Simulations of Biomolecules Using Molecular Dynamics

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2 Free Energy Simulations Free energy perturbation simulations involve a kind of "computer alchemy". These simulations model processes that cannot be carried out experimentally. In these simulations, the focus is on the determination of relative free energy differences of two processes, illustrated in Figure 2 for the binding of two ligands to the same molecular compound. ΔG 1 M + L 1 M -L 1 ΔG 2 M + L 2 M -L 2 Figure 2. Free energies for binding of two ligands, L 1 and L 2 , to molecule M. For the processes shown in Figure 2, the free energies denoted by ΔG 1 and ΔG 2 correspond to the binding of two different ligands (L 1 and L 2 ) to the same molecule (M) to form complexes (M-L 1 and M-L 2 , respectively). The relative free energy of binding, Δ(ΔG), is given by the relation Δ(ΔG) = ΔG 2 – ΔG 1 . (1) Processes 1 and 2 generally are difficult to simulate using molecular dynamics, so to determine Δ(ΔG) computationally, an alternative route is used, as illustrated in Figure 3. ΔG 1 M + L 1 M -L 1 M + L 2 M -L 2 ΔG 2 Figure 3. Free energy cycle for binding of two ligands, L 1 and L 2 , to molecule M. Processes 3 and 4 mutate ligand 1 into ligand 2 in the unbound and bound states. Since the Gibbs free energy is a thermodynamic state function, we have the equivalence Δ(ΔG) = ΔG 2 – ΔG 1 = ΔG 4 – ΔG 3 . (2) Obviously, processes 3 and 4 cannot be performed experimentally since they generally involve mutation of one atom into another or one functional group into another. However, computationally, all that the mutations require are modifications of the force field used to represent the system. To carry out the simulations, known as free energy perturbation simulations, a perturbation parameter λ is defined such that λ=0 corresponds to the processes involving ligand 1 and λ=1 corresponds to the processes involving ligand 2. The force field V can then be defined as V = λ V 2 + (1 – λ) V 1 , (3)
where V 1 corresponds to a force field that includes parameters for ligand 1 and V 2 corresponds to a force field that includes parameters for ligand 2. Simulations begin with the system in a state corresponding to ligand 1 (λ=0) and a molecular dynamics run is performed in which the parameter λ is slowly changed from 0 to 1. At the end of the simulation, the system has mutated into a state corresponding to ligand 2. A simulation of this type is performed for processes 3 and 4 to yield the free energy differences ΔG 3 and ΔG 4 . The relative free energy difference, Δ(ΔG), is then calculated using Eq. (2). 3 Examples of Free Energy Simulations Many of the first examples of free energy simulations were performed in order to study the binding of ions to small organic macrocyclic systems. In 1986, a free energy simulation was carried out to determine the relative free energies for binding of chloride and bromide to the macrocycle SC24 (Figure 4) in water [T. P. Lybrand, J. A. McCammon, and G. Wipff, Proc. Nat. Acad. Sci. 83, 833 (1986)]. Figure 4A. The anion binding macrocycle SC24. Figure 4B. Stylized form of the proposed anion binding structure of SC24. In the simulations, a periodic box containing between 191 and 214 water molecules was used to represent the solvent. The simulations were carried out at 300 K using the SHAKE algorithm to constrain C-H bonds and a large time step of 4 fs was employed. After equilibration, the simulations were run for 30 ps. The calculated relative free energy of binding of Br – relative to Cl – was Δ(ΔG) = 4.2 ± 0.4 kcal/mol. This result indicates that Br – binds less favorably in the center of SC24 because of its larger size. The calculated result is in excellent agreement with the experimental value of 4.3 kcal/mol. Another hallmark free energy simulation of macrocycle ion binding involved the study of Na + and K + interacting with 18-crown-6 in solution (Figure 5) [L. X. Dang and P. A. Kollman, J. Am. Chem. Soc. 112, 5716 (1990)]. O O O K + O O O Figure 5. 18-crown-6 binding K + .
Page 1: Chemistry 380.37 Fall 2011 Dr. Jean

Simulations of Biomolecules Using Molecular Dynamics

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