Comparing Conformational Ensembles Using the Kullback–Leibler Divergence Expansion

Abstract

We present a thermodynamical approach to identify changes in macromolecular structure and dynamics in response to perturbations such as mutations or ligand binding, using an expansion of the Kullback-Leibler Divergence that connects local population shifts in torsion angles to changes in the free energy landscape of the protein. While the Kullback-Leibler Divergence is a known formula from information theory, the novelty and power of our implementation lies in its formal developments, connection to thermodynamics, statistical filtering, ease of visualization of results, and extendability by adding higher-order terms. We present a formal derivation of the Kullback-Leibler Divergence expansion and then apply our method at a first-order approximation to molecular dynamics simulations of four protein systems where ligand binding or pH titration is known to cause an effect at a distant site. Our results qualitatively agree with experimental measurements of local changes in structure or dynamics, such as NMR chemical shift perturbations and hydrogen-deuterium exchange mass spectrometry. The approach produces easy-to-analyze results with low background, and as such has the potential to become a routine analysis when molecular dynamics simulations in two or more conditions are available. Our method is implemented in the MutInf code package and is available on the SimTK website at https://simtk.org/home/mutinf.

Dynamic architecture of a protein kinase

“…In the case of glycine, the carbonyl oxygen was used as a second atom because this atom is most distal from the Cα atom. The matrix of mutual information among these atoms was calculated with the MutInf software package (30,95). To construct the probability densities required to obtain the mutual information values, we used uniformly binned histograms with 24 bins per degree-of-freedom.…”

Crystal Cryocooling Distorts Conformational Heterogeneity in a Model Michaelis Complex of DHFR

“…To fill this gap, the program CONTACT identifies steric clashes and reliefs between adjacent alternate conformations in multiconformer models to build networks of energetically coupled residues, which are often relevant to protein function (van den Bedem et al, 2013). Similarly, the program MutInf (McClendon et al, 2012) identifies statistically correlated torsion angles in traditional molecular dynamics simulations, and could be adapted to identify correlations in crystallographic ensemble models (Burnley et al, 2012). …”

Finding Semirigid Domains in Biomolecules by Clustering Pair-Distance Variations

“…Now, we evaluate which atoms migrate. For quantification, we resort to the Kullback-Leibler-distance [36, 37]: $\begin{array}{c}{\text{KLD}}_i=\underset{m=1}{\overset{N_{\text{clust}}}{{\displaystyle false\sum }}}{p}_{im}·\log \frac{p_{im}}{1/N_{\text{clust}}}.\end{array}$1/ N clust represents the assumed background probability if the assignment of atom i to any of the clusters were equally probable. p im is the actual probability for atom i to belong to cluster m , estimated from an average over cluster memberships c im obtained from clustering subsets of the trajectory: $\begin{array}{c}{p}_{im}=\langle c_{im}\rangle .\end{array}$ Large values of KLD i indicate that atom i stays predominantly in the same cluster throughout the trajectory.…”