Dramatic reduction of dimensionality in large biochemical networks owing to strong pair correlations.

Large multi-dimensionality of high-throughput datasets pertaining to cell signalling and gene regulation renders it difficult to extract mechanisms underlying the complex kinetics involving various biochemical compounds (e.g. proteins and lipids). Data-driven models often circumvent this difficulty by using pair correlations of the protein expression levels to produce a small number (fewer than 10) of principal components, each a linear combination of the concentrations, to successfully model how cells respond to different stimuli. However, it is not understood if this reduction is specific to a particular biological system or to nature of the stimuli used in these experiments. We study temporal changes in pair correlations, described by the covariance matrix, between concentrations of different molecular species that evolve following deterministic mass-action kinetics in large biologically relevant reaction networks and show that this dramatic reduction of dimensions (from hundreds to less than five) arises from the strong correlations between different species at any time and is insensitive to the form of the nonlinear interactions, network architecture, and to a wide range of values of rate constants and concentrations. We relate temporal changes in the eigenvalue spectrum of the covariance matrix to low-dimensional, local changes in directions of the system trajectory embedded in much larger dimensions using elementary differential geometry. We illustrate how to extract biologically relevant insights such as identifying significant timescales and groups of correlated chemical species from our analysis. Our work provides for the first time, to our knowledge, a theoretical underpinning for the successful experimental analysis and points to a way to extract mechanisms from large-scale high-throughput datasets.