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

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.