Chemistry Reference
In-Depth Information
significant changes in many properties of the trajectory. The trade-off between
SDEL and normal and MTS MD is that trajectories obtained with such a large
step size are only approximate; molecular motions that occur on a scale short-
er than the step size are filtered out from the trajectories. Also, the initial and
final configurations of the system under study must be known because this is a
boundary value algorithm. This means that the algorithm cannot be used to
predict the final conformation of a molecular system such as a protein, and,
consequently, confines the applicability of the algorithm to situations in which
the initial and final configurations are known by experiment or modeling. This
is not an unbearable limitation because in many chemical events we are inter-
ested in determining how a system changes from a reactant state to a product
state. The algorithm can be used, for example, to describe folding mechan-
isms,
107
i.e., how a protein folds to its native conformation starting from an
unfolded structure.
The Onsager-Machlup action methodology has a critical disadvantage:
the total time of the trajectory is needed in advance. Also, low-resolution tra-
jectories do not approach a physical limit when the step size increases, in con-
trast to SDEL as will be shown below.
Like the Onsager-Machlup action method, the SDEL algorithm is based
on the classical action. However, in this case the starting point is the action
S
parameterized according to the length of the trajectory:
108
Y
f
ð
p
2
S
¼
ð
E
V
ð
Y
ÞÞ
dl
½
30
Y
u
where
Y
u
and
Y
f
(lower and upper limits of integration) are the mass-weighted
coordinates
p
X
Þ
of the initial and final conformation of the system,
respectively,
E
is the total energy,
V
is the potential energy of the system,
and
dl
is an infinitesimal mass-weighted arc length element for the path con-
necting
Y
u
and
Y
f
. Using the least-action principle of classical mechanics, one
obtains a classical trajectory connecting these two states of the system when a
stationary solution for the action is computed, i.e.,
ð
Y
¼
0 (the action is
not necessarily a minimum
108,109
). These trajectories are calculated differently
from usual MD simulations. First, the trajectory is obtained using double
boundary conditions, where the initial and final coordinates of the system
are required as input. In contrast, the initial positions and velocities (usually
chosen randomly from a Boltzmann distribution) are needed in a standard MD
algorithm. Second, the trajectory in Eq. [30] is parameterized as a function of
length and not as a function of time. Finally, the total energy of the trajectory
is fixed in the SDEL formulation. This contrasts with an MD trajectory where
the total time is fixed once the step size and the number of steps are con-
strained in the calculation.
d
S
=
d
Y
¼
