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).
* Menu:
* Syntax:: Syntax of MOVE and MC commands
* Description:: Description of MOVE and MC commands
* Examples:: Examples of MOVE and MC commands
* Data Structures:: Data structures shared by the MOVE and MC commands
* Shortcomings:: Known problems and limitations
* References:: Some references of use
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 ] [ NLIMit 1 ] [ LABEL move-label ] -
[ opt-spec ] [ mini-spec ] [ hmc-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 ]
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 ]
[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 ] -
[ IARMfrq 0 ] [ IDOMcfrq 0 ]
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. A move instance is
counted for each selected heavy atom with at least
one hydrogen attached (whether or not the hydrogens
are selected does not matter).
TORS 2 The two selections define the middle atoms (JK in
IJKL) of the rotatable torsions. If the FEWEr keyword
is set to 1, 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); only 0 and
1 are implemented at present. In general, NLIMit 1
is recommended since it speeds up the root finding
process and moves with large changes to the torsions
are likely 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.
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).
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.
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).
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.
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.
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.
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.
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.
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.
NSAVc The frequency of writing out the trajectory.
If NSAVc is 0, no coordinates are written.
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.
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.
EXAMPLE
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 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.
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.
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. All non-bonded calculations can be
either atom- or group-based. Terms not listed above are not included in the
present implementation.
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.
REFERENCES
All studies that employ the MOVE and MC commands should reference:
Dinner, A. R. (1999) Monte Carlo Simulations of Protein Folding.
Ph.D. Thesis (Harvard University, Cambridge, MA).
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.
Dinner, A. R. (2000) Local deformations of polymers with nonplanar rigid
main chain internal coordinates. J. Comp. Chem., in press.
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
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