Molecular Dynamics of the Proline Switch and Its Role in Crk Signaling

The Crk adaptor proteins play a central role as a molecular timer for the formation of protein complexes including various growth and differentiation factors. The loss of regulation of Crk results in many kinds of cancers. A self-regulatory mechanism for Crk was recently proposed, which involves domain–domain rearrangement. It is initiated by a cis–trans isomerization of a specific proline residue (Pro238 in chicken Crk II) and can be accelerated by Cyclophilin A. To understand how the proline switch controls the autoinhibition at the molecular level, we performed large-scale molecular dynamics and metadynamics simulations in the context of short peptides and multidomain constructs of chicken Crk II. We found that the equilibrium and kinetic properties of the macrostates are regulated not only by the local environments of specified prolines but also by the global organization of multiple domains. We observe the two macrostates (cis closed/autoinhibited and trans open/uninhibited) consistent with NMR experiments and predict barriers. We also propose an intermediate state, the trans closed state, which interestingly was reported to be a prevalent state in human Crk II. The existence of this macrostate suggests that the rate of switching off the autoinhibition by Cyp A may be limited by the relaxation rate of this intermediate state

Improving Prediction Accuracy of Binding Free Energies and Poses of HIV Integrase Complexes Using the Binding Energy Distribution Analysis Method with Flattening Potentials

To accelerate conformation sampling of slow dynamics from receptor or ligand, we introduced flattening potentials on selected bonded and nonbonded intramolecular interactions to the binding energy distribution analysis method (BEDAM) for calculating absolute binding free energies of protein–ligand complexes using an implicit solvent model and implemented flattening BEDAM using the asynchronous replica exchange (AsyncRE) framework for performing large scale replica exchange molecular dynamics (REMD) simulations. The advantage of using the flattening feature to reduce high energy barriers was exhibited first by the <i>p</i>-xylene-T4 lysozyme complex, where the intramolecular interactions of a protein side chain on the binding site were flattened to accelerate the conformational transition of the side chain from the <i>trans</i> to the <i>gauche</i> state when the <i>p</i>-xylene ligand is present in the binding site. Much more extensive flattening BEDAM simulations were performed for 53 experimental binders and 248 nonbinders of HIV-1 integrase which formed the SAMPL4 challenge, with the total simulation time of 24.3 μs. We demonstrated that the flattening BEDAM simulations not only substantially increase the number of true positives (and reduce false negatives) but also improve the prediction accuracy of binding poses of experimental binders. Furthermore, the values of area under the curve (AUC) of receiver operating characteristic (ROC) and the enrichment factors at 20% cutoff calculated from the flattening BEDAM simulations were improved significantly in comparison with that of simulations without flattening as we previously reported for the whole SAMPL4 database. Detailed analysis found that the improved ability to discriminate the binding free energies between the binders and nonbinders is due to the fact that the flattening simulations reduce the reorganization free energy penalties of binders and decrease the overlap of binding free energy distributions of binders relative to that of nonbinders. This happens because the conformational ensemble distributions for both the ligand and protein in solution match those at the fully coupled (complex) state more closely when the systems are more fully sampled after the flattening potentials are applied to the intermediate states

How Kinetics within the Unfolded State Affects Protein Folding: An Analysis Based on Markov State Models and an Ultra-Long MD Trajectory

Understanding how kinetics in the unfolded state affects protein folding is a fundamentally important yet less well-understood issue. Here we employ three different models to analyze the unfolded landscape and folding kinetics of the miniprotein Trp-cage. The first is a 208 μs explicit solvent molecular dynamics (MD) simulation from D. E. Shaw Research containing tens of folding events. The second is a Markov state model (MSM-MD) constructed from the same ultralong MD simulation; MSM-MD can be used to generate thousands of folding events. The third is a Markov state model built from temperature replica exchange MD simulations in implicit solvent (MSM-REMD). All the models exhibit multiple folding pathways, and there is a good correspondence between the folding pathways from direct MD and those computed from the MSMs. The unfolded populations interconvert rapidly between extended and collapsed conformations on time scales ≤40 ns, compared with the folding time of ∼5 μs. The folding rates are independent of where the folding is initiated from within the unfolded ensemble. About 90% of the unfolded states are sampled within the first 40 μs of the ultralong MD trajectory, which on average explores ∼27% of the unfolded state ensemble between consecutive folding events. We clustered the folding pathways according to structural similarity into “tubes”, and kinetically partitioned the unfolded state into populations that fold along different tubes. From our analysis of the simulations and a simple kinetic model, we find that, when the mixing within the unfolded state is comparable to or faster than folding, the folding waiting times for all the folding tubes are similar and the folding kinetics is essentially single exponential despite the presence of heterogeneous folding paths with nonuniform barriers. When the mixing is much slower than folding, different unfolded populations fold independently, leading to nonexponential kinetics. A kinetic partition of the Trp-cage unfolded state is constructed which reveals that different unfolded populations have almost the same probability to fold along any of the multiple folding paths. We are investigating whether the results for the kinetics in the unfolded state of the 20-residue Trp-cage is representative of larger single domain proteins

NMR Relaxation in Proteins with Fast Internal Motions and Slow Conformational Exchange: Model-Free Framework and Markov State Simulations

Calculating NMR relaxation effects for proteins with dynamics on multiple time scales generally requires very long trajectories based on conventional molecular dynamics simulations. In this report, we have built Markov state models from multiple MD trajectories and used the resulting MSM to capture the very fast internal motions of the protein within a free energy basin on a time scale up to hundreds of picoseconds and the more than 3 orders of magnitude slower conformational exchange between macrostates. To interpret the relaxation data, we derive new equations using the model-free framework which includes two slowly exchanging macrostates, each of which also exhibits fast local motions. Using simulations of HIV-1 protease as an example, we show how the populations of slowly exchanging conformational states as well as order parameters for the different states can be determined from the NMR relaxation data

Contingency and Entrenchment of Drug-Resistance Mutations in HIV Viral Proteins

The ability of HIV-1 to rapidly mutate leads to antiretroviral therapy (ART) failure among infected patients. Drug-resistance mutations (DRMs), which cause a fitness penalty to intrinsic viral fitness, are compensated by accessory mutations with favorable epistatic interactions which cause an evolutionary trapping effect, but the kinetics of this overall process has not been well characterized. Here, using a Potts Hamiltonian model describing epistasis combined with kinetic Monte Carlo simulations of evolutionary trajectories, we explore how epistasis modulates the evolutionary dynamics of HIV DRMs. We show how the occurrence of a drug-resistance mutation is contingent on favorable epistatic interactions with many other residues of the sequence background and that subsequent mutations entrench DRMs. We measure the time-autocorrelation of fluctuations in the likelihood of DRMs due to epistatic coupling with the sequence background, which reveals the presence of two evolutionary processes controlling DRM kinetics with two distinct time scales. Further analysis of waiting times for the evolutionary trapping effect to reverse reveals that the sequences which entrench (trap) a DRM are responsible for the slower time scale. We also quantify the overall strength of epistatic effects on the evolutionary kinetics for different mutations and show these are much larger for DRM positions than polymorphic positions, and we also show that trapping of a DRM is often caused by the collective effect of many accessory mutations, rather than a few strongly coupled ones, suggesting the importance of multiresidue sequence variations in HIV evolution. The analysis presented here provides a framework to explore the kinetic pathways through which viral proteins like HIV evolve under drug-selection pressure

Contingency and Entrenchment of Drug-Resistance Mutations in HIV Viral Proteins

The ability of HIV-1 to rapidly mutate leads to antiretroviral therapy (ART) failure among infected patients. Drug-resistance mutations (DRMs), which cause a fitness penalty to intrinsic viral fitness, are compensated by accessory mutations with favorable epistatic interactions which cause an evolutionary trapping effect, but the kinetics of this overall process has not been well characterized. Here, using a Potts Hamiltonian model describing epistasis combined with kinetic Monte Carlo simulations of evolutionary trajectories, we explore how epistasis modulates the evolutionary dynamics of HIV DRMs. We show how the occurrence of a drug-resistance mutation is contingent on favorable epistatic interactions with many other residues of the sequence background and that subsequent mutations entrench DRMs. We measure the time-autocorrelation of fluctuations in the likelihood of DRMs due to epistatic coupling with the sequence background, which reveals the presence of two evolutionary processes controlling DRM kinetics with two distinct time scales. Further analysis of waiting times for the evolutionary trapping effect to reverse reveals that the sequences which entrench (trap) a DRM are responsible for the slower time scale. We also quantify the overall strength of epistatic effects on the evolutionary kinetics for different mutations and show these are much larger for DRM positions than polymorphic positions, and we also show that trapping of a DRM is often caused by the collective effect of many accessory mutations, rather than a few strongly coupled ones, suggesting the importance of multiresidue sequence variations in HIV evolution. The analysis presented here provides a framework to explore the kinetic pathways through which viral proteins like HIV evolve under drug-selection pressure

Large Scale Affinity Calculations of Cyclodextrin Host–Guest Complexes: Understanding the Role of Reorganization in the Molecular Recognition Process

Host–guest inclusion complexes are useful models for understanding the structural and energetic aspects of molecular recognition. Due to their small size relative to much larger protein–ligand complexes, converged results can be obtained rapidly for these systems thus offering the opportunity to more reliably study fundamental aspects of the thermodynamics of binding. In this work, we have performed a large scale binding affinity survey of 57 β-cyclodextrin (CD) host–guest systems using the binding energy distribution analysis method (BEDAM) with implicit solvation (OPLS-AA/AGBNP2). Converged estimates of the standard binding free energies are obtained for these systems by employing techniques such as parallel Hamiltonian replica exchange molecular dynamics, conformational reservoirs, and multistate free energy estimators. Good agreement with experimental measurements is obtained in terms of both numerical accuracy and affinity rankings. Overall, average effective binding energies reproduce affinity rank ordering better than the calculated binding affinities, even though calculated binding free energies, which account for effects such as conformational strain and entropy loss upon binding, provide lower root-mean-square errors when compared to measurements. Interestingly, we find that binding free energies are superior rank order predictors for a large subset containing the most flexible guests. The results indicate that, while challenging, accurate modeling of reorganization effects can lead to ligand design models of superior predictive power for rank ordering relative to models based only on ligand–receptor interaction energies

A Stochastic Solution to the Unbinned WHAM Equations

The weighted histogram analysis method (WHAM) and unbinned versions such as the multistate Bennett acceptance ratio (MBAR) and unbinned WHAM (UWHAM) are widely used to compute free energies and expectations from data generated by independent or coupled parallel simulations. Here we introduce a replica exchange-like algorithm (RE-SWHAM) that can be used to solve the UWHAM equations stochastically. This method is capable of analyzing large data sets generated by hundreds or even thousands of parallel simulations that are too large to be “WHAMMED” using standard methods. We illustrate the method by applying it to obtain free energy weights for each of the 240 states in a simulation of host–guest ligand binding containing ∼3.5 × 10⁷ data elements collected from 16 parallel Hamiltonian replica exchange simulations, performed at 15 temperatures. In addition to using much less memory, RE-SWHAM showed a nearly 80-fold improvement in computational time compared with UWHAM

Distance between like and unlike-charge pairs as a function of the statistical coupling parameter, .

<p>The statistical coupling parameter is a fitting parameter that describes the statistical interaction energy between pairs of states. Since the Bethe mean field pair correlation model is a good approximation for this data, a negative indicates that a pair of states is enhanced (positively correlated), while a positive indicates that a pair of states is suppressed (negatively correlated). Using simple electrostatics, we observe that like-charge patterns (blue) are mostly suppressed while unlike-charge patterns (red) are enhanced. The sign of is able to correctly predict the charge patterns for of the top 35 most significantly correlated charge pairs out of a total of 135 pairs. The p-value for the statistical significance of this result is . The reason there are 135 pairs is as follows: For each pair of residues, there are 4 possible sets of like and unlike charge combinations, resulting in a total of 612 like/unlike charge pair combinations. However, not all pairs exist in the database or are significantly correlated. Filtering results in 135 pairs with probability greater than 0.001%.</p

Comparison of the sequence probabilities in the tail of the Lee database and the pair correlation model with the sequence probabilities in the Stanford database.

<p>The probabilities of sequences under the pair correlation model, , predicted using the Bethe approximation, are plotted as a function of the sequence probabilities from the Lee database, . Sequences with a probability of 0 in the Lee database, i.e. unobserved sequences, are plotted to the left of the abscissa break. Every sequence is shaded using a color gradient corresponding to , which represents the number of times the sequence occurs in the Stanford database, relative to its probability in the Lee database. Sequences that occur frequently in the Stanford database as compared to the Lee database have a higher ratio and are shaded red, while the sequences that do not occur as frequently in the Stanford database as compared to the Lee database have a lower ratio and are shaded blue. Sequences that are shaded green have equal probabilities in both databases. Sequences unobserved in the Lee database (leftmost row in the graph), but observed in the Stanford database have a ratio that is artificially set to equal 4, which corresponds to the color red. Unobserved Lee sequences that are also unobserved in the Stanford database are shaded green because . Sequences with probabilities are shaded according to the average value of for a window of 10 sequences around the sequence of interest. Sequences with probabilities or are not shown. The indices (0), (1), (2), etc mark the locations of sequences observed zero, once, twice (etc) in the Lee database. Each dot corresponds to a unique sequence.</p