Singular value decomposition of genome-scale mRNA lengths distribution reveals asymmetry in RNA gel electrophoresis band broadening

Abstract

We describe the singular value decomposition (SVD) of yeast genome-scale mRNA lengths distribution data measured by DNA microarrays. SVD uncovers in the mRNA abundance levels data matrix of genes x arrays, i.e., electrophoretic gel migration lengths or mRNA lengths, mathematically unique decorrelated and decoupled "eigengenes." The eigengenes are the eigenvectors of the arrays x arrays correlation matrix, with the corresponding series of eigenvalues proportional to the series of the "fractions of eigen abundance." Each fraction of eigen abundance indicates the significance of the corresponding eigengene relative to all others. We show that the eigengenes fit "asymmetric Hermite functions," a generalization of the eigenfunctions of the quantum harmonic oscillator and the integral transform which kernel is a generalized coherent state. The fractions of eigen abundance fit a geometric series as do the eigenvalues of the integral transform which kernel is a generalized coherent state. The "asymmetric generalized coherent state" models the measured data, where the profiles of mRNA abundance levels of most genes as well as the distribution of the peaks of these profiles fit asymmetric Gaussians. We hypothesize that the asymmetry in the distribution of the peaks of the profiles is due to two competing evolutionary forces. We show that the asymmetry in the profiles of the genes might be due to a previously unknown asymmetry in the gel electrophoresis thermal broadening of a moving, rather than a stationary, band of RNA molecules.

Fig. 1.

Eigengenes of the yeast genome-scale mRNA lengths distribution data. (a) Raster display of v̂^T, the abundance of X = 30 eigengenes in 30 arrays, corresponding to 30 gel slices, with overabundance (red), no change in abundance (black), and underabundance (green) around the “ground state” of abundance, which is captured by the first eigengene 〈1|v̂^T. The inflection points of the eigengenes approximately sample a graph of the asymmetric parabolic potential kx²/2, where k = k₁ for x ≤ 0 and k = k₂ = k₁/4 for x ≥ 0 (blue) at unit intervals. (b) Bar chart of the 30 fractions of eigen abundance {Ω_n}, which approximately fit a graph of the exponential function of n, {cλⁿ} (blue).

Fig. 2.

Line-joined graphs of the abundance levels of the 1st (red) through 10th (violet) eigengenes of the yeast genome-scale mRNA lengths distribution data, {〈n|v̂^T} for n = 1, … , 10, approximately fit dashed graphs of the 0th (red) through 9th (violet) asymmetric Hermite functions of Eq. 4 with correlations ranging from 0.51 to 0.87. The inflection points of the eigengenes approximately sample a dashed graph of the asymmetric parabolic potential at unit intervals.

Fig. 3.

Line-joined graphs of the measured profiles of abundance levels of the yeast genes VMA7 (red), ADE12 (green), and HIS4 (blue) approximately fit dashed graphs of the asymmetric Gaussian exp[−a(x − p)²], where a = a₁ for x ≤ p and a = a₂ = a₁/4 for x ≥ p, and where the Gaussian peak is set at the gel migration lengths of 88, 70, and 56 mm, respectively, with the corresponding correlations of 0.97, 0.97, and 0.88.

Fig. 4.

Asymmetric generalized coherent state model of the yeast genome-scale mRNA lengths distribution data of Eq. 9. Line-joined graph of the arithmetic mean of the profiles of mRNA abundance levels of the P genes of Eq. 8 approximately fits a dashed graph of the asymmetric Gaussian exp(−bp²), which models the distribution of the peaks of the profiles of the genes, where b = b₁ for x ≤ 0 and b = b₂ = b₁/4 for x ≥ 0 and where the Gaussian peak is set at the gel migration length of 80 mm. Graphs of the asymmetric Gaussian exp[−a(x − p)²] model the profiles of the genes, where the Gaussian peaks are set at 100 mm (violet) through 46 mm (blue), and where the Gaussian amplitudes are determined by the distribution of the peaks, which models the relative abundance of each Gaussian peak among all genes.

Fig. 5.

Eigengenes of the discretized approximated asymmetric generalized coherent state of Eq. 10. (a) Raster display of the 30 eigengenes in 30 arrays, the inflection points of which approximately sample a graph of the asymmetric parabolic potential (blue) at unit intervals. (b) Bar chart of the 30 fractions of eigen abundance, which approximately fit a graph of the exponential function of n (blue).

Fig. 6.

Line-joined graphs of the abundance levels of the 1st (red) through 10th (violet) eigengenes of the yeast genome-scale mRNA lengths distribution data, {〈n|v̂^T} for n = 1, … , 10, approximately fit line-joined graphs of the abundance levels of the 1st through 10th eigengenes (black) of the discretized approximated asymmetric generalized coherent state of Eq. 10 with correlations ranging from 0.76 to 0.93. The inflection points of the eigengenes approximately sample a dashed graph of the asymmetric parabolic potential at unit intervals.