gene i in pattern k, P_kj is the strength of mutant j in pattern k, and ε_ij is the noise for gene i in mutant j estimated by the Rosetta error model. BD estimates A and P by a Markov chain Monte Carlo (MCMC) simulation. For the fixed noise estimate, ε, A and P are inferred from the marginal probability distribution

p (D | A,P) p (A,P)     [2]

where p(A,P) is the prior probability and p (D | A,P) is given by the likelihood such that

log ⁡ p ( D | A , P ) = − ∑ i ∑ j { 1 2 ε i j 2 ( D i j − ∑ k = 1 K A i k P k j ) 2 } .       [ 3 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@682F@

The prior is used here to require positivity and to minimize structure in the estimates of A and P 46.

The analysis with BD is similar mathematically to an analysis with singular value decomposition (SVD) or with principal component analysis (PCA), since all methods estimate two matrices that together reconstruct the data. In both PCA and SVD, orthogonality conditions force each row of P to be linearly independent, deriving P from either the data matrix (SVD) or the covariance matrix (PCA) using deterministic algorithms. Since patterns of expression related to biological processes will not generally be independent, BD uses the MCMC approach to avoid orthogonality. The resulting rows of P are usually easier to relate to biological processes than those from SVD or PCA.

BD was run at each posited dimensionality, K (as in equation 1), between 3 and 25, generating a mean estimate and an uncertainty (i.e., standard deviation from samples of the posterior probability distribution) for each element of A and P. The dimensionality is equivalent to the number of patterns, as the patterns can be viewed as nonorthogonal basis vectors for D.

The results of Bayesian Decomposition across the different estimated dimensionalities, K, were compared with ClutrFree in order to visualize stable basis vectors and a persistence measurement on them 49. Each independent BD analysis provides a tree level, with each pattern represented by a node (see Figure 4). The analysis with the fewest patterns is placed at the top of the tree, and additional levels of analyses are added creating a tree from fewer to larger numbers of patterns. Connections in the tree are created in a greedy way. Level N+1 (e.g., 4) is connected to level N (e.g., 3) by finding the node in level N+1 with the highest correlation to a node in level N. The correlation is given by the Pearson correlation for the strength of the assignment of mutants to a pattern (i.e., between P_kj for the nodes). These nodes are connected and removed. From the remaining nodes, the maximum correlation between a node at level N+1 and one at N is found again, and this process is repeated until only a single node remains at level N+1. This node is then connected to the node at level N that yields the highest correlation.

Following our previous work 49, we use a measure of persistence to quantify the robustness of a pattern across the variation of the number of patterns. An example calculation is shown in Figure 5. The assignment of a mutant to a pattern (i.e., a node) is binarized based on the mean and uncertainty of the assignment of the mutant to the pattern determined by the MCMC sampling, using a requirement that the mutant be assigned to the pattern at greater than 3σ above zero. In Figure 5, it is assumed there are four mutants in a pattern, thus there are four binarized values. Then at each node, each mutant is compared for consistency in presence of the mutant in linked nodes within the tree. For example, for the highlighted node in Figure 5, the first mutant is present in the node above and the node below, so it is present in all 3 connected nodes. For the second mutant it is present in 2, and the third and fourth mutants are absent. The average persistence for the node is therefore (3+2+0+0)/4 = 1.25 as noted in Figure 5. For branches, the mutant status is only required to agree in a single branch to be counted. The average persistence at a dimension is then the average of the persistence for all nodes at that dimension (i.e., a row in Figure 4).

We assessed the dimensionality of the data using measurements of persistence. The persistence was measured for analyses from 3 – 25 patterns and the dimension chosen where a significant drop occurred in an otherwise slow monotonic decline, which was expected due to the branching nature of the tree. Figure 2 shows the significant drop between 15 and 16 dimensions, so 15 patterns were chosen for further analysis.

To assign ontological terms to the genes contained in basis vectors, we annotated our data using the gene ontologies from the Comprehensive Yeast Genome Database (CYGD) hosted at the Munich Information center for Protein Sequences (MIPS) 5172. The analysis here utilizes the Functional Catalog (FunCat) format that describes each gene using a hierarchical ontological model.

Similar to persistence, we defined enhancement as a measure of the over-representation, or under representation, of a gene function in a subset of the data 47. It is the ratio of the frequency of occurrence of genes annotated by a particular ontological term in the pattern to the frequency of occurrence of the same term in the whole dataset,

e ( t , p ) = g p / n p G / N ,       [ 4 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGLbqzcqGGOaakcqWG0baDcqGGSaalcqWGWbaCcqGGPaqkcqGH9aqpdaWcaaqaaiabdEgaNnaaBaaaleaacqWGWbaCaeqaaOGaei4la8IaemOBa42aaSbaaSqaaiabdchaWbqabaaakeaacqWGhbWrcqGGVaWlcqWGobGtaaGaeiilaWIaaCzcaiaaxMaacqGGBbWwcqaI0aancqGGDbqxaaa@441D@

with g_p being the number of genes annotated with the term t in pattern p, n_p the total number of genes in the pattern p, G the number of genes annotated by the term t in the data, and N being the total number of genes in the dataset. In addition, we apply a hypergeometric test to estimate a p-value for each term. Function was then determined by inspection of enhanced ontological terms.

The genes were also analyzed for the patterns determined to be related to signaling pathways by exploration of the ten genes most strongly associated with the pattern. Each gene was analyzed using the Saccharomyces Genome Database 73 to determine whether it was known to be associated with mating or filamentation processes and to determine if it was directly regulated by the Ste12p transcription factor.