Modeling conformational ensembles of slow functional motions in Pin1-WW

Hierarchical Nyström methods for constructing Markov state models for conformational dynamics

Markov state models (MSMs) have become a popular approach for investigating the conformational dynamics of proteins and other biomolecules. MSMs are typically built from numerous molecular dynamics simulations by dividing the sampled configurations into a large number of microstates based on geometric criteria. The resulting microstate model can then be coarse-grained into a more understandable macrostate model by lumping together rapidly mixing microstates into larger, metastable aggregates. However, finite sampling often results in the creation of many poorly sampled microstates. During coarse-graining, these states are mistakenly identified as being kinetically important because transitions to/from them appear to be slow. In this paper, we propose a formalism based on an algebraic principle for matrix approximation, i.e., the Nyström method, to deal with such poorly sampled microstates. Our scheme builds a hierarchy of microstates from high to low populations and progressively applies spectral clustering on sets of microstates within each level of the hierarchy. It helps spectral clustering identify metastable aggregates with highly populated microstates rather than being distracted by lowly populated states. We demonstrate the ability of this algorithm to discover the major metastable states on two model systems, the alanine dipeptide and trpzip2 peptide.

Dynamics of an intrinsically disordered protein reveal metastable conformations that potentially seed aggregation

Amyloid fibril deposits of the intrinsically disordered hIAPP peptide are found in 95% of type II diabetes patients, and the aggregation of this peptide is suggested to induce apoptotic cell-death in insulin-producing β-cells. Understanding the structure and dynamics of the hIAPP monomer in solution is thus important for understanding the nucleation of aggregation and the formation of oligomers. In this study, we identify the metastable conformational states of the hIAPP monomer and the dynamics of transitioning between them using Markov state models constructed from extensive molecular dynamics simulations. We show that the overall structure of the hIAPP peptide is random coil-like and lacks a dominant folded structure. Despite this fact, our model reveals a large number of reasonably well-populated metastable conformational states (or local free energy minima) having populations of a few percent or less. The time scales for transitioning between these states range from several microseconds to milliseconds. In contrast to folded proteins, there is no kinetic hub. More strikingly, a few states contain significant amounts of β-hairpin secondary structure and extended hydrophobic surfaces that are exposed to the solvent. We propose that these states may facilitate the nucleation of hIAPP aggregation through a significant component of the conformational selection mechanism, because they may increase their populations upon aggregation by promoting hydrophobic interactions and at the same time provide a flat geometry to seed the ordered β-strand packing of the fibrils.

Network visualization of conformational sampling during molecular dynamics simulation

Effective data reduction methods are necessary for uncovering the inherent conformational relationships present in large molecular dynamics (MD) trajectories. Clustering algorithms provide a means to interpret the conformational sampling of molecules during simulation by grouping trajectory snapshots into a few subgroups, or clusters, but the relationships between the individual clusters may not be readily understood. Here we show that network analysis can be used to visualize the dominant conformational states explored during simulation as well as the connectivity between them, providing a more coherent description of conformational space than traditional clustering techniques alone. We compare the results of network visualization against 11 clustering algorithms and principal component conformer plots. Several MD simulations of proteins undergoing different conformational changes demonstrate the effectiveness of networks in reaching functional conclusions.

Transient domain interactions enhance the affinity of the mitotic regulator Pin1 toward phosphorylated peptide ligands

The mitotic regulator Pin1 plays an important role in protein quality control and age-related medical conditions such as Alzheimer disease and Parkinson disease. Although its cellular role has been thoroughly investigated during the past decade, the molecular mechanisms underlying its function remain elusive. We provide evidence for interactions between the two domains of Pin1. Several residues displayed unequivocal peak splits in nuclear magnetic resonance spectra, indicative of two different conformational states in equilibrium. Pareto analysis of paramagnetic relaxation enhancement data demonstrates that the two domains approach each other upon addition of a nonpeptidic ligand. Titration experiments with phosphorylated peptides monitored by fluorescence anisotropy and chemical shift perturbation indicate that domain interactions increase Pin1's affinity toward peptide ligands. We propose this interplay of the domains and ligands to be a general mechanism for a large class of two-domain proteins.

Conformational equilibrium of N-myristoylated cAMP-dependent protein kinase A by molecular dynamics simulations

The catalytic subunit of protein kinase A (PKA-C) is subject to several post- or cotranslational modifications that regulate its activity both spatially and temporally. Among those, N-myristoylation increases the kinase affinity for membranes and might also be implicated in substrate recognition and allosteric regulation. Here, we investigated the effects of N-myristoylation on the structure, dynamics, and conformational equilibrium of PKA-C using atomistic molecular dynamics simulations. We found that the myristoyl group inserts into the hydrophobic pocket and leads to a tighter packing of the A-helix against the core of the enzyme. As a result, the conformational dynamics of the A-helix are reduced and its motions are more coupled with the active site. Our simulations suggest that cation-π interactions among W30, R190, and R93 are responsible for coupling these motions. Two major conformations of the myristoylated N-terminus are the most populated: a long loop (LL conformation), similar to Protein Data Bank (PDB) entry 1CMK , and a helix-turn-helix structure (HTH conformation), similar to PDB entry 4DFX , which shows stronger coupling between the conformational dynamics observed at the A-helix and active site. The HTH conformation is stabilized by S10 phosphorylation of the kinase via ionic interactions between the protonated amine of K7 and the phosphate group on S10, further enhancing the dynamic coupling to the active site. These results support a role of N-myristoylation in the allosteric regulation of PKA-C.

To milliseconds and beyond: challenges in the simulation of protein folding

Quantitatively accurate all-atom molecular dynamics (MD) simulations of protein folding have long been considered a holy grail of computational biology. Due to the large system sizes and long timescales involved, such a pursuit was for many years computationally intractable. Further, sufficiently accurate forcefields needed to be developed in order to realistically model folding. This decade, however, saw the first reports of folding simulations describing kinetics on the order of milliseconds, placing many proteins firmly within reach of these methods. Progress in sampling and forcefield accuracy, however, presents a new challenge: how to turn huge MD datasets into scientific understanding. Here, we review recent progress in MD simulation techniques and show how the vast datasets generated by such techniques present new challenges for analysis. We critically discuss the state of the art, including reaction coordinate and Markov state model (MSM) methods, and provide a perspective for the future.

A fast parallel clustering algorithm for molecular simulation trajectories

We implemented a GPU-powered parallel k-centers algorithm to perform clustering on the conformations of molecular dynamics (MD) simulations. The algorithm is up to two orders of magnitude faster than the CPU implementation. We tested our algorithm on four protein MD simulation datasets ranging from the small Alanine Dipeptide to a 370-residue Maltose Binding Protein (MBP). It is capable of grouping 250,000 conformations of the MBP into 4000 clusters within 40 seconds. To achieve this, we effectively parallelized the code on the GPU and utilize the triangle inequality of metric spaces. Furthermore, the algorithm's running time is linear with respect to the number of cluster centers. In addition, we found the triangle inequality to be less effective in higher dimensions and provide a mathematical rationale. Finally, using Alanine Dipeptide as an example, we show a strong correlation between cluster populations resulting from the k-centers algorithm and the underlying density. © 2012 Wiley Periodicals, Inc.

Equilibrium fluctuations of a single folded protein reveal a multitude of potential cryptic allosteric sites

Cryptic allosteric sites--transient pockets in a folded protein that are invisible to conventional experiments but can alter enzymatic activity via allosteric communication with the active site--are a promising opportunity for facilitating drug design by greatly expanding the repertoire of available drug targets. Unfortunately, identifying these sites is difficult, typically requiring resource-intensive screening of large libraries of small molecules. Here, we demonstrate that Markov state models built from extensive computer simulations (totaling hundreds of microseconds of dynamics) can identify prospective cryptic sites from the equilibrium fluctuations of three medically relevant proteins--β-lactamase, interleukin-2, and RNase H--even in the absence of any ligand. As in previous studies, our methods reveal a surprising variety of conformations--including bound-like configurations--that implies a role for conformational selection in ligand binding. Moreover, our analyses lead to a number of unique insights. First, direct comparison of simulations with and without the ligand reveals that there is still an important role for an induced fit during ligand binding to cryptic sites and suggests new conformations for docking. Second, correlations between amino acid sidechains can convey allosteric signals even in the absence of substantial backbone motions. Most importantly, our extensive sampling reveals a multitude of potential cryptic sites--consisting of transient pockets coupled to the active site--even in a single protein. Based on these observations, we propose that cryptic allosteric sites may be even more ubiquitous than previously thought and that our methods should be a valuable means of guiding the search for such sites.

Exposing the Moving Parts of Proteins with NMR Spectroscopy

Nuclear Magnetic Resonance (NMR) spectroscopy is a powerful tool for investigating the dynamics of biomolecules since it provides a description of motion that is comprehensive, site-specific, and relatively non-invasive. In particular, the study of protein dynamics has benefited from sustained methodological advances in NMR that have expanded the scope and time scales of accessible motion. Yet, many of these advances may not be well known to the more general physical chemistry community. Accordingly, this Perspective provides a glimpse of some of the more powerful methods in liquid state NMR that are helping reshape our understanding of functional motions of proteins.

Comparing Conformational Ensembles Using the Kullback-Leibler Divergence Expansion

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.

Quasi-anharmonic analysis reveals intermediate states in the nuclear co-activator receptor binding domain ensemble

The molten globule nuclear receptor co-activator binding domain (NCBD) of CREB binding protein (CBP) selectively recruits transcription co-activators (TCAs) during the formation of the transcription preinitiation complex. NCBD:TCA interactions have been implicated in several cancers, however, the mechanisms of NCBD:TCA recognition remain uncharacterized. NCBD:TCA intermolecular recognition has challenged traditional investigation as both NCBD and several of its corresponding TCAs are intrinsically disordered. Using 40μs of explicit solvent molecular dynamics simulations, we relate the conformational diversity of ligand-free NCBD to its bound configurations. We introduce two novel techniques to quantify the conformational heterogeneity of ligand-free NCBD, dihedral quasi-anharmonic analysis (dQAA) and hierarchical graph-based diffusive clustering. With this integrated approach we find that three of four ligand-bound states are natively accessible to the ligand-free NCBD simulations with root-mean squared deviation (RMSD) less than 2Å These conformations are accessible via diverse pathways while a rate-limiting barrier must be crossed in order to access the fourth bound state.

Longer simulations sample larger subspaces of conformations while maintaining robust mechanisms of motion

Recent studies suggest that protein motions observed in molecular simulations are related to biochemical activities, although the computed time scales do not necessarily match those of the experimentally observed processes. The molecular origin of this conflicting observation is explored here for a test protein, cyanovirin-N (CV-N), through a series of molecular dynamics simulations that span a time range of three orders of magnitude up to 0.4 μs. Strikingly, increasing the simulation time leads to an approximately uniform amplification of the motional sizes, while maintaining the same conformational mechanics. Residue fluctuations exhibit amplitudes of 1-2 Å in the nanosecond simulations, whereas their average sizes increase by a factor of 4-5 in the microsecond regime. The mean-square displacements averaged over all residues (y) exhibit a power law dependence of the form y ∝ x(0.26) on the simulation time (x). Essential dynamics analysis of the trajectories, on the other hand, demonstrates that CV-N has robust preferences to undergo specific types of motions that already can be detected at short simulation times, provided that multiple runs are performed and carefully analyzed.

Markov state model reveals folding and functional dynamics in ultra-long MD trajectories

Two strategies have been recently employed to push molecular simulation to long, biologically relevant time scales: projection-based analysis of results from specialized hardware producing a small number of ultralong trajectories and the statistical interpretation of massive parallel sampling performed with Markov state models (MSMs). Here, we assess the MSM as an analysis method by constructing a Markov model from ultralong trajectories, specifically two previously reported 100 μs trajectories of the FiP35 WW domain (Shaw, D. E. Science 2010, 330, 341-346). We find that the MSM approach yields novel insights. It discovers new statistically significant folding pathways, in which either beta-hairpin of the WW domain can form first. The rates of this process approach experimental values in a direct quantitative comparison (time scales of 5.0 μs and 100 ns), within a factor of ∼2. Finally, the hub-like topology of the MSM and identification of a holo conformation predicts how WW domains may function through a conformational selection mechanism.

Probing molecular kinetics with Markov models: metastable states, transition pathways and spectroscopic observables

Markov (state) models (MSMs) have attracted a lot of interest recently as they (1) can probe long-term molecular kinetics based on short-time simulations, (2) offer a way to analyze great amounts of simulation data with relatively little subjectivity of the analyst, (3) provide insight into microscopic quantities such as the ensemble of transition pathways, and (4) allow simulation data to be reconciled with measurement data in a rigorous and explicit way. Here we sketch our current perspective of Markov models and explain in short their theoretical basis and assumptions. We describe transition path theory which allows the entire ensemble of protein folding pathways to be investigated and that combines naturally with Markov models. Experimental observations can be naturally linked to Markov models with the dynamical fingerprint theory, by which experimentally observable timescales can be equipped with an understanding of the structural rearrangement processes that take place at these timescales. The concepts of this paper are illustrated by a simple kinetic model of protein folding.

A role for both conformational selection and induced fit in ligand binding by the LAO protein

Molecular recognition is determined by the structure and dynamics of both a protein and its ligand, but it is difficult to directly assess the role of each of these players. In this study, we use Markov State Models (MSMs) built from atomistic simulations to elucidate the mechanism by which the Lysine-, Arginine-, Ornithine-binding (LAO) protein binds to its ligand. We show that our model can predict the bound state, binding free energy, and association rate with reasonable accuracy and then use the model to dissect the binding mechanism. In the past, this binding event has often been assumed to occur via an induced fit mechanism because the protein's binding site is completely closed in the bound state, making it impossible for the ligand to enter the binding site after the protein has adopted the closed conformation. More complex mechanisms have also been hypothesized, but these have remained controversial. Here, we are able to directly observe roles for both the conformational selection and induced fit mechanisms in LAO binding. First, the LAO protein tends to form a partially closed encounter complex via conformational selection (that is, the apo protein can sample this state), though the induced fit mechanism can also play a role here. Then, interactions with the ligand can induce a transition to the bound state. Based on these results, we propose that MSMs built from atomistic simulations may be a powerful way of dissecting ligand-binding mechanisms and may eventually facilitate a deeper understanding of allostery as well as the prediction of new protein-ligand interactions, an important step in drug discovery.