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 quantitatively 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.

```
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]
```

The following table describes the keywords which apply to only four
dimension dynamics & minimization. The remaining parameters are described in
*Dynamics* and *Energy Manipulations: Minimization and Dynamics*.

```
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).
```

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

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

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