Simulations of Biomolecules Using Molecular Dynamics

Chemistry 380.37
Fall 2011
Dr. Jean M. Standard
November 30, 2011
Simulations of Biomolecules Using Molecular Dynamics
Introduction
The first molecular dynamics simulation of a biomolecule was a simulation of bovine pancreatic trypsin inhibitor
(BPTI) in 1977 [J. A. McCammon, B. R. Gelin, and M. Karplus, Nature 267, 585 (1977)]. BPTI was chosen for
simulation because it is relatively small (58 residues) and a high-resolution x-ray crystal structure was available at
the time (Figure 1). The simulation was carried out in vacuum and was 9.2 ps in length.
Figure 1. Atomic model and ribbon diagram of BPTI.
Eleven years later, a molecular dynamics simulation of BPTI was carried out in solution [M. Levitt and R. Sharon,
Proc. Nat. Acad. Sci. 85, 7557 (1988)]. This simulation, including water molecules, consisted of approximately
8700 atoms (892 protein atoms and 7821 solvent atoms) and was 210 ps in length with a step size of 2 fs. The
force fields employed in these early gas and solution phase simulations were quite crude compared to those employed
in similar simulations today.
Thanks to the enormous increase in computing power over the last 10 years, biomolecular simulations today can be
performed on systems of hundreds of thousands of atoms and for times on the order of nanoseconds (some
approaching microseconds). Molecular dynamics simulations may be employed to study a wide variety of
behaviors of biomolecules, from the dynamics of binding of carbon monoxide to hemoglobin to the mechanism of
the passage of ions through the gramicidin A ion channel. An excellent review article on molecular dynamics
simulations is M. Karplus and J. A. McCammon, Nature Structure Biology 9, 646 (2002). Another excellent
resource, with more in depth articles is a special issue of the journal Accounts of Chemical Research dedicated to
biomolecular simulations (Acc. Chem. Res. 35, 2002). One important example of a current type of biomolecular
simulation, free energy simulation, is discussed below.

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 + .

4
In the simulation of 18-crown-6 cation binding, the Amber force field was employed along with the SHAKE
algorithm. A 1.5 fs time step was employed and the simulations were carried out for 50 ps at 300 K. The solvent
consisted of a periodic box of 343 methanol molecules. The simulations produce the result Δ(ΔG) = –3.5 ± 1.3
kcal/mol for the free energy difference of binding K + relative to Na + . That the free energy of binding K + is lower than
the free energy of binding Na + indicates that it is more favorable to bind K + due to the better fit of K + in the 18-
crown-6 cavity. The simulations agree favorably with the experimental result of –2.5 kcal/mol.
The first application of free energy simulations to biomolecules was by Wong and McCammon in 1986 [C. F.
Wong and J. A. McCammon, J. Am. Chem. Soc. 108, 3830 (1986)]. In this study, the relative binding affinity of
two inhibitors (benzamidine and p-fluorobenzamidine) to trypsin was determined. The simulations were performed
at 300 K using a primitive force field and the SHAKE algorithm. Simulation length ranged from 22-64 ps with a
step size of 2 fs in a periodic box of water molecules. The results for free energy simulations of the two inhibitors
was Δ(ΔG) = 3.8 ± 2.2 kJ/mol compared to an experimental result of 2.1 kJ/mol.
Reviews of free energy simulations can be found in Volume 35 of Accounts of Chemical Research (2002).