4 Dimension dynamics: Description and Discussion
The energy embedding technique entails placing a molecule into a
higher spatial dimension {Crippen,G.M. & Havel,T.F. (1990)
J.Chem.Inf.Comput.Sci. Vol 30, 222-227}. The possibility of surmounting
energy barriers with these added degrees of freedom may lead to lower
energy minima. Here, this is accomplished by molecular dynamics in four
dimensions. Specifically, another cartesian coordinates was added
to the usual X, Y, and Z coordinates in the LEAPfrog VERLet algorithm.
To employ 4D energy embedding, the energy function and force field
in CHARMM was modified to include fourth dimension coordinates. An
additional harmonic energy function has been included to control the
extent to which a molecule is embedded. This is quantatitatively done by
altering the value of its force constant, initially given by the parameter
K4DI.
The 4D energy embedding procedure can be broken down into three
parts: 4D coordinate generation, relaxation, and back projection. Fourth
dimensional coordinates can be generated in several ways. An energy, E4FILL,
in the Fourth dimension can be specified with random coordinates generated
as to sum up to the 4D harmonic energy that a user specifies (i.e. E4FILL 50.0
will give coordinates such that the total sums approximately 50.0 Kcal).
This method may seem a bit abrupt since a molecule is suddently "thrown"
into a higher dimension, hence, molecular dynamics can be used to
allow a molecule to more slowly obtain fourth dimension coordinates.
This is done by specifying an initial 4D temperature, FSTT4, with subsequent
velocities assigned accordingly. Finally, both these methods may be applied
simultaneously. Relaxation involves allowing the molecule to explore the
potential energy surface and is essentially equilibration. Alternatively,
minimization in 4D can be done with the steepest descent algorithm followed
by 4D dynamics. Now all that remains is to project this structure back into
three dimensions. This last step is thus termed the back projection and is
achieved by increasing the fourth dimensional force constant linearly
from its initial value of K4DI to MULTK4*K4DI step-wise over the period INC4D
to DEC4D. This results in a stronger force, confining the 4th dimension
coordinates to smaller values (i.e. eventually back to 3D).
A problem inherent in the final step of 4D energy embedding is that
"sometimes all projections lead to a bad final conformation" {Crippen,G.M &
Havel,T.F.(1990)J.Chem.Inf.Comput.Sci.Vol 30,222-227}. Thus, the structure
is rotated into its principal axis of intertia (center of mass) both before
and after its back projection. When this step is applied the message
ROTATION APPLIED TO PRINCIPAL AXES
will appear. Dynamc4.src is essentially dynamc.src in 4 dimensions. Note
that even though qeuler still exists in dynamc4.src it has not yet been
tested. Also, the usual shake algorithm will only be applied to
3-dimensional space.
* Syntax / Syntax of the 4 dimension dynamics command
* Description / Description of the keywords and options
* Recommended / Recommended input options and values
* Discussion / Running 4 dimension dynamics
* Output / Output from a 4 dimension dynamics run
Top
Options common 4D dynamics & minimization
The following table describes the keywords which apply to only four
dimension dynamics & minimization. The remaining parameters are described in
dynamc.doc and minimiz.doc.
FOURdimensions [INC4d int] [DEC4d int] [K4DI real] [MULTK4 real] -
[ SKBO ] [ SKAN ] [ SKDI ] [ SKVD ] [ SKEL ] [ SKCO ] -
[FIL4 [E4FILL real ] ] [ SHAKe ]
Keyword Default Purpose
INC4D NSTEP The step number (specifically, the time in a
dynamics run) at which the back projection from
4 to 3 dimensions will begin. Note the default
value of NSTEP will result in no back projection.
DEC4D NSTEP The step number at which the back projection from
4 to 3 dimensions will end.
K4DI 50.0 The initial force constant for the 4th dimensional
harmonic energy term.
MULTK4 1.0 The factor by which K4DI will increase linearly from
INC4D to DEC4D.
FSTT4 FIRSTT The initial temperature, in the 4th dimension, at which the
velocities have to be assigned to begin the dynamics run.
If an equal amount of kinetic energy is needed in all 4
dimensions, the default value should be used. This is
because the velocities are all assigned independently in
accordance to the initial temperature.
FNLT4 FINALT The desired final (equilibrium) temperature, in the 4th
dimension, for the system. A final temperature of zero
degrees is recommended during a back projection (from
INC4D to DEC4D).
IEQ4 IEQFRQ The step frequency for assigning or scaling the 4th
dimension velocities to FNLT4 temperature during the
equilibration stage of the dynamics run.
IHT4 IHTFRQ The step frequency for heating the molecule in the 4th
dimension, in increments of TIN4 degrees in the heating
portion of a dynamcis run.
TIN4 TEMINC The temperature increment to be given to the system every
IHT4 steps. Important in the 4th dimension heating stage.
TWH4 TWINDH The temperature deviation from FNLT4 to be allowed on the
high temperature side. Used only during 4th dimension
equilibration.
TWL4 TWINDL The temperature deviation from FNLT4 to be allowed on the
low temperature side. Used only during 4th dimension
equilibration.
ICH4 ICHECW The option for checking to see if the average 4th
dimension temperature of the system lies within the
allotted temperature window (between FNLT4+TWH4 and
FNLT4-TWL4) every IEQ4 steps.
FIL4 The flag to fill the 4th dimension coordinates. The
harmonic energy potential of these coordinates will sum
to E4FILL. If not present (recommended), the 4th
dimension coordinates are set to zero and the system will
'go into the 4th dimension' as a result of their
initial velocities.
E4FILL 0.0 The total harmonic potential energy from which the initial
4th dimension coordinates will be calculated. Only used
when the flag FIL4 is present.
SKBO Flag to skip 4th dimension bond energies (i.e.only
compute bond energies in 3 dimensions).
SKAN Flag to skip 4th dimension angle energies.
SKDI Flag to skip 4th dimension proper dihedral energies.
SKVD Flag to skip 4th dimension Van der Waals energies.
SKEL Flag to skip 4th dimension electrostatic energies.
SKCO Flag to skip 4th dimension restraint (so restraining Forces
are calculated in 3D only).
SHAKe Command to place all 4D W's into same W every iteration
(NOTE:energy not conserved). The 4D forces are not normally
mass weighted, but if SHA4 is used then they are. Maybe it
should be a 4D option in the future.
Other Commands:
CONS FIX4 ... Used in analogy to the FIX command to FIX 4th D coordinates
with CONS (meaning one can FIX something in 3D only).
SCALar FDEQ (0.0) The equilibrium value(s) that the 4th D function will use as
the center of the harmonic. Used for restraining the
4th D to non zero values (i.e. forcing a system into
the 4th Di). It should be set with the SCALAR
option for individual atoms (if one wants to set different
atoms into different 4th D coordinate minima).
(1/2)*K4d*W**2, where W=FDIM(I)-FDEQ(I)
SCALar FDIM (0.0) The coordinate(s) (in analogy to X,Y, & Z) of the 4th D.
It should be set with the SCALAR option for individual atoms
(if one wants to set different atoms into different 4th D
coordinates).
Top
Recommended CHARMM input for 4d dynamics
1) Beginning with a 3d structure and no 4d coordinates, a structure is
equilibrated in 4d and then back projected (forced back) to 3d.
DYNAMCS LEAP VER4 START K4DI 50.0 NSTEP 20000 -
TIMESTEP .001 FSTT4 300.0 FNLT4 300.0 CUTBN 8.0 -
IHTFRQ 0 IEQFRQ 100 IEQ4 100 NPRINT 10 -
IUNREA -1 IUNWRI 16 -
IHBFRQ 25 FIRSTT 1000.0 FINALT 1000.0 TEMINC 0.0 TIN4 0.0
DYNAMCS LEAP VER4 RESTART NPRE 0 NSTEP 15000 -
K4DI 50.0 INC4D 0 DEC4D 15000 MULTK4 10.0 -
TIMESTEP .001 FSTT4 300.0 FNLT4 300.0 CUTBN 8.0 -
IHTFRQ 0 IEQFRQ 100 IEQ4 100 NPRINT 10 -
IUNREA 16 IUNWRI 17 -
IHBFRQ 25 FIRSTT 1000.0 FINALT 100.0 TEMINC 3.0 TIN4 1.0
2) Beginning with a 4d structure with 10.0 Kcal initially in the 4th
dimension.
DYNAMCS LEAP VER4 START K4DI 50.0 NSTEP 20000 -
FIL4 E4FILL 10.0 -
TIMESTEP .001 FSTT4 300.0 FNLT4 300.0 CUTBN 8.0 -
IHTFRQ 0 IEQFRQ 100 IEQ4 100 NPRINT 10 -
IUNREA -1 IUNWRI 16 -
IHBFRQ 25 FIRSTT 1000.0 FINALT 1000.0 TEMINC 0.0 TIN4 0.0
3) Fixing the 4th D coordinates of some bulk solvent and setting the
solute coordinates "out" in 4D space and along with its equilibrium
value. Following this the energy is determined.
CONS FIX4 SELE SEGID BULK END
SCALAR FDIM SET 10.0 SELE SEGID SOLV END
FOUR K4DI 50.0 SKBO SKAN SKDI SKCO
ENERGY
