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# replica (c41b1)

Replica: Commands which deal with replication of the molecular system: Replica.

# <caves>-Aug-18-1993 (Leo Caves) Initial release.

# REPLICA/PATH method added by B. Brooks March 1994.

# Feynmann Path Integral Methods added

by B. Roux, K. Hinsen and Marc Souaille, June 1997.

The commands described in this node are associated with the replication of

regions of the PSF,

for replication of regions of the PSF has been implemented to support a class

of methods which seek to improve the sampling of a (usually small) region of

the molecular system, by selective replication. Such methods include LES

(Locally Enhanced Sampling [Elber and Karplus 1990, J. Amer. Chem. Soc. 112,

9161-9175]) and MCSS (Multiple Copy Simultaneous Search [Miranker and Karplus

1991, Proteins 11, 29-34]).

The Replica Path Method as applied to QM, MM and QM/MM reaction paths is

described in the following paper and should be cited when applied...

H. Lee Woodcock, M. Hodoscek, P. Sherwood, Y. S. Lee, H. F. Schaefer, and

B. R. Brooks; Exploring the QM/MM Replica Path Method: A Pathway

Optimization of the Chorismate to Prephenate Claisen Rearrangement

Catalyzed by Chorismate; Theor. Chem. Acc. 2003; 109 (3); 140-148.

the Nudged Elastic Band method as implemented in CHARMM is built upon the

Replica Path functionality therefore the above paper and the following paper

(which describes the NEB implementation and improvements in minimization

techniques) should be cited when applied...

J. W. Chu, B. L. Trout and B. R. Brooks; A super-linear minimization scheme

for the nudged elastic band method; J. Chem. Phys. 2003; 119(24);

12708-12717.

* Syntax | Syntax of the replication commands

* Usage | Description of command usage

* Implementation | A brief description of the anatomy of replication

* Restrictions | Restrictions on usage

* Examples | Supplementary examples of the use of REPLica

* Path | Replica Path Method

* OffPath | Off-Path Optimzation / Simulation

* Pathint | Path Integral Calculation using REPLica

# <caves>-Aug-18-1993 (Leo Caves) Initial release.

# REPLICA/PATH method added by B. Brooks March 1994.

# Feynmann Path Integral Methods added

by B. Roux, K. Hinsen and Marc Souaille, June 1997.

The commands described in this node are associated with the replication of

regions of the PSF,

**»**struct Generate. A facilityfor replication of regions of the PSF has been implemented to support a class

of methods which seek to improve the sampling of a (usually small) region of

the molecular system, by selective replication. Such methods include LES

(Locally Enhanced Sampling [Elber and Karplus 1990, J. Amer. Chem. Soc. 112,

9161-9175]) and MCSS (Multiple Copy Simultaneous Search [Miranker and Karplus

1991, Proteins 11, 29-34]).

The Replica Path Method as applied to QM, MM and QM/MM reaction paths is

described in the following paper and should be cited when applied...

H. Lee Woodcock, M. Hodoscek, P. Sherwood, Y. S. Lee, H. F. Schaefer, and

B. R. Brooks; Exploring the QM/MM Replica Path Method: A Pathway

Optimization of the Chorismate to Prephenate Claisen Rearrangement

Catalyzed by Chorismate; Theor. Chem. Acc. 2003; 109 (3); 140-148.

the Nudged Elastic Band method as implemented in CHARMM is built upon the

Replica Path functionality therefore the above paper and the following paper

(which describes the NEB implementation and improvements in minimization

techniques) should be cited when applied...

J. W. Chu, B. L. Trout and B. R. Brooks; A super-linear minimization scheme

for the nudged elastic band method; J. Chem. Phys. 2003; 119(24);

12708-12717.

* Syntax | Syntax of the replication commands

* Usage | Description of command usage

* Implementation | A brief description of the anatomy of replication

* Restrictions | Restrictions on usage

* Examples | Supplementary examples of the use of REPLica

* Path | Replica Path Method

* OffPath | Off-Path Optimzation / Simulation

* Pathint | Path Integral Calculation using REPLica

Top

Syntax of PSF Replication commands

[SYNTAX: REPLication commands]

REPLica { [segid] [NREPlica integer] [SETUP] [atom-selection] [COMP] }

{ RESEt }

segid:== Basename for replica segment identifiers.

atom-selection:== (

{ OFF }

RPATh { [KRMS real] [KANGle real] [COSMax real] [MASS] [WEIGht] other-spec}

other-spac:== [ KMIN real RMIN real ]

[ KMAX real RMAX real ]

[ EVWIdth real ] [ CYCLic ]

[ROTAte ] [TRANslate ]

[NOROtate] [NOTRanslate]

[ NEBA ] [ KNEB real ]

[ NEBF ] [ ETAN ] [ CIMG ] [ PPMF ] [ ANAL ]

Syntax of PSF Replication commands

[SYNTAX: REPLication commands]

REPLica { [segid] [NREPlica integer] [SETUP] [atom-selection] [COMP] }

{ RESEt }

segid:== Basename for replica segment identifiers.

atom-selection:== (

**»**select .){ OFF }

RPATh { [KRMS real] [KANGle real] [COSMax real] [MASS] [WEIGht] other-spec}

other-spac:== [ KMIN real RMIN real ]

[ KMAX real RMAX real ]

[ EVWIdth real ] [ CYCLic ]

[ROTAte ] [TRANslate ]

[NOROtate] [NOTRanslate]

[ NEBA ] [ KNEB real ]

[ NEBF ] [ ETAN ] [ CIMG ] [ PPMF ] [ ANAL ]

Top

Description of REPLica command usage

1) (The implicit GENERate subcommand)

This command performs the essential act of replication. Its action is to

replicate (to a degree specified by NREPlica, default: 2) (a subset of) the

molecular system, as specified in the (primary) atom-selection (default: all).

All atomic properties and topological attributes of the region are replicated

(for a full list,

replica of the primary atom selection constitutes a new segment in (and

appended to) the PSF, however the atom and residue names and the residue

identifiers of the primary atom selection are carried over.

The implicit generation subcommand optionally accepts a segment identifier

(segid). The length of segid must be such that when concatenated with the

(integer representing the) maximum number of replicas specified for generation,

it does not exceed 4 characters. If omitted, then replica segment identifiers

will simply be set to the replica number. At present no check is made for

duplicate segment identifiers, so choose with care. The command is designed to

operate in a manner similar to the GENErate command from the main parser.

The effect of the replication command may be classified into two areas:

structure and interactions. Structurally, as mentioned above, the command

performs the necessary book-keeping work for CHARMM, in order that each

individual replica is functionally equivalent to the region of the structure

specified in the atom selection. ie. in the case where the atomic positions of

an individual replica are the same as the primary atom selection (as they will

be immediately after issuing the REPLica command), the energy and forces of the

individual replica and the appropriate region of the primary system are

identical (there is an important corollary to this statement which is now

discussed).

In the area of discussing the interactions of replicas it is useful to

introduce the concept of a subsystem. Before issuing a REPLica command, there

is considered to be one subsystem, the primary subsystem, to which all atoms

belong. Upon issuing the REPLIca command a new subsystem is generated, which

consists of replicas of a subset of the primary subsystem (as specified in the

atom selection). In this case there are now two subsystems.

The simple cases specifying interactions of subsystems and replicas may now

be stated:

* Replicas within a subsystem do NOT interact.

* Replicas belonging to different subsystems do interact.

In CHARMM, the interaction rules of replicas are applied in the non-bonded list

generation routines, through appropriate group/atom exclusions. You will notice

some diagnostic messages from the list generation routines indicating the

number of group/atom interactions excluded on the basis of replication. In

following the rules of interaction of replicas it is important to note that a

given replication of a subset of the primary subsystem, results in a new

subsystem. Thus the subset of the primary subsystem and its individual replicas

are now in different subsystems and are thus will interact. For this reason,

the replication action is usually followed by an immediate removal of the atoms

of the subset of the primary subsystem, through a call to DELEte *note

dele:» chmdoc/struct Delete). This leaves all replicas of the specified

region in a single subsystem, arranged as contiguous segments appended to the

current PSF.

A note on renormalization of energy and forces:

In the original implementation of REPLica in a developmental version of CHARMM

at Harvard, there exists a close coupling of the REPLica command and the

energy/force evaluation routines. In the current REPLica implementation in the

standard CHARMM distribution, appropriate energy/force scaling for the system

in question may be achieved through the use of the BLOCK facility of CHARMM see

for very flexible method of handling replica interactions. Note that if the

primary system is FIXed and that only one replicated subsystem is present (the

case in many MCSS applications) then normalization of energy/forces is NOT

required.

Example:

In the following section of CHARMM command script, a segment named PROT is

generated from a sequence read from a coordinate file. A couple of selection

definitions are made which together identify the sidechain atoms of residue

12. In the REPLIca command, 4 copies of the sidechain are generated and placed

in three new segments A1 to A4 at the end of the PSF. Next the selections are

redefined (as REPLica has altered the PSF and this corrupts existing selections

made with the DEFIne command). These (newly redefined) selections are made to

remove the sidechain atoms in the primary system that were selected for

replication. Next BLOCK is used to setup the scaling of energy and forces in

this system with a primary and a single replicated subsystem. In the call to

BLOCK, 2 blocks are requested. By default BLOCK places all atoms of the system

in block 1, so the first action is to redefine the replicated subsystem to

block 2. Next we simply set up the desired interaction matrix. Primary

subsystem self interactions are simply set to unity (no scaling). Interactions

within each replica are set to 0.25 (the reciprocal of the number of

replicas). Primary <--> replicated subsystem interactions are similarly scaled

by 0.25. (Note that the REPLica interface to the non-bonded list generation

routines removes all inter-replica (intra-subsystem) interactions.) Finally,

the masses of the replicated atoms are scaled by 0.25, by using the SCALar

commands. (Note that mass-scaling may not be desirable as it has been

demonstrated that in the original LES framework, the thermal properties of the

replicas are such that at thermal equilibrium, the mapping of replicas back to

the "physical" system (with a single copy) results in too high a temperature.

The overestimation of the temperature in the physical system is a factor of N

in the simplest case of a uniform "weighting" of all replicas by a factor of

1/N, where N is the number of replicas employed in the simulation. This effect

is an active field of research, though a solution for systems where only

equilibrium properties are desired is to either scale up the masses of the

replicas by a factor of N, or to selectively rescale the velocities of the

replicas.)

...

! { read sequence and generate segment }

READ SEQU COOR UNIT 11

GENErate PROT

! { define some useful atom selections }

DEFIne backbone SELEct TYPE N .OR. TYPE CA .OR. TYPE C .OR. -

TYPE HN .OR. TYPE HA .OR. TYPE CB END

DEFIne disorder SELEct (SEGID PROT .AND. RESId 12) .AND. .NOT. backbone END

! { replicate the selected sidechain four times }

REPLIcate A NREPlica 4 SELEct ( disorder ) END

! { redefine as REPLIca has changed PSF and this trashes SELEction

! definitions }

DEFIne backbone SELEct TYPE N .OR. TYPE CA .OR. TYPE C .OR. -

TYPE HN .OR. TYPE HA .OR. TYPE CB END

DEFIne disorder SELEct (SEGID PROT .AND. RESId 12) .AND. .NOT. backbone END

DELEte ATOM SELEct ( disorder ) END

DEFIne replicas SELEct SEGId A* END

! { set up an appropriate interaction matrix }

BLOCK 2

CALL 2 SELEct ( replicas ) END

COEF 1 1 1.0

COEF 2 2 0.25

COEF 2 1 0.25

END

! { note masses can be modified if desired through the SCALar commands }

! { note that this may not always be desirable --- see comments above }

SCALar MASS MULt 0.25 SELEct replicas END

... load/generate some coordinates and proceed..

2) The RESEt subcommand.

The RESEt subcommand has the effect of reducing all current subsystems to a

single primary subsystem. This is accomplished by simply switching off the

as much as the REPLica state must be restored through appropriate calls to the

REPLica command. This command is there to support the use of REPLica for

simple replication of PSF elements for which subsequent REPLIca handling is not

required.

Example:

The following example begins by building a PSF containing a single CO

molecule. An immediate call to REPLica requests the generation of 256 replicas

(with SEGId's of R1 to R256) of the primary subsystem (the CO molecule with the

SEGId CO). Next the original CO molecule is removed. The final command,

switches off the CHARMM's replica handling, leaving a PSF with 256 CO's which

interact with each other. This may seem like a redundant command given the

ability to generate a long sequence with commands like READ SEQU COOR or a

little copy and paste with your favorite editor, but remember that REPLica can

handle replication of ANY subset of the PSF, reducing the need for tampering

with RTF definitions and creating new PATCh residues (PRES's).

READ SEQUence CARDS

* a single carbon monoxide molecule

CO

GENErate CO ! generate the primary system

REPLica R NREP 256 SELEct SEGId CO END ! replicate

DELEte ATOM SELEct SEGId CO END ! remove primary system

REPLica RESEt ! reduce replicates to primary

Description of REPLica command usage

1) (The implicit GENERate subcommand)

This command performs the essential act of replication. Its action is to

replicate (to a degree specified by NREPlica, default: 2) (a subset of) the

molecular system, as specified in the (primary) atom-selection (default: all).

All atomic properties and topological attributes of the region are replicated

(for a full list,

**»**replica Implementation). Eachreplica of the primary atom selection constitutes a new segment in (and

appended to) the PSF, however the atom and residue names and the residue

identifiers of the primary atom selection are carried over.

The implicit generation subcommand optionally accepts a segment identifier

(segid). The length of segid must be such that when concatenated with the

(integer representing the) maximum number of replicas specified for generation,

it does not exceed 4 characters. If omitted, then replica segment identifiers

will simply be set to the replica number. At present no check is made for

duplicate segment identifiers, so choose with care. The command is designed to

operate in a manner similar to the GENErate command from the main parser.

The effect of the replication command may be classified into two areas:

structure and interactions. Structurally, as mentioned above, the command

performs the necessary book-keeping work for CHARMM, in order that each

individual replica is functionally equivalent to the region of the structure

specified in the atom selection. ie. in the case where the atomic positions of

an individual replica are the same as the primary atom selection (as they will

be immediately after issuing the REPLica command), the energy and forces of the

individual replica and the appropriate region of the primary system are

identical (there is an important corollary to this statement which is now

discussed).

In the area of discussing the interactions of replicas it is useful to

introduce the concept of a subsystem. Before issuing a REPLica command, there

is considered to be one subsystem, the primary subsystem, to which all atoms

belong. Upon issuing the REPLIca command a new subsystem is generated, which

consists of replicas of a subset of the primary subsystem (as specified in the

atom selection). In this case there are now two subsystems.

The simple cases specifying interactions of subsystems and replicas may now

be stated:

* Replicas within a subsystem do NOT interact.

* Replicas belonging to different subsystems do interact.

In CHARMM, the interaction rules of replicas are applied in the non-bonded list

generation routines, through appropriate group/atom exclusions. You will notice

some diagnostic messages from the list generation routines indicating the

number of group/atom interactions excluded on the basis of replication. In

following the rules of interaction of replicas it is important to note that a

given replication of a subset of the primary subsystem, results in a new

subsystem. Thus the subset of the primary subsystem and its individual replicas

are now in different subsystems and are thus will interact. For this reason,

the replication action is usually followed by an immediate removal of the atoms

of the subset of the primary subsystem, through a call to DELEte *note

dele:» chmdoc/struct Delete). This leaves all replicas of the specified

region in a single subsystem, arranged as contiguous segments appended to the

current PSF.

A note on renormalization of energy and forces:

In the original implementation of REPLica in a developmental version of CHARMM

at Harvard, there exists a close coupling of the REPLica command and the

energy/force evaluation routines. In the current REPLica implementation in the

standard CHARMM distribution, appropriate energy/force scaling for the system

in question may be achieved through the use of the BLOCK facility of CHARMM see

**»**block . The combination of REPLica and BLOCK providesfor very flexible method of handling replica interactions. Note that if the

primary system is FIXed and that only one replicated subsystem is present (the

case in many MCSS applications) then normalization of energy/forces is NOT

required.

Example:

In the following section of CHARMM command script, a segment named PROT is

generated from a sequence read from a coordinate file. A couple of selection

definitions are made which together identify the sidechain atoms of residue

12. In the REPLIca command, 4 copies of the sidechain are generated and placed

in three new segments A1 to A4 at the end of the PSF. Next the selections are

redefined (as REPLica has altered the PSF and this corrupts existing selections

made with the DEFIne command). These (newly redefined) selections are made to

remove the sidechain atoms in the primary system that were selected for

replication. Next BLOCK is used to setup the scaling of energy and forces in

this system with a primary and a single replicated subsystem. In the call to

BLOCK, 2 blocks are requested. By default BLOCK places all atoms of the system

in block 1, so the first action is to redefine the replicated subsystem to

block 2. Next we simply set up the desired interaction matrix. Primary

subsystem self interactions are simply set to unity (no scaling). Interactions

within each replica are set to 0.25 (the reciprocal of the number of

replicas). Primary <--> replicated subsystem interactions are similarly scaled

by 0.25. (Note that the REPLica interface to the non-bonded list generation

routines removes all inter-replica (intra-subsystem) interactions.) Finally,

the masses of the replicated atoms are scaled by 0.25, by using the SCALar

commands. (Note that mass-scaling may not be desirable as it has been

demonstrated that in the original LES framework, the thermal properties of the

replicas are such that at thermal equilibrium, the mapping of replicas back to

the "physical" system (with a single copy) results in too high a temperature.

The overestimation of the temperature in the physical system is a factor of N

in the simplest case of a uniform "weighting" of all replicas by a factor of

1/N, where N is the number of replicas employed in the simulation. This effect

is an active field of research, though a solution for systems where only

equilibrium properties are desired is to either scale up the masses of the

replicas by a factor of N, or to selectively rescale the velocities of the

replicas.)

...

! { read sequence and generate segment }

READ SEQU COOR UNIT 11

GENErate PROT

! { define some useful atom selections }

DEFIne backbone SELEct TYPE N .OR. TYPE CA .OR. TYPE C .OR. -

TYPE HN .OR. TYPE HA .OR. TYPE CB END

DEFIne disorder SELEct (SEGID PROT .AND. RESId 12) .AND. .NOT. backbone END

! { replicate the selected sidechain four times }

REPLIcate A NREPlica 4 SELEct ( disorder ) END

! { redefine as REPLIca has changed PSF and this trashes SELEction

! definitions }

DEFIne backbone SELEct TYPE N .OR. TYPE CA .OR. TYPE C .OR. -

TYPE HN .OR. TYPE HA .OR. TYPE CB END

DEFIne disorder SELEct (SEGID PROT .AND. RESId 12) .AND. .NOT. backbone END

DELEte ATOM SELEct ( disorder ) END

DEFIne replicas SELEct SEGId A* END

! { set up an appropriate interaction matrix }

BLOCK 2

CALL 2 SELEct ( replicas ) END

COEF 1 1 1.0

COEF 2 2 0.25

COEF 2 1 0.25

END

! { note masses can be modified if desired through the SCALar commands }

! { note that this may not always be desirable --- see comments above }

SCALar MASS MULt 0.25 SELEct replicas END

... load/generate some coordinates and proceed..

2) The RESEt subcommand.

The RESEt subcommand has the effect of reducing all current subsystems to a

single primary subsystem. This is accomplished by simply switching off the

as much as the REPLica state must be restored through appropriate calls to the

REPLica command. This command is there to support the use of REPLica for

simple replication of PSF elements for which subsequent REPLIca handling is not

required.

Example:

The following example begins by building a PSF containing a single CO

molecule. An immediate call to REPLica requests the generation of 256 replicas

(with SEGId's of R1 to R256) of the primary subsystem (the CO molecule with the

SEGId CO). Next the original CO molecule is removed. The final command,

switches off the CHARMM's replica handling, leaving a PSF with 256 CO's which

interact with each other. This may seem like a redundant command given the

ability to generate a long sequence with commands like READ SEQU COOR or a

little copy and paste with your favorite editor, but remember that REPLica can

handle replication of ANY subset of the PSF, reducing the need for tampering

with RTF definitions and creating new PATCh residues (PRES's).

READ SEQUence CARDS

* a single carbon monoxide molecule

CO

GENErate CO ! generate the primary system

REPLica R NREP 256 SELEct SEGId CO END ! replicate

DELEte ATOM SELEct SEGId CO END ! remove primary system

REPLica RESEt ! reduce replicates to primary

Top

Notes on Implementation of REPLica in CHARMM.

This node is of primary directed at CHARMM developers, but may be of

interest to the curious user.

Structurally, the call to REPLica handles all atomic and topological properties

of atoms in the primary atom selection. Properties that are replicated include

group/residue membership, atom-code, IUPAC name, partial charge, parameter type

code, fixed atom flag, X,Y,Z and W for main and comparison and reference

coordinates, the forces DX,DY and DZ, the friction coefficient FBETa, and the

harmonic constraint. Topological entries include bond, angle, dihedral,

improper terms, explicit non-bonded exclusion flags and H-bond donor and

acceptor arrays. Optionally, IC table entries for the primary selection are

replicated.

For interactions, the handling of replicas in CHARMM has been implemented using

a very simple data structure which allows for a simple and efficient interface

to the central CHARMM routines. Essentially, subsystem and replica identities

are maintained through the use of linked lists. On the first call to REPLica,

the primary system (the existing PSF) is initialized to be subsystem 1 (repID),

consisting of 1 replica. Each call to REPLica, establishes a new subsystem,

and each replica requested is distinguished by a separate replica number. The

replica number is maintained at both the group (repNoG) and atom (repNoA) level

for efficiency in the non-bonded list generation routines.

In the following schematic we see the state of the data structure in which

there is a primary system consisting of 4 atoms. The threefold replication of

atoms 2 and 3 (which form a distinct group in the primary system) is shown. The

replication forms a new subsystem (repID). Each replicated group gets a

distinct flag representing the individual replica, as do the replicated atoms.

These flags index into the repID array which contains the subsystem membership

flags. In this way the subsystem/replica membership is easily established

through knowledge of the group or atom number.

Atom Name repID repNoG repNoA Comments

1 N 1 1 1 | Primary subsystem

2 CA 1 |

3 C 1 |

4 O 1 |

5 CA 2 2 2 & Replicated substem (NREP=3)

6 C 2 &

7 CA 3 3 &

8 C 3 &

9 CA 4 4 &

10 C 4 &

An schematic of the replica exclusion code in the non-bond list generation

is now given for an atom pair i and j.

IF ( ( repNoA(i) .NE. repNoA(j) ) .AND.

( repID(repNoA(i)).EQ.repID(repNoA(j)) ) ) THEN

EXCLUDE PAIR (i,j) in list

ELSE

INCLUDE PAIR (i,j) in list

ENDIF

There is another component of the REPLica data structure which is a array

(byatom) of "weights". These weights in general reflect the degree of

replication of the subsystem to which the atom belongs, but may be changed

through SCALar commands (SCALar WEIGht SET..). This array was used in the

developmental version of CHARMM with REPLicas as the interface to the

energy/force routines for correct normalization. In the current standard CHARMM

release, this array exists, but is redundant. Currently it will be filled by a

value of the reciprocal of the number of replicas requested for any subsystem.

It has been retained for some degree of flexibility in future releases. At

present it may be used as an additional array for book-keeping.

Notes on Implementation of REPLica in CHARMM.

This node is of primary directed at CHARMM developers, but may be of

interest to the curious user.

Structurally, the call to REPLica handles all atomic and topological properties

of atoms in the primary atom selection. Properties that are replicated include

group/residue membership, atom-code, IUPAC name, partial charge, parameter type

code, fixed atom flag, X,Y,Z and W for main and comparison and reference

coordinates, the forces DX,DY and DZ, the friction coefficient FBETa, and the

harmonic constraint. Topological entries include bond, angle, dihedral,

improper terms, explicit non-bonded exclusion flags and H-bond donor and

acceptor arrays. Optionally, IC table entries for the primary selection are

replicated.

For interactions, the handling of replicas in CHARMM has been implemented using

a very simple data structure which allows for a simple and efficient interface

to the central CHARMM routines. Essentially, subsystem and replica identities

are maintained through the use of linked lists. On the first call to REPLica,

the primary system (the existing PSF) is initialized to be subsystem 1 (repID),

consisting of 1 replica. Each call to REPLica, establishes a new subsystem,

and each replica requested is distinguished by a separate replica number. The

replica number is maintained at both the group (repNoG) and atom (repNoA) level

for efficiency in the non-bonded list generation routines.

In the following schematic we see the state of the data structure in which

there is a primary system consisting of 4 atoms. The threefold replication of

atoms 2 and 3 (which form a distinct group in the primary system) is shown. The

replication forms a new subsystem (repID). Each replicated group gets a

distinct flag representing the individual replica, as do the replicated atoms.

These flags index into the repID array which contains the subsystem membership

flags. In this way the subsystem/replica membership is easily established

through knowledge of the group or atom number.

Atom Name repID repNoG repNoA Comments

1 N 1 1 1 | Primary subsystem

2 CA 1 |

3 C 1 |

4 O 1 |

5 CA 2 2 2 & Replicated substem (NREP=3)

6 C 2 &

7 CA 3 3 &

8 C 3 &

9 CA 4 4 &

10 C 4 &

An schematic of the replica exclusion code in the non-bond list generation

is now given for an atom pair i and j.

IF ( ( repNoA(i) .NE. repNoA(j) ) .AND.

( repID(repNoA(i)).EQ.repID(repNoA(j)) ) ) THEN

EXCLUDE PAIR (i,j) in list

ELSE

INCLUDE PAIR (i,j) in list

ENDIF

There is another component of the REPLica data structure which is a array

(byatom) of "weights". These weights in general reflect the degree of

replication of the subsystem to which the atom belongs, but may be changed

through SCALar commands (SCALar WEIGht SET..). This array was used in the

developmental version of CHARMM with REPLicas as the interface to the

energy/force routines for correct normalization. In the current standard CHARMM

release, this array exists, but is redundant. Currently it will be filled by a

value of the reciprocal of the number of replicas requested for any subsystem.

It has been retained for some degree of flexibility in future releases. At

present it may be used as an additional array for book-keeping.

Top

The only absolute requirement for this command is that a PSF of the molecular

system be present prior to the call to REPLIca.

All non-bonded list generation options are currently supported, however

IMAGES and EXTENDED electrostatics are currently not supported.

Please note that the replica group flags follow the group membership of the

primary atom selection, therefore take care not to split groups in a selection

if group-based energy evaluations are to be subsequently used.

Run-time attributes of the system such as SHAKE constraints, BLOCK membership

and SBOUND flags will not be replicated. (Re)Issue such commands after

replication has been performed.

It must be noted that currently the replica handling mechanisms of CHARMM are

generated through the run-time use of the REPLica command. i.e. the REPLica

data structure is not currently incorporated in the standard system PSF or able

to be saved to an external file for restoring its state. The philosophy is

that all necessary attributes of the replicas are contained in the primary

system PSF and that it is therefore only necessary to keep that explicitly. Of

course, the coordinates of the individual replicas must be saved.

The only absolute requirement for this command is that a PSF of the molecular

system be present prior to the call to REPLIca.

All non-bonded list generation options are currently supported, however

IMAGES and EXTENDED electrostatics are currently not supported.

Please note that the replica group flags follow the group membership of the

primary atom selection, therefore take care not to split groups in a selection

if group-based energy evaluations are to be subsequently used.

Run-time attributes of the system such as SHAKE constraints, BLOCK membership

and SBOUND flags will not be replicated. (Re)Issue such commands after

replication has been performed.

It must be noted that currently the replica handling mechanisms of CHARMM are

generated through the run-time use of the REPLica command. i.e. the REPLica

data structure is not currently incorporated in the standard system PSF or able

to be saved to an external file for restoring its state. The philosophy is

that all necessary attributes of the replicas are contained in the primary

system PSF and that it is therefore only necessary to keep that explicitly. Of

course, the coordinates of the individual replicas must be saved.

Top

Supplementary examples.

Replication of PHE 22 and 33 and TYR 35 of BPTI

These examples illustrate two ways of setting up replicated subsystems.

In both cases replicas of the sidechains are created from CG outwards.

In the first example three calls to REPLica are made, one for each sidechain,

which create 5 replicas for each subsystem.

In the second example, one call to REPLica is made, which replicates

all three of the sidechains, to create one replicated subsystem containing

five 5 replicas of the triad.

In each case an appropriate interaction matrix for the subsystems is

created with the use of the BLOCK command.

Example 1:

3 replicated subsystems: 5 copies of each individual sidechain in each.

REPLicate A NREPlica 5 SETUP -

SELEct (SEGId 4PTI .AND. RESId 22) .AND. .NOT. -

(type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

REPLicate B NREPlica 5 SETUP -

SELEct (SEGId 4PTI .AND. RESId 33) .AND. .NOT. -

(type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

REPLicate C NREPlica 5 SETUP -

SELEct (SEGId 4PTI .AND. RESId 35) .AND. .NOT. -

(type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

! DELETE the necessary regions of the primary sub-system

DELEte ATOM -

SELEct (SEGId 4PTI .AND. (RESI 22 .OR. RESI 33 .OR. RESI 35)) .AND. -

.NOT. (type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

DEFIne phe22 SELEct SEGId A* END

DEFIne phe33 SELEct SEGId B* END

DEFIne tyr35 SELEct SEGId C* END

! set up the correct energy/force scaling.

! the default coefficient is one.

BLOCK 4

CALL 2 SELEct phe22 END ! assign replicated subsystems to blocks

CALL 3 SELEct phe33 END

CALL 4 SELEct tyr35 END

COEF 2 1 0.2 ! primary <-> replicated subsystems

COEF 3 1 0.2

COEF 4 1 0.2

COEF 2 2 0.2 ! replicated subsystem self-terms

COEF 3 3 0.2

COEF 4 4 0.2

COEF 3 2 0.04 ! replicated <-> replicated subsystems

COEF 4 2 0.04

COEF 4 3 0.04

END

Example 2:

1 replicated subsystem: 5 replicas consisting of the 3 different sidechains

REPLicate A NREPlica 5 SETUP -

SELEct (SEGId 4PTI .AND. (RESId 22 .OR. RESID 33 .OR. RESID 35) ) -

.AND. .NOT. (type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

! DELETE the necessary regions of the primary sub-system

DELEte ATOM -

SELEct (SEGId 4PTI .AND. (RESI 22 .OR. RESI 33 .OR. RESI 35)) .AND. -

.NOT. (type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

! set up the correct energy/force scaling.

BLOCK 2

CALL 2 SELEct SEGId A* END

COEF 1 2 0.2

COEF 2 2 0.2

END

Supplementary examples.

Replication of PHE 22 and 33 and TYR 35 of BPTI

These examples illustrate two ways of setting up replicated subsystems.

In both cases replicas of the sidechains are created from CG outwards.

In the first example three calls to REPLica are made, one for each sidechain,

which create 5 replicas for each subsystem.

In the second example, one call to REPLica is made, which replicates

all three of the sidechains, to create one replicated subsystem containing

five 5 replicas of the triad.

In each case an appropriate interaction matrix for the subsystems is

created with the use of the BLOCK command.

Example 1:

3 replicated subsystems: 5 copies of each individual sidechain in each.

REPLicate A NREPlica 5 SETUP -

SELEct (SEGId 4PTI .AND. RESId 22) .AND. .NOT. -

(type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

REPLicate B NREPlica 5 SETUP -

SELEct (SEGId 4PTI .AND. RESId 33) .AND. .NOT. -

(type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

REPLicate C NREPlica 5 SETUP -

SELEct (SEGId 4PTI .AND. RESId 35) .AND. .NOT. -

(type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

! DELETE the necessary regions of the primary sub-system

DELEte ATOM -

SELEct (SEGId 4PTI .AND. (RESI 22 .OR. RESI 33 .OR. RESI 35)) .AND. -

.NOT. (type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

DEFIne phe22 SELEct SEGId A* END

DEFIne phe33 SELEct SEGId B* END

DEFIne tyr35 SELEct SEGId C* END

! set up the correct energy/force scaling.

! the default coefficient is one.

BLOCK 4

CALL 2 SELEct phe22 END ! assign replicated subsystems to blocks

CALL 3 SELEct phe33 END

CALL 4 SELEct tyr35 END

COEF 2 1 0.2 ! primary <-> replicated subsystems

COEF 3 1 0.2

COEF 4 1 0.2

COEF 2 2 0.2 ! replicated subsystem self-terms

COEF 3 3 0.2

COEF 4 4 0.2

COEF 3 2 0.04 ! replicated <-> replicated subsystems

COEF 4 2 0.04

COEF 4 3 0.04

END

Example 2:

1 replicated subsystem: 5 replicas consisting of the 3 different sidechains

REPLicate A NREPlica 5 SETUP -

SELEct (SEGId 4PTI .AND. (RESId 22 .OR. RESID 33 .OR. RESID 35) ) -

.AND. .NOT. (type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

! DELETE the necessary regions of the primary sub-system

DELEte ATOM -

SELEct (SEGId 4PTI .AND. (RESI 22 .OR. RESI 33 .OR. RESI 35)) .AND. -

.NOT. (type N .or. type CA .or. type C .or. -

type O .or. type HN .or. type HA .or. type CB) END

! set up the correct energy/force scaling.

BLOCK 2

CALL 2 SELEct SEGId A* END

COEF 1 2 0.2

COEF 2 2 0.2

END

Top

Replica Path Method

The replica/path method allows the positions between sequential

replicas to be restrained. This allows minimization and simulated

annealing methods to be used to search for transition states.

B. Brooks, NIH, March 1994

This code currently requires exactly one replicated subsystem with at

least three replicas.

The nudged elastic band (NEB) method [H. Jonsson, G. Mills, and

K.W. Jacobsen, "Nudged Elastic Band Method for Finding Minimum Energy

Paths of Transitions", in "Classical and Quantum Dynamics in Condensed

Phase Simulations", World Scientific, 1998] is implemented as part of

the replica path method. The energy function in this method does not

correspond to the forces (does not pass TEST FIRST) because of the

projections involved. Only simple minimization/quenching schemes can

be used for the path optimization.

P. Maragakis, Harvard, June 2002

The nudged elastic band (NEB) method [H. Jonsson, G. Mills, and

K.W. Jacobsen, "Nudged Elastic Band Method for Finding Minimum Energy

Paths of Transitions", in "Classical and Quantum Dynamics in Condensed

Phase Simulations", World Scientific, 1998] is implemented within the

framework of the replica path method. ABNR and SD can be used with this

method. Ecount is used to calculate the energy of each replica in the path.

Note that this NEB method uses the keyword NEBF, it is different from NEBA.

See Chu, Trout, and Brooks, JCP Vol 119, pp. 12708-12717,2003 for more details

Note that GRMS will not be zero even when NEB minimization is converged.

Instead, the OFF-path and along path gradient should be close to zero.

The print of OFF-path and path gradient can be turned on by the "DEBU"

keyword of abnr.

Jhih-Wei Chu, MIT, 2003

Syntax:

RPATh OFF ! clear the replica/path energy restraint

RPATh [ KRMS real ] [ KANGle real ] [ COSMax real ] [MASS] [WEIGht]

[ KMAX real RMAX real ] [ EVWIdth real ] [CYCLic]

[ ROTAte ] [ TRANslate ]

[ NOROtate ] [ NOTRanslate ]

[ NEBA ] [ KNEB real ]

[ NEBF ] [ ETAN ] [ CIMG ] [ PPMF ] [ ANAL ]

MASS - Use mass weighting in rms determination.

WEIGht - Use the main weighting array for weighting the rms vector.

KRMS - The rms deviation force constant (Kcal/mol/A**2)

KANGle - The COS(angle) deviation force constant (Kcal/mol)

COSMax - The value of COS(theta) below which the vectors are restrained.

RMAX - The maximum rms deviation per replica step

KMAX - The force constant for exceeding the maximum step (Kcal/mol/A**2)

EVWIdth- Width of switching region in the rms bestfit rotation (FROTU)

(Angstrom**2) which is used when degenerate eigenvalues occur.

Current recommended value: 0.001, default: 0.0)

ROTAte - Do a best fit rotation for every replica step

NOROt - Don't "

TRANsl - Do a best fit translation for every replica step

NOTRans- Don't "

NEBA - Use the nudged elastic band approach

KNEB - The nudged elastic band spring constant

NEBF - Use the nudged elastic band (can work with ABNR. use KRMS

instead of KNEB.)

ETAN - Jonsson's energy based tangent estimation,

JCP, 113, 9978-9985, 2000

CIMG - Jonsson's climbing image for transition state refinement,

JCP, 113, 9901, 2000

PPMF - Print the force along the tangent direction of tha path of each

replica at each energy call.

ANAL - Print statistics of NEB calculations after some minimization

or MD steps.

Replica/Path energy functions:

Erms = sum { 0.5* Krms * ( rms - <rms> )**2 } I=1,NREP-1

I I

where rms is the weighted rms deviation between replica I and I+1

I

Erms = sum { 0.5* Kmaxrms * ( rms - Rmaxrms )**2 } I=1,NREP-1

I I

Eang = sum { 0.5* Kang * ( cosmax - cos(theta) )**2 } (cos(theta)<cosmax)

I

= 0.0 when cos(theta) >= cosmax I=1,NREP-2

where cos(theta) is determined from the dotproduct of weighted deviation

vectors between replicas I and I+1 and between I+1 and I+2.

By default, this restraint uses absolute positions. This is only appropriate

if a subset of the atoms is replicated. The ROTAtions and TRANslations options

should be used if the replicated atoms have significant freedom to move.

An example of use:

! { read sequence and generate segment }

READ SEQU COOR UNIT 11

GENErate PROT

! { define atom region in which to search for transition state }

DEFINE active sele segid prot .and. resid 15 : 19 end

! { replicate the selected residues 20 times }

REPLIcate A NREPlica 20 SELEct ( active ) END

! { redefine is necessary }

DEFINE active sele segid prot .and. resid 15 : 19 end

! { read product coordinates }

OPEN read card unit 12 name products.crd

READ coor card unit 12 COMP

! { read reactant coordinates }

OPEN read card unit 12 name reactants.crd

READ coor card unit 12

COOR orient rms mass sele .not. ( active .or segid A* ) end

! { setup initial guess coordinates for all intermediates }

set 1 1

set a 0.0

label loop

coor duplicate sele active end sele segid A@1 end

coor duplicate sele active end sele segid A@1 end comp

coor average fact @a sele segid A@1 end

incr a by 0.05

incr 1 by 1

if @1 .lt. 20.5 goto loop

DELEte ATOM SELEct active END

DEFIne replicas SELEct SEGId A* END

! { average the non active reactant and product atoms }

COOR average sele .not. replicas end

COOR copy comp

! { set up an appropriate interaction matrix }

BLOCK 2

CALL 2 SELEct replicas END

COEF 1 1 1.0

COEF 2 2 0.05

COEF 2 1 0.05

END

! { specify residue 17 as more inportant in the weighting }

SCALAR wmain set 1.0

SCALAR wmain set 4.0 sele replicas .and. resid 17 end

! invoke the path code

RPATH KRMS 100.0 KANGle 100.0 COSMax 0.5 MASS WEIGHT ROTAtion TRANSlations

! { fix the endpoints }

cons fix sele segid a1 .or. segid a20 end

minimize abnr nstep 100

! {.... perhaps simulated annealing using MD ...}

! { plot energy as a function of the path }

open write card unit 20 name energy.dat

set 1 1

label eloop

BLOCK 2

CALL 1 sele all end

CALL 2 sele replicas .and. .not. segid A@1 end

COEF 1 1 1.0

COEF 2 1 0.0

COEF 2 2 0.0

END

ENERGY

write title unit 20

* @1 ?energy

incr 1 by 1

if @1 .lt. 20.5 goto loop

.... more analysis ...

STOP

Replica Path Method

The replica/path method allows the positions between sequential

replicas to be restrained. This allows minimization and simulated

annealing methods to be used to search for transition states.

B. Brooks, NIH, March 1994

This code currently requires exactly one replicated subsystem with at

least three replicas.

The nudged elastic band (NEB) method [H. Jonsson, G. Mills, and

K.W. Jacobsen, "Nudged Elastic Band Method for Finding Minimum Energy

Paths of Transitions", in "Classical and Quantum Dynamics in Condensed

Phase Simulations", World Scientific, 1998] is implemented as part of

the replica path method. The energy function in this method does not

correspond to the forces (does not pass TEST FIRST) because of the

projections involved. Only simple minimization/quenching schemes can

be used for the path optimization.

P. Maragakis, Harvard, June 2002

The nudged elastic band (NEB) method [H. Jonsson, G. Mills, and

K.W. Jacobsen, "Nudged Elastic Band Method for Finding Minimum Energy

Paths of Transitions", in "Classical and Quantum Dynamics in Condensed

Phase Simulations", World Scientific, 1998] is implemented within the

framework of the replica path method. ABNR and SD can be used with this

method. Ecount is used to calculate the energy of each replica in the path.

Note that this NEB method uses the keyword NEBF, it is different from NEBA.

See Chu, Trout, and Brooks, JCP Vol 119, pp. 12708-12717,2003 for more details

Note that GRMS will not be zero even when NEB minimization is converged.

Instead, the OFF-path and along path gradient should be close to zero.

The print of OFF-path and path gradient can be turned on by the "DEBU"

keyword of abnr.

Jhih-Wei Chu, MIT, 2003

Syntax:

RPATh OFF ! clear the replica/path energy restraint

RPATh [ KRMS real ] [ KANGle real ] [ COSMax real ] [MASS] [WEIGht]

[ KMAX real RMAX real ] [ EVWIdth real ] [CYCLic]

[ ROTAte ] [ TRANslate ]

[ NOROtate ] [ NOTRanslate ]

[ NEBA ] [ KNEB real ]

[ NEBF ] [ ETAN ] [ CIMG ] [ PPMF ] [ ANAL ]

MASS - Use mass weighting in rms determination.

WEIGht - Use the main weighting array for weighting the rms vector.

KRMS - The rms deviation force constant (Kcal/mol/A**2)

KANGle - The COS(angle) deviation force constant (Kcal/mol)

COSMax - The value of COS(theta) below which the vectors are restrained.

RMAX - The maximum rms deviation per replica step

KMAX - The force constant for exceeding the maximum step (Kcal/mol/A**2)

EVWIdth- Width of switching region in the rms bestfit rotation (FROTU)

(Angstrom**2) which is used when degenerate eigenvalues occur.

Current recommended value: 0.001, default: 0.0)

ROTAte - Do a best fit rotation for every replica step

NOROt - Don't "

TRANsl - Do a best fit translation for every replica step

NOTRans- Don't "

NEBA - Use the nudged elastic band approach

KNEB - The nudged elastic band spring constant

NEBF - Use the nudged elastic band (can work with ABNR. use KRMS

instead of KNEB.)

ETAN - Jonsson's energy based tangent estimation,

JCP, 113, 9978-9985, 2000

CIMG - Jonsson's climbing image for transition state refinement,

JCP, 113, 9901, 2000

PPMF - Print the force along the tangent direction of tha path of each

replica at each energy call.

ANAL - Print statistics of NEB calculations after some minimization

or MD steps.

Replica/Path energy functions:

Erms = sum { 0.5* Krms * ( rms - <rms> )**2 } I=1,NREP-1

I I

where rms is the weighted rms deviation between replica I and I+1

I

Erms = sum { 0.5* Kmaxrms * ( rms - Rmaxrms )**2 } I=1,NREP-1

I I

Eang = sum { 0.5* Kang * ( cosmax - cos(theta) )**2 } (cos(theta)<cosmax)

I

= 0.0 when cos(theta) >= cosmax I=1,NREP-2

where cos(theta) is determined from the dotproduct of weighted deviation

vectors between replicas I and I+1 and between I+1 and I+2.

By default, this restraint uses absolute positions. This is only appropriate

if a subset of the atoms is replicated. The ROTAtions and TRANslations options

should be used if the replicated atoms have significant freedom to move.

An example of use:

! { read sequence and generate segment }

READ SEQU COOR UNIT 11

GENErate PROT

! { define atom region in which to search for transition state }

DEFINE active sele segid prot .and. resid 15 : 19 end

! { replicate the selected residues 20 times }

REPLIcate A NREPlica 20 SELEct ( active ) END

! { redefine is necessary }

DEFINE active sele segid prot .and. resid 15 : 19 end

! { read product coordinates }

OPEN read card unit 12 name products.crd

READ coor card unit 12 COMP

! { read reactant coordinates }

OPEN read card unit 12 name reactants.crd

READ coor card unit 12

COOR orient rms mass sele .not. ( active .or segid A* ) end

! { setup initial guess coordinates for all intermediates }

set 1 1

set a 0.0

label loop

coor duplicate sele active end sele segid A@1 end

coor duplicate sele active end sele segid A@1 end comp

coor average fact @a sele segid A@1 end

incr a by 0.05

incr 1 by 1

if @1 .lt. 20.5 goto loop

DELEte ATOM SELEct active END

DEFIne replicas SELEct SEGId A* END

! { average the non active reactant and product atoms }

COOR average sele .not. replicas end

COOR copy comp

! { set up an appropriate interaction matrix }

BLOCK 2

CALL 2 SELEct replicas END

COEF 1 1 1.0

COEF 2 2 0.05

COEF 2 1 0.05

END

! { specify residue 17 as more inportant in the weighting }

SCALAR wmain set 1.0

SCALAR wmain set 4.0 sele replicas .and. resid 17 end

! invoke the path code

RPATH KRMS 100.0 KANGle 100.0 COSMax 0.5 MASS WEIGHT ROTAtion TRANSlations

! { fix the endpoints }

cons fix sele segid a1 .or. segid a20 end

minimize abnr nstep 100

! {.... perhaps simulated annealing using MD ...}

! { plot energy as a function of the path }

open write card unit 20 name energy.dat

set 1 1

label eloop

BLOCK 2

CALL 1 sele all end

CALL 2 sele replicas .and. .not. segid A@1 end

COEF 1 1 1.0

COEF 2 1 0.0

COEF 2 2 0.0

END

ENERGY

write title unit 20

* @1 ?energy

incr 1 by 1

if @1 .lt. 20.5 goto loop

.... more analysis ...

STOP

Top

Off-Path Optimzation / Simulation (Extension to the Replica Path Method)

The off-path simulation technique allows users to compute the Potential of

Mean Force (PMF) of a particular reaction. This is accomplished by restraining

simulations to run in orthogonal planes of a pre-computed replica/path. Using

distributed computing these simulations can be run in parallel thus increasing

the sampling abilty of simulations.

Syntax:

RPATh [KRMS real] [ KMAX real] [RMAX real] [OPTImize] [ANALysis] [CURVCorr]

[Additional RPATh keywords]

KRMS - The RMS deviation force constant (Kcal/mol/A**2)

KMAX - This force constant is applied if the simulation path moves too

far away from the reference path.

RMAX - The max RMS distance the simulation path is allowed to move away

from the reference path.

OPTImize - Turn on the off-path procedure

ANALysis - Do the analysis of the off-path procedure (This must be called

after the minimization/simulation is performed)

CURVCorr - Apply a curvature correction during the off-path procedure

Example:

--------

RPATh KRMS 5000.0 KMAX 2000.0 RMAX 0.10 ROTA TRANS WEIGHT CYCLIC OPTIMIZE

Path definition:

----------------

_j_

/ \ (Simulation Path)

/ j \

/ / \ \

/ / \ \

i i k k

(Ref. Path)

Off-Path Details:

-----------------

Rij = RMSd(j(sim) --> i(ref))

Rjk = RMSd(j(sim) --> k(ref))

Rjj = RMSd(j(sim) --> j(ref))

Erms = Sum { 0.5 * Krms * (Rij - Rjk)**2 }

- This is applied to keep Rij = Rjk

- This is added to the EPATHR term

Ermax = Sum { 0.5 * Kmax * (Rjj - Rmax)**2 }

- Is applied if Rjj > Rmax

- This is added to the EPATHA term

Using this procedure allows an approximate PMF to be computed via determination

of the work needed to move from plane to plane during the simulation.

CURVCorr:

---------

The curvature correction acts as an additional restraint to prevent the

simulation path from wondering into nearby deep wells and skewing the PMF

downward. This is accomplished by scaling the force projection by the ratio of

the forward and backward distance with respect to Rij and Rjk. To use this add

the keyword CURVCorr to the RPATh command.

ANALysis:

---------

After performing an off-path optimization / simulation you can then run the

command...

RPATh ANALysis

This will print out the PMF obtained from the simulation. Two columns will be

printed: WORKTOT and ETOT. This is the work (ETOT) it takes to move from

simulation plane to simulation plane as you move through the pathway. The work

it takes to go half way from point to plane is also printed (WORKTOT). Given

long enough simulations are run, these are good approximations to the pathway

PMF.

Off-Path Optimzation / Simulation (Extension to the Replica Path Method)

The off-path simulation technique allows users to compute the Potential of

Mean Force (PMF) of a particular reaction. This is accomplished by restraining

simulations to run in orthogonal planes of a pre-computed replica/path. Using

distributed computing these simulations can be run in parallel thus increasing

the sampling abilty of simulations.

Syntax:

RPATh [KRMS real] [ KMAX real] [RMAX real] [OPTImize] [ANALysis] [CURVCorr]

[Additional RPATh keywords]

KRMS - The RMS deviation force constant (Kcal/mol/A**2)

KMAX - This force constant is applied if the simulation path moves too

far away from the reference path.

RMAX - The max RMS distance the simulation path is allowed to move away

from the reference path.

OPTImize - Turn on the off-path procedure

ANALysis - Do the analysis of the off-path procedure (This must be called

after the minimization/simulation is performed)

CURVCorr - Apply a curvature correction during the off-path procedure

Example:

--------

RPATh KRMS 5000.0 KMAX 2000.0 RMAX 0.10 ROTA TRANS WEIGHT CYCLIC OPTIMIZE

Path definition:

----------------

_j_

/ \ (Simulation Path)

/ j \

/ / \ \

/ / \ \

i i k k

(Ref. Path)

Off-Path Details:

-----------------

Rij = RMSd(j(sim) --> i(ref))

Rjk = RMSd(j(sim) --> k(ref))

Rjj = RMSd(j(sim) --> j(ref))

Erms = Sum { 0.5 * Krms * (Rij - Rjk)**2 }

- This is applied to keep Rij = Rjk

- This is added to the EPATHR term

Ermax = Sum { 0.5 * Kmax * (Rjj - Rmax)**2 }

- Is applied if Rjj > Rmax

- This is added to the EPATHA term

Using this procedure allows an approximate PMF to be computed via determination

of the work needed to move from plane to plane during the simulation.

CURVCorr:

---------

The curvature correction acts as an additional restraint to prevent the

simulation path from wondering into nearby deep wells and skewing the PMF

downward. This is accomplished by scaling the force projection by the ratio of

the forward and backward distance with respect to Rij and Rjk. To use this add

the keyword CURVCorr to the RPATh command.

ANALysis:

---------

After performing an off-path optimization / simulation you can then run the

command...

RPATh ANALysis

This will print out the PMF obtained from the simulation. Two columns will be

printed: WORKTOT and ETOT. This is the work (ETOT) it takes to move from

simulation plane to simulation plane as you move through the pathway. The work

it takes to go half way from point to plane is also printed (WORKTOT). Given

long enough simulations are run, these are good approximations to the pathway

PMF.

Top

Kinetic Energy Potential

The curvatures along a path may be controlled by adding kinetic energy

potentials. If only kinetic energy potentials are used, a straight line gives

the optimal answer. Minimizing the sum of kinetic energy potentials and

potential energies gives minimium Hamiltonian path. Adding potential energy

significantly enhances the convergence speed of path optimization on rugged

potential energy surface. The force constants of kinetic energy potentials

can be tunned to give negligible effects on reaction barriers but maintaining

efficiency. The weighting of each atom is inherited from the setups of rpath.

For each segment of replica, the kinetic energy potential is:

half * kpki * sum { wi * (ri_I+1 - ri_I)**2) }.

Determined by the length of segment, a time step associated with kinetic energy

potential can be obtained to computed temperature.

Syntax:

RPATh [ PKIN ] [ KPKI real ] [ ISOK ] [PTEM TEMP REAL ] [ WETH ] -

[ PKNU ] -

[Additional RPATh keywords]

PKIN - Actives the use of kinetic energy potential

KPKI - The force constant of kinetic energy potentials

The unit is (kcal/mole/A**2) if the unit of weighting

array is ignored.

ISOK - Maintaining constant kinetic energy (isokinetic) during

optimization. force constants are adjusted according to segment

length.

PTEM - Maintaining constant kinetic energy (isokinetic) based on

a specied value of temperature.

WETH - Using work-energy theorem to adjust the force constants of

kinetic energy potential. Force constants are changed to

compensate the changes in potential energy to maintain total

energy.

PKNU - The component of the gradient of kinetic energy potential along

the direction of potential energy gradient that is perpendicular

to the path is projected for minimial purturbation for the

optimization perpendicular to the path

EXAMPLE:

Add kinetic energy potential to path energy.

RPATH KRMS 100.0 MASS PKIN KPKI 1.0

Maintain constant kinetic energy energy.

RPATH KRMS 100.0 MASS PKIN KPKI 1.0 ISOK

Maintain constant kinetic energy energy with a temperature of 300.0

RPATH KRMS 100.0 MASS PKIN KPKI 1.0 ISOK PTEM TEMP 300.0

Kinetic Energy Potential

The curvatures along a path may be controlled by adding kinetic energy

potentials. If only kinetic energy potentials are used, a straight line gives

the optimal answer. Minimizing the sum of kinetic energy potentials and

potential energies gives minimium Hamiltonian path. Adding potential energy

significantly enhances the convergence speed of path optimization on rugged

potential energy surface. The force constants of kinetic energy potentials

can be tunned to give negligible effects on reaction barriers but maintaining

efficiency. The weighting of each atom is inherited from the setups of rpath.

For each segment of replica, the kinetic energy potential is:

half * kpki * sum { wi * (ri_I+1 - ri_I)**2) }.

Determined by the length of segment, a time step associated with kinetic energy

potential can be obtained to computed temperature.

Syntax:

RPATh [ PKIN ] [ KPKI real ] [ ISOK ] [PTEM TEMP REAL ] [ WETH ] -

[ PKNU ] -

[Additional RPATh keywords]

PKIN - Actives the use of kinetic energy potential

KPKI - The force constant of kinetic energy potentials

The unit is (kcal/mole/A**2) if the unit of weighting

array is ignored.

ISOK - Maintaining constant kinetic energy (isokinetic) during

optimization. force constants are adjusted according to segment

length.

PTEM - Maintaining constant kinetic energy (isokinetic) based on

a specied value of temperature.

WETH - Using work-energy theorem to adjust the force constants of

kinetic energy potential. Force constants are changed to

compensate the changes in potential energy to maintain total

energy.

PKNU - The component of the gradient of kinetic energy potential along

the direction of potential energy gradient that is perpendicular

to the path is projected for minimial purturbation for the

optimization perpendicular to the path

EXAMPLE:

Add kinetic energy potential to path energy.

RPATH KRMS 100.0 MASS PKIN KPKI 1.0

Maintain constant kinetic energy energy.

RPATH KRMS 100.0 MASS PKIN KPKI 1.0 ISOK

Maintain constant kinetic energy energy with a temperature of 300.0

RPATH KRMS 100.0 MASS PKIN KPKI 1.0 ISOK PTEM TEMP 300.0

Top

Discretized Feynman Path Integral Method

The REPLica command can be used together with the PINT command to compute

averaged observables of a quantum system. This computation approach

exploit the isomorphism between the discretized form of Feynmann path integrals

representation of the density matrix with an effective classical system

obeying Boltzmann statistics of a canonical ensemble at temperature T

(see D. Chandler and P.G. Wolynes, J. Chem. Phys. 74 (1981) 4078).

Molecular dynamics simulations of the effective classical system

are valid for obtaining ensemble averages, although they do not provide

information on the time-dependent quantum dynamics of the system.

Roughly speaking, the quantum delocalisation of each atom of the system

is represented in terms of a ring polymer or necklace of beads. These beads

are treated as classical particles. A discretization of the path integral

with 20 to 30 fictitious particles is usually adequate in studies of

proton transfer, although for a protein one might want to use a much smaller

number of beads. For proper use, read carefully this documentation until the

end as well as the accompagnying stream file.

B. Roux & M. Souaille, Montreal, June 1997

Following the path integral approach, each nucleus is replaced in the

effective classical system by a ring polymer, or necklace, of Nbeads

fictitious particles with a harmonic spring between nearest neighbors

along the ring. For the sake of simplicity, in the current implementation

in CHARMM, each atom is represented by the the same number of beads.

The creation of the beads is achieved by the command REPLICA (see example

below). The collection of beads of a given atom has the structure of a

necklace: each bead interacts with two neighbours and the last bead interacts

with the first. The energy of the ring polymers is a sum of harmonic terms

between consecutive beads along a necklace:

spring energy = -((Kb*T*P/(2*LAMBDA**2))*|r-r'|**2

where r and r' are the position vectors of the two beads and LAMBDA is

the thermal wavelength of the quantum particule (of mass M) represented by

the necklace, LAMBDA=HBAR**2/(M*Kb*T). These interactions are added in the

The interaction between two quantum atoms A and B is represented as follows:

the necklace of A interacts with the necklace of B in a ONE TO ONE

correspondence: each bead of A interacts with ONE AND ONLY ONE bead of B by

means of the classical CHARMM potential energy function scaled by 1/Nbeads.

There is NO such interaction between the beads belonging to the same necklace.

Moreover, if only a part of the whole system is treated quantum mechanically,

(this MUST be an entire segment) the beads of an atom A of such a subsystem

interacts with all the classical atoms by means of the classical CHARMM

potential energy function scaled by 1/Nbeads. The attribution of the

diffferent interactions as well as their scaling is achieved by the command

BLOCK. The resulting potential energy of the effective classical system is:

U_eff({R}_1, {R}_2,...,{R}_p,...,{R}_Nbeads) =

(Spring energy of ring polymers)

+ U({R}_1)/Nbeads

+ U({R}_2)/Nbeads

+ ...

+ U({R}_p)/Nbeads

+ ...

+ U({R}_Nbeads)/Nbeads

where {R}_p represents all the coordinates of the $p$-th REPLICA and

U({R}_p) represents the full CHARMM energy of the p-th REPLICA.

The configurational sampling of the effective classical system may be performed

using Langevin molecular dynamics. The choice of Langevin dynamics is dictated

by the need to avoid the non-ergodicity of path integral molecular dynamics

simulations based on the microcanonical ensemble. Of course the friction

constant and the masses used in the dynamics are immaterial as far as the

convergence toward a canonical ensemble is concerned but the spring constant

in PINT are calculated from the AMASS array, so those cannot be changed

carelessly.

Alternatively, the configuration space can now be sampled with Monte Carlo.

At present, the allowed path integral moves are displacements of single beads

(RTRN BYATom) or movements of seven sequential beads such that the path

integral spring lengths remain unchanged (CROT PIMC) (» mc ). Use of

Monte Carlo with path integrals requires addition of the keyword MC to the

PINT command.

The following stream file allows to treat the molecule SEGID as a quantum

system with Nbeads.

----------------------------------------------------------------------------

* Stream file for path integral calculations

* Before calling, set the following variables:

* Nbeads number of beads

* SEGID segid of the quantum molecule

* TEMP temperature

! Define replicas and delete original

REPLICA @segid nreplica @nbeads select segid @segid show end setup

delete atom select segid @segid show end

! Set up the correct energy/force scaling

set scale = 1.0

divide scale by @nbeads

BLOCK 2

CALL 2 SELEct ( segid @{segid}* ) show end

COEF 1 2 @scale

COEF 2 2 @scale

END

! Add springs

pint temp @temp beads @nbeads select segid @{segid}1 end -

select none end

------------------------------------------------------------------------------

Discretized Feynman Path Integral Method

The REPLica command can be used together with the PINT command to compute

averaged observables of a quantum system. This computation approach

exploit the isomorphism between the discretized form of Feynmann path integrals

representation of the density matrix with an effective classical system

obeying Boltzmann statistics of a canonical ensemble at temperature T

(see D. Chandler and P.G. Wolynes, J. Chem. Phys. 74 (1981) 4078).

Molecular dynamics simulations of the effective classical system

are valid for obtaining ensemble averages, although they do not provide

information on the time-dependent quantum dynamics of the system.

Roughly speaking, the quantum delocalisation of each atom of the system

is represented in terms of a ring polymer or necklace of beads. These beads

are treated as classical particles. A discretization of the path integral

with 20 to 30 fictitious particles is usually adequate in studies of

proton transfer, although for a protein one might want to use a much smaller

number of beads. For proper use, read carefully this documentation until the

end as well as the accompagnying stream file.

B. Roux & M. Souaille, Montreal, June 1997

Following the path integral approach, each nucleus is replaced in the

effective classical system by a ring polymer, or necklace, of Nbeads

fictitious particles with a harmonic spring between nearest neighbors

along the ring. For the sake of simplicity, in the current implementation

in CHARMM, each atom is represented by the the same number of beads.

The creation of the beads is achieved by the command REPLICA (see example

below). The collection of beads of a given atom has the structure of a

necklace: each bead interacts with two neighbours and the last bead interacts

with the first. The energy of the ring polymers is a sum of harmonic terms

between consecutive beads along a necklace:

spring energy = -((Kb*T*P/(2*LAMBDA**2))*|r-r'|**2

where r and r' are the position vectors of the two beads and LAMBDA is

the thermal wavelength of the quantum particule (of mass M) represented by

the necklace, LAMBDA=HBAR**2/(M*Kb*T). These interactions are added in the

The interaction between two quantum atoms A and B is represented as follows:

the necklace of A interacts with the necklace of B in a ONE TO ONE

correspondence: each bead of A interacts with ONE AND ONLY ONE bead of B by

means of the classical CHARMM potential energy function scaled by 1/Nbeads.

There is NO such interaction between the beads belonging to the same necklace.

Moreover, if only a part of the whole system is treated quantum mechanically,

(this MUST be an entire segment) the beads of an atom A of such a subsystem

interacts with all the classical atoms by means of the classical CHARMM

potential energy function scaled by 1/Nbeads. The attribution of the

diffferent interactions as well as their scaling is achieved by the command

BLOCK. The resulting potential energy of the effective classical system is:

U_eff({R}_1, {R}_2,...,{R}_p,...,{R}_Nbeads) =

(Spring energy of ring polymers)

+ U({R}_1)/Nbeads

+ U({R}_2)/Nbeads

+ ...

+ U({R}_p)/Nbeads

+ ...

+ U({R}_Nbeads)/Nbeads

where {R}_p represents all the coordinates of the $p$-th REPLICA and

U({R}_p) represents the full CHARMM energy of the p-th REPLICA.

The configurational sampling of the effective classical system may be performed

using Langevin molecular dynamics. The choice of Langevin dynamics is dictated

by the need to avoid the non-ergodicity of path integral molecular dynamics

simulations based on the microcanonical ensemble. Of course the friction

constant and the masses used in the dynamics are immaterial as far as the

convergence toward a canonical ensemble is concerned but the spring constant

in PINT are calculated from the AMASS array, so those cannot be changed

carelessly.

Alternatively, the configuration space can now be sampled with Monte Carlo.

At present, the allowed path integral moves are displacements of single beads

(RTRN BYATom) or movements of seven sequential beads such that the path

integral spring lengths remain unchanged (CROT PIMC) (» mc ). Use of

Monte Carlo with path integrals requires addition of the keyword MC to the

PINT command.

The following stream file allows to treat the molecule SEGID as a quantum

system with Nbeads.

----------------------------------------------------------------------------

* Stream file for path integral calculations

* Before calling, set the following variables:

* Nbeads number of beads

* SEGID segid of the quantum molecule

* TEMP temperature

! Define replicas and delete original

REPLICA @segid nreplica @nbeads select segid @segid show end setup

delete atom select segid @segid show end

! Set up the correct energy/force scaling

set scale = 1.0

divide scale by @nbeads

BLOCK 2

CALL 2 SELEct ( segid @{segid}* ) show end

COEF 1 2 @scale

COEF 2 2 @scale

END

! Add springs

pint temp @temp beads @nbeads select segid @{segid}1 end -

select none end

------------------------------------------------------------------------------