# Adaptive Biasing Force Algorithm¶

## Introduction¶

Adaptive Biasing Force is a variant of a flat histogram method. Like many other methods that seek uniform sampling over CV space such as Metadynamics, it adaptively biases the simulation until such diffusive sampling is achieved. However, unlike metadynamics, ABF does not estimate the free energy surface. Rather, it directly estimates the derivative of the free energy in CV directions - the generalized force on that CV by the system.

In practice, this translates to histogramming coordinates in CV space with an instantaneous estimation of the free energy derivative. This instantaneous estimate fluctuates around the true, global free energy derivative at that point, but the average quickly converges to the real value. Then, the free energy derivatives can be integrated much like Thermodynamic Integration to get the free energy surface.

Thus, ABF gives a vector field and not a free energy surface.

An excellent write-up on the method can be found here.

Details on the specific implementation used in SSAGES can be found here.

An integrator for 1D and 2D surfaces are provided in SSAGES/Tools/ABF_integrator (requires numpy, scipy and matplotlib). ABF_integrator.py -i <inputfile> -o <outputname> –periodic1 <True/False> –periodic2 <True/False> –interpolate <integer> –scale <float>

## Options & Parameters¶

Adaptive Biasing Force Method

• Calculate the generalized force on CVs at each timestep
• Bias with the negative of the estimated generalized force
• Define a CV range. Outside of the CV range, there will be no bias, and no histogram hits will be collected.
• Can restart from a previous run. Simply include Fworld_cvX and Nworld outputs in your working directory at runtime. IMPORTANT! If you want to start a clean run, or if you’ve made changes to the grid, these files must NOT BE PRESENT in the working directory, otherwise SSAGES will give you a warning, or worse, you will merge your histogram into another one on accident.
• Can optionally define a restraint range. Outside this range, a harmonic restraint of user-chosen spring constant(s) will drive the CV(s) back into the range. This range should be WIDER than the CV range by at least one bin size in each direction. To disable restraints, enter a spring constant k equal to or less than zero. If restraints are used on a periodic system, one can define the periodic boundaries, so that minimum image convention to CVs can be applied. (CV_periodic_boundary_upper/lower_bounds). For example, on a -pi to pi CV, if the CV is restrained to -3.14 to -2.36 and the CV crosses the -3.14 boundary to 3.14, this will ensure the restraint is applied correctly back towards -3.14 rather than a large force applied to bring it from 3.14 all the way to -2.36.

How to define the ABF Method: "type" : "ABF"

cvs
array of integers. This array selects which CVs this method will operate on. Index starts from 0.
CV_lower_bounds
array of doubles (nr of CVs) long. This array defines the minimum values for the CVs for the range in which the method will be used in order.
CV_upper_bounds
array of doubles (nr of CVs) long. This array defines the minimum values for the CVs for the range in which the method will be used in order.
CV_bins
array of doubles (nr of CVs) long. This array defines the number of histogram bins in each CV dimension in order.
CV_restraint_minimums
array of doubles (nr of CVs) long. This array defines the minimum values for the CV restraints in order.
CV_restraint_maximums
array of doubles (nr of CVs) long. This array defines the maximum values for the CV restraints in order.
CV_restraint_spring_constants
array of doubles (nr of CVs) long. This array defines the spring constants for the CV restraints in order. Enter a value equal to or less than zero to turn restraints off.
CV_isperiodic
array of booleans (nr of CVs) long. This array defines whether a given CV is periodic for restraint purposes. This is only used to apply minimum image convention to CV restraints. Can be safely set to false even for periodic CVs if no restraints are being used. If ANY CV is set to periodic, then CV_periodic_boundary_lower_bounds and CV_periodic_boundary_upper_bounds must be provided for ALL CVs. Values entered for non-periodic CVs are not used.
CV_periodic_boundary_lower_bounds
array of doubles (nr of CVs) long. This array defines the lower end of the period. This only matters if CV_isperiodic is true for the CV.
CV_periodic_boundary_upper_bounds
array of doubles (nr of CVs) long. This array defines the upper end of the period. This only matters if CV_isperiodic is true for the CV.
timestep
double. The timestep of the simulation. Units depend on the conversion factor that follows. This must be entered correctly, otherwise generalized force estimate will be incorrect.
minimum_count
integer. Number of hits in a histogram required before the full bias is active for that bin. Below this value, the bias linearly decreases to equal 0 at hits = 0. Default = 200, but user should provide a reasonable value for their system.
mass_weighing
boolean Turns on/off mass weighing of the adaptive force. Default is off. Keep off if your system has massless sites such as in TIP4P water.
output_file
string. Default = F_out Name of the file to save Adaptive Force Vector Field information to - this is what’s useful
Fworld_output_file
string. Default = Fworld_cv Name of the file to backup the raw Fworld output to for restarts. There will be separate outputs for each CV.
Nworld_output_file
string. Default = Nworld Name of the file to backup the raw Nworld output to for restarts.
output_frequency
integer. Saves the histogram of generalized force every this many timesteps.
unit_conversion
double. Unit conversion from d(momentum)/d(time) to force for the simulation. For LAMMPS using units real, this is 2390.06 (gram.angstrom/mole.femtosecond^2 -> kcal/mole.angstrom) For GROMACS, this is 1.
frequency
1. OPTIONAL Leave at 1.

## Example input¶



“methods” : [{
“type” : “ABF”, “cvs” : [0,1], “CV_lower_bounds” : [-3.14, -3.14], “CV_upper_bounds” : [3.14,3.14], “CV_bins” : [21,21], “CV_restraint_minimums” : [-5,-5], “CV_restraint_maximums” : [5,5], “CV_restraint_spring_constants” : [0,0], “CV_isperiodic” : [false,false], “timestep” : 0.002, “minimum_count” : 50, “filename” : “F_out”, “backup_frequency” : 1000, “unit_conversion” : 1, “frequency” : 1

}]

## Output¶

The main output of the method is stored in a file specified in ‘filename’. This file will contain the Adaptive Force vector field printed out every ‘backup_frequency’ steps and at the end of a simulation. The method outputs a vector field, with vectors defined on each point on a grid that goes from (CV_lower_bounds) to (CV_upper_bounds) of each CV in its dimension, with (CV_bins) of grid points in each dimension. For example, for 2 CVs defined from (-1,1) and (-1,0) with 3 and 2 bins respectively would be a 3x2 grid (6 grid points). The printout is in the following format: 2*N number of columns, where N is the number of CVs. First N columns are coordinates in CV space, the N+1 to 2N columns are components of the Adaptive Force vectors. An example for N=2 is:

CV1 Coord CV2 Coord d(A)/d(CV1) d(A)/d(CV2)
-1 -1 -1 1
-1 0 2 1
0 -1 1 2
0 0 2 3
1 -1 2 4
1 0 3 5

## Tutorial¶

Alanine Dipeptide

For LAMMPS (must be built with RIGID and MOLECULE packages) To build RIGID and MOLECULE:

2. Do:
make yes-RIGID
make yes-MOLECULE

1. Go to your build folder and make.

Find the following input files in Examples/User/ABF/Example_AlanineDipeptide:

• in.ADP_ABF_Example(0-1) (2 files)
• example.input
• ADP_ABF_1walker.json
• ADP_ABF_2walkers.json
1. Put the contents of ABF_ADP_LAMMPS_Example folder in your ssages build folder
2. For a single walker example, do:
./ssages ADP_ABF_1walker.json.json


For 2 walkers, do:

mpirun -np 2 ./ssages ADP_ABF_2walkers.json


For GROMACS:

Optional:

• adp.gro
• topol.top
• nvt.mdp

Required:

• example_adp(0-1).tpr (2 files)
• ADP_ABF_1walker.json
• ADP_ABF_2walkers.json
1. Put the contents of ABF_ADP_Gromacs_Example in your ssages build folder
2. For a single walker example, do:
./ssages ABF_ADP_1walker.json


For 2 walkers, do:

mpirun -np 2 ./ssages ABF_ADP_2walkers.json


These will run using the pre-prepared input files in .tpr format. If you wish to prepare the input files yourself using GROMACS tools (if compiled with -DGROMACS=YES):

/build/hooks/gromacs/gromacs/bin/gmx_mpi grompp -f nvt.mdp -p topol.top -c adp.gro -o example_adp0.tpr
/build/hooks/gromacs/gromacs/bin/gmx_mpi grompp -f nvt.mdp -p topol.top -c adp.gro -o example_adp1.tpr


Be sure to change the seed in .mdp files for random velocity generation, so walkers can explore different places on the free energy surface.

Multiple walkers initiated from different seeds will explore different regions and will all contribute to the same adaptive force.

After the run is finished, you can check that your output matches the sample outputs given in the examples folders:

1. Copy ABF_integrator.py (requires numpy, scipy and matplotlib) into your build folder.
2. Run the integrator:
python ABF_integrator.py --periodic1 True --periodic2 True --interpolate 200

1. This will output a contour map, a gradient field and a heatmap. Compare these to the sample outputs.

Sodium Chloride

For LAMMPS (must be built with KSPACE and MOLECULE packages) To build RIGID and MOLECULE:

2. Do:
make yes-KSPACE
make yes-MOLECULE

1. Go to your build folder and make.

Find the following input files in Examples/User/ABF/Example_NaCl/ABF_NaCl_LAMMPS_Example:

• in.NaCl_ADP_example(0-1) (2 files)
• data.spce
• ADP_NaCl_1walker.json
• ADP_NaCl_2walkers.json
1. Put the contents of ABF_NaCl_LAMMPS_Example folder in your ssages build folder
2. For a single walker example, do:
./ssages ADP_NaCl_1walker.json.json


For 2 walkers, do:

mpirun -np 2 ./ssages ADP_NaCl_2walkers.json


For GROMACS:

Optional:

• NaCl.gro
• topol.top
• npt.mdp

Required:

• example_NaCl(0-1).tpr (2 files)
• ADP_NaCl_1walker.json
• ADP_NaCl_2walkers.json
1. Put the contents of ABF_NaCl_Gromacs_Example in your ssages build folder
2. For a single walker example, do:
./ssages ABF_NaCl_1walker.json


For 2 walkers, do:

mpirun -np 2 ./ssages ABF_NaCl_2walkers.json


These will run using the pre-prepared input files in .tpr format. If you wish to prepare the input files yourself using GROMACS tools (if compiled with -DGROMACS=YES):

/build/hooks/gromacs/gromacs/bin/gmx_mpi grompp -f npt.mdp -p topol.top -c NaCl.gro -o example_NaCl0.tpr
/build/hooks/gromacs/gromacs/bin/gmx_mpi grompp -f npt.mdp -p topol.top -c NaCl.gro -o example_NaCl1.tpr


Be sure to change the seed in .mdp files for random velocity generation, so walkers can explore different places on the free energy surface.

Multiple walkers initiated from different seeds will explore different regions and will all contribute to the same adaptive force.

After the run is finished, you can check that your output matches the sample outputs given in the examples folders:

1. Copy ABF_integrator.py (requires numpy, scipy and matplotlib) into your build folder.
2. Run the integrator:
python ABF_integrator.py

1. This will output a Potential of Mean Force graph. Compare this to the sample output.

## Developers¶

Emre Sevgen Hythem Sidky