New theoretical techniques are being developed and characterized. These efforts are usually coupled with software development, and involve the systematic testing and evaluation of new ideas. Estimating accurate binding free energy calculations by constant-pH simulation Solution pH is one of the most important environmental factors that regulate enzymatic activities and protein-ligand binding. Recently, from the community-wide blind binding free energy calculation experiment (SAMPL3), it was identified that determining a correct protonation state is critical in obtaining accurate binding free energy estimates. However, conventional binding free energy methods have almost exclusively depended on a fixed charge model in which protonation states and partial charges are determined before a free energy calculation and remain constant during the calculation. However, this approach cannot reflect the effect of solution pH and existence of multiple protonation states properly. To address this problem, we developed a new binding free energy calculation scheme that includes the effects of multiple protonation states. In this scheme, any kind of conventional binding free energy calculation can be combined with our newly developed constant-pH method. Free energy changes associated with constraining multiple protonation states into one and allowing multiple protonation states from a single state are added to a binding free energy calculated with a fixed charge model. We verified that our new scheme accurately reproduces a pH-dependent binding free energy profile. Developing new replica-exchange methodology We have utilized the new 2-dimensional replica exchange method that we implemented in CHARMM to perform 2D-EDS-HREM calculations on a variety of systems. In this method, exchanges are performed between different EDS potentials as well as different pH conditions. This method preserves the semi-grand canonical ensemble of the system being studied. We have shown that the number of protonation state transitions is significantly improved, especially at pH values that are far away from the pKa of the titratable groups of the system being studied. This increased sampling improves the accuracy of the method. Accuracy at these extreme pH values is especially important for accurately calculating titration curves. We have shown that this method works well calculating the pKa values of electrostatically coupled groups. We have also made the interesting discovery that the increased sampling in protonation-state space also enhances conformational sampling, especially of the orientations of titratable residues. Reservoir pH replica exchange Reservoir pH replica exchange (R-pH-REM) is a new method for constant pH simulations. Within the framework of the R-pH-REM method, the problem of conformational sampling is decoupled from the constant pH simulation problem. The method relies on pre-generation of one or two structural reservoirs of conformations that correspond to end states at pH values where titratable groups are fully protonated and fully deprotonated. We demonstrate that the complicated problem of sampling of the conformational/protonation equilibrium can be separated into two separate steps: a sampling step, where regular or enhanced MD simulations with fixed charges are used to generate reservoirs of conformations, and a coupling step, where the reservoir is coupled to pH-REM simulations to obtain adequate conformational sampling at different pH values. Free energy methods with ab initio QM/MM methods Free energy simulations are an important tool in the arsenal of computational biophysics, allowing the calculation of thermodynamic properties of binding or enzymatic reactions. Recently developed multiscale free energy simulations between molecular mechanics (MM) and quantum mechanics (QM) show a clear way to significantly improve the accuracy of free energy simulations. However, the reweighting process commonly used in multiscaling is very sensitive to the phase space overlap between the two Hamiltonians. One of the major factors that leads to lack of phase space overlap are the bonded terms. In particular, very small deviations of the equilibrium bond length or bond angle can lead to large potential energy fluctuations. One of the most efficient strategies to deal with this problem is to use constraints on the bonds and angles and to employ normal mode analysis to account for the differences of the bonded terms. Since the Hessian can be approximated with MM, while the gradients can be obtained from QM for only about 30% of additional costs, this approach can determine the right equilibrium bond lengths and angles in just one step of Newton Raphson minimization. By also accounting for the vibrational entropy and the Jacobian factor, the right free energy difference is obtained. A generalized self-guided Langevin dynamics simulation method Langevin dynamics (SGLD) was developed for enhanced conformational search. This method makes rare events that otherwise not accessible by regular dynamics simulation observable with current available computing resources. A major challenge when applying SGMD/SGLD method in simulation studies is that how to quantitatively measure ensemble deviation. By analyzing the characters of the guiding force and SGLD simulation behavior, we derived a thermodynamic relation between a regular simulation and a self-guided simulation. By separating low frequency part from high frequency part in a property, we are able to quantitative describe the enhancement in low frequency motion and the alternation in conformational distribution. A virtual mixture simulation approach to study multi-state equilibrium: Application to constant pH simulation in explicit water This method constructs a virtual mixture of multiple states (VMMS) to sample the conformational space of all chemical states simultaneously. The VMMS system consists of multiple subsystems, one for each state. The subsystem contains a solute and a solvent environment. The solute molecules in all subsystems share the same conformation but have their own solvent environments. Transition between states is implicated by the change of their molar fractions. Simulation of a VMMS system allows efficient calculation of relative free energies of all states, which in turn determine their equilibrium molar fractions. A direct application of the VMMS method is for constant pH simulation to study protonation equilibrium. SSDQO in MPOLE We have implemented a mixed PME/switching strategy for treating a single site water model that includes terms through octopole. There are two variations, an exact dipole-octopole term and the approximate dipole-octopole term from the previous code, that included only the fully contracted terms and neglected coupling to the internuclear vector. QM Protocol Optimization for studying free energy of solvation using NBB with implicit solvents We utilized the NBB method of Koenig and Boresch to evaluate the solvation free energy of the blind portion of molecules from the SAMPL4 data set. Using various DFT methods along with implicit solvent models, we were able to establish protocols for reliably reproducing experimentally determined solvation free energy data to better than 1 kcal/mol for the entire test set, and to 1 kT for the subset of species without variable protonation states. Efficient Treatment of Induced Dipoles Method for approximately, yet accurately, computing dipolar polarization in molecular simulation. Method was derived rigorously from first principles. It should speed up polarization simulations, while having better numerical stability than iterative methods.
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