Entropy tree networks of residue dynamics encode protein allostery

Abstract

Proteins can sense signals and-in a process called allostery-transmit information to distant sites. Such information is often not encoded by a protein's average structure, but rather by its dynamics in a way that remains unclear. We show that maximum information tree networks learned from microseconds-long molecular dynamics simulations provide mechanistically-detailed maps of information transmission within proteins in a ligand- and mutation-sensitive manner. On a PDZ domain and the entire human steroid receptor family, these networks quantitatively predict functionally relevant experimental datasets spanning multiple scales, including allosteric sensitivity across a saturation mutagenesis library, calorimetric binding entropies, and phylogenetic trees. These results suggest that a sparse network of entropic couplings encodes the dynamics-to-function map; functional reprogramming and diversification by ligand binding and evolution can modify this network without changing protein structure.

figure 1:. ciMIST infers maximum information spanning trees linking single-residue conformations into a global statistical model.

All frames of a molecular dynamics ensemble are mapped to internal coordinates. For each residue, a separate von Mises mixture model is fit to the observed residue configurations, parameterized using bond and dihedral angles. Mixture components for each residue are coarse-grained by applying DBSCAN, which identifies residue conformations by merging components whose densities overlap, and maps all low-probability components to a single disordered state. Conformational probabilities and mutual informations are computed for each pair of residues residue, and the maximum information spanning tree is then found.

figure 2:. Connecting molecular dynamics to experimental signatures of allostery in PDZ3.

A) Structure of the third PDZ domain (PDZ3) from PSD95 bound to the CRIPT peptide. B) Maximum information spanning trees for PDZ3 (top) and $\text{PDZ}3\Delta 7\text{ct}$ in apo (left) and CRIPT-bound (right) states. Sphere volumes of $\text{C}\alpha$ atoms (shades of green) are proportional to residue marginal entropies, while volumes of links (shades of brown) are proportional to mutual information. C) Contributions of individual residues to the conformational entropy difference between apo and CRIPT-bound states. Regions of of the protein are color-coded as in (A). D) 20 randomly-selected snapshots from simulations of PDZ3 (top) and $\text{PDZ}3\Delta 7\text{ct}$ (bottom) with secondary structural elements colored by contribution to the binding conformational entropy. CRIPT is classified as a single element. E) Prediction of allosteric mutational propensities in saturation mutagenesis from ciMIST residue thermodynamic features using a Bayesian GLM. Residue marginal entropies and summed mutual informations with tree neighbors from the CRIPT-bound state of $\text{PDZ}3\Delta 7$-ct were used as predictors. F) Posterior densities for whole-protein conformational entropy differences calculated by ciMIST, entropies measured with ITC (50), and conformational entropy differences estimated by NMR (54). ITC and NMR posteriors are Gaussians with reported means and standard deviations. $T=298K$.

figure 3:. Entropic effects of 22 stabilizing mutations in ERα.

A) View of a single subunit of $\text{ER}\alpha$ with functional annotation. B) ciMIST networks for the $\text{ER}\alpha$ (top) and a $\text{prsER}\alpha$ (bottom) bound to genistein (left, pink sticks) and estradiol (right, purple sticks). Sphere volumes of $\text{C}\alpha$ atoms (shades of green) are proportional to residue marginal entropies, while link volumes (shades of brown) are proportional to mutual information. Note that while simulations were of homodimers, ciMIST was applied to the aggregate statistics of both subunits. C) Differences in residue entropy contributions between genistein- and estradiol-bound states for $\text{ER}\alpha$ (top) and the $\text{prsER}\alpha$ (bottom). C* denotes positions of cysteine-serine substitutions made for calorimetry (64) but not in our simulations. Parentheses denote positions not present in the calorimetry experiment. Regions are color-coded as in (A) D) Connections in ciMIST networks between PHE461 and other residues for human $\text{ER}\alpha$ ensembles. E) Correlation between differences in contributions of protein segments to total conformational entropy and average percent differences in deuteration levels (3, 63) between estradiol and genistein-bound states of human $\text{ER}\alpha$. F) Estimated posterior densities of entropy differences from ITC (64) and calculated using ciMIST. $T=298K$. Uncertainties for ITC are estimated using the standard deviation of three replicates. $T=298K$.

figure 4:. Redistribution of entropy in the unliganded (apo) state of the Y537S estrogen receptor α.

A) Average structure observed in simulations of human $\text{ER}\alpha$-LBD homodimer and the Y537s variant. Structures were computed by superimposing all frames onto the first and then computing the average positions of atoms over all frames. B) Maximum information spanning trees for the unliganded wild-type and Y537S variants of $\text{ER}\alpha$. C) 20 randomly selected snapshots from MD simulations of unliganded wild-type and Y537S variants of $\text{ER}\alpha$. Secondary structural elements are colored by their contributions to the difference in entropy between mutant and wild-type states, $T\left(S_{\text{Y537S}}-S_{\text{wild-type}}\right)$. $T\Delta S_{\text{conf}}$ is the ciMIST calculation of difference in conformational entropy of Y537S from the wild-type. $T=298K$.

figure 5:. Entropic allostery in estrogen receptor α.

A) 20 randomly selected snapshots from simulations of $\text{ER}\alpha$-LBD homodimers bound to estradiol (purple), 4-hydroxytamoxifen (lime green), genistein (pink), lasofoxifene (forest green). Secondary structural elements are colored by their contribution to the difference in entropy from the wild-type apo state. $T\Delta S_{\text{conf}}$ is the ciMIST calculation of change in conformational entropy relative to the apo state, with posterior standard deviation as the stated uncertainty. $T\Delta S_{ITC}$ denotes experimentally-measured binding entropy (64), for which uncertainties are standard deviations of three replicates. B) First three principal components of changes in entropy contributions of the set of liganded states shown here as well as the Y537S apo state from the wild-type apo state. C) Projection of residue entropy contribution vectors for $\text{ER}\alpha$ states onto the first two principal components. D) Correlation between average contributions of residues in protein segments to binding conformational entropy and average percent difference in deuteration levels between apo and liganded states (3, 63). The gray band demarcates 95% prediction intervals. $T=298K$.

figure 6:. Evolutionary divergence of entropic contributions in the steroid receptor ligand-binding domains.

A) Maximum information spanning trees for human steroid receptor ligand-binding domains bound to endogenous ligands. Structures are oriented as $\text{ER}\alpha$ is in figure 3A. Sphere volumes of $\text{C}\alpha$ atoms (shades of green) are proportional to residue marginal entropies, while volumes of links (shades of brown) are proportional to mutual information. B) Human steroid receptor phylogeny and application of the UPGMA clustering algorithm to four different similarity matrices calculated for human steroid receptors. C) The first two principal components of the aligned entropy contributions.