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Free Energy Perturbation Calculations

The PERTurbe command allows the scaling of energy between PSFs for use in energy analysis, comparisons, slow growth free energy simulations, widowing free energy simulation, and for slow growth homology modeling. This is a rather flexible implementation of free energy perturbation that allows connectivity to change. Also, three energy restraint terms (harmonic, dihedral and NOE) and the SKIP command flags are subject to change which allows a flexible way in which to compute free energy differences between different conformations. This code in implemented in a non-intrusive manner and works with all minimizers and integrators. SHAKe can now be applied to bonds which are mutated as well; an appropriate constraint corrections is calculated automatically in these cases.

Syntax of Free Energy Perturbation Commands

PERTurb  [OFF] [INBFrq int nonbond-specs] [RESEt] [MMFP] soft-core-spec
         atom-selection [INBFrq  0] [CHEM]               ]

soft-core-spec::= [SCL0 SCR0 real] [SCL1 SCR1 real]

The PERT OFF command disables the free energy routines and the current (lambda=1) PSF is used for subsequent commands.

When OFF is not specified, this command saves the current PSF as the lambda=0 state. The atom-selection indicated which atoms have changed. This is to make the calculation run more efficiently. If only a small percent of the atoms have changed, this doubles the performance. The nonbond specs are included to make sure the nonbond exclusion lists are properly setup. This then allows the connectivity to change during the simulation. INBFrq should not be set to zero here unless the exclusion lists have already been setup in a previous command.

Adding the MMFP keyword signals that the user intends to change MMFP restraints as part of an alchemical transformation. If it is omitted, the MMF-potentials are used as constant energy terms as was the case in CHARMM versions up to c30a2(x).

The soft-core-spec specifies whether the shift-based soft-core potential applies to the the simulations. The soft-core only applies on the repulsive part (as defined by the WCA separation) of the Lennard-Jones potential. To get meaningful free energy, the soft-core potential only is to be used when the electrostatic and WCA attractive interaction between the selected atoms and the rest of the system are turned off by using the SCALar CHARge and SCALar WCAD. There are two sets of parameters that correspond to the energy evaluation in the initial and final state. SCL0 specifies that soft-core potential is used for the initial state. SCR0 is the corresponding parameter that controls the strength of the soft-core potential for the initial state. SCL1 and SCR0 are the corresponding values for the final state. When SCR* equals to zero, all WCA repulsive interaction of the selected atoms is off. When SCR* equals to one, full WCA repulsive interaction of the selected atoms is on. A typical usage of the soft-core potential is to compute the free energy from non-interaction state to repulsively interaction state for a given selection of atoms. The simulation can be conducted with a series of Widom insertion/deletion stages. The following values of the SCR* are typical to perform the simulation in stages.

Window SCR0 SCR1
1 0 0.2
2 0.2 0.3
3 0.3 0.4
4 0.4 0.5
5 0.5 0.6
6 0.6 0.7
7 0.7 0.8
8 0.8 0.9
9 0.9 1.0

Because of the number of stages, the repulsive potential change in each stage is so small that a single Widom insertion or deletion (the sampling is done at the two end points, no intermediate windows required) is sufficient for the free energy in each stages. Notice this implementation of soft-core potential only works with FEP or WHAM (exponential average). Once the selected atoms interact with the rest of the system with full repulsion, it is straightforward to compute free energy contributions from WCA attraction and electrostatics. For details, see Deng Y. and Roux B., J. Phys. Chem. B, 108 (42) 16567-16576

Note

The WCA separation is implemented for:

  • group based slow (fast off) vdW routine,
  • atom based slow routine with distance switch cutoff (fast off),
  • group based fast (fast gene) vdW routine and
  • atom based fast (fast gene) vdW energy with distance switch cutoff.
ENERgy   ...   [ RESET ]  [ free-energy-step-spec ]
DYNAmics ...              [ PUNIt integer         ]   [WHAM integer]
MINImize ...

  [ RESET ]     ! Resets all all accumulation data and counters.
                 (automatic for the first PERT or after a PERT OFF command)

free-energy-step-spec::=
 [PWINdow [LAMBda real] ] [PSTArt int] [PSTOp int] [LSTArt real] [LSTOp real] -
 [PSLOwgrowth           ]

         [PINCrement int]  [PEQUilibrate int]  [LAVErage]  [LINCrement real]


  [PWINdow   ]  ! specifies the windowing algorithm (default)
  [PSLOwgrowth] ! specifies the slow growth algorithm

  [LAMBda real] ! specifies the lambda value for windowing methods or for
                  energy or minimization calculations.

  [PSTArt int]  ! starting dynamics step number for accumulation (default 1)
  [PSTOp  int]  ! ending dynamics step number for accumulation (default 0)

  [LSTArt real] ! specifies the starting lambda value (default 0.0)
  [LSTOp  real] ! specifies the final lambda value (default 1.0)

  [PINCrement int] ! specifies number of steps to next window (auto mode).
  [PEQUil int]  ! specifies number of steps for equilibration (auto mode).
  [LAVErage]    ! specifies that lambda = (LSTART+LSTOP)/2    (auto mode).
  [LINCrement]  ! Specifies the lambda increment between windows (auto mode).

  [PSSP]        ! use soft core potentials for interactions in reac.
                ! and product list.  This option is remembered. With
                ! the PSSP keyword, two parameters, ALAM and DLAM can
                ! be set.

  [ALAM real]   ! Separation parameter for elec. interaction (defaults to 5A^2)

  [DLAM real]   ! Separation parameter for LJ interaction (defaults to 5A^2)

  [NOPSsp]      ! Turn off use of soft core interactions.

  [ END   ]     ! Turns off the free energy code

The PSTArt and PSTOp values are relative to the number of dynamics steps since PERT command was first enabled, or if a PERT RESEt command is used (regardless of whether the DYNAmics commands is run more than once or whether the dynamic run involved the use of restart files).

By specifying the auto mode parameters (PINCrement, PEQUilibrate, LINCrement), a new window will start at the conclusion of the current window with modified parameters. Also, in auto mode, the run will terminate at the end of a window where the LSTOp value is 1.0 or 0.0.

The new commands PSSP/NOPSsp and the optional parameters ALAM and DLAM control the interactions between soft core potentials and PERT. After you specify PSSP in an energy, minimization or Dynamics command, soft core LJ and elec. interactions are used in all reactant and product nonbonded list terms. The separation parameters for elec. and LJ interactions can be set with the ALAM and DLAM options, the default of 5A^2 should be reasonable. The option is remembered, i.e., after the first invocation of PSSP all further calls to EPERT use soft core interactions. To turn this off, use the NOPSsp keyword. Use of softcore interactions is also turned off after a PERT RESEt or a PERT OFF command. For example (assuming that PERT has been already set up):

ENER LAMB 0.5 ! calc. system energy for L=0.5
ENER LAMB 0.5 PSSP ! calc. system energy for L=0.5 using soft core potentials
ENER LAMB 0.6 ! calc. system energy for L=0.6 using soft core potentials
              ! since the PSSP option is remembered
ENER LAMB 0.6 NOPS ! calc. system energy for L=0.6, use of soft
              ! core pots. turned off.

The PERT/PSSP code supports only thermodynamic integration and slow-growth. Please ignore all results starting with TP> when using PSSP.

The PUNIt option allows the free-energy-step-spec to be specified more than once and acts as a scheduler for a particular simulation. The format of the PUNIt file is;

* title
*
repeat-lines-of(free-energy-step-specs)
END

The end is optional and terminates the free energy run. Lack of an END (i.e. and end-of-file or blank lines) will put PERT into auto mode which will continue until LSTOP becomes 1.0 or 0.0 (based on the LINCrement value).

The WHAM option allows to write a formatted file to post-process with the Weighted Histogram Analysis Method (see below).

Description of PERT Commands

The PERTurb command copies and saves the current PSF and restraint data for harmonic, NOE and dihedral restraints to the initial (lambda=0) saved state. The SKIP command flags are also saved to allow linear scaling of entire energy terms.

The structure may then be modified or perturbed with patches, SCALar commands, with the DELEte command, or by generating or reading a new PSF.

The Basic mode of operation is;

....
PERTurb        ! Define the lambda=0 state.
PATCH ....     ! Define the lambda=1 state.
DYNAmics ....  ! Run MD on intermediate surface...
....

The PSF in use when dynamics or energy minimization is invoked becomes the final (lambda=1) state. The actual energy computed is a linear combination of these two endpoints.

The PATCH command may be replaced with any other command that modifies the PSF. Some examples which modify the PSF;

SCALAR CHARGE SET -0.55 SELE ATOM A 1 O* SHOW END ! change a charge

DELETE ATOM SELE ALL END
READ SEQUENCE ....
GENERATE ...    ! generate a new different PSF.

DELETE CONNECTIVITY ....  ! modify the PSF by changing the connectivity.
                ! see a word of warning below!

SCALAR TYPE SET 14 SELE ATOM A 1 O SHOW END ! change the vdw atom type

OPEN ....     ! Read a new PSF
READ PSF ...

It is not required that the PSF be modified. If one wants to carry out coordinate perturbation only, it is sufficient to modify the harmonic restraints, the NOE restraints, or the dihedral restraints. In this way, it is possible to calculate the free energy differences between different conformers. (However, this option should not be used with simultaneous change of SHAKE constraints)

Note that with this implementation, because two PSFs are used, that the connectivity may change. The use of 1-4 interactions and nonbond exclusions is fully supported. This allows this method to be used for examining changes that involve bond changes, such as cystine bridge formation. [Note added by Stefan Boresch (stefan@mdy.univie.ac.at) Changes in connectivity that involve bond breaking or forming are highly problematic and may not converge. This is explained in detail in Boresch & Karplus, J. Phys. Chem. B 1999, 103, 103-136. The flexibility made possible by the implementation of PERT puts the responsibility of what can be done and what not on the user!]

The advanced mode of operation is;

....
PERTurb
PATCH ....
DYNAmics ....
....
PERTurb
PATCH ....
DYNAmics ....
....
PERTurb
PATCH ....
DYNAmics ....
....

In this way, several changes can be affected in a single CHARMM run. For example, the first patch might be the removal of charges, and the second patch could correspond to a change in atom size, and the third patch could simply consist of modifying dihedral restraints so as to affect a conformational change. The free energy differences and fluctuations will be calculated for each window as well as the total for all previous windows.

RESTRICTIONS:

The number of atoms in both sets must match! If the system of interest has different numbers of atoms, then dummy atoms must be used. The mapping of atoms between the first and last structure is one to one.

The following CHARMM features are not currently supported for use with free energy perturbation;

INTEraction_energy

These commands will continue to work, but will only use the final (lambda=1) structure. Most other energy related CHARMM features are supported.

The IMAGE/CRYSTAL facility has been supported now for some time; however, IMAGE/CRYSTAL needs to be set up after the PERT command!!!

The following CHARMM energy related features cannot be modified with the PERT command (e.g. cannot be part of what is changing, and are only determined by the final state).

  • HBON - hydrogen bond energy
  • ST2 - ST2 energy
  • CIC - internal coordinate constraint energy
  • CDRO - quartic droplet potential energy
  • USER - user supplied energy (USERLINK)
  • RXNF - Reaction field energy
  • IMNB - image van der Waal energy
  • IMEL - image electrostatic energy
  • IMHB - image hydrogen bond energy
  • IMST - image ST2 energy
  • SBOU - solvent boundary energy
  • UREY - Urey Bradley energy terms
  • XTLV - Crystal vdw terms
  • XTLE - Crystal electrostatics

Extended electrostatics is implemented within PERT and can be used with the following CHARMM commands:

NBONDS GROUP SWITCH CDIE VDW VSWITCH EXTEND GRAD QUAD -
CTONNB 12.0 CTOFNB 12.0 CUTNB 12.0 WMIN 1.2 WRNMXD 1.2 EPS 1.0

Note

The ctonnb, ctofnb and cutnb values should be the same when implementing extended electrostatics in PERT to prevent problems with mixing of usage of switching functions and extended electrostatics

References

  • M Mezei and D.L. Beveridge, in Annals of the NYAS, “Free Energy Simulations” 482 (1986)
  • T. P. Straatsma Ph.D. Thesis “Free Energy Evaluation by Molecular Dynamics Simulations”
  • Kollman, P. A.; et al. J. Am. Chem. Soc. 1987, 109, 1607.
  • Kollman, P. A.; et al. J. Am. Chem. Soc. 1987, 109, 6283.
  • Kollman, P. A.; et al. J. Chem. Phys. 1989, 91, 7831.
  • Bell, C. D.; Harvey, S. C., J. Phys. Chem. 1986, 90, 6595.
  • van Gunsteren, W.F. et al. in: Computer Simulation of Biomolecular Systems: Theoretical and Experimental Applications, vol. 2, eds. van Gunsteren W.F. and Weiner P.K. (Escom, Leiden, 1994), p. 349

Examples

The input file

* A SIMPLE TEST RUN FOR PERT
*
bomlev -1
OPEN READ FILE UNIT 1 NAME ~/c22pt/toph19.mod
READ      RTF   UNIT  1
OPEN READ FILE UNIT 2 NAME ~/c22pt/param19.mod
READ      PARAMETER  UNIT  2
READ      SEQUENCE  CARD
*  FIRST SEQUENCE FOR SECOND DERIVATIVE TEST
*
    2
AMN CBX
GENERATE  A
GENERATE  B DUPLICATE A

OPEN UNIT 3 READ CARD NAME perttest.crd
READ COOR CARD UNIT 3

! modify the charge for the lambda=0 state
SCALAR CHARGE SET -0.55 SELE ATOM A 1 O* SHOW END

! minimize initial state so initial forces will be small.
MINI ABNR NSTEP 100 CTOFNB 12.0 CUTNB 14.0

PERT  ! save all PSF data for the lambda=0 state

! modify the charge again for the lambda=1 state
SCALAR CHARGE SET -0.15 SELE ATOM A 1 O* SHOW END

! carry out free energy run from first to final state

OPEN READ CARD UNIT 88 NAME perttest.punit
DYNA VERLET STRT  NSTEP 12000 TIMESTEP 0.001 -
    IPRFRQ 100 IHTFRQ 0 IEQFRQ 100 NTRFRQ 2000  -
    IUNCRD 50  ISEED 314159  -
    NPRINT 100 NSAVC 0 NSAVV 0 INBFRQ 25 IHBFRQ 0 -
    CTOFNB 12.0 CUTNB 14.0 -
    FIRSTT 300.0 FINALT 300.0 TEMINC 100.0   -
    IASORS 0 IASVEL 1 ISCVEL 0 ICHECW 1 TWINDH 20.0 TWINDL -20.0 -
    PUNIT 88

PERT OFF
energy ! just a check at lamda=1
STOP

The punit file

* PUNIT FILE FOR SIMPLE TEST CASE
* use window method with 2000 steps of equilibration
* and 8000 steps of analysis for each of 10 evenly spaces
* windows.
*
  LSTART 0.0  LAMBDA 0.0  LSTOP 0.05  PSTART  12000  PSTOP  20000  PWIND
  LSTART 0.05 LAMBDA 0.1  LSTOP 0.15  PSTART  22000  PSTOP  30000  PWIND
  LSTART 0.15 LAMBDA 0.2  LSTOP 0.25  PSTART  32000  PSTOP  40000  PWIND
  LSTART 0.25 LAMBDA 0.3  LSTOP 0.35  PSTART  42000  PSTOP  50000  PWIND
  LSTART 0.35 LAMBDA 0.4  LSTOP 0.45  PSTART  52000  PSTOP  60000  PWIND
  LSTART 0.45 LAMBDA 0.5  LSTOP 0.55  PSTART  62000  PSTOP  70000  PWIND
  LSTART 0.55 LAMBDA 0.6  LSTOP 0.65  PSTART  72000  PSTOP  80000  PWIND
  LSTART 0.65 LAMBDA 0.7  LSTOP 0.75  PSTART  82000  PSTOP  90000  PWIND
  LSTART 0.75 LAMBDA 0.8  LSTOP 0.85  PSTART  92000  PSTOP 100000  PWIND
  LSTART 0.85 LAMBDA 0.9  LSTOP 0.95  PSTART 102000  PSTOP 110000  PWIND
  LSTART 0.95 LAMBDA 1.0  LSTOP 1.0   PSTART 112000  PSTOP 120000  PWIND
  END

Or equivalently using auto mode:

* PUNIT FILE FOR SIMPLE TEST CASE
* use window method with 2000 steps of equilibration
* and 8000 steps of analysis for each of 10 evenly spaces
* windows.
*
 LSTART 0.0  LAMBDA 0.0  LSTOP 0.05  PSTART  12000  PSTOP  20000  PWIND PEQUIL 2000 PINCR 10000 LINCR 0.1

Or also equivalently as:

* PUNIT FILE FOR SIMPLE TEST CASE
* use window method with 2000 steps of equilibration
* and 8000 steps of analysis for each of 10 evenly spaces
* windows.
*
  LSTART 0.0  LAMBDA 0.0  LSTOP 0.05  PSTART  12000  PSTOP  20000  PWIND
  LSTART 0.05 LAMBDA 0.1  LSTOP 0.15  PEQUIL   2000  PINCR  10000  PWIND
  LSTART 0.15 LAMBDA 0.2  LSTOP 0.25  PEQUIL   2000  PINCR  10000  PWIND
  LSTART 0.25 LAMBDA 0.3  LSTOP 0.35  PEQUIL   2000  PINCR  10000  PWIND
  LSTART 0.35 LAMBDA 0.4  LSTOP 0.45  PEQUIL   2000  PINCR  10000  PWIND
  LSTART 0.45 LAMBDA 0.5  LSTOP 0.55  PEQUIL   2000  PINCR  10000  PWIND
  LSTART 0.55 LAMBDA 0.6  LSTOP 0.65  PEQUIL   2000  PINCR  10000  PWIND
  LSTART 0.65 LAMBDA 0.7  LSTOP 0.75  PEQUIL   2000  PINCR  10000  PWIND
  LSTART 0.75 LAMBDA 0.8  LSTOP 0.85  PEQUIL   2000  PINCR  10000  PWIND
  LSTART 0.85 LAMBDA 0.9  LSTOP 0.95  PEQUIL   2000  PINCR  10000  PWIND
  LSTART 0.95 LAMBDA 1.0  LSTOP 1.0   PEQUIL   2000  PINCR  10000  PWIND
  END

An annotated output example

This output is a short excerpt from perttest.out where the input lines start with “|” and output lines start with ”.....|”.

The CHARMM command is:

| CHARMM>    DYNA VERLET STRT  NSTEP 12000 TIMESTEP 0.001 -
| CHARMM>        IPRFRQ 100 IHTFRQ 0 IEQFRQ 100 NTRFRQ 2000  -
| CHARMM>        IUNCRD 50  ISEED 314159  -
| CHARMM>        NPRINT 100 NSAVC 0 NSAVV 0 INBFRQ 25 IHBFRQ 0 -
| CHARMM>        CTOFNB 12.0 CUTNB 14.0 -
| CHARMM>        FIRSTT 300.0 FINALT 300.0 TEMINC 100.0   -
| CHARMM>        IASORS 0 IASVEL 1 ISCVEL 0 ICHECW 1 TWINDH 20.0 TWINDL -20.0 -
| CHARMM>        PUNIT 88

and the relevent punit data is in perttest.punit:

|* PUNIT FILE FOR SIMPLE TEST CASE
|* use window method with 2000 steps of equilibration
|* and 8000 steps of analysis for each of 10 evenly spaces
|* windows.
|*
| LSTART 0.0  LAMBDA 0.0  LSTOP 0.05  PSTART  1200 -
|   PSTOP  2000  PWIND PEQUIL 200 PINCR 1000 LINCR 0.1

The output starting at line 1618 (in the middle of the dynamics command) is:

.....| PERTURBATION> Free energy perturbation results:

This indicates that a “window” was just completed.

.....| PERTURBATION> results, LSTART=    0.050000  LSTOP=    0.150000  LLAST=    0.100000 Number of steps used=       800

This says that the window started at lambda=0.05 and ended at 0.15. The window “center” was at lambda=0.1 A total of 800 steps was used for collecting averages and fluctuations.

.....| PERTURBATION> result: EXPAVE=0.155456E+01 EXPFLC=0.225213E+00 DIFAVE=   -0.256530 DIFFLC=    0.088960

The values:

  • EXPAVE is the time average of e^{(E_f(t)-E_i(t)-E_f(0)+E_i(0))/kT}
  • EXPFLC is the fluctuation of this value about its average
  • DIFAVE is the time average of E_f(t)-E_i(t)-E_f(0)+E_i(0)
  • DIFFLC is the fluctuation of this value about its average

Note

this value should not be much larger than kT for a good window schedule. If this value is too large, then smaller window lambda steps should be used. The value here (0.09) indicates that a much larger window would have been OK. ef(t) is the energy at lambda=LSTOP, ei(t) is the energy at lambda=LSTART ef(0) is the initial energy at LSTOP, ei(0) is the initial energy at LSTART

.....| PERTURBATION> TP Windowing result, EPRTOT=    1.392400  EFORWARD=    0.914282 EPREF=    1.177303

This is the old format. In the new format the values EPRTOT,EFORWARD, EBACKWARD,EPREFF,EPREFB are reported where the forward energy is from LLAST to LSTOP and the backward from LLAST to LSTART. Separating the window into two halves (double wide sampling) improves the accuracy of the the TP method.

.....| PERTURBATION> TI Windowing result, EPRTOT=    1.400218  EFORWARD=    0.920773 EPREF=    1.177303
  • EPRTOT is the total energy for this window and all previous (since a PERT RESET).
  • EFORWARD is the energy for this current window.
  • EPREF is the initial energy difference (ef(0)-ei(0))
.....| PERTRES> LSTART=     0.05000 LSTOP=     0.15000 EPRTOT=     1.40022  EFORWARD=     0.92077 EPREF=     1.17730 DIFAVE=    -0.25653 DIFFLC=     0.08896

This is the same data on a one line format. To use this, grep (search) for “PERTRES”.

.....| PERTURBATION> Averages for the last      800  steps:
.....|PAVE DYN: Step         Time      TOTEner        TOTKe       ENERgy  TEMPerature
.....|PAVE PROP:             GRMS      VEREner        VERKe       EHFCor        VIRKe
.....|PAVE EXTERN:        VDWaals         ELEC       HBONds                      USER
.....|PAVE PRESS:            VIRE      VIRI           PRESSE      PRESSI    VOLUme
.....| ----------       ---------    ---------    ---------    ---------    ---------
.....|PAVE>      800      0.00000      0.92077     -0.00039      0.92077      0.00000
                                                   <delF*v>       <delE>
.....|PAVE PROP>          0.00000      0.00000      0.00000      0.00000      0.00000
.....|PAVE EXTERN>        0.00000      0.92077      0.00000                   0.00000
.....|PAVE PRESS>         0.00000      0.00000      0.00000      0.00000      0.00000

This is the average values for this window (TI results).

Note

the <delF*v> is a “correction” term that shows how well the window is equilibrated. This value should be close to zero and much smaller than its fluctuation. If it is not, then the assumptions required for a free energy calculation using windowing are not met. This can occur if there is a “snapping” event with releases energy in an irreversible manner, or if the system is not at equilibrium. For a slow growth “window”, this is a correction term which should be multiplied by the estimated delay of the equilibration at the current step and then added to the total. For example, if the configuration distribution lags behind the energy potential by 100 steps, this value should be scaled by 100. When forcing a “continuous” change through slow growth, there tends to be a delay since the structure does not respond instantly to the potential.

.....| ----------       ---------    ---------    ---------    ---------    ---------
.....| PERTURBATION> Fluctuations for the last      800  steps:
.....|PFLC>      800      0.00000      0.08911      0.00985      0.08896      0.00000
.....|PFLC PROP>          0.00000      0.00000      0.00000      0.00000      0.00000
.....|PFLC EXTERN>        0.00000      0.08896      0.00000                   0.00000
.....|PFLC PRESS>         0.00000      0.00000      0.00000      0.00000      0.00000

This is the fluctuation data for the current window (TI results).

.....| ----------       ---------    ---------    ---------    ---------    ---------
.....| PERTURBATION> TOTALS since last reset:
.....|PTOT>      800      0.00000      1.40022     -0.00038      1.40022      0.00000
.....|PTOT PROP>          0.00000      0.00000      0.00000      0.00000      0.00000
.....|PTOT EXTERN>        0.00000      1.40022      0.00000                   0.00000
.....|PTOT PRESS>         0.00000      0.00000      0.00000      0.00000      0.00000

This is the total for this window and all previous (since the last PERT RESET).

.....| ----------       ---------    ---------    ---------    ---------    ---------
.....|
.....| PERTURBATION> EOF on punit file: PERT in auto mode.

This indicates that no more data was found on the PUNIT file.

.....| PERTURBATION> Free energy perturbation calculation continues.

Now we start a new window.

.....| PERTURBATION> PSTART=        3200  PSTOP=        4000

We will run equilibration until step 3200, then we will collect data until step 4000.

.....| PERTURBATION> LSTART=    0.150000  LSTOP=    0.250000  LAMBDA=    0.200000

This indicates the boundaries of the new window.

.....| PERTURBATION> Windowing will be used.

We are using windowing (fixed lambda values), instead of slow growth.

Constraints

If SHAKE is applied to bond terms which are changed as the result of an alchemical mutation a constraint correction is calculated where required in slow-growth mode and TI in windowing mode. The exponential formula in windowing mode is not supported. The user has to beware of subtle problems regarding a possible “moment of inertia” term that may be or may be not included in this correction (S. Boresch & M. Karplus, to be published) In order for the constraint correction to work properly, attention has to be given to the following points:

  1. SHAKE must not be applied to angle terms

  2. the PARA option has to be used (it’s not clear, how to support reference coordinates in the context of an alchemical mutation)

  3. the SHAKE command has to issued after the PERT command. (This is similar to setting up IMAGEs in connection with PERT). A typical input will look something like

      PERT SELE ... END
      !change psf; after ALL changes have been made
    SHAKe BOND PARA
    DYNA ... ! carry out MD simulations etc.
      PERT OFF
  4. One should not mix situations where a constraint correction for SHAKE is necessary with the use of harmonic, NOE and dihedral restraints to calculate conformational free energy differences.

Items (2) and (3) can lead to error messages in situations where there is actually no problem, e.g. you just want to apply SHAKe to your solvent which is not affected by the mutation, so you specify SHAKe before PERT and “bomb”. Nevertheless, I thought better safe than sorry and if wanted one can override the warnings with a BOMLEV -3. Item (4) is simply untested.

WHAM

The post-processing Weighted Histogram Analysis Method (WHAM) can be used to help reaching better statistical convergence on free energy perturbations calculations. The approach represents a generalization of the histogram method developed by Ferrenberg and Swendsen (1989). The central idea, which goes back to the maximum overlap method developed by Bennet (1976) to estimate free energy differences, consists in constructing an optimal estimate of the unbiased distribution function as a weighted sum over the data extracted from all the simulations and determining the functional form of the weight factors that minimizes the statistical error. The WHAM approach can be used to calculate the PMF along coordinates (Kollman, 1992; Boczko, 1993; Brooks, 1993; Roux, 1995) and can also be used to post-process free energy perturbation calculations in which no PMF is desired.

Assuming that the simulations are performed (as in PERT) with a potential function given by a linear switching from E_0 to E_1, that is,

E_k = (\lambda_{k-1})*E_1 + (\lambda_k)*E_0

The free energy constants F_k corresponding to any \lambda_k window are given by,

e^{-F_k/kBT} = \sum_i \left( \sum_t \frac{ \mathrm{Top}_i(t) }{ \mathrm{Bot}_i(t) } \right)

where

\mathrm{Top}_i(t) = e^{-E_k[X_i(t)]/kBT}

and

\mathrm{Bot}_i(t) = \sum_j  { Ntime(j) * e^{+F_j/kBT-E_j[x_i(t)]/kBT} }

where E_j[x_i(t)] = (\lambda_j-1)*E_1[x_i(t)] + (\lambda_j)*E_0[x_i(t)] is the potential energy function of the j-th window evaluated with the configuration taken from the i-th simulation. The WHAM equations for the F_k must be solved iteratively.

The syntax of the command is:

WHAM MAXWindow <integer>  MAXTime <integer> unit <integer> -
     tolerance <real>  nstep <integer> [guess] -
     ioffset <integer>   nskip <integer> {lambda <real> lambda <real> ...}

where

  • MAXWindow is the total number of windows
  • MAXTime is the total number of time-step configuration for each window
  • unit is the unit of the formatted file containing all the information
  • tolerance is the tolerance on the F_k to reach convergence
  • nstep is the maximum number of iterations on the WHAM equations
  • guess to flag that an initial guess is provided for the F_k. Those are read directly from the input stream with one line per window in the format [ window <integer> F() <real> ]
  • nskip use only every nskip data point to reach convergence (faster)
  • lambda give any value of lambda for which you want the free energy (a list)
  • ioffset reference energy level at window number “ioffset”.

The file containing the information can be written by PERT during dynamics if the keyword WHAM is used in the dynamics command (see above). In principle, the WHAM could also be used with non-linear perturbations, but then the code in PERT would have to evaluate several energies since those could not be obtained by lambda interpolations.

The total free energy of is stored in ‘WHAMFE’ substitution viriable.

Some references on WHAM:

  • S. Kumar, D. Bouzida, R.H. Swendsen, P.A. Kollman, and J.M. Rosenberg. J. Comp. Chem. 13, 1011–1021 (1992).
  • E.M. Boczko and C.L. Brooks III. J. Phys. Chem. 97, 4509–4513 (1993).
  • A.M. Ferrenberg and R.H. Swendsen. Phys. Rev. Lett. 63, 1195–1198 (1989).
  • C.M. Bennet. J. Comp. Phys. 22, 245–268 (1976).
  • C.L. Brooks III and L. Nilsson. J. Am. Chem. Soc. 115, 11034–11035 (1993).
    1. Roux, Comp. Phys. Comm. 91, 275-282 (1995).

PERT/PSSP

Some details concerning the implementation of the PERT/PSSP code, including present limitations:

Introduction

The PERT free energy module of CHARMM is based on a linear dependence on the coupling parameter. While simplifying implementation, this approach is prone to van der Waals endpoint problems. One widely used method to overcome the van der Waals endpoint problem is the use of soft core Lennard Jones and electrostatic interactions for those energy terms that cause problems. This capability has been added, following Zacharias, Straatsma and McCammon, J. Chem. Phys. 1994, 100, 9025.

Outline of the implementation

In the following L denotes the coupling parameter lambda. Subscripts _i and _f denote initial and final state respectively. The variables de (ALAM) and dv (DLAM) can be set by the user; reasonable defaults (5 A^2) are used.

The functional form of the soft core routines in combination with PERT is as follows:

U_LJ(L) = U_LJ,0 +

                     A_f                      B_f
    L     * (-------------------  -  --------------------) +
              (r^2 + dv*(1-L))^6      (r^2 + dv*(1-L))^3

                     A_i                      B_i
    (1-L) * (-------------------  -  --------------------) +
              (r^2 + dv*L)^6             (r^2 + dv*L)^3


U_ELEC(L) = U_ELEC,0 +

                   qi_f*qj_f
    L     * (----------------------) +
              sqrt(r^2 + de*(1-L))


                   qi_i*qj_i
    (1-L) * (----------------------)
                sqrt(r^2 + de*L)

Of course, the effects of tapering functions (SHIF, SWIT etc.) have to be taken into account properly; this is particularly important for the calculation of the forces and of the derivative dU/dL

In principle, soft core potentials are only required for atoms that ‘vanish’ at one of the endpoints (i.e., dummy atoms). In this implementation, a simpler approach was used: When PERT is activated, CHARMM uses three nonbonded lists (six with IMAGE/PERT): (1) one for the “environment” part (the part of the system that remains unchanged), (2) one for the “reactant” part (interactions between initial state atoms themselves and initial state atoms and the environment), and (3) one for the “product” part (interactions between final state atoms themselves and final state atoms and the environment). All reactant and product list interactions are calculated using soft core potentials. Since (see equations above) at the endpoints the soft core expressions reduce to normal interactions, use of the soft core potentials is equivalent to a modified path, but the overall result of the free energy simulation is unchanged. (Note: the effect on free energy components has not been explored systematically!)

Obviously, using soft core potentials breaks the standard scheme how PERT calculates dU(L)/dL since instead of

U(L) = U_0 + (1-L)*U_i + L*U_f ! standard PERT

we now have

U(L) = U_0 + (1-L)*U_i(L) + L*U_f(L) ! PERT/PSSP

with corresponding differences for dU/dL. One sees that the standard PERT scheme gives approximately “half” of the required derivative, but we still need the terms (1-L)*[dU_i(L)/dL] and L*[dU_f(L)/dL]. The modified energy routines I use do these additional calculations.

Summing up, modifications of the code were necessary in the following three areas:

(a) Additions to the parser: PSSP/NOPSsp keywords, ALAM, DLAM parameters; initializations and resets

(b) Modifications to subroutine EPERT itself, making (i) sure that the correct energy subroutines are called if PSSP is active, and (ii) that the additional contributions to dU/dL are temporarily stored and added to the LJ and elec free energy contributions calculated in the standard PERT way. To achieve (i), subroutines FASTST, EVDW (enbonda.src) and EGROUP (enbondg.src) were modified slightly as well.

(c) Special purpose nonbonded energy routines (based on the standard slow energy routines) have been added to the file epert.src. Currently only a subset of nonbonded options is supported (see below)

All new nonlocal variables (no arrays are needed!) have been added in pert.fcm (epert.fcm is unchanged)

The outline given here together with the comments in the code should make the inner workings of the PERT/PSSP code clear.

To quickly ‘grep’ for all changes, seach for lines(comments) starting with Cpssp

For a description of how to use the new functionality (activated by the PSSP keyword), see the modified PERT documentation and the new test cases.

Comments and present limitations

To the best of my knowledge, all reported uses of soft core potentials in free energy simulations have been based on thermodynamic integration (TI), not the “exponential formula” (“thermodynamic perturbation”, “free energy perturbation (FEP)”). The present implementation also supports only TI. While the output claims to give values obtained with the exponential formula (TP> lines in output file), the reported values ARE W R O N G if soft core potentials are used. This is similar to all cases when the constraint correction is needed, which also only works with TI !!! At present, it is not clear whether the exponential formula can be supported easily.

Only a limited subset of nonbonded options is supported at presented. Nonsupported options are hopefully caught and should make CHARMM die. Nevertheless, check against the following list:

At present, the following limitations apply:

  • Only constant dielectric (CDIE)

  • For group based cutoffs (GROU / VGROU), the following nonbonded options are supported at present:

    VDW (LJ)                 ELEC
    -----------------------------
      VSWI                   SWIT
  • For atom based cutoffs (ATOM / VATO), the following nonbonded options are supported at present:

    VDW (LJ)                 ELEC
    -----------------------------
      VSWI                   SHIF
                             FSWI
                             EWAL (tradional or PME)

Finally, note that there is no support for parallel architectures.

Outlook / TODO

  • Support parallel architectures (someone else needs to do this, since I have no hardware) – this should probably postponed until a merge with the standard slow energy routines
  • All PERT/PSSP specific modifications could easily be put behind a separate compilation flag (e.g., ##IF PSSP) if this were desired.
  • Support additional tapering functions: While it doesn’t seem necessary to support all combinations of nonbonded options in CHARMM, support for FSHIft and RDIE is planned.
  • Use the slow energy routines instead of special purpose routines in epert.src. Further, maybe a merge with the existing soft core routines in CHARMM (intended primarily for docking) is possible. This would lead to a unified, general and flexible soft core facility. Also, this would cure (maybe?) (most of) the parallel code incompatibilities...
  • Consider support of the exponential formula; if this cannot be done easily, remove the output lines to avoid confusion.

PATCHING DUMMY SIDECHAINS FOR FREE ENERGY PERTURBATIONS

The command MKPRES has been introduced to write a PATCH for adding a dummy sidechain onto a backbone with the goal of performing free energy calculations. The command generates the list of needed dihedral angles and non-bonded exclusions. Only the cross internal energy terms between the dummy sidechain and the bakbone are introduced, the rest is generated from the normal generate command. Basically, it is ok to use such a mixed topology (single vs dual) by branching at the carbon CB. With this treatment, all the bonds and angles are kept, some dihedrals may need to be turned off for statistical consistency of the reference state (this requires some thinking by the user, sorry...). It can be shown that free energy differences calculated with these end-points are correct (even though the individual free energies are themselves different than those with ideal gas of free particle in which all internal energy terms are turned off). Proline is not supported by this method. Glycine might be ok, but remain vigilant.

One can introduce dummy atoms, which retain all the covalent interactions, in a “hybrid residue”, in such a way that the influence of the bonded interactions with dummy atoms do not influence the final free energy change. The simulations thus can be done using a transformation protocol in which all covalent bond contributions are maintained invariant throughout the calculations; only the nonbonded interactions are varied.

The hybrid method is a scheme which retains some features of both the single and dual topology techniques. The dummy atoms, which are covalently linked to the protein in question, have no nonbonded interactions at one or the other of the two end point reference states. The potential energy function describing the transformation is constructed such that all internal energy terms are invariant with respect to the thermodynamic coupling parameter lambda. This simulation procedure therefore has similarities with both the “single topology” and “dual topology” methods.

The coupling of the dummy atoms to the real atoms cancels out exactly from the calculated free energy differences. The equivalence holds as long as the coupling between the dummy atoms and the rest of the system satisfies certain conditions. First, there can be only one bond between the dummy atoms of a mutated residue and the real atoms in the rest of the system, because multiple bonds would add spurious coupling between the real atoms. Second, to avoid spurious coupling between the dummy atoms and the rest of the system, there cannot be multiple bond angles and dihedral torsion angles between the dummy atoms of a transformed residue and more than two real atoms in the rest of the system.

The theoretical arguments explaining the approach have been elaborated in the following references:

  • Boresch, S.; Karplus, M. J. Phys. Chem. A 1999, 103, 103-118.
  • Boresch, S.; Karplus, M. J. Phys. Chem. A 1999, 103, 119-136.
  • Shobana S.,B. Roux, and Olaf S. Andersen, J. Phys. Chem. B 2000, 104, 5179-5190

Syntax:

MKPRES {PatchName} unit <integer> atom_selection atom_selection atom_selection atom_selection

The four atom_selections correspond to the following pieces of the psf:

  • first: is the invariant backbone of state 0
  • second: is the mutated sidechain for state 0
  • third: is the invariant backbone of state 1 (normally this should correspond identically to the first selection)
  • fourth: is the mutated sidechain for state 1

Here is an example for liking a dummy valine to an alanine:

set Residue1 = ALA
set Residue2 = VAL

!first state
read sequence card
*  residue1
*
  3
ALA @Residue1 ALA
generate SEG1 setup

! second state (must have complete second segment to generate internal
! energy terms of second state)
read sequence card
*  residue2
*
  3
ALA @Residue2 ALA
generate SEG2 setup

define BACK select type CA .or. type HA* .or. type N .or. type HN .or. -
       type C .or. type O .or. type HT* .or. type OT* .or. type CB show end

open write card unit 10 name mkpres.rtf
write title unit 10
** Patch for alchemical mutation of ALA to VAL
**
*

MKPRES @PatchName unit 10 -
       select segid SEG1 .and. back end -
       select segid SEG1 .and. resid 2 .and. (.not. back ) end -
       select segid SEG2 .and. back end -
       select segid SEG2 .and. resid 2 .and. (.not. back ) end

Two patches are written in unit 10. The first one is for real alanin/dummy valine while the second patch turns the system into dummy alanine/real valine. Please check your patch before lengthy calculations!

Details about the use of MFF-potentials in PERT

The MMFP keyword makes it possible to modify (add/remove) MMF-potentials during an alchemical PERT mutation. If the keyword was specified when activating PERT, the original call to the MMFP routines in the constant-energy section of EPERT (epert.src) is omitted. Instead, two new calls to the MMFP routines, one in the lambda=0 section and one in the lambda=1 section, are used. It is an optional command, so the original use of the MMFP potentials in PERT (as constant energy terms, independent from lambda) is still available (and the default behaviour). The possibility of lambda-dependent MMF-potentials should facilitate free energy simulations where auxiliary restraints are needed, such as calculations of absolute binding free energies.

Note

It is important to choose a MAXGEO value (see note geo:(chmdoc/mmfp.doc)) appropriate for the sum of all GEO restraints that will be set up *before and after the call to PERT.

The testcase ‘pert-mmfp.inp’ demonstrates the new functionality.

Chemical

Experimental support of “chemical paths” in PERT free energy simulations (i.e., simulations in which a solute is decoupled from solvent or in which a ligand is decoupled from the protein). CHARMM has to be compiled with the CHEMPERT keyword in pref.dat for this functionality to be available.

The CHEM keyword on the PERT commandline changes how energies and forces are computed when PERT is active. (Note that there are no provisions to turn this mode off other than specifying PERT OFF!)

Its intended use is as follows: Suppose you have a solute in SEGI SOLU and water in SEGI WAT. Now specify

fast off ! important, otherwise SCALAR RSCA has no effect!!

PERT sele segi SOLU end CHEM

scalar char set 0. sele segi SOLU end ! turn off solute charges
scalar rsca set 0. sele segi SOLU end ! turn off solute LJ interactions

Then lambda=0 corresponds to a fully interacting system and lambda=1 corresponds to water plus the “solute” in the gas phase. However, within the SOLU segment (the solute) all interactions, in particular the intramolecular nonbonded terms, are computed normally.

If the CHEM keyword were missing, then at lambda=1 no solute intramolecular nonbonded interactions would be needed, requiring a separate calculation correcting for this.

The code (should) work(s) with most nonbonded options, including EWALD and PSSP soft-cores (note that these impose there own limitations which nonbonded options can be used). The EWALD support is admittedly a wild hack. Quite generally I do recommend to test correct functionality by applying tests similarly to the new testcases chempert1.inp and chempert2.inp adapted to one’s particular system one wishes to study.

Long-range correction

Since c29 CHARMM supports an isotropic Lennard-Jones long range correction (lrc nonbonded option, ##LRVDW keyword in pref.dat). So far, there is no support for this correction in connection with PERT. In principle, this support has now been added; both the energy, as well as the virial correction are supported. HOWEVER, because of the way the LRC interfaces with the energy routines, the correction is NOT computed correctly if LJ interactions are turned off by scalar RSCA set 0. Thus, in the presumably most interesting case, simulations where a solute or a ligand is decoupled from the rest of the system (cf. CHEMical PERT above), the LRCorrection cannot be computed in the most obvious manner. It is hoped that full support can be added, but this will entail a rewrite of how the LRC is computed in general.

PERT-BLOCK Lambda Dynamics

New code is added in current version so that PERT and BLOCK lambda-dynamcs can be used together. Keyword QLDM of lambda-dynamcs will trigger the mixture of PERT/BLOCK. No additional setup is required as long as the traditional PERT and BLOCK setup is kept. FAST OFF is recommended for this setup. The acdemic usage of this method will be published shortly. – H. Li and W. Yang