Geometry of gene expression dynamics

Motivation: A gene expression trajectory moves through a high dimensional space where each axis represents the mRNA abundance of a different gene. Genome wide gene expression has a dynamic structure, especially in studies of development and temporal response. Both visualization and analyses of such data require an explicit attention to the temporal structure. Results: Using three cell cycle trajectories from Saccharomyces cerevisiae to illustrate, we present several techniques which reveal the geometry of the data. We import phase-delay time plots from chaotic systems theory as a dynamic data visualization device and show how these plots capture important aspects of the trajectories. We construct an objective function to find an optimal two-dimensional projection of the cell cycle, demonstrate that the system returns to this plane after three different initial perturbations, and explore the conditions under which this geometric approach outperforms standard approaches such as singular value decomposition and Fourier analysis. Finally, we show how a geometric analysis can isolate distinct parts of the trajectories, in this case the initial perturbation versus the cell cycle.