Binding Energy Distribution Analysis Method: Hamiltonian Replica Exchange with Torsional Flattening for Binding Mode Prediction and Binding Free Energy Estimation.

Molecular dynamics modeling of complex biological systems is limited by finite simulation time. The simulations are often trapped close to local energy minima separated by high energy barriers. Here, we introduce Hamiltonian replica exchange (H-REMD) with torsional flattening in the Binding Energy Distribution Analysis Method (BEDAM), to reduce energy barriers along torsional degrees of freedom and accelerate sampling of intramolecular degrees of freedom relevant to protein-ligand binding. The method is tested on a standard benchmark (T4 Lysozyme/L99A/p-xylene complex) and on a library of HIV-1 integrase complexes derived from the SAMPL4 blind challenge. We applied the torsional flattening strategy to 26 of the 53 known binders to the HIV Integrase LEDGF site found to have a binding energy landscape funneled toward the crystal structure. We show that our approach samples the conformational space more efficiently than the original method without flattening when starting from a poorly docked pose with incorrect ligand dihedral angle conformations. In these unfavorable cases convergence to a binding pose within 2-3 Å from the crystallographic pose is obtained within a few nanoseconds of the Hamiltonian replica exchange simulation. We found that torsional flattening is insufficient in cases where trapping is due to factors other than torsional energy, such as the formation of incorrect intramolecular hydrogen bonds and stacking. Work is in progress to generalize the approach to handle these cases and thereby make it more widely applicable.

How long does it take to equilibrate the unfolded state of a protein?

How long does it take to equilibrate the unfolded state of a protein? The answer to this question has important implications for our understanding of why many small proteins fold with two state kinetics. When the equilibration within the unfolded state U is much faster than the folding, the folding kinetics will be two state even if there are many folding pathways with different barriers. Yet the mean first passage times (MFPTs) between different regions of the unfolded state can be much longer than the folding time. This seems to imply that the equilibration within U is much slower than the folding. In this communication we resolve this paradox. We present a formula for estimating the time to equilibrate the unfolded state of a protein. We also present a formula for the MFPT to any state within U, which is proportional to the average lifetime of that state divided by the state population. This relation is valid when the equilibration within U is very fast as compared with folding as it often is for small proteins. To illustrate the concepts, we apply the formulas to estimate the time to equilibrate the unfolded state of Trp-cage and MFPTs within the unfolded state based on a Markov State Model using an ultra-long 208 microsecond trajectory of the miniprotein to parameterize the model. The time to equilibrate the unfolded state of Trp-cage is ∼100 ns while the typical MFPTs within U are tens of microseconds or longer.

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.

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.

Elucidating the energetics of entropically driven protein-ligand association: calculations of absolute binding free energy and entropy.

The binding of proteins and ligands is generally associated with the loss of translational, rotational, and conformational entropy. In many cases, however, the net entropy change due to binding is positive. To develop a deeper understanding of the energetics of entropically driven protein-ligand binding, we calculated the absolute binding free energies and binding entropies for two HIV-1 protease inhibitors Nelfinavir and Amprenavir using the double-decoupling method with molecular dynamics simulations in explicit solvent. For both ligands, the calculated absolute binding free energies are in general agreement with experiments. The statistical error in the computed ΔG(bind) due to convergence problem is estimated to be ≥2 kcal/mol. The decomposition of free energies indicates that, although the binding of Nelfinavir is driven by nonpolar interaction, Amprenavir binding benefits from both nonpolar and electrostatic interactions. The calculated absolute binding entropies show that (1) Nelfinavir binding is driven by large entropy change and (2) the entropy of Amprenavir binding is much less favorable compared with that of Nelfinavir. Both results are consistent with experiments. To obtain qualitative insights into the entropic effects, we decomposed the absolute binding entropy into different contributions based on the temperature dependence of free energies along different legs of the thermodynamic pathway. The results suggest that the favorable entropic contribution to binding is dominated by the ligand desolvation entropy. The entropy gain due to solvent release from binding site appears to be more than offset by the reduction of rotational and vibrational entropies upon binding.

Insights into the dynamics of HIV-1 protease: a kinetic network model constructed from atomistic simulations.

The conformational dynamics in the flaps of HIV-1 protease plays a crucial role in the mechanism of substrate binding. We develop a kinetic network model, constructed from detailed atomistic simulations, to determine the kinetic mechanisms of the conformational transitions in HIV-1 PR. To overcome the time scale limitation of conventional molecular dynamics (MD) simulations, our method combines replica exchange MD with transition path theory (TPT) to study the diversity and temperature dependence of the pathways connecting functionally important states of the protease. At low temperatures the large-scale flap opening is dominated by a small number of paths; at elevated temperatures the transition occurs through many structurally heterogeneous routes. The expanded conformation in the crystal structure 1TW7 is found to closely mimic a key intermediate in the flap-opening pathways at low temperature. We investigated the different transition mechanisms between the semi-open and closed forms. The calculated relaxation times reveal fast semi-open ↔ closed transitions, and infrequently the flaps fully open. The ligand binding rate predicted from this kinetic model increases by 38-fold from 285 to 309 K, which is in general agreement with experiments. To our knowledge, this is the first application of a network model constructed from atomistic simulations together with TPT to analyze conformational changes between different functional states of a natively folded protein.

Free energy profile of RNA hairpins: a molecular dynamics simulation study.

RNA hairpin loops are one of the most abundant secondary structure elements and participate in RNA folding and protein-RNA recognition. To characterize the free energy surface of RNA hairpin folding at an atomic level, we calculated the potential of mean force (PMF) as a function of the end-to-end distance, by using umbrella sampling simulations in explicit solvent. Two RNA hairpins containing tetraloop cUUCGg and cUUUUg are studied with AMBER ff99 and CHARMM27 force fields. Experimentally, the UUCG hairpin is known to be significantly more stable than UUUU. In this study, the calculations using AMBER force field give a qualitatively correct description for the folding of two RNA hairpins, as the calculated PMF confirms the global stability of the folded structures and the resulting relative folding free energy is in quantitative agreement with the experimental result. The hairpin stabilities are also correctly differentiated by the more rapid molecular mechanics-Poisson Boltzmann-surface area approach, but the relative free energy estimated from this method is overestimated. The free energy profile shows that the native state basin and the unfolded state plateau are separated by a wide shoulder region, which samples a variety of native-like structures with frayed terminal basepair. The calculated PMF lacks major barriers that are expected near the transition regions, and this is attributed to the limitation of the 1-D reaction coordinate. The PMF results are compared with other studies of small RNA hairpins using kinetics method and coarse grained models. The two RNA hairpins described by CHARMM27 are significantly more deformable than those represented by AMBER. Compared with the AMBER results, the CHARMM27 calculated DeltaG(fold) for the UUUU tetraloop is in better agreement with the experimental results. However, the CHARMM27 calculation does not confirm the global stability of the experimental UUCG structure; instead, the extended conformations are predicted to be thermodynamically stable in solution. This finding is further supported by separate unrestrained CHARMM27 simulations, in which the UUCG hairpin unfolds spontaneously within 10 ns. The instability of the UUCG hairpin originates from the loop region, and propagates to the stem. The results of this study provide a molecular picture of RNA hairpin unfolding and reveal problems in the force field descriptions for the conformational energy of certain RNA hairpin.

Insights into affinity and specificity in the complexes of alpha-lytic protease and its inhibitor proteins: binding free energy from molecular dynamics simulation.

We report the binding free energy calculation and its decomposition for the complexes of alpha-lytic protease and its protein inhibitors using molecular dynamics simulation. Standard mechanism serine protease inhibitors eglin C and OMTKY3 are known to have strong binding affinity for many serine proteases. Their binding loops have significant similarities, including a common P1 Leu as the main anchor in the binding interface. However, recent experiments demonstrate that the two inhibitors have vastly different affinity towards alpha-lytic protease (ALP), a bacterial serine protease. OMTKY3 inhibits the enzyme much more weakly (by approximately 10(6) times) than eglin C. Moreover, a variant of OMTKY3 with five mutations, OMTKY3M, has been shown to inhibit 10(4) times more strongly than the wild-type inhibitor. The underlying mechanisms for the unusually large difference in binding affinities and the effect of mutation are not well understood. Here we use molecular dynamics simulation with molecular mechanics-Poisson Boltzmann/surface area method (MM-PB/SA) to investigate quantitatively the binding specificity. The calculated absolute binding free energies correctly differentiate the thermodynamic stabilities of these protein complexes, but the magnitudes of the binding affinities are systematically overestimated. Analysis of the binding free energy components provides insights into the molecular mechanism of binding specificity. The large DeltaDeltaG(bind) between eglin C and wild type OMTKY3 towards ALP is mainly attributable to the stronger nonpolar interactions in the ALP-eglin C complex, arising from a higher degree of structural complementarity. Here the electrostatic interaction contributes to a lesser extent. The enhanced inhibition in the penta-mutant OMTKY3M over its wild type is entirely due to an overall improvement in the solvent-mediated electrostatic interactions in the ALP-OMTKY3M complex. The results suggest that for these protein-complexes and similar enzyme-inhibitor systems (1) the binding is driven by nonpolar interactions, opposed by overall electrostatic and solute entropy contributions; (2) binding specificity can be tuned by improving the complementarity in electrostatics between two associating proteins. Binding free energy decomposition into contributions from individual protein residues provides additional detailed information on the structural determinants and subtle conformational changes responsible for the binding specificity.

Molecular Dynamics and Free Energy Study of the Conformational Equilibria in the UUUU RNA Hairpin.

A series of molecular dynamics (MD) simulations was performed to elucidate the thermodynamic basis for the relative stabilities of hairpin, duplex, and single stranded forms of the 5'-CGC(UUUU)GCG-3' oligonucleotide. According to a recent NMR study this sequence exhibits dynamic conformational equilibrium in aqueous solution in the vicinity of room temperature. Free energy calculations using the molecular mechanics-Poisson Boltzmann-surface area (MM-PB/SA) approach support a shift in the conformational equilibrium from duplex to hairpin as the temperature is increased from 276 to 300 K, in agreement with the NMR results. The effect of added salt on the relative stabilities of RNA conformers is also reproduced by our calculations. The calculated ΔH° for the equilibrium between hairpin and single stranded forms is estimated to be -23.4 kcal/mol, in reasonable agreement with experimental values. Our results reveal that the conformational equilibrium strongly depends on the solute entropy and the electrostatic interactions modulated by added salt. Simulations of hairpin loop conducted at two different temperatures converged to the same lowest energy loop conformation. This conformer is stabilized by favorable van der Waals interactions as a result of U5-U6-U7 base stacking, a hydrogen bond between the U4 base and the phosphate linking U6 and U7, and hydrogen bonds involving the 2'OH groups at U4 and U6. However, the sugar pucker of the four uridines in the lowest energy conformer is different from that reported by a NMR study. While the NMR study found that U5 and U7 adopt the C2'-endo conformation, the simulation results suggests that overall the structure with the U5 and U7 in the C3'-endo conformation is thermodynamically more stable than the structure containing the C2'-endo pucker by approximately 8 kcal/mol. Calculations based on the MM-PB/SA scheme show that although the electrostatic solvation free energy favors the C2'-endo conformation for the U5 and U7 riboses, it is offset by the less favorable intramolecular electrostatic and van der Waals energies. To enhance the conformational sampling, a replica exchange molecular dynamics (REMD) simulation was conducted in a generalized Born (GB) continuum solvent for the hairpin loop. This simulation indicates that the stable loop structure observed in the explicit solvent simulations corresponds to the free energy minimum. It also reveals that while the U4, U5, and U6 sugar rings are predominantly in the C3'-endo conformation, there is considerable variation in the sugar pucker of the U7 ribose ring.

Molecular basis for the Cu2+ binding-induced destabilization of beta2-microglobulin revealed by molecular dynamics simulation.

According to experimental data, binding of the Cu(2+) ions destabilizes the native state of beta2-microglobulin (beta2m). The partial unfolding of the protein was generally considered an early step toward fibril formation in dialysis-related amyloidosis. Recent NMR studies have suggested that the destabilization of the protein might be achieved through increased flexibility upon Cu(2+) binding. However, the molecular mechanism of destabilization due to Cu(2+), its role in amyloid formation, and the relative contributions of different potential copper-binding sites remain unclear. To elucidate the effect of ion ligation at atomic detail, a series of molecular dynamics simulations were carried out on apo- and Cu(2+)-beta2m systems in explicit aqueous solutions, with varying numbers of bound ions. Simulations at elevated temperatures (360 K) provide detailed pictures for the process of Cu(2+)-binding-induced destabilization of the native structure at the nanosecond timescale, which are in agreement with experiments. Conformational transitions toward partially unfolded states were observed in protein solutions containing bound copper ions at His-31 and His-51, which is marked by an increase in the protein vibrational entropy, with TDeltaS(vibr) ranging from 30 to 69 kcal/mol. The binding of Cu(2+) perturbs the secondary structure and the hydrogen bonding pattern disrupts the native hydrophobic contacts in the neighboring segments, which include the beta-strand D2 and part of the beta-strand E, B, and C and results in greater exposure of the D-E loop and the B-C loop to the water environment. Analysis of the MD trajectories suggests that the changes in the hydrophobic environment near the copper-binding sites lower the barrier of conformational transition and stabilize the more disordered conformation. The results also indicate that the binding of Cu(2+) at His-13 has little effect on the conformational stability, whereas the copper-binding site His-31, and to a lesser extent His-51, are primarily responsible for the observed changes in the protein conformation and dynamics.