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

TL;DR

This paper demonstrates that strong pair correlations in large biochemical networks lead to a dramatic reduction in effective dimensionality, enabling simpler modeling of complex cellular responses across various systems.

Contribution

It provides a theoretical explanation for the observed dimensionality reduction in biochemical networks due to strong correlations, independent of specific network details or stimuli.

Findings

Dimensionality reduces from hundreds to fewer than five due to correlations.

Eigenvalue spectrum analysis reveals low-dimensional trajectory changes.

Method enables extraction of biological insights like time scales and correlated groups.

Abstract

Large multidimensionality of high-throughput datasets pertaining to cell signaling and gene regulation renders it difficult to extract mechanisms underlying the complex kinetics involving various biochemical compounds (e.g., proteins, lipids). Data-driven models often circumvent this difficulty by using pair correlations of the protein expression levels to produce a small numbers (<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 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 <5) arises from the strong correlations between different species at any time and is in sensitive of the form of the nonlinear interactions, network architecture and values of rate constants and concentrations over a wide range. We relate temporal changes in the eigenvalue spectrum of the covariance matrix to low-dimensional, local changes in directions of the trajectory embedded in much larger dimensions using elementary differential geometry. We illustrate how to extract biologically relevant insights such as identifying significant time scales and groups of correlated chemical species from our analysis. Our work provides for the first time a theoretical underpinning for the successful experimental analysis and points to way to extract mechanisms from large- scale high throughput data sets.