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s4.method
.
In the PGMC method, conformations of a polypeptide are generated
by rotating backbone (
) and/or sidechain (
) dihedral angles for
individual amino acids. These dihedral angles are shown in
Figure 
for arginine, which has the largest number of
freely-rotating dihedrals. The conformations are not chosen randomly,
but are selected from probability grids calculated from a selected
subset of proteins from the Brookhave Protein Database. Each grid
is an 
-dimensional matrix, where is the number of
dihedrals involved. For instance, backbone sampling involves
two-dimensional grids, and each point on the grid is the
probability of chosing a particular pair. The grids have
spacing, where 
= 5, 10, 15, 30, or 60. Therefore, grids
have 
points, where 
. The probabilities were
derived by partitioning every pair in a set of high-resolution
protein crystal structures into -degree bins. The probabilities
(
) are normalized so that
Sidechain probability grids have varying dimensionality, depending
upon the number of dihedrals needed to specify the conformation. This
ranges from = 1 for small sidechains like serine and threonine,
to = 5 for arginine. For alanine and glycine, = 0.
Our standard approach for doing Monte Carlo simulations uses these
probability grids to generate trial conformations and assesses these
conformations using the Metropolis criterion[59]. The
protein is assigned an initial conformation, usually by rotating all
backbone and sidechain dihedrals to the highest-probability grid
values. A new conformation is generated by modifying one amino acid,
which is chosen at random. Depending on the nature of the simulation,
either a new main-chain conformation is chosen from the
probability grid, or a new side-chain conformation is chosen from the
probability grid. The potential energy of the new conformation
(
) is calculated, using a standard forcefield such as
DREIDING[60] or AMBER[61] and is compared to
the energy of the previous conformation (
). Conformation is
either accepted or rejected according to the Metropolis
criterion[59]. If the new structure is lower in energy
(
, where 
), it is accepted.
If it is higher in energy, the probability of acceptance, 
, is
defined by:
where 
is the Boltzmann constant and 
is the simulation
temperature. A random number 
is generated between 0.0 and 1.0.
If 
, the new conformation is accepted. Otherwise, the
structure is rejected and the previous structure is restored. This is
the Metropolis criterion for accepting or rejecting a new structure, designed
to ensure a Boltzmann distribution of conformations[59].
This criterion means, for example, that there is a 50%chance of accepting
a new structure if 
. At 300 K, this value
is 0.413 kcal/mol. At higher temperatures, the probability of
accepting a bad structure increases.
We are generally interested in finding the lowest-energy conformation
of a given peptide. This can be done, theoretically[62],
by the simulated annealing method: starting at a high simulation
temperature, and slowly cooling the system in a process called
simulated annealing. However, it is usually not possible to know
beforehand exactly what cooling rate is necessary to achieve the
global minimum. Considerable work has been done in optimizing the
heating and cooling process in Monte Carlo simulations of
peptides[64][63]. We have employed both constant
temperature and simulated annealing in our studies of protein
conformations.
Next: The Protein Database
Up: Dynamic and Stochastic Protein
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