CHARMM c28a3 mc.doc

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                               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

File: mc ]-[ Node: Syntax
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                      Syntax for MOVE and MC commands

[Syntax MOVE < ADD | DELEte | EDIT | READ | WRITe > ]

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     ]

           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          ( 1+) >   ! Concerted torsion rotations

rig-unit ::= < BYREsidue | BYALl >

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 ] 

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    ]

prev ::= previous value



[Syntax MC]

MC [ NSTEps  0 ] [ ISEEd prev ] [ TEMPerature 300.0 ] -
   [ INBFrq  0 ] [ IMGFrq   0 ] [ IECHeck         0 ] -
   [ IUNCrd -1 ] [ NSAVc    0 ] [ IMULti         -1 ] [ IACCept 0 ] -
   [ IARMfrq 0 ] [ IDOMcfrq 0 ] [ ACECut       0.01 ] [ PICK    0 ] 

File: mc ]-[ Node: Description
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     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 (CART), rotations of individual 
torsions (TORS), and 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).  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.

     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.

     CART            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).

     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.

     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.

     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 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 Cartesian moves (RTRN and CART) allow anisotropic 

     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.

     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

    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.


     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.

     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.

     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 

     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 compatability 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 = [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.

File: mc ]-[ Node: Examples
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     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.
              ARMP 0.2 ARMA 0.8 ARMB 0.1 DOMCF 2.0 ANISo 1
              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.
              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.

File: mc ]-[ Node: Data Structures
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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
     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.

File: mc ]-[ Node: Shortcomings
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     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.

     The energy terms considered are bonds, angles, Urey-Bradley, dihedrals, 
impropers, vdw, electrostatic, image vdw, image electrostatic, asp-EEF1, 
asp-ACE/ACS, NOE constraints, and user.  All non-bonded calculations can be 
either atom- or group-based.  Terms not listed above (e.g., explicit H-bond 
terms) are not included in the present implementation.

     No attempt has been made to see if the image structure in MC works with
the CRYStal command.

     Group-based calculations scale poorly with the size of the system
in the present implementation due to the structure of the CHARMM exclusion
list and the group non-bonded routines.  

File: mc ]-[ Node: References
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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.

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.

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.

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.

Okamoto, Y. and Hansmann, U. H. E. (1995) Thermodynamics of helix-coil 
     transitions studied by multicanonical algorithms.  J. Phys. Chem. 99,

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.

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