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or MCFA [
107
,
113
]. These methods can selectively handle implicit constraints
through appropriate projection matrices in the EOM. Nonetheless, most have
focused on the torsional degrees of freedom (DOF) which affect the conformation
of a system. The general state space EOM for internal coordinate constrained MD
can be written as
Q
Þ Q
Q
t ¼
M
ð
Q
þ
CQ
;
;
(14)
where
corresponds to the vector of generalized forces (e.g., torques),
M
denotes
the articulated body inertia matrix,
C
denotes the nonlinear velocity dependent
terms of force (e.g., Coriolis, centrifugal and gyroscopic forces), and
Q
t
; Q
correspond to the generalized coordinates that define the state of the system. It
then follows that the dynamics of motion for a microcanonical ensemble is obtained
by solving for the hinge accelerations, access to increased integration time-steps,
faster exchange between low- and high-frequency modes for high temperature
dynamics, and faster and smoother sampling of the PES (conformational space),
among others. Our rigid body MD approaches, with atomistic and coarse-grain
force fields, are currently used to predict the conformational evolution of helical
domains in GPCR protein bundles (see Fig.
13
):
Q
;
Q
:
Q
Q
M
1
¼
ð
Q
Þ t
CQ
;
(15)
Fig. 13 Beta2 GPCR helix 7 final structure (shown in ribbons representation) after 100 ps of
constrained MD-NVT, using Comodyn with a 5 fs timestep and the Dreiding force field, shows a
kink about a proline amino-acid group. The kink is also observed during full-atom MD-NVT at
roughly the same timescale. The original starting structure is shown in transparency, for both the
full-atomistic and ribbon representations. The coarse-grain representation involved 127 clusters
for a total of 133 DOF in the internal coordinate representation of the equations of motion
(compared to the 1,170 DOF for the atomistic model)
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