# pnm (c38b1)

Plastic Network Model (PNM)

--------------------------------------------
Jingzhi Pu (pu@tammy.harvard.edu)
Paul Maragakis (Paul.Maragakis@deshaw.com)
Martin Karplus (marci@tammy.harvard.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 | PNM Command Options
* Limitations | Limitations of the current code
* Examples | Usage example
* Installation | How to install PNM in CHARMM environment.
* Status | Status of the PNM code and future development
* References | References for the PNM method

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Description of the PNM method

For a system undergone conformational transitions that connect
multiple free energy basins around metastable conformers, the
system can be represented by a plastic network, based on which
reaction pathways and dynamics between these stable states can
be conveniently studied. In particular, 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
newtork model (ENM), G11 and G22 can be calculated as a harmonic
deformation of each conformer with respect to its equlibrium
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 eplison^2]
G = ------------------------------------------------
2

Besides the use by itself to study conformational change,
the PNM method can also be added as a ridigification potential
in conjunction to an all-atom force field. This is sometimes
useful in an all-atom simulation with accelerated molecular
dynamics (for example, targeted molecular dynamics), where
local unfolding occurs due to the use of a trajectory much
shorter than the timescale of the physical event. By using this
kind of combination, the simulation stability can be achieved
with a rigidifying potential that smoothly connects two
conformational states (PNM), while the full atomic details
of interactions that a coarse-grained representation lack,
can be obtained as well.

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Syntax of the PNM commands

PNMInitialize { [ COEA real ] } [ atom-selection ]
{ [ COEB real ] }
{ [ CUTA real ] }
{ [ CUTB real ] }
{ [ ZERA real ] }
{ [ ZERB real ] }
{ [ PMIX real ] }
{ [ REMO] [atom-selection] [atom-selection]}

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

PNMI: turn on a PNM calculation; specified before ENERGY is called
to invoke the PNM option.

COEA: spring constants for conformational state A

COEB: spring constants for conformational state B

CUTA: network cutoff for conformational state A

CUTB: network cutoff for conformational state B

ZERA: zero of free energy of conformational state A

ZERB: zero of free energy of conformational state B

PMIX: the mixing constant in PNM (epsilon)

REMO: remove the connections between two groups of atoms
specified in an double selection following this keyword

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

^_
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Limitations

The current PNM code is only limited for two conformational states.
The implementation assumes that the atomic coordinates that define
the two equilibrium conformations are present in the main and the
the comparison coordinate set, respectively, at the point where
the PNMI command is invoked (once the PNM is initialized, these
coordinate information are not used by PNM any more, thus can be
modified for other purposes). A script following a similar procedure
given below can be used for setting up a PNM:

...

! First make a copy of the main and comparison coordinate
!--------------------------------------------------------
scalar x store 1
scalar y store 2
scalar z store 3

scalar xcomp store 4
scalar ycomp store 5
scalar zcomp store 6

! read in coordinate file (PNM state 1)
!--------------------------------------
open read form unit 1 name "pnm1.crd"
close unit 1

! read in coordinate file (PNM state 2)
!--------------------------------------
open read form unit 1 name "pnm2.crd"
read coord comp card unit 1
close unit 1

! PNM initialization
!-------------------
PNMI coea 0.2 coeb 0.2 cuta 12.0 cutb 12.0 -
zera 0.0 zerb 0.0 pmix 1.0 sele prot end -
remo sele crown end sele gamma end

! restore the main and comparison set, as if nothing happened
!------------------------------------------------------------
scalar x recall 1
scalar y recall 2
scalar z recall 3

scalar xcomp recall 4
scalar ycomp recall 5
scalar zcomp recall 6

...

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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 performs a TMD simulation for one beta subunit in
F1-ATPase from a open to a closed conformation, i.e, beta_E -> beta_TP,
which corresponds to bind an ATP. The coarse-grained potential used for
PNM is defined by CA atom positions in two protein conformations
(taken from PDB:1BMF, chain E for beta_E and chain F for beta_TP).
The system is first equilibrated at the open state (beta_E) for 1000
steps, then transformed to the closed conformation (beta_TP) during a
1000 steps TMD simualtion, and finally equlibrated at the end state for
anther 1000 steps. Energy and first derivatives are also tested.

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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, the
'PNM' keyword needs to be specified in the build/host-machine-type/pref.dat
file. The modification of the pref.dat file can be done by providing the
'+PNM' argument to the installation script "install.com" when CHARMM
is installed:

./install.com host-machine-type size +PNM

Here the '+PNM' option will adds the 'PNM' keyword into the pref.dat
file for a given compilation.

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Status of the PNM code

Several aspects of the code will be improved in the future,
and new functionalities are under development:

1. Parallelization of the code
2. Treating bonded and non-boned connections differently
3. Generalization to a PNM with more than two metastable basins
4. Multi-scaled molecular dynamics that switch between a coarse-grained
PNM representation and an all-atom representation.

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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., work in progress.