Singular value decomposition for genome-wide expression data processing and modeling

Abstract

We describe the use of singular value decomposition in transforming genome-wide expression data from genes x arrays space to reduced diagonalized "eigengenes" x "eigenarrays" space, where the eigengenes (or eigenarrays) are unique orthonormal superpositions of the genes (or arrays). Normalizing the data by filtering out the eigengenes (and eigenarrays) that are inferred to represent noise or experimental artifacts enables meaningful comparison of the expression of different genes across different arrays in different experiments. Sorting the data according to the eigengenes and eigenarrays gives a global picture of the dynamics of gene expression, in which individual genes and arrays appear to be classified into groups of similar regulation and function, or similar cellular state and biological phenotype, respectively. After normalization and sorting, the significant eigengenes and eigenarrays can be associated with observed genome-wide effects of regulators, or with measured samples, in which these regulators are overactive or underactive, respectively.

Figure 1

Normalized elutriation eigengenes. (a) Raster display of v̂_N^T, the expression of 14 eigengenes in 14 arrays. (b) Bar chart of the fractions of eigenexpression, showing that |γ₁〉_N and |γ₂〉_N capture about 20% of the overall normalized expression each, and a high entropy d = 0.88. (c) Line-joined graphs of the expression levels of |γ₁〉_N (red) and |γ₂〉_N (blue) in the 14 arrays fit dashed graphs of normalized sine (red) and cosine (blue) of period T = 390 min and phase θ = 2π/13, respectively.

Figure 2

Normalized elutriation expression in the subspace associated with the cell cycle. (a) Array correlation with |α₁〉_N along the y-axis vs. that with |α₂〉_N along the x-axis, color-coded according to the classification of the arrays into the five cell cycle stages, M/G₁ (yellow), G₁ (green), S (blue), S/G₂ (red), and G₂/M (orange). The dashed unit and half-unit circles outline 100% and 25% of overall normalized array expression in the |α₁〉_N and |α₂〉_N subspace. (b) Correlation of each gene with |γ₁〉_N vs. that with |γ₂〉_N, for 784 cell cycle regulated genes, color-coded according to the classification by Spellman et al. (3).

Figure 3

Genes sorted by relative correlation with |γ₁〉_N and |γ₂〉_N of normalized elutriation. (a) Normalized elutriation expression of the sorted 5,981 genes in the 14 arrays, showing traveling wave of expression. (b) Eigenarrays expression; the expression of |α₁〉_N and |α₂〉_N, the eigenarrays corresponding to |γ₁〉_N and |γ₂〉_N, displays the sorting. (c) Expression levels of |α₁〉_N (red) and |α₂〉_N (green) fit normalized sine and cosine functions of period Z ≡ N − 1 = 5,980 and phase θ ≈ 2π/13 (blue), respectively.

Figure 4

Rotated normalized α factor, CLB2, and CLN3 eigengenes. (a) Raster display of v̂_RN^T, where |γ₁〉_RN = R̂₂R̂₁|γ₁〉_N, |γ₂〉_RN = R̂₁|γ₂〉_N, and |γ₃〉_RN = R̂₂|γ₃〉_N. (b) |γ₁〉_RN, |γ₂〉_RN and |γ₃〉_RN capture 20% of the overall normalized expression each. (c) Expression levels of |γ₁〉_RN (red) and |γ₂〉_RN (blue) fit dashed graphs of normalized sine (red) and cosine (blue) of period T/2 = 66 min and phase π/4, respectively, and |γ₃〉_RN (green) fits dashed graph of normalized sine of period T = 112 min and phase −π/8, from t = 7 to t = 119 min during the cell cycle.

Figure 5

Rotated normalized α factor, CLB2, and CLN3 expression in the subspace associated with the cell cycle. (a) Array correlation with |α₁〉_RN along the y-axis vs. that with |α₂〉_RN along the x-axis, color-coded according to the classification of the arrays into the five cell cycle stages, M/G₁ (yellow), G₁ (green), S (blue), S/G₂ (red), and G₂/M (orange). The dashed unit and half-unit circles outline 100% and 25% of overall normalized array expression in the |α₁〉_RN and |α₂〉_RN subspace. (b) Correlation of each gene with |γ₁〉_RN vs. that with |γ₂〉_RN, for 638 cell cycle regulated genes, color-coded according to the classification by Spellman et al. (3).

Figure 6

Genes sorted by relative correlation with |γ₁〉_RN and |γ₂〉_RN of rotated normalized α factor, CLB2, and CLN3. (a) Normalized expression of the sorted 4,579 genes in the 22 arrays, showing traveling wave of expression from t = 0 to 119 min during the cell cycle and standing waves of expression in the CLB2- and CLN3-overactive arrays. (b) Eigenarrays expression; the expression of |α₁〉_RN and |α₂〉_RN, the eigenarrays corresponding to |γ₁〉_RN and |γ₂〉_RN, displays the sorting. (c) Expression levels of |α₁〉_RN (red) and |α₂〉_RN (green) fit normalized sine and cosine functions of period Z ≡ N − 1 = 4,578 and phase π/8 (blue), respectively.