Calculating Transfer Entropy from Variance-Covariance Matrices Provides Insight into Allosteric Communication in ERK2

Transfer entropy methods provide an approach to understanding asymmetric information flow in coupled systems, with particular application to understanding allosteric interactions in biomolecular systems. Transfer entropy analysis holds the potential to reveal pathways or networks of residues that are coupled in their information flow and thus give new insights into folding and binding dynamics. Most current methods for calculating transfer entropy require very long simulations and almost equally long calculations of joint probability histograms to compute the information transfer that make these methods either functionally intractable or statistically unreliable. Available approximate methods based on graph and network theory approaches are rapid but lose sensitivity to the chemical nature of the biomolecules and thus are not applicable in mutation studies. We show that reliable estimates of the transfer entropy can be obtained from the variance-covariance matrix of atomic fluctuations, which converges quickly and retains sensitivity to the full chemical profile of the biomolecular system. We validate our method on ERK2, a well-studied kinase involved in the MAPK signaling cascade for which considerable computational, experimental, and mutation data are available. We present the results of transfer entropy analysis on data obtained from molecular dynamics simulations of wild type active and inactive ERK2, along with mutants Q103A, I84A, L73P, and G83A. We show that our method is consistent with the results of computational and experimental studies on ERK2, and we provide a method for interpreting networks of interconnected residues in the protein from a perspective of allosteric coupling. We introduce new insights about possible allosteric activity of the extreme N-terminal region of the kinase, which to date has been under-explored in the literature and may provide an important new direction for kinase studies. We also describe evidence that suggests activation may occur by different paths or routes in different mutants. Our results highlight systematic advantages and disadvantages of each method for calculating transfer entropy and show the important role of transfer entropy analysis for understanding allosteric behavior in biomolecular systems.