gamus (c46b2)

Gaussian Mixture Adaptive Umbrella Sampling (GAMUS)

GAMUS is a hybrid of adaptive umbrella sampling (see
» adumb syntax ) and metadynamics which is suited to
identifying free energy minima for multidimensional reaction coordinates.
Like adaptive umbrella sampling this method attempts to calculate the free
energy surface in terms of designated reaction coordinates, and uses the
negative of this as a biasing potential to enhance the sampling.
The distribution of reaction coordinates is expressed in terms of mixtures
of Gaussians whose size and shape are optimized to fit the data as closely
as possible. This is similar to the use of Gaussians in metadynamics to fill
free energy basins but is more flexible. No grids or histograms are used,
which reduces the memory and statistical requirements of the method and
allows the efficient exploration of free energy surfaces in 3-5 dimensions.
The method is described in

P. Maragakis, A. van der Vaart, and M. Karplus. J. Phys. Chem. B 113, 4664
(2009). doi:10.1021/jp808381s

Please report problems to Justin Spiriti at jspiriti@usf.edu and/or
Arjan van der Vaart at avandervaart@usf.edu.


* Syntax | Syntax of the GAMUS commands
* Function | Purpose of each of the commands
* Caveats | Some limitations of the GAMUS method
* Installation | Installation of GAMUS within CHARMM
* Examples | Usage example of the GAMUS module


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Syntax

[SYNTAX GAMUS functions]

Syntax:

GAMUs DIHE 4X(atom-spec)

GAMUs INIT [TEMP real] [RUNI int] [WUNI int] [IFRQ int]

GAMUS FIT DATA int NGAUSS int NREF int NITR int SIGMA real GAMMA real

GAMUS REWEight NSIM int QUNI int PUNI int NGAUss int [NREF int] [NITR int]
[SIGMa real] [GAMMa real] [WEIGHT int]

GAMUS WRITE UNIT int

GAMUS CLEAR

GAMUS INFO [VERBose]

where: atom-spec ::= { segid resid iupac }
{ resnumber iupac }


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0. Introduction

A GAMUS simulation cycles through three stages:
1. Molecular dynamics with a biasing potential equal to minus the free energy
surface estimated from the previous simulation.
2. Determination of the partition function ratios Z_i/Z_0 for each biased
simulation relative to the unbiased simulation. This is done using the
multistate acceptance ratio method (MARE), which optimizes the likelihood of
observing the work values associated with the transitions between biasing
potentials (see P. Maragakis et al. J. Phys. Chem. B 112, 6168 (2008))
3. Reweighting of the data and fitting of a new mixture of Gaussians to the
distribution of reaction coordinates encountered in previous simulations.
This is done using the expectation-maximization (E-M) algorithm (see A. P.
Dempster et al. J. R. Stat. Soc. B 39, 1 (1977) and Bowers et al.
Comput. Phys. Commun. 164, 311 (2008)). Starting from an initial guess, this
algorithm iteratively refines the weights, means, and variance-covariance
matrices of the Gaussians in order to maximize the likelihood of observing the
data given the fitted distribution. The E-M algorithm is started several
times from randomly chosen starting points; the fit with the best likelihood
is chosen. From this fit, the new free energy surface and biasing potential
may be obtained.

GAMUS produces three sets of input/output files. GAMUS potential files contain
the weights, means, and variance-covariance matrices of the Gaussians used to
define the biasing potential, as well as the value of the Bayesian knowledge
parameter gamma. These files have the following format:

34 5 ! Number of Gaussians, number of coordinates
-25 ! log(gamma)

then follows the natural logarithms of the weights, the means, and the
inverses of the variance-covariance matrices, see subroutine READGAMUS in
gamus/gamus.src
Energy-coordinate files contain the values of the reaction coordinates
encountered during the simulation, as well as the values of the biasing
potential. Weight files contain the values of the reaction coordinates
as well as the natural logarithms of the weights needed to recover a canonical
Boltzmann distribution.


1. GAMUS DIHE

Define a dihedral angle as a reaction coordinate for GAMUS.


2. GAMUS INIT

Initializes the GAMUS potential. TEMP specifies the temperature of the
simulation. RUNI and WUNI give unit numbers for the biasing potential to be
used and the energy-coordinate file to be written during the simulation.
An initial potential file is needed for the first GAMUS simulation; this
should specify 0 gaussians and a value of ln(gamma) equal to -D*ln(360)
where D is the number of reaction coordinates.
IFRQ gives the frequency of recording the values of the reaction coordinates.
This should be infrequent enough that consecutive values are uncorrelated
(about 0.2 ps in most cases).


3. GAMUS CLEAR

Turns GAMUS off and clears all associated data structures.


4. GAMUS REWEight

Performs MARE and the E-M algorithm in order to calculate a new biasing
potential from previous potentials and energy-coordinate files. The biasing
potential files must be opened as a continuous sequence of NSIM units starting
with unit PUNI; likewise, the energy-coordinate files must be opened as
a continuous sequence of units starting with QUNI.

Keyword Default Purpose

NSIM n/a Number of previous simulations to include

PUNI n/a Initial biasing potential unit

QUNI n/a Initial energy-coordinate file unit

NGAUss n/a Number of Gaussians to be used (should gradually increase
with the number of simulations. It is suggested to add
about 1-2 Gaussians to the fit per ns of simulation.)

NREF 20 Number of refinements using the E-M algorithm from randomly
chosen starting points.

NITR 200 Maximum number of iterations of the E-M algorithm per
refinement

SIGMa 5 Minimum size of a Gaussian in any direction (in degrees)

SMAX 90 Maximum size of a Gaussian in any direction (in degrees)

GAMMa -1000 Cutoff for the Bayesian prior (see below)

WEIGht -1 Unit number for writing weights of the frames (as their
natural logarithms) together with values of the reaction
coordinates

WORK -1 Unit number for writing work values input into MARE
(primarily for debugging purposes)

The E-M optimization can have a tendency for Gaussians to collapse around
individual data points. For this reason, a minimum size of the Gaussian
in each direction is imposed. When a Gaussian collapses to this minimum
in all directions, the message "gaussian number N is of minimum size in
all directions" is produced. The SMAX option is used to impose a maximum size
of a Gaussian in order to prevent periodicity assumptions from being violated.
If a large number of these messages appear, it may mean that too many
Gaussians are being used for the fit, or that the cap on ln(gamma) is too
small (see below).

The value of gamma is chosen based on the probability of obtaining the least
probable sampled data point. A cap on the value of ln(gamma) can be used to
restrict the extrapolation of the free energy surface in order to avoid deep
artificial minima in the free energy surface (as described in Maragakis et al.
JPC B 113, 4664 (2009)). This is recommended if more than two reaction
coordinates are used. The value of the cap should be chosen so that the free
energy differences ln(Zi/Z0) average around zero. It should be noted that
the biasing potential is limited to kT ln(gamma) so setting too low a cap
can limit the part of the free energy surface that is sampled.


5. GAMUS FIT

Uses the E-M algotithm to fit weighted values of the reaction coordinates
found in the unit specified by DATA to a mixture of Gaussian distributions.
The NGAUSS, NREF, NITR, SIGMA and GAMMA parameters are specified as described
above for GAMUS REWEight. The first column in the file must be for the natural
logarithms of the weights; the remaining columns are for the values of the
reaction coordinates.


6. GAMUS WRITE

Writes the current GAMUS potential to a unit specified by UNIT. This usually
follows a GAMUS REWEIGHT or GAMUS FIT command that generates a new GAMUS
potential (see above).


7. GAMUS INFO

Prints out the weights and mean values of all the Gaussians in the current
GAMUS potential. If the VERBose option is specified, each variance-covariance
matrix will also be diagonalized to find the principal axes of each Gaussian
and the width of the Gaussian along each axis.


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Caveats

1. The expectation-maximization algorithm fits the probability density, not
the free energy. Since the probability density is lower near free energy
barriers, and since the free energy is proportional to the logarithm of
the probability density, the statistical errors in estimating the free energy
surface are greater near barriers. Consequently, GAMUS does a much better job
of identifying and locating free energy basins and of determining their shapes
and relative free energies than it does of estimating free energy barriers.

2. The time necessary to fit a set of Gaussians increases linearly with the
number of Gaussians to be fitted. Consequently, fitting can become very
expensive if many Gaussians are used. In addition, using too many Gaussians
can result in some of the Gaussians collapsing around individual data points
("gaussian number N is of minimum size in all directions" message).
Consequently, it is suggested to add Gaussians slowly during the run;
a rate of about 1-2 additional Gaussians per ns of
simulation is recommended.

3. Because of the extrapolation involved in determining the biasing potential
from the free energy surface, it is possible for the fitting to introduce
artificial minima in the free energy surface. These artificial minima
should go away in long enough simulations. Adjusting the cap value of
ln(gamma) can help with this, as described above.


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Installation

GAMUS requires an implementation of LAPACK in order to perform linear algebra
as part of the E-M and MARE algorithms.

To compile CHARMM with GAMUS included add the option '--with-gamus'
to the configure command line, for example:

./configure --with-gamus

Since the name and path of the LAPACK library can vary from one system to
another, cmake will attempt to find a suitable LAPACK installation
and linker options. These options will be added to the link command line.
If the options are not determined correctly, one may set the cmake variables
LAPACK_LINKER_FLAGS and LAPACK_LIBRARIES to the required linker flags (excluding -l and -L) and to the libraries (using full path name) to link against to use LAPACK respectively. The cmake variables may be set using the '-D' option
to the configure command. If all else fails, one may set
the environment variable LDFLAGS before running the configure command
with the correct options, such as -L and -l options.


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Examples

The main loop for cycling through steps 1-3 above is encoded in the script.
During each cycle molecular dynamics is invoked twice: once for equilibration
and once for sampling. GAMUS is also invoked twice: once for each section of
molecular dynamics. At the end of the molecular dynamics simulations,
the GAMUS REWEIGHT command is used to reweight the data from all previous
simulations and fit this to a mixture of Gaussians using the E-M algorithm.
This results in a new GAMUS potential, which is used for the next cycle of
molecular dynamics simulation.

The script structure is as follows:

! first read in the force field, PSF, set up implicit or explicit solvent, etc.
! write an initial GAMUS potential, specifying the number of reaction
! coordinates (4 in this case)

calc initgamma = -4 * ln( 360.0 )
open unit 1 write formatted name @9gamus-1.dat
write title unit 1
* 0 4
* @initgamma

close unit 1


set index = 1
label gamusloop

calc oldindex = @index - 1
open unit 1 read formatted name @9restart-col-@oldindex.rst
read coor dynr curr unit 1
close unit 1

! here we equilibrate
open unit 29 read card name @9restart-col-@oldindex.rst
open unit 30 write card name @9restart-eq-@index.rst

!this file contains the GAMUS potential
open unit 44 read card name @9gamus-@index.dat
! in this file CHARMM will record biasing potential and reaction
! coordinate values encountered during the simulation.
! We write to a different file from "gamuse..." to keep this from being
! used later by the MARE and GMM fits
open unit 45 write card name @9gamusex-@INDEX-q-@INDEX.dat
open unit 46 write unform name @9gamus-eq-@index.dcd
! This activates the GAMUS biasing potential and specifies the reaction
! coordinate (4 dihedral angles)
gamus init temp @temp runi 44 wuni 45 ifrq @gamusfreq
gamus dihe pep 1 c pep 2 n pep 2 ca pep 2 c
gamus dihe pep 2 n pep 2 ca pep 2 c pep 3 n
gamus dihe pep 2 c pep 3 n pep 3 ca pep 3 c
gamus dihe pep 3 n pep 3 ca pep 3 c pep 4 n
dynamics ... iunrea 29 iunwri 30 iuncrd 46 !perform dynamics for equilibration
close unit 29
close unit 30
close unit 44
close unit 45
close unit 46
gamus clear
! and now we collect statistics

open unit 29 read card name @9restart-eq-@index.rst
open unit 30 write card name @9restart-col-@index.rst
! We do everything over again, this time writing in a file that is
! used by the MARE and GMM fits.
! The definition of the reaction coordinate must match the one given above.
open unit 44 read card name @9gamus-@index.dat
open unit 45 write card name @9gamuse-@INDEX-q-@INDEX.dat
open unit 46 write unform name @9gamus-col-@index.dcd
gamus init temp 300.0 runi 44 wuni 45 ifrq @gamusfreq
gamus dihe pep 1 c pep 2 n pep 2 ca pep 2 c
gamus dihe pep 2 n pep 2 ca pep 2 c pep 3 n
gamus dihe pep 2 c pep 3 n pep 3 ca pep 3 c
gamus dihe pep 3 n pep 3 ca pep 3 c pep 4 n
! prnlev -6
dynamics ... iunrea 29 iunwri 30 iuncrd 46 !perform dynamics for sampling
close unit 29
close unit 30
close unit 44
close unit 45
close unit 46


! now use the new commands to reweight the potential
calc newindex = @index + 1
calc ngauss = 4 + @index !This calculates the number of Gaussians to be used for the fit.
set i = 1
label loop
calc u1 = 10 + @i
calc u2 = 100 + @i
open unit @u1 read formatted name @9gamus-@i.dat
open unit @u2 read formatted name @9gamuse-@i-q-@i.dat
incr i by 1
if i .le. @index then goto loop

open unit 7 write formatted name @9weights-@index
open unit 8 write formatted name @9gamus-@newindex.dat
gamus reweight nsim @index puni 11 quni 101 ngauss @ngauss nref 4 nitr 200 sigma 5.0 gamma -25.0 weights 7
gamus write unit 8
close unit 8
close unit 7

set i = 1
label loop2
calc u1 = 10 + @i
calc u2 = 100 + @i
close unit @u1
close unit @u2
incr i by 1
if i .le. @index then goto loop2
gamus clear

incr index by 1
if index .le. 4 then goto gamusloop

gamus clear