Monte Carlo
The Monte Carlo commands in CHARMM have been designed to allow construction
and use of an almost arbitrary move set with only a few atom selections.
This goal is accomplished by providing a pre-defined set of move types which
can be combined to specify the allowed movements of an arbitrary CHARMM
molecule. Speed and flexibility are gained by separating the bookkeeping
associated with a move (MOVE subcommands) from the actual application of
that move to the molecule (MC).
* Syntax / Syntax of MOVE and MC commands
* Description / Description of MOVE and MC commands
* Examples / Examples of MOVE and MC commands
* SA-MC simulations / How to use the SA-MC algorithm with this module
* Data Structures / Data structures shared by the MOVE and MC commands
* Shortcomings / Known problems and limitations
* References / Some references of use

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Syntax for MOVE and MC commands
[Syntax MOVE < ADD | DELEte | EDIT | READ | WRITe | LINK > ]
MOVE ADD 1{ MVTP move-type } nsele{ SELE...END } -
[ WEIGht 1.0 ] [ DMAX 1.0 ] [ TFACtor 1.0 ] -
[ FEWEr 0/1 ] [ NLIMit 1 ] [ LABEL move-label ] -
[ opt-spec ] [ mini-spec ] [ hmc-spec ]
[ samc-spec ]
where nsele, the number of SELE...END statements,
depends on move-type
move-type (nsele)::= < RTRN rig-unit ( 1 ) | ! Rigid translations
RROT rig-unit ( 1+) | ! Rigid rotations
CART ( 1 ) | ! Single atom displacements
TORS ( 2 ) | ! Simple torsion rotations
CROT [PIMC] ( 1+) | ! Concerted torsion rotations
HMC ( 1 ) | ! Hybrid Monte Carlo
VOLU rig-unit ( 1 ) > ! Volume scalings
rig-unit ::= < BYREsidue | BYALl | BYHEavy | BYATom >
opt-spec ::= [ ARMP -1.0 ] [ ARMA 0.0 ] [ ARMB 0.0 ] -
[ DOMCf -1.0 ] [ ANISotropic 0 ]
mini-spec::= [ MINI < SD (default) | CONJ > ] -
[ NSTEps 0 ] [ NPRInt 0 ] [STEP 0.2 ] -
[ TOLEner 0.0 ] [ TOLGrad 0.0 ] [TOLStep 0.0 ] -
[ INBFrq -1 ]
hmc-spec ::= [ NMDSteps 0 ] [ TIMEstep 0.0 ] -
[ MEUPdate 0 ] [ MEFActor 0.0 ] [ MEWEight 0.0 ]
samc-spec::= [ SAMC ] [ WEPSilon 0.0 ] -
[ MEPSilon 0 ] [ NEPSilon 0 ]
MOVE DELEte < MVINdex move-index | LABEL move-label > -
MOVE EDIT < MVINdex move-index | LABEL move-label > -
[ WEIGht prev ] [ DMAX prev ] [ TFACtor prev ] -
[ NLIMit prev ] [ opt-spec ] [ mini-spec ] -
[ hmc-spec ]
prev ::= previous value
MOVE WRITE [UNIT -1]
MOVE READ [UNIT -1] [APPEnd 1]
MOVE LINK < MVI1 move-index | LAB1 move-label > -
[ MVI2 move-index ] [ LAB2 move-label ] [ GCMC ]
[Syntax MC]
MC [ NSTEps 0 ] [ ISEEd prev ] [ IACCept 0 ] [ PICK 0 ] -
[ TEMPerature 300.0 ] [ PRESsure 0.0 ] [ VOLUme prev ] [ ACECut 0.01 ] -
[ INBFrq 0 ] [ IMGFrq 0 ] [ IECHeck 0 ] -
[ IUNCrd -1 ] [ NSAVc 0 ] [ IMULti -1 ] -
[ RESTart ] [ IUNRead -1 ] [ IUNWrite -1 ] [ ISVFrq 0 ] -
[ IARMfrq 0 ] [ IDOMcfrq 0 ] [ gcmc-spec ] [ wl-spec ] -
[ samc-opt ]
gcmc-spec ::= [ GCBF 0.0 ] [ MUEX 0.0 ] [ DENSity 0.0 ] [ NGCTry 0 ] -
[ GCCUt 0.0 ] [ NOTBias 1 ] [ RGRId -1.0 ] [ INSPhere ] -
[ INSX 0.0 ] [ INSY 0.0 ] [ INSZ 0.0 ] [ INSR 0.0 ] -
[ XMIN 0.0 ] [ YMIN 0.0 ] [ ZMIN 0.0 ] -
[ XMAX 0.0 ] [ YMAX 0.0 ] [ ZMAX 0.0 ]
wl-spec ::= [ IWLRead -1 ] [ IWLWrite -1 ] [ NWLFrq 0 ] [ WLINcr 0.0 ] -
[ WLTOl 0.0 ] [ WLUPdate 0.0 ]
samc-opt ::= [ SAMC ] [ UNB ] [ IUNB -1 ]

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MOVE
The MOVE subcommands are associated with construction of the move set.
The primary MOVE subcommand is MOVE ADD, which determines all of the
locations in a subset of atoms to which a move type can be applied. For
each location (or "move instance"), MOVE ADD also determines the rotation
axes and centers, the moving atoms, and the relevant bonded terms. Thus,
each call of MOVE ADD results in a group of move instances of the same move
type (the number of instances is stored in the substitution variable ?NMVI).
By repeatedly calling the MOVE ADD command, the user can employ several
different types of moves in conjunction, which typically yields the most
efficient and complete sampling.
The available pre-defined move types are rigid translations (RTRN), rigid
rotations (RROT), single atom displacements (RTRN), rotations of individual
torsions (TORS), concerted rotation of seven (or, in the case of a chain end,
six) torsions (CROT) to deform the system locally (Dinner, 2000; Dinner, 1999;
Go and Scheraga, 1970; Dodd et al., 1993; Leontidis et al., 1994), hybrid
Monte Carlo propagations (HMC) (Duane et al., 1987; Mehlig et al., 1992),
and volume scaling moves for constant pressure simulations (Eppenga and
Frenkel, 1984). Each of these can be applied to an arbitrary subset of atoms
using standard CHARMM SELE...END statements.
MVTP rig-unit nsele Description
---- -------- ----- -----------
RTRN BYALl 1 The entire atom selection is rigidly translated.
RTRN BYREsidue 1 The residue containing each selected atom is
rigidly translated. If more than one atom in
a residue is selected, each counts as a separate
move instance.
RTRN BYHEavy 1 Each heavy atom and its associated hydrogen atoms
are rigidly translated.
RTRN BYATom 1 Each instance is a displacement of a single atom by
a random vector distributed uniformly in an ellipsoid
(see the description of the ANISotropic keyword).
For historic reasons, the CART keyword is a synonym
for RTRN BYATom, but use of the former is discouraged
since the moves are not actually based on Cartesian
coordinates.
RROT BYALl 1-2 The entire first atom selection specifies the rigid
body of atoms to be rotated, and each of the atoms in
the second atom selection is an allowed rotation
center. The second selection need not be a subset of
the first, so rotations around atoms outside the
rigid body can occur. If no second atom selection is
given (or one is given, but no atoms are selected),
the rotations are made around the center of mass of
the first atom selection.
RROT BYREsidue 1 There is only a single atom selection, and each
selected atom is a center of rotation (around a
randomly selected axis) for its residue. If more
than one atom in a residue is selected, each counts
as a separate move instance.
RROT BYHEavy 1 The hydrogens attached to a heavy atom are rigidly
rotated around the heavy atom. If the FEWEr keyword
is set to 1 (the default), a move instance is counted
for each selected heavy atom with at least one hydrogen
atom attached (whether or not the hydrogens are selected
does not matter). If the FEWEr keyword is set to 0,
all heavy atoms are counted to permit straightforward
linking with RTRN BYHEavy move groups.
TORS 2 The two selections define the middle atoms (JK in
IJKL) of the rotatable torsions. If the FEWEr keyword
is set to 1 (default is 0), the directionality of the
selection will be ignored and each rotatable bond will
be included only once in the move set (such as to rotate
the end with fewer atoms). Otherwise, each rotatable
bond will be included either once or twice depending on
whether the atom selections match the bond in only one
direction (JK) or both (JK and KJ). Only torsions in
the PSF are enumerated.
CROT 1+ The first atom selection defines the "backbone"
along which the 7 (or in the case of a chain end, 6)
dihedrals lie. Each additional pair of selections
defines non-rotatable bonds. The first bond in a set
of 6 or 7 is the driver torsion. Non-rotatable bonds
are not allowed at the third or fifth bonds following
the driver (counting only rotatable ones). Note that
there is no checking for whether bonds selected to be
rotatable are indeed so. NLIMit is the number of
torsions in addition to the driver torsion that are
restricted by the maximum rotation (DMAX); values of
0 to 5 are possible. In general, setting NLIMit
greater than or equal to 1 is recommended since it
speeds up the root finding process and moves with large
changes to the torsions tend to be rejected anyway.
Concerted rotations of path integral "polymers" require
the PIMC keyword.
HMC 1 The selected atoms are propagated with the specified
TIMEstep for NMDSteps molecular dynamics steps. The
change in total energy is used for the acceptance
criterion. SHAKE constraints are respected. The
standard fixed atom list is ignored, but note that,
in the present implementation, selections that move
only small parts of the system will be inefficient.
The non-bond list update during dynamics is separate
from that used in MC and is controlled by a common
variable set by the DYNAmics command. To suppress
updates of the non-bond list, it is necessary to
issue a dummy dynamics statement prior to MC:
DYNAmics NSTEps 0 INBFrq 0 NSAVC 0.
If IACCept=2, the dynamics take place on a transformed
potential (Andricioaei and Straub, 1996; Andricioaei
and Straub, 1997); use of the Tsallis method with HMC
requires that the TSALLIS keyword be included in
pref.dat during compilation.
If IACCept=0, sampling can be enhanced using the
method introduced in Andricioaei et al. (2003)
by setting MEFAactor, MEUPdate, and MEWEight to
non-zero values. MEFActor is the csi multiplicative
factor above Eq. 18 in the above paper, MEUPdate is
the frequency of updating the bias vector, and
MEWEight is the weight each new dynamics step carries
in the bias vector (dt/tl, where dt is the timestep
and tl is the averaging period).
VOLUme rig-unit 1 Volume scaling moves. Changes in ln V are selected
uniformly from the allowed range, and the scaling is
around the image center. The possible rigid units
are the same as for RTRN and RROT. If a "mixed"
scaling move is desired (e.g., solvent atoms are
scaled by residue while solute atoms are scaled
individually), it is necessary to couple two or more
"pure" scaling moves (see MOVE LINK).
GCMC 1 Insertion and deletion moves for grand canonical
Monte Carlo. Move instances are identified in the
same way as RTRN BYREsidue above. In other words,
there is one instance per selected atom in a grand
canonical molecule (which must be a single residue);
selecting more than one atom in a residue will waste
memory. Moreover, translation and rotation moves of
grand canonical molecules should be linked to the
GCMC move group to avoid wasting time moving inactive
atoms (see MOVE LINK). See MC and the Examples
section below for further details on grand canonical
Monte Carlo.
In addition, MOVE ADD associates with each group of move instances a set
of parameters.
The values of the following parameters are used in all MC calls.
WEIGht The relative weight of that group of move instances in
the complete move set. The probability of picking a
group of move instances with weight w_i is w_i/(sum_j w_j)
where (sum_j w_j) is the total of all the WEIGht values.
DMAX The initial maximum displacement of each instance in a
group. Translations are in angstroms and rotations are in
degrees. In cases where anisotropic automatic optimization
is to be performed, the initial assignment is isotropic.
TFACtor A multiplicative factor to scale the TEMPerature in the
acceptance criterion. TFACtor is not used in assigning
the initial velocities in HMC moves.
LABEL An optional tag for the group of move instances.
Only the first four characters are retained. All sets of
move instances are also given an integer index which can
be used instead.
The following optional parameters are associated with automatic
optimization of the volumes from which individual move instances are chosen
(the timestep in HMC moves). The two available methods are the Acceptance
Ratio Method (ARM) and Dynamically Optimized Monte Carlo (DOMC); both are
described in detail by Bouzida et al. (1992). The latter has the advantage
that it allows optimization of anisotropic volumes.
ARMP ARM target probability of move instance acceptance.
ARMA, ARMB Parameters to avoid taking the logarithm of zero in ARM:
DMAX(new) = DMAX(old)*ln(ARMA*ARMP+ARMB)/ln(ARMA*obsP+ARMB)
where obsP is the observed probability of accepting that
move instance.
DOMCF The F factor in DOMC:
DMAX(new) = DOMCF*SQRT[(d2ave*TEMP)/Eave]
where d2ave is the observed average square of the
displacement and Eave is the observed average change in
energy (both averages are done over all moves, not just those
accepted). DOMCF is used for the anisotropic version of
this equation as well. In the event that the square
root of a negative number must be taken, the routine
branches to ARM optimization, so ARMA, ARMB, and ARMP
should be set even if one plans on using DOMC.
ANISotropic DOMC anisotropic optimization of the volume from which the
moves are chosen. If ANISotropic is 0, it is off (isotropic)
and, if ANISotropic is non-zero, it is on. At present,
only 3D translation moves (RTRN and CART) allow anisotropic
optimization.
The parameters NSTEps, NPRInt, STEP, TOLEner, TOLGrad, TOLstep, and
INBFrq are associated with minimization prior to application of the acceptance
criterion (Li and Scheraga, 1987) and have the same meanings as for
MINImization (see minimiz.doc). Note that the INBFrq used for minimization
(set in MOVE) is distinct from that used for Monte Carlo (set in MC) even
though the command-line keywords are the same; moreover, INBFrq and NPRInt
access common variables associated with minimization directly and thus are
not stored with the rest of the move set by MOVE WRITE. During the
minimization phase of a move, all atoms that have not been constrained with
CONS FIX are considered mobile. At present, the minimization algorithms
available are steepest descents (SD) and conjugate gradients (CONJ);
in the case of CONJ, the following parameters are fixed: NCGC = 100,
PCUT = 0.9999, and TOLIter = 100. It is important that the user keep in
mind that MC with minimization does not satisfy the detailed balance
condition (microscopic reversibility) and thus should be used only for
conformational searching, not calculating equilibrium averages. Minimization
following HMC moves is not allowed.
The optional parameters SAMC, WEPSilon, MEPSilon and NEPSilon, are used for Spatial
Averaging MC simulations (SA-MC) (Refs): see the section 'SPATIAL AVERAGING SIMULATIONS'
below for more details. For each move instance the user can decide to enable
the SA-MC algorithm by putting the SAMC keyword, and then precise the 3 parameters
WEPSilon, MEPSilon, and NEPSilon.
MOVE DELEte allows the user to delete a group of move instances. The
group to be deleted is the first that matches the four-character tag specified
by LABEL or the integer specified by MVINdex; if there is a conflict, the
LABEL is used.
MOVE EDIT allows one to change the values of the parameters associated
with a group of move instances. The matching rules are the same as those for
MOVE DELEte (as a result, the LABEL parameter itself cannot be changed with
MOVE EDIT). Any parameter not specified retains its current value. If a
positive value is specified for DMAX, all move instances will be reset to
that (isotropic) value; if a negative value (default) is specified, the
maximum displacements retain their current values. If DMAX is not specified
and the ANISotropic flag changes such that anisotropy is no longer allowed
(when it was previously), the maximum displacements are assigned as the
geometric mean of the eigenvalues of the matrix used to calculate the allowed
ellipsoid from the unit sphere.
MOVE WRITe writes out the current move set to a formatted file opened
with OPEN WRITe CARD.
MOVE READ reads in a move set. If APPEnd is 0, existing moves
are eliminated; otherwise they are preserved and the new moves are appended.
MOVE ADD calls can follow to expand the move set further.
MOVE LINK links two existing moves such that they are always performed
together before applying the acceptance criterion. For example, one might
wish to perform both a rigid body translation and a rigid body rotation of
a butane molecule in the same MC step. The first move group [specified by
either its label (LAB1) or index (MVI1)] remains "active", while the second
move group becomes "slaved" to the first. In other words, a move from the
second group can no longer be selected by itself (as a result, only the WEIGht
parameter of the first move group matters). At present, move instances within
the groups are matched by indices, so the two move groups must have the same
numbers of instances. MOVE LINK can be called repeatedly to create a chain
of moves. In the example mentioned above, one might also want to change the
central dihedral of the butane molecule in the same MC step. In the second
MOVE LINK call, the second move group from the first call would become the
first move group, and the new move group would be the second:
MOVE LINK LAB1 RTRN LAB2 RROT !Resulting chain is RTRN->RROT
MOVE LINK LAB1 RROT LAB2 DIHE !Resulting chain is RTRN->RROT->DIHE
Moves can be decoupled (in the reverse order by which they were linked) by
specifying only a single move label (LAB1) or index (IND1):
MOVE LINK LAB1 RROT !Resulting chain is RTRN->RROT
MOVE LINK LAB1 RTRN !All moves are treated separately
If minimization before applying the acceptance criterion is desired, it must
be associated with the first move group in the chain (RTRN in the example);
other minimization parameters will be ignored. By the same token, only the
TFACtor for the first move group will be used. Linking is not allowed with
Hybrid Monte Carlo moves (HMC).
The GCMC keyword is used to specify that a move group involves changes to
the coordinates of molecules that are inserted and deleted during grand
canonical simulations. MOVE LINK GCMC suppresses moving "ghost" molecules,
and thus saves simulation time. In contrast to other calls to MOVE LINK,
no chain of move groups is constructed, and the non-GCMC move group is still
active (not slaved).
Please note that the MOVE LINK command is presently under development.
Consequently, the syntax might change in future versions. Moreover, its
compatibility with certain other features, such as automatic optimization
of the move limits, is not guarranteed.
MC
The MC command performs the loop over the appropriate number of Monte
Carlo steps. Each step consists of (1) randomly picking a group of move
instances (weighted), (2) randomly picking an instance from that group
(unweighted), (3) calculating the energetic contribution of the moving
atoms and their images, (4) applying the move, (5) calculating the energetic
contribution in the new configuration, (6) applying the acceptance criterion,
(7) if necessary updating the statistics for automatic optimization of the
move limits, and finally (8) performing any desired I/O (at present, only
trajectory writing is enabled).
NSTEps The number of loop iterations. Each pick of a single move
instance and subsequent application of the acceptance
criterion counts.
ISEEd The seed for the random number generator. If it is not
specified, it is unchanged, so that a script can be seeded
once initially and then loop over an MC command and yield
different behavior with each call.
TEMPerature The absolute temperature in degrees Kelvin.
PRESsure The pressure in atmospheres.
VOLUme The starting volume for constant pressure simulations.
It is only necessary to specify if the images are created
by an IMAGe TRANsformation rather than the CRYStal command.
INBFrq The non-bond list update frequency.
If INBFrq = 0, the list is not updated.
If INBFrq < 0, a heuristic is applied every -INBFrq steps;
the list is updated if any atom during a checking step moved
more than 0.5*(CUTNB - CTOFNB).
Note that a call to ENERgy or UPDAte must be made before
MC to initialize parameters for non-bond list generation.
IMGFrq The image list update frequency.
An image update will force a non-bond list update.
If IMGFrq = 0, the list is not updated.
If IMGFrq < 0, the list is updated if a heuristic non-bond
list update is done; this option should be used only if
INBFrq is also negative.
IECHeck The total energy check frequency.
If IECHeck = 0, the energy is not checked.
If IECHeck < 0, the energy is checked if a heuristic non-bond
list update is done; this option should be used only if
INBFrq is also negative.
The difference between the MC running total and the current
total is printed in the Delta-E column of the table.
NSAVc The frequency of writing out the trajectory.
If NSAVc is 0, no coordinates are written.
IUNCrd The I/O unit for trajectory writing.
RESTart If present, this keyword indicates that the run is a restart.
IUNRead The I/O unit from which to read the restart information.
IUNWrite The I/O unit to which to write the restart information.
ISVFrq The frequency of writing the restart information.
IARMfrq The frequency of updating the move size by ARM. Note that
this counter runs separately for each move instance.
If IARMfrq is 0, the move size is not updated.
IDOMcfrq The frequency of updating the move size by DOMC. Note that
this counter runs separately for each move instance.
If IDOMcfrq is 0, the move size is not updated.
If both IARMfrq and IDOMcfrq are non-zero, IARMfrq takes
priority.
PICK Flag for method of selecting moves from the move set:
0 = Random move group and random instance (default)
1 = Try each move group and instance sequentially
2 = Random move group but sequential instances within
a group
At present, the PICK flag is considered to be an unsupported
feature and may be changed without backwards compatibility in
future versions.
IACCept The acceptance criterion to be used.
If IACCept is 0, Boltzmann (Metropolis) weighting is used.
If IACCept is 1, multicanonical (constant entropy) weighting
is used (in which case TEMPerature is ignored).
If IACCept is 2, Tsallis (generalized) weighting is used.
If IACCept is 3, Wang-Landau version of the multicanonical
algorithm is used (recommended over IACCept=1).
If IACCept is 4, Spatial Averaging (SA-MC) can be used.
see the dedicated section below for more details.
The following optional parameters are specific to particular non-canical
acceptance criteria.
EMIN The estimated minimum energy of the system in Tsallis MC.
QTSAllis The Tsallis q parameter (see Andricioaei and Straub, 1997).
IMULti The I/O unit for reading in the multicanonical weights.
The file format (subject to change) is:
CHARMM title
Emin Emax Nbin
i E_i ln[n(E_i)]
.
.
.
Nbin E_Nbin ln[n(E_Nbin)]
Note that MC closes this file, so that it must be reopened
before each MC call with multicanonical weighting.
IWLRead The I/O unit from which to read the initial guesses of the
histogram and free energy for Wang-Landau MC. The file must
begin with a CHARMM title (lines starting with "*") followed
by the data in three columns: (1) the indices of the arrays
holding the accumulated histogram and free energy surface,
(2) minus the free energy divided by kT (-bF), and (3) the
accumulated histogram (n). In other words,
CHARMM title
i -bF_i n_i
.
.
.
Nbin -bF_Nbin n_Nbin
IWLWrite The I/O unit to which to write the histogram and free energy
for Wang-Landau MC. The format for the result file is similar
to that for the initial guess, except that it does not have a
title section.
NWLFrq The frequency of checking the flatness of the histogram for
Wang-Landau MC. A value on the order of 100000 is recommended.
WLINcrement The initial value of the amount added to the free energy at
each step of a Wang-Landau MC simulation. It will be halved
automatically when the criterion specified by WLUPdate is met.
WLUPdate The criterion for checking if the accumulated histogram is
flat. CHARMM MC compares this number with the ratio of
the standard deviation of the accumulated histogram to its
average. A smaller WLUP will give more accurate results for
the free energy but will require more simulation time.
Reasonable choices are typically between 0.01 and 0.05.
WLTOlerance If WLIN reaches WLTO, a Wang-Landau simulation is considered
converged and will stop prior to reaching the specified number
of MC steps. A value on the order of 10^(-8) is recommended.
The following optional parameters are associated with grand canonical
Monte Carlo simulations. Two algorithms for facilitating insertions can
be used. The cavity bias method (Mezei, 1980) generates a set of candidate
insertion positions at each GCMC step randomly, determines the ratio P_N =
cavity sites/total sites, and picks a cavity site for insertion. The
acceptance probability is adjusted based on the average of P_N to satisfy
detailed balance. Alternatively, one can keep track of the cavities with
a grid (Mezei, 1987). The grid-based method is more memory intensive and
slower at lower density but is generally more efficient at higher density
and in confined geometries such as within the binding pocket of a protein.
See Woo et al. (2004) for a discussion of the methods.
MUEX Excess chemical potential.
DENSity Desired density.
GCBFactor B = mu/kT + ln<N>, which sets the excess chemical potential.
If DENSity is greater than zero, GCBFactor will be calculated
from MUEX and DENSity.
NGCTry Number of points to generate in cavity-biased simulations
that do not employ a grid. Simple cavity bias is used
automatically if NGCTry is greater than zero.
GCCUt Cutoff distance used for evaluating whether a point in space
corresponds to a cavity in cavity-biased simulations.
RGRId The grid spacing for grid-based cavity-biased insertion.
Grid-based insertion is used automatically if RGRId is
greater than zero
INSPhere Restrict insertion to a sphere of radius INSR centered on
INSX, INSY, INSZ. Otherwise, insertions are made in the box
spanning XMIN < X < XMAX, YMIN < Y < YMAX, and
ZMIN < Z < ZMAX.
NOTBias The number of orientations to attempt when inserting.
The parameter ACECut allows approximation of the ACE screening energy
term to accelerate MC simulations which employ the ACE/ACS solvation model.
In calculating the total screening energy, as in molecular dynamics, one
performs two summations: the first determines the Born radii (b_i) and self
energies of the atoms and the second determines the screening energy given
the Born radii. In MC, this scheme becomes inefficient. One typically moves
only a small part of the system in each step, but one must update all the
pairwise interactions (between atoms i and j) in which b_i, b_j, or both
change (even if the distance between i and j remains the same). In CHARMM,
this problem is overcome by neglecting the contribution to the change in
screening energy from atom pairs in which both S_i and S_j are less than
ACECut, where S_i = Sum_k.ne.i [E_ik^self/(tau*q_i^2)] (see equations 22, 28,
and 31 of Schaefer and Karplus, 1996). For peptides, a choice of ACECut =
0.01/Angstrom has been found to yield close to the maximum increase in speed
with errors of less than 0.001 kcal/mol/step. Note that HMC moves and moves
involving minimization employ the standard ACE energy routines and thus
calculate the ACE energy exactly.
The following optional parameters are associated with Spatial Averaging
simulations (SA-MC), see the detailed section below for more details.
SAMC Keyword enabling the use of the SA-MC algorithm.
UNB Keyword enabling storage of the unbiasing factor.
IUNB The unit of the file to which the unbiasing factor is stored.
(1 column text file previously opened by the user).

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EXAMPLE OF A STANDARD MC SIMULATION
No special actions must be taken during PSF generation to run an MC
simulation. Essentially, input files set up for dynamics can be turned into
MC input files by replacing the DYNAmics call with a series of MOVE ADD calls
(or a MOVE READ call) followed by a MC call. For example, to simulate a
peptide in water, one could add to the CHARMM script:
.
.
.
! Standard PSF generation and coordinate input above
! Create the MC move set
! Allow waters to move by rigid translations and rotations.
! Allow anisotropic optimization of the volume from which the
! translation vectors are chosen.
MOVE ADD MVTP RTRN BYREsidue WEIGht 2.0 DMAX 0.10 SELE (TYPE OH2) END -
ARMP 0.2 ARMA 0.8 ARMB 0.1 DOMCF 2.0 ANISo 1
MOVE ADD MVTP RROT BYREsidue WEIGht 2.0 DMAX 30.0 SELE (TYPE OH2) END -
ARMP 0.2 ARMA 0.8 ARMB 0.1 DOMCF 2.0 ANISo 0
! Allow all torsions to move by simple rotations
MOVE ADD MVTP TORS WEIGht 0.1 DMAX 30.0 FEWEr 1 -
SELE ALL END SELE ALL END
! Allow the backbone to move by concerted rotations with non-rotatable
! peptide bonds and N-CA proline bonds. If disulfides are present, the
! cysteine phi and psi should be restricted too.
MOVE ADD MVTP CROT WEIGht 0.5 DMAX 10.0 NLIMit 1 LABEL PEPTide -
SELE ((TYPE N).OR.(TYPE CA).OR.(TYPE C)) END -
SELE (TYPE C) END SELE (TYPE N) END -
SELE (RESNAME PRO .AND. TYPE CA) END -
SELE (RESNAME PRO .AND. TYPE N) END
! Seed the generator (for this example, this could be done below)
MC ISEEd 391004
OPEN WRITE UNFOrmatted UNIT 32 NAME example.trj
! Do 20000 steps at 300 K, writing energy 100 steps.
! Update the non-bonded list every 200 and
! the maximum displacements every 5 picks of that move instance
MC IACCept 0 NSTEp 20000 TEMP 300 -
INBFrq 200 IECHeck 400 IMGFrq 400 IDOMcfrq 10 -
IUNC 32 NSAVc 100
In this example, there are four groups of move instances (one for
each MOVE ADD call).
It should be mentioned that it is also possible to use moves in MC
apart from those which can be generated by MOVE ADD since the MOVE READ
command does not do any checking as it reads in the necessary move set
information. For example, it is straightforward to make rigid rotations
around a pseudo-dihedral simply by changing the pivot and moving atom lists
of a dihedral rotation. An understanding of the following section
(Data Structures) will aid in manual move creation.
GRAND CANONICAL SIMULATIONS
Some additional actions are necessary for grand canonical Monte Carlo
simulations.
Grand canonical atoms are designated as active through the GCMCon
array, which can be manipulated with the SCALAR command. A value
of 1 indicates that an atom is active and a value of 0 indicates
that an atom is inactive. It is suggested that there be roughly twice
as many grand canonical molecules as anticipated will be active on
average to accomodate fluctuations.
Atoms that block grand canonical insertions in the cavity-based
schemes described above are also initialized through a SCALAR array,
GCBLocker. A value of 1 indicates that an atom is a blocker, and
a value value of 0 indicates that it is not. Time can be saved by
excluding hydrogens.
.
.
.
! Generate PSF allowing for extra molecules.
READ SEQUENCE TIP3 432
GENERATE MAIN SETUP NOANGLE NODIHEDRAL
! Read coordinates of starting structure (216 molecules here).
OPEN READ CARD UNIT 1 NAME tip216.crd
READ COOR CARD UNIT 1
CLOSE UNIT 1
! Copy coordinates to uninitialized atoms. This is important for
! molecular liquids to define the internal structure.
COOR DUPLicate SELEct IRES 1:216 END SELEct IRES 217:432 END
! Set the active and blocking atoms
SCALar GCMC SET 1.0 SELEct IRES 1:216 END
SCALar GCMC SET 0.0 SELEct IRES 217:432 END
SCALar GCBLocker SET 1.0 SELEct TYPE OH2 END
ENERGY
! Create the MC move set
! Rigid translations
MOVE ADD MVTP RTRN BYREsidue WEIGht 1.0 DMAX 0.25 -
SELE (TYPE OH2) END LABEl DISP
! Rigid rotations
MOVE ADD MVTP RROT BYREsidue WEIGht 1.0 DMAX 30.0 -
SELE (TYPE OH2) END LABEl ROTA
! Insertion and deletion
MOVE ADD MVTP GCMC WEIGht 1.0 SELE (TYPE OH2) END LABEl GCMC
! Link the GCMC moves to the rotations and displacements to avoid moving
! inactive molecules.
MOVE LINK GCMC LAB1 GCMC LAB2 DISP
MOVE LINK GCMC LAB1 GCMC LAB2 ROTA
! Link the translations and rotations to each other for greater efficiency
MOVE LINK LAB1 DISP LAB2 ROTA
OPEN WRITE UNFOrmatted UNIT 32 NAME gcmc.trj
! Do 1000000 steps at 298 K, writing energy 1000 steps.
! Grid-based insertions with 10 attempted orientations are used.
MC NSTEp 1000000 TEMPerature 298.0 ISEEd 302430 -
INBFrq 100 IECHeck 1000 IMGFrq 100 IUNC 32 NSAVc 1000 -
MUEX -5.8 DENS 0.03342 -
RGRId 0.25 GCCUt 2.5 NOTB 10 -
XMIN -18.856 YMIN -18.856 ZMIN -18.856 -
XMAX 18.856 YMAX 18.856 ZMAX 18.856
The trajectory saved contains all of the grand canonical molecules.
The inactive coordinates are set to the initialization flag (9999.0D0) before
being written. When using the trajectory file, read the trajectory and then
delete the inactive molecules:
DELEte ATOM SELEct .BYRES. PROP X .GT. 9998.0 END
The list of active atoms is common, and thus GCMC can be combined readily with
other CHARMM features such as dynamics. In addition, it is worth noting that
the marriage of GCMC and images is not an entirely happy one; errors arising
from insufficiently frequent image updates will be minimized by making the
region in which insertion is allowed well within the primary system.

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SPATIAL AVERAGING SIMULATIONS
Spatial averaging (SA-MC) is an efficient algorithm dedicated
to the study of rare-event problems: at the heart of this method is the
realization that from the equilibrium density a related, modified probability
density candidate can be constructed through a suitable transformation. This new
density is more highly connected than the original density which increases the
probability for transitions between neighboring states which in turn speeds up
the sampling. Practically, several new configurations are generated at each step
as following :
(1) Starting from a trial configuration X_0 of the system, a Gaussian distribution
of MEPSilon sets of NEPSilon configurations with standard deviation WEPSilon,
centered around X_0 is generated.
(2) The randomly chosen MC MOVE --- such as translation or rotation --- is then
applied to all MEPSilon*NEPSilon configurations and the corresponding energies
are estimated.
(3) A modified acceptance criterion (see Refs) is calculated by using those
energies.
So SA-MC uses a biased acceptance criterion: if the user is interested in some
thermodynamic properties the results have to be unbiased. Two optional parameters
(LUNB and IUNB) are used for storing in a given text file an unbiasing ratio which
can be used in a later post processing and unbiasing step (see SA-MC reference below).
Enabling SA-MC is a two steps procedure: first, for all or some of the move
instances in the input file, the user can enable SA-MC by adding the SAMC keyword,
and then give a value for WEPSilon (Gaussian width), MEPSilon (number of sets) and
NEPSilon (number of configurations):
.
.
.
! Create the MC move set by a series of calls to MOVE ADD
MOVE ADD MVTP RTRN BYATom WEIGht 1.0 DMAX 0.15 LABEL TR -
ARMP 0.40 ARMA 0.4 ARMB 0.4 DOMCf 3.0 -
SAMC WEPS 0.5 MEPS 10 NEPS 10 -
SELE ALL END
MOVE ADD MVTP TORS WEIGht 1.0 DMAX 25.0 FEWEr 1 LABEL DIHE -
ARMP 0.20 ARMA 0.4 ARMB 0.4 DOMCf 5.0 -
SAMC WEPS 0.25 MEPS 5 NEPS 5 -
SELE ALL END SELE ALL END
.
.
.
Then on the main "MC [...]" command line the user needs to set the
IACCept parameter to 4, and to add the SAMC keyword once more time. If the UNB
keyword is also added, unbiasing data is stored in the unit file IUNB
(note that this unit has to be opened before, see the following example):
.
.
.
OPEN WRITE UNFOrmatted UNIT 2 NAME test.dcd
OPEN WRITE FORMatted UNIT 10 NAME unbiasing.dat
MC TEMPerature 300.0 NSTEps 1000 IECHeck 1 IACC 4 PICK 0 -
ISEED 9921142 222455522 11142255 6221444 -
IUNCrd 2 NSAVc 1 ACECut 0.0 -
IDOM 10 -
SAMC UNB IUNB 10
.
.
.
Compatibility of the SA-MC implementation with features of the MC module (Jan. 2014):
Moves:
RTRN YES
RROT YES
TORS YES
CROT NO
HMC NO
VOLU NO
GCMC NO
MOVE LINK feature NO
Move limits optimisations
ARM and DOM optim. YES
ANISotropic optim. NO
Ensembles :
NVT (default) YES
NPT (with PRES) NO
Wang-Landau NO

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Data Structures
MOVE ADD establishes each of the following pointers for all move types.
Each is a pointer to a dynamically allocated array that is n-instance elements
long, where n-instance is equal to the number of move instances in that group.
In all cases, if the array does not apply to a particular move, it is not
allocated.
MDXP This array contains the information about the limits of the
move. For isotropic or one-dimensional moves, it is simply
an n-instance-long array of REAL*8 elements containing the
maximum displacement. If the displacements are to be drawn
from an anisotropic volume, the array is a list of pointers,
each of which points to an array of 9 REAL*8 elements which
make up the matrix that transforms the unit sphere into the
appropriate ellipsoid.
IBLSTP A list of n-instance pointers, each of which points to
the list of bonded terms changing under that move instance.
For each element in the four-element array QBND (bonds=1,
angles=2, dihedrals=3, impropers=4) that is true, there is
an element listing the index of the final element containing
indices of that bonded term type followed by the list of
terms themselves. This list is then followed by a similar
one for the next bonded term type with QBND(i) set to true.
For example, if bonds 3, 8, and 10 and angles 16 and 17
were changing, the QBND array would contain (T T F F) and the
list would contain (4 3 8 10 7 16 17).
Urey-Bradley terms are handled with the lists generated for
angle terms, so do not get their own entries.
IPIVTP This array keeps track of any pivot or special atoms.
If there is only one pivot atom, then it is stored in the
array. If there are multiple (e.g., 2 for a TORS move
and 14 for a CROT move), the list stores a pointer to
a list containing the pivot atoms.
IMVNGP This array contains a compact list of the moving atoms.
Each element contains a pointer to a list of the following
form. The first element in the list is 1 more than the
number of rigid groups (NG). Elements 2 to NG contain the
index of the last array element with information about that
rigid group. The atoms in a rigid group are stored as
the first and last atoms in a contiguous range of atom indices.
In addition, it is worth commenting on the CHARMM fixed atom list (IMOVE)
here. MC does NOT use the fixed atom list in selecting atoms to move; rather,
atoms are held in place by judicious construction of the move set. However,
CONS FIX can be used to save memory because MC constructs the symmetrized
non-bonded list that it uses for energy calculations from the standard (upper
triangle) non-bonded list. Care must be exercised when using this feature to
avoid errors arising from moving atoms in the fixed atom list since no checking
is done.

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Shortcomings
In the interest of computational efficiency, Monte Carlo calls specific
energy routines directly, rather than through the main ENERGY routine. As a
result, not all energy terms are supported. Those that are supported are
bonds, angles, Urey-Bradley, dihedrals, impropers, vdw, electrostatic,
image vdw, image electrostatic, QM/MM (MOPAC only), path integral, asp-EEF1,
asp-ACE/ACS, NOE constraints, and user (also note that the user must edit
both usersb.src and mcuser.src for the user energy to work correctly). All
non-bonded calculations can be either atom- or group-based. Terms not listed
above are not included in the present implementation.
Only atom-based non-bonded lists can be used in grand canonical simulations.
No warnings are printed for attempts to move a bonded (or patched)
residue by rigid translation and rotation.
Attempts to move cross-linked residues will break MOVE ADD if
MVTP is CROT. If there is a large drift in the bond energies when
bonds lengths and angles are fixed, it is probably because a non-rotatable
bond (for example, the N-CA bond in proline) is being rotated by CROT.
Someday, a flag might be provided to choose between automatic elimination
of such moves and CROT-type relaxation of the cross-link (correct Jacobian
weighting is necessary to meet the detailed balance condition in the latter),
but such a flag is not on the immediate agenda of the MC developer.

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REFERENCES
All studies that employ the MOVE and MC commands should reference:
Hu, J., Ma, A. and Dinner, A. R. (2006) Monte Carlo simulations of
biomolecules: The MC module in CHARMM. J. Comp. Chem. 27, 203-216.
In addition, studies that employ the CROT moves should reference:
Dinner, A. R. (2000) Local deformations of polymers with nonplanar rigid
main chain internal coordinates. J. Comp. Chem., 21, 1132-1144.
Grand canonical simulation studies should reference:
Woo, H.-J., Dinner, A. R. and Roux, B. (2004) Grand canonical Monte Carlo
simulation of water in protein environments. J. Chem. Phys., in press.
Studies that employ the momentum-enhanced hybrid MC should reference:
Andricioaei, I., Dinner, A. R. and Karplus, M. (2003) Self-guided enhanced
sampling methods for thermodynamic averages. J. Chem. Phys., 118,
1074-1084.
Studies that employ Wang-Landau MC should reference:
Ma, A., Nag, A. and Dinner, A. R. (2006) Dynamic coupling between coordinates
in a model for biomolecular isomerization. J. Chem. Phys. 124, 144911.
Calvo, F. (2002) Sampling along reaction coordinates with the Wang-Landau
method. Mol. Phys. 100, 3421-3427.
Wang, F. and Landau, D. P. (2001) Efficient, multiple-range random walk
algorithm to calculate the density of states. Phys. Rev. Lett. 86, 2050-2053.
Studies that employ SA-MC should reference:
Doll, J. D., Gubernatis, J. E., Plattner, N., Meuwly, M., Dupuis, P., Wang, H. (2009)
A spatial averaging approach to rare-event sampling. J. Chem. Phys., 131
Plattner, N., Doll, J. D., Meuwly, M. (2010) Spatial averaging for small molecule
diffusion in condensed phase environments. J. Chem. Phys., 133 .
Hedin, F., Meuwly, M., Plattner, N., Doll, J. D. (2014) Spatial Averaging: Sampling
Enhancement for Exploring Configurational Space of Atomic Clusters
and Biomolecules. JCTC, DOI: 10.1021/ct500529w
The following references may also be of interest:
Andricioaei, I. and Straub, J. (1997) On Monte Carlo and molecular dynamics
methods inspired by Tsallis statistics: Methodology, optimization, and
application to atomic clusters. J. Chem. Phys. 107, 9117-9124.
Andricioaei, I. and Straub, J. (1996) Generalized simulated annealing
algorithms using Tsallis statistics: Application to conformational
optimization of a tetrapeptide. Phys. Rev. E 53, R3055-R3058.
Berg, B. A. and Neuhaus, T. (1992) Multicanonical ensemble: A new approach
to simulate first-order phase transitions. Phys. Rev. Lett. 68, 9-12.
Bouzida, D., Kumar, S. and Swendsen, R. H. (1992) Efficient Monte Carlo
methods for the computer simulation of biological molecules.
Phys. Rev. A 45, 8894-8901.
Dodd, L. R., Boone, T. D. and Theodorou, D. N. (1993) A concerted
rotation algorithm for atomistic Monte Carlo simulation of polymer
melts and glasses. Mol. Phys. 78, 961-996.
Duane, S., Kennedy, A. D., Pendleton, B. J. and Roweth, D. (1987) Hybrid
Monte Carlo. Phys. Lett. B 195, 216-222.
Eppenga, R. and Frenkel, D. (1984) Monte Carlo study of the isotropic and
nematic phases of infinitely thin hard platelets. Mol. Phys. 52,
1303-1334.
Go, N. and Scheraga, H. A. (1970) Ring closure and local conformational
deformations of chain molecules. Macromolecules 3, 178-187.
Leontidis, E., de Pablo, J. J., Laso, M. and Suter, U. W. (1994)
A critical evaluation of novel algorithms for the off-lattice Monte Carlo
simulation of condensed polymer phases. Adv. Polymer Sci. 116, 285-318.
Lee, J. (1993) New Monte Carlo algorithm: Entropic sampling.
Phys. Rev. Lett. 71, 211-214.
Li, Z. and Scheraga, H. A. (1987) Monte Carlo-minimization approach to the
multiple-minima problem in protein folding. Proc. Natl. Acad. Sci. USA
84, 6611-6615.
Mehlig, B., Heermann, D. W. and Forrest, B. M. (1992) Hybrid Monte Carlo
method for condensed-matter systems. Phys. Rev. B 45, 679-685.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and
Teller, E. (1953) Equation of state calculations by fast computing
machines. J. Chem. Phys. 21, 1087-1092.
Mezei, M. (1980) A cavity-biased (T, V, mu) Monte Carlo method for the
simulation of fluids. Mol. Phys. 40, 901-906.
Mezei, M. (1987) Grand-canonical ensemble Monte Carlo study of dense liquid
Lennard-Jones, soft spheres and water. Mol. Phys. 61, 565-582.
Okamoto, Y. and Hansmann, U. H. E. (1995) Thermodynamics of helix-coil
transitions studied by multicanonical algorithms. J. Phys. Chem. 99,
11276-11287.
Schaefer, M. and Karplus, M. (1996) A comprehensive analytical treatment of
continuum electrostatics. J. Phys. Chem. 100, 1578-1599.
Tsallis, C. (1988) Possible generalization of Bolzmann-Gibbs statistics.
J. Stat. Phys. 52, 479-487.