# pnm (c48b1)

Plastic Network Model (PNM)

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

Jingzhi Pu

Paul Maragakis (Paul.Maragakis@deshaw.com)

Martin Karplus

Victor Ovchinnikov (ovchinnv/at/georgetown/dot/edu)

The PNM module provides an implementation of the plastic network

model (Maragakis and Karplus, 2005) for studying conformational

changes at a coarse-grained level.

* Description | Description of the PNM method

* Syntax | Syntax of the PNM commands

* Options | Command-line Options

* Examples | Usage examples

* Installation | Compiling CHARMM with PNM

* Status | Status of the code

* References | References

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

Jingzhi Pu

Paul Maragakis (Paul.Maragakis@deshaw.com)

Martin Karplus

Victor Ovchinnikov (ovchinnv/at/georgetown/dot/edu)

The PNM module provides an implementation of the plastic network

model (Maragakis and Karplus, 2005) for studying conformational

changes at a coarse-grained level.

* Description | Description of the PNM method

* Syntax | Syntax of the PNM commands

* Options | Command-line Options

* Examples | Usage examples

* Installation | Compiling CHARMM with PNM

* Status | Status of the code

* References | References

Top

Description of the PNM method

For a system with multiple energy basins, with each basin described by an

elastic network model, a combined energy function can be represented

by the plastic network model (PNM). PNMs can be used to model pathways

and dynamics between metastable states. For example, for a system

expressed as a PNM of two conformations (labeled 1 and 2),

we can construct a phenomenological energy Hamiltonian in a

diabatic representation (a 2 x 2 matrix):

H = [ G11 G12 ]

[ G21 G21 ]

where G11 and G22 are the configurational free energy functionals

for conformer 1 and 2, respectively. Following Tirion's elastic

network model (ENM), G11 and G22 can be calculated as a harmonic

deformation of each conformer with respect to its equilibrium

network configuration:

G11 = G_0(1) + 1/2 Sum { D_ab(1) C(1) [r_ab(1) - r_ab,0(1)]^2 }

a,b

G22 = G_0(2) + 1/2 Sum { D_ab(2) C(2) [r_ab(1) - r_ab,0(2)]^2 }

a,b

where G_0(i) represents the equilibrium free energy of the

conformer i; D_a,b(i) is the network connectivity matrix for the

conformer i, whose element is 1 for an atom pair (a,b) if their

distance is smaller than a cutoff distance, 0 otherwise; C(i)

is a uniform elastic constant for elastic i; r_ab(i) and

r_ab,0(i) denote the distance between atoms (a,b) in the

network i obtained from the instantaneous and equilibrium

positions, respectively.

Given a constant coupling term (G12=G21=epsilon), the adiabatic

free energy (G) of the PNM is expressed as the lowest eigen-energy

of the eigen-states that diagonalize the above energy Hamiltonian:

(G11 + G22) - sqrt[(G11 - G22)^2 + 4 epsilon^2]

G = ------------------------------------------------

2

More generally, for a system with n>2 metastable basins and n

corresponding elastic energy functions, the above Hamiltonian matrix will

be of size n x n. As for the case with n=2, the PNM energy corresponds to the

smallest eigenvalue of the matrix, which can be obtained by numerical matrix

diagonalization.

Alternative mixing models that do not involve diagonalization are also

possible. One such model available in the present implementation (as of

comprised of N elastic models, the network free energy is computed as:

G = - 1 / \beta * log ( \Sum_{i=1}^n exp [ - \beta Gii ] )

Note that the inverse temperature \beta will generally be much higher

than the room temperature value ( ~ 1.7 ) to facilitate transitions

between the constituent ENMs on relatively short simulation timescales.

Values \beta are typically be explored by trial and error.

In addition to its utility for modeling conformational changes

at a coarse-grained level, the PNM potential function can be added

as an additional (`rigidification`) potential to standard force fields. This can

be useful in an all-atom transition simulations, whereby the two (or more)

conformational states are connected via coarse-grained PNM potentials,

while the fine-grain interatomic forces are computed by the underlying force

field.

The syntax of PNM invocation has been changed in CHARMM v. c39 to allow

greater functionality and flexibility.

Description of the PNM method

For a system with multiple energy basins, with each basin described by an

elastic network model, a combined energy function can be represented

by the plastic network model (PNM). PNMs can be used to model pathways

and dynamics between metastable states. For example, for a system

expressed as a PNM of two conformations (labeled 1 and 2),

we can construct a phenomenological energy Hamiltonian in a

diabatic representation (a 2 x 2 matrix):

H = [ G11 G12 ]

[ G21 G21 ]

where G11 and G22 are the configurational free energy functionals

for conformer 1 and 2, respectively. Following Tirion's elastic

network model (ENM), G11 and G22 can be calculated as a harmonic

deformation of each conformer with respect to its equilibrium

network configuration:

G11 = G_0(1) + 1/2 Sum { D_ab(1) C(1) [r_ab(1) - r_ab,0(1)]^2 }

a,b

G22 = G_0(2) + 1/2 Sum { D_ab(2) C(2) [r_ab(1) - r_ab,0(2)]^2 }

a,b

where G_0(i) represents the equilibrium free energy of the

conformer i; D_a,b(i) is the network connectivity matrix for the

conformer i, whose element is 1 for an atom pair (a,b) if their

distance is smaller than a cutoff distance, 0 otherwise; C(i)

is a uniform elastic constant for elastic i; r_ab(i) and

r_ab,0(i) denote the distance between atoms (a,b) in the

network i obtained from the instantaneous and equilibrium

positions, respectively.

Given a constant coupling term (G12=G21=epsilon), the adiabatic

free energy (G) of the PNM is expressed as the lowest eigen-energy

of the eigen-states that diagonalize the above energy Hamiltonian:

(G11 + G22) - sqrt[(G11 - G22)^2 + 4 epsilon^2]

G = ------------------------------------------------

2

More generally, for a system with n>2 metastable basins and n

corresponding elastic energy functions, the above Hamiltonian matrix will

be of size n x n. As for the case with n=2, the PNM energy corresponds to the

smallest eigenvalue of the matrix, which can be obtained by numerical matrix

diagonalization.

Alternative mixing models that do not involve diagonalization are also

possible. One such model available in the present implementation (as of

comprised of N elastic models, the network free energy is computed as:

G = - 1 / \beta * log ( \Sum_{i=1}^n exp [ - \beta Gii ] )

Note that the inverse temperature \beta will generally be much higher

than the room temperature value ( ~ 1.7 ) to facilitate transitions

between the constituent ENMs on relatively short simulation timescales.

Values \beta are typically be explored by trial and error.

In addition to its utility for modeling conformational changes

at a coarse-grained level, the PNM potential function can be added

as an additional (`rigidification`) potential to standard force fields. This can

be useful in an all-atom transition simulations, whereby the two (or more)

conformational states are connected via coarse-grained PNM potentials,

while the fine-grain interatomic forces are computed by the underlying force

field.

The syntax of PNM invocation has been changed in CHARMM v. c39 to allow

greater functionality and flexibility.

Top

Syntax of the PNM commands

PNM < [{ INITialize int }] |

[{ DONE }] |

[{ NEWModel }] |

[{ EXP <on|true|t|yes|off|false|f|no> [TEMP real] }] |

[{ ADD [FORC real]

[CUT real]

[ZERO real]

[PMIX real]

[atom-selection]

[REMO atom-selection atom-selection]

[COMP] }] |

[ { PARA <on|true|t|yes|off|false|f|no> } ]

[ { HELP } ] >

Note: the PNM energy can be conditionally skipped by the 'SKIP' command

in CHARMM, e.g., 'SKIP PNME' will remove the PNM energy and related

force from the energy calculation.

Syntax of the PNM commands

PNM < [{ INITialize int }] |

[{ DONE }] |

[{ NEWModel }] |

[{ EXP <on|true|t|yes|off|false|f|no> [TEMP real] }] |

[{ ADD [FORC real]

[CUT real]

[ZERO real]

[PMIX real]

[atom-selection]

[REMO atom-selection atom-selection]

[COMP] }] |

[ { PARA <on|true|t|yes|off|false|f|no> } ]

[ { HELP } ] >

Note: the PNM energy can be conditionally skipped by the 'SKIP' command

in CHARMM, e.g., 'SKIP PNME' will remove the PNM energy and related

force from the energy calculation.

Top

PNM Command Options

1) INIT : initialize PNM module with a maximum number of

networks (default=2); can be called multiple times

2) DONE : finalize PNM module; can be called multiple times

3) ADD : add a new elastic model with the following optional parameters:

FORC real : elastic spring constant (default 2)

CUT real : cutoff for constructing nearest neighbors (default 10)

ZERO : equilibrium value of elastic energy function (default 0)

PMIX : the mixing constant (eps) for this elastic model (default 0.5)

the (i,j) off-diagonal entry in the Hamiltonian matrix is

computed as Gij = 1/2 ( eps_i + eps_j )

REMO : remove the connections between two groups of atoms

specified in a double selection following this keyword

(all atoms selected by default)

atom-selection: define the atom selection that PNM nodes reside on

COMP : when specified, equilibrium coordinates for the elastic

network will be taken from the comparison set

(the main set is used by default)

4)NEWM : construct a new PNM model (in addition to the ones present, if any);

the elastic models that are ADDed after this command will be part of the

new PNM model (e.g. they correspond to a separate Hamiltonian matrix);

with this functionality it is possible to simulate multiple interacting

molecules, each described by a separate PNM.

5)EXP : specify whether the exponential mixing is to be used for the

active network model (default : EXP = off) and supply an optional

temperature in K (default: 300K)

6)PARA : turn on or off parallel force computation in PNM (on by default)

7)HELP : print usage syntax

PNM Command Options

1) INIT : initialize PNM module with a maximum number of

networks (default=2); can be called multiple times

2) DONE : finalize PNM module; can be called multiple times

3) ADD : add a new elastic model with the following optional parameters:

FORC real : elastic spring constant (default 2)

CUT real : cutoff for constructing nearest neighbors (default 10)

ZERO : equilibrium value of elastic energy function (default 0)

PMIX : the mixing constant (eps) for this elastic model (default 0.5)

the (i,j) off-diagonal entry in the Hamiltonian matrix is

computed as Gij = 1/2 ( eps_i + eps_j )

REMO : remove the connections between two groups of atoms

specified in a double selection following this keyword

(all atoms selected by default)

atom-selection: define the atom selection that PNM nodes reside on

COMP : when specified, equilibrium coordinates for the elastic

network will be taken from the comparison set

(the main set is used by default)

4)NEWM : construct a new PNM model (in addition to the ones present, if any);

the elastic models that are ADDed after this command will be part of the

new PNM model (e.g. they correspond to a separate Hamiltonian matrix);

with this functionality it is possible to simulate multiple interacting

molecules, each described by a separate PNM.

5)EXP : specify whether the exponential mixing is to be used for the

active network model (default : EXP = off) and supply an optional

temperature in K (default: 300K)

6)PARA : turn on or off parallel force computation in PNM (on by default)

7)HELP : print usage syntax

Top

Examples of using PNM

An example is provided in the test suite to demonstrate the usage

of the PNM command:

pnm_test1.inp

This testcase shows a MD simulation for one beta subunit in the open

conformation of the F1-ATPase. The coarse-grained potential used for

PNM is defined by CA atom positions in the conformation taken from

PDB:1BMF. The system is simulated for 1000 MD steps. Energy and first

derivatives are also tested.

pnm_test2.inp

A test case designed to demonstrate the flexibility of the PNM code.

Sets up and performs a MD simulation of a hexameric AAA protein machine,

with each monomer described by a separate PNM with six constituent ENMs

(for a total of 36 ENMs). The six individual monomers interact via LJ

and electrostatic forces. The BLOCK module is used to switch off

unwanted monomer self-interactions.

pnm_test3.inp

Identical to pnm_test3.inp except that the exponential version of

mixing is used.

Examples of using PNM

An example is provided in the test suite to demonstrate the usage

of the PNM command:

pnm_test1.inp

This testcase shows a MD simulation for one beta subunit in the open

conformation of the F1-ATPase. The coarse-grained potential used for

PNM is defined by CA atom positions in the conformation taken from

PDB:1BMF. The system is simulated for 1000 MD steps. Energy and first

derivatives are also tested.

pnm_test2.inp

A test case designed to demonstrate the flexibility of the PNM code.

Sets up and performs a MD simulation of a hexameric AAA protein machine,

with each monomer described by a separate PNM with six constituent ENMs

(for a total of 36 ENMs). The six individual monomers interact via LJ

and electrostatic forces. The BLOCK module is used to switch off

unwanted monomer self-interactions.

pnm_test3.inp

Identical to pnm_test3.inp except that the exponential version of

mixing is used.

Top

Installation of PNM

Currently, the PNM module is not activated by default in a standard

installation. To compile the PNM code under the CHARMM environment,

execute the following commands.

./configure --add pnm

make -C build/cmake install

Installation of PNM

Currently, the PNM module is not activated by default in a standard

installation. To compile the PNM code under the CHARMM environment,

execute the following commands.

./configure --add pnm

make -C build/cmake install

Top

Status of the PNM code

The PNM energy calculations are parallelized under the atom decomposition

model in CHARMM. However, because of the simplicity of the elastic energy

networks that constitute the PNMs and the communication overhead associated

with parallelization, the serial version of the code may actually run

faster for some (usually smaller) systems.

Status of the PNM code

The PNM energy calculations are parallelized under the atom decomposition

model in CHARMM. However, because of the simplicity of the elastic energy

networks that constitute the PNMs and the communication overhead associated

with parallelization, the serial version of the code may actually run

faster for some (usually smaller) systems.

Top

References

[1]. Maragakis, P.; Karplus, M. J. Mol. Biol. 2005, 352, 807.

[2]. Tirion, M. M. Phys. Rev. Lett. 1996, 77, 1905.

[3]. Schlitter, J.; Engels, M.; Kruger, P. J. Mol. Graphics 1994, 12, 84.

[4]. Pu, J.; Karplus, M., PNAS, 2008, 105, 1192

References

[1]. Maragakis, P.; Karplus, M. J. Mol. Biol. 2005, 352, 807.

[2]. Tirion, M. M. Phys. Rev. Lett. 1996, 77, 1905.

[3]. Schlitter, J.; Engels, M.; Kruger, P. J. Mol. Graphics 1994, 12, 84.

[4]. Pu, J.; Karplus, M., PNAS, 2008, 105, 1192