# fourd (c41b2)

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

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

Syntax for the Dynamics Command

DYNAmics { [LEAPfrog] } VER4 {STRT } {[TIMEstp real]} [NSTEp integer] -

{ [LANGevin] } {STARt } {[FIL4dimension]}

{RESTart} {[SKBOnd]} {[SKANgle]} {[SKDIhedral]}

{[SKVDerWaals]} {[SKELectrostatics]}

four dimension-spec nonbond-spec hbond-spec frequency-spec -

unit-spec temperature-spec options-spec

hbond-spec::= updated as in normal LEAPfrog VERLet.

nonbond-spec::= updated in 4 dimensions.

four dimension-spec::= [K4DInitial real] [INC4Dforce integer]

[DEC4Dforce integer] [MULTK4di real]

[E4FILLcoordinates real]

frequency-spec::= [INBFrq integer] [IEQFrq integer] [IHBFrq integer]

[IHTFrq integer] [IPRFrq integer] [NPRInt integer]

[NSAVC integer] [NSAVV integer] [NTRFrq integer]

[ILBFrq integer] [ISVFRQ integer]

[IEQ4 integer] [IHT4 integer]

unit-spec::= [IUNCrd integer] [IUNRea integer] [IUNVel integer]

[IUNWri integer] [KUNIt integer] [CRAShu integer]

[BACKup integer]

temperature-spec::= [FINAlt real] [FIRStt real] [TEMInc real]

[TSTRuc real] [TWINDH real] [TWINDL real]

[FNLT4 real] [FSTT4 real] [TIN4 real]

[TWH4 real] [TWL4 real]

options-spec::= [IASOrs integer] [IASVel integer] [ICHEcw integer]

[ISCAle integer] [ISCVel integer] [ISEEd integer]

[SCALe real] [NDEGg integer] [RBUFfer real]

[AVERage] [ECHEck real] [TOL real]

[ICH4 integer]

Syntax for the Dynamics Command

DYNAmics { [LEAPfrog] } VER4 {STRT } {[TIMEstp real]} [NSTEp integer] -

{ [LANGevin] } {STARt } {[FIL4dimension]}

{RESTart} {[SKBOnd]} {[SKANgle]} {[SKDIhedral]}

{[SKVDerWaals]} {[SKELectrostatics]}

four dimension-spec nonbond-spec hbond-spec frequency-spec -

unit-spec temperature-spec options-spec

hbond-spec::= updated as in normal LEAPfrog VERLet.

nonbond-spec::= updated in 4 dimensions.

four dimension-spec::= [K4DInitial real] [INC4Dforce integer]

[DEC4Dforce integer] [MULTK4di real]

[E4FILLcoordinates real]

frequency-spec::= [INBFrq integer] [IEQFrq integer] [IHBFrq integer]

[IHTFrq integer] [IPRFrq integer] [NPRInt integer]

[NSAVC integer] [NSAVV integer] [NTRFrq integer]

[ILBFrq integer] [ISVFRQ integer]

[IEQ4 integer] [IHT4 integer]

unit-spec::= [IUNCrd integer] [IUNRea integer] [IUNVel integer]

[IUNWri integer] [KUNIt integer] [CRAShu integer]

[BACKup integer]

temperature-spec::= [FINAlt real] [FIRStt real] [TEMInc real]

[TSTRuc real] [TWINDH real] [TWINDL real]

[FNLT4 real] [FSTT4 real] [TIN4 real]

[TWH4 real] [TWL4 real]

options-spec::= [IASOrs integer] [IASVel integer] [ICHEcw integer]

[ISCAle integer] [ISCVel integer] [ISEEd integer]

[SCALe real] [NDEGg integer] [RBUFfer real]

[AVERage] [ECHEck real] [TOL real]

[ICH4 integer]

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).

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

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