# pnm (c46b2)

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

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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.

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Syntax of the PNM commands

PNM < [{ INITialize int }] |
[{ DONE }] |
[{ NEWModel }] |
[{ EXP <on|true|t|yes|off|false|f|no> [TEMP 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.

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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

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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.

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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.

make -C build/cmake install

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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.

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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