Thermodynamic pathway used to calculate the absolute binding free energy of a ligand (benzene) to an enzyme (T4 lysozyme). (1) The ligand is extracted from the binding site into vacuum along a fourth spatial dimension; spatial restraints on the heavy atoms of the ligand limit the ligand’s mobility once it leaves the binding site. (2) The free energy change associated with the removal of the restraints is accounted for, leaving the ligand inside a single boundary that mimics a standard state. (3) The hydration free energy of the ligand is computed by inserting the ligand into a water droplet along the fourth dimension; note that this step does not depend on solute concentration; this calculation is performed with a harmonic potential that keeps the ligand near the center of the water droplet.
The 4D force acting on the benzene ligand computed using DR sampling and Boltzmann-weighted jumping on a busy cluster (simulation protocol 3; see the Simulation Protocol section above). Each column shows, for a particular , the force acting on the ligand as a function of the progression of the simulation. Note that since each replica moves in a random-walk fashion along , the data for each particular discrete position represent a composite of all replicas that visited that position. Data are shown for all simulated positions (note the nonlinear scale). The thick line represents the average sampled force taken from the second half of the data at each and serves to guide the eye.
Sampling distribution over (note nonlinear scale) resulting from all replicas. The number of counts sampled was normalized such that the desired number of counts is 1 for replica 1. The thick line represents the desired sampling profile. The thin line is the profile attained using DR sampling and Boltzmann-weighted jumping on a busy cluster (simulation protocol 3; see the Simulation Protocol section above). Note that some replicas at large values were stopped much earlier than others. The outlier point (at ) resulted from data corruption that occurred as a result of a full disk and should not be considered an artifact of DR sampling.
Average force along the -axis as a function of . Results are shown for the four simulation protocols. Only the range from is shown. Beyond , the four curves are nearly identical and smoothly taper to zero.
coordinates (note the nonlinear scale) of two representative replicas as a function of the progression of the simulation. Random-walk behavior is demonstrated. Note that some replicas at large values were stopped much earlier than others as this region requires much less sampling. The protocol prevents replicas from entering the region of suspended replicas as shown.
Main-chain positional shifts in the benzene-bound complex relative to the apoprotein. Alignment was performed on the basis of main-chain atoms. The value plotted for each residue is the RMSD of the shifts in the three backbone atoms. The -terminal domains (residues 1–79) were not used in the alignment and are not included in the figure. Plots derived from the crystal structures (Ref. 43) (solid) as well as the simulation results (dashed) are shown. Alignment and distance calculations were performed using the PROFIT program (Ref. 59).
Free energy of extraction of benzene from T4 lysozyme calculated from a block of sample data ( in duration for each replica) vs the amount of equilibration time that had elapsed before that block was taken. Results are shown for the four simulation protocols: (a) Independent simulations, DR sampling with (b) Monte Carlo moves, (c) Boltzmann-weighted jumping on a busy cluster, and (d) Boltzmann-weighted jumping on a free cluster. Best fit curves of the form were applied (see Table II for parameters , , and ).
Force acting inward by the spherical restraints as a function of restraint radius and the free energy change associated with the expansion process. The individual free energies for expanding the restraints are stacked on top of each other. The free energy sum is .
Example Boltzmann-weighted jump calculation for replica 3.
Summary of the free energy data for the calculation of absolute binding free energy of benzene to T4 lysozyme using simulation protocols (1)–(4)—see the Simulation Protocol section. Units are in kcal/mol unless otherwise indicated.
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