def _process_phase(i, phase):
all_forward = []
all_reverse = []
for gen_id in range(_max_gen(RUN)):
fe, err, f_works, r_works = bootstrap_BAR(
RUN, phase, gen_id, n_bootstrap)
d[f"{phase}_fe_GEN{gen_id}"] = fe
d[f"{phase}_dfe_GEN{gen_id}"] = err
all_forward.extend(f_works)
all_reverse.extend(r_works)
sns.kdeplot(all_forward,
shade=True,
color="cornflowerblue",
ax=axes[i])
sns.rugplot(
all_forward,
ax=axes[i],
color="cornflowerblue",
alpha=0.5,
label=f"forward : N={len(f_works)}",
)
sns.rugplot(
all_forward,
ax=axes[i],
color="darkblue",
label=f"forward (gen0) : N={len(f_works)}",
)
sns.rugplot(
[-x for x in all_reverse],
ax=axes[i],
color="mediumvioletred",
label=f"reverse (gen0) : N={len(r_works)}",
)
sns.kdeplot([-x for x in all_reverse],
shade=True,
color="hotpink",
ax=axes[i])
sns.rugplot(
[-x for x in all_reverse],
ax=axes[i],
color="hotpink",
alpha=0.5,
label=f"reverse : N={len(r_works)}",
)
axes[i].set_title(phase)
# TODO add bootstrapping here
d[f"{phase}_fes"] = BAR(np.asarray(all_forward),
np.asarray(all_reverse))
def _get_bar_free_energy(works: np.ndarray) -> PointEstimate:
"""
Compute the BAR free energy
Parameters
----------
works : (N,) ndarray
1-D array of records containing fields "forward" and "reverse"
Returns
-------
PointEstimate
BAR free energy point estimate and standard error
"""
from pymbar import BAR
delta_f, ddelta_f = BAR(works["forward"], works["reverse"])
return PointEstimate(point=delta_f, stderr=ddelta_f)
def _bootstrap(
gens: List[GenAnalysis],
n_bootstrap: int,
clones_per_gen: int,
gen_number: int,
) -> List[float]:
fes = []
for _ in range(n_bootstrap):
random_indices = np.random.choice(clones_per_gen, gen_number)
subset_f = [
gen.works[i].forward for i in random_indices for gen in gens
]
subset_r = [
gen.works[i].reverse for i in random_indices for gen in gens
]
fe, _ = BAR(np.asarray(subset_f), np.asarray(subset_r))
fes.append(fe * KT_KCALMOL)
return fes
def check_harmonic_oscillator_ncmc(ncmc_nsteps=50, ncmc_integrator="VV"):
"""
Test NCMC switching of a 3D harmonic oscillator.
In this test, the oscillator center is dragged in space, and we check the computed free energy difference with BAR, which should be 0.
"""
# Parameters for 3D harmonic oscillator
mass = 39.948 * unit.amu # mass of particle (argon)
sigma = 5.0 * unit.angstrom # standard deviation of harmonic oscillator
collision_rate = 5.0 / unit.picosecond # collision rate
temperature = 300.0 * unit.kelvin # temperature
platform_name = 'Reference' # platform anme
NSIGMA_MAX = 6.0 # number of standard errors away from analytical solution tolerated before Exception is thrown
# Compute derived quantities.
kT = kB * temperature # thermal energy
beta = 1.0 / kT # inverse energy
K = kT / sigma**2 # spring constant
tau = 2 * math.pi * unit.sqrt(mass / K) # time constant
timestep = tau / 20.0
platform = openmm.Platform.getPlatformByName(platform_name)
# Create a 3D harmonic oscillator with context parameter controlling center of oscillator.
system = openmm.System()
system.addParticle(mass)
energy_expression = '(K/2.0) * ((x-x0)^2 + y^2 + z^2);'
force = openmm.CustomExternalForce(energy_expression)
force.addGlobalParameter('K', K.in_unit_system(unit.md_unit_system))
force.addGlobalParameter('x0', 0.0)
force.addParticle(0, [])
system.addForce(force)
# Set the positions at the origin.
positions = unit.Quantity(np.zeros([1, 3], np.float32), unit.angstroms)
functions = {'x0': 'lambda'} # drag spring center x0
from perses.annihilation.ncmc_integrator import NCMCVVAlchemicalIntegrator, NCMCGHMCAlchemicalIntegrator
if ncmc_integrator == "VV":
ncmc_insert = NCMCVVAlchemicalIntegrator(
temperature,
system,
functions,
direction='insert',
nsteps=ncmc_nsteps,
timestep=timestep) # 'insert' drags lambda from 0 -> 1
ncmc_delete = NCMCVVAlchemicalIntegrator(
temperature,
system,
functions,
direction='delete',
nsteps=ncmc_nsteps,
timestep=timestep) # 'insert' drags lambda from 0 -> 1
elif ncmc_integrator == "GHMC":
ncmc_insert = NCMCGHMCAlchemicalIntegrator(
temperature,
system,
functions,
direction='insert',
collision_rate=9.1 / unit.picoseconds,
nsteps=ncmc_nsteps,
timestep=timestep) # 'insert' drags lambda from 0 -> 1
ncmc_delete = NCMCGHMCAlchemicalIntegrator(
temperature,
system,
functions,
direction='delete',
collision_rate=9.1 / unit.picoseconds,
nsteps=ncmc_nsteps,
timestep=timestep) # 'insert' drags lambda from 0 -> 1
else:
raise Exception(
"%s not recognized as integrator name. Options are VV and GHMC" %
ncmc_integrator)
# Run NCMC switching trials where the spring center is switched with lambda: 0 -> 1 over a finite number of steps.
w_f = collect_switching_data(system,
positions,
functions,
temperature,
collision_rate,
timestep,
platform,
ncmc_integrator=ncmc_insert,
ncmc_nsteps=ncmc_nsteps,
direction='insert')
w_r = collect_switching_data(system,
positions,
functions,
temperature,
collision_rate,
timestep,
platform,
ncmc_integrator=ncmc_delete,
ncmc_nsteps=ncmc_nsteps,
direction='delete')
from pymbar import BAR
[df, ddf] = BAR(w_f, w_r, method='self-consistent-iteration')
print('%8.3f +- %.3f kT' % (df, ddf))
if (abs(df) > NSIGMA_MAX * ddf):
msg = 'Delta F (%d steps switching) = %f +- %f kT; should be within %f sigma of 0' % (
ncmc_nsteps, df, ddf, NSIGMA_MAX)
msg += '\n'
msg += 'w_f = %s\n' % str(w_f)
msg += 'w_r = %s\n' % str(w_r)
raise Exception(msg)
def check_alchemical_hybrid_elimination_bar(topology_proposal, old_positions, new_positions, ncmc_nsteps=50, n_iterations=50, NSIGMA_MAX=6.0, geometry=False):
"""
Check that the hybrid topology, where both endpoints are identical, returns a free energy within NSIGMA_MAX of 0.
Parameters
----------
topology_proposal
positions
ncmc_nsteps
NSIGMA_MAX
Returns
-------
"""
#make the hybrid topology factory:
factory = HybridTopologyFactory(topology_proposal, old_positions, new_positions)
platform = openmm.Platform.getPlatformByName("CUDA")
hybrid_system = factory.hybrid_system
hybrid_topology = factory.hybrid_topology
initial_hybrid_positions = factory.hybrid_positions
#alchemical functions
functions = {
'lambda_sterics' : '2*lambda * step(0.5 - lambda) + (1.0 - step(0.5 - lambda))',
'lambda_electrostatics' : '2*(lambda - 0.5) * step(lambda - 0.5)',
'lambda_bonds' : 'lambda',
'lambda_angles' : 'lambda',
'lambda_torsions' : 'lambda'
}
w_f = np.zeros(n_iterations)
w_r = np.zeros(n_iterations)
#make the alchemical integrators:
forward_integrator = NCMCGHMCAlchemicalIntegrator(temperature, hybrid_system, functions, nsteps=ncmc_nsteps, direction='insert')
forward_context = openmm.Context(hybrid_system, forward_integrator, platform)
print("Minimizing for forward protocol...")
forward_context.setPositions(initial_hybrid_positions)
for parm in functions.keys():
forward_context.setParameter(parm, 0.0)
openmm.LocalEnergyMinimizer.minimize(forward_context, maxIterations=10)
initial_state = forward_context.getState(getPositions=True, getEnergy=True)
print("The initial energy after minimization is %s" % str(initial_state.getPotentialEnergy()))
initial_forward_positions = initial_state.getPositions(asNumpy=True)
equil_positions = simulate_hybrid(hybrid_system,functions, 0.0, initial_forward_positions)
print("Beginning forward protocols")
#first, do forward protocol (lambda=0 -> 1)
with progressbar.ProgressBar(max_value=n_iterations) as bar:
for i in range(n_iterations):
equil_positions = simulate_hybrid(hybrid_system, functions, 0.0, equil_positions)
forward_context.setPositions(equil_positions)
forward_integrator.step(ncmc_nsteps)
w_f[i] = -1.0 * forward_integrator.getLogAcceptanceProbability(forward_context)
bar.update(i)
del forward_context, forward_integrator
reverse_integrator = NCMCGHMCAlchemicalIntegrator(temperature, hybrid_system, functions, nsteps=ncmc_nsteps, direction='delete')
print("Minimizing for reverse protocol...")
reverse_context = openmm.Context(hybrid_system, reverse_integrator, platform)
reverse_context.setPositions(initial_hybrid_positions)
for parm in functions.keys():
reverse_context.setParameter(parm, 1.0)
openmm.LocalEnergyMinimizer.minimize(reverse_context, maxIterations=10)
initial_state = reverse_context.getState(getPositions=True, getEnergy=True)
print("The initial energy after minimization is %s" % str(initial_state.getPotentialEnergy()))
initial_reverse_positions = initial_state.getPositions(asNumpy=True)
equil_positions = simulate_hybrid(hybrid_system,functions, 1.0, initial_reverse_positions, nsteps=1000)
#now, reverse protocol
print("Beginning reverse protocols...")
with progressbar.ProgressBar(max_value=n_iterations) as bar:
for i in range(n_iterations):
equil_positions = simulate_hybrid(hybrid_system,functions, 1.0, equil_positions)
reverse_context.setPositions(equil_positions)
reverse_integrator.step(ncmc_nsteps)
w_r[i] = -1.0 * reverse_integrator.getLogAcceptanceProbability(reverse_context)
bar.update(i)
del reverse_context, reverse_integrator
from pymbar import BAR
[df, ddf] = BAR(w_f, w_r)
print("df = %12.6f +- %12.5f kT" % (df, ddf))
for k in range(K):
stdevs[k, k] = 0
print(stdevs)
print("==============================================")
print(" Testing computeBAR ")
print("==============================================")
nonzero_indices = numpy.array(list(range(K)))[Nk_ne_zero]
Knon = len(nonzero_indices)
for i in range(Knon - 1):
k = nonzero_indices[i]
k1 = nonzero_indices[i + 1]
w_F = u_kln[k, k1, 0:N_k[k]] - u_kln[k, k, 0:N_k[k]] # forward work
w_R = u_kln[k1, k, 0:N_k[k1]] - u_kln[k1, k1, 0:N_k[k1]] # reverse work
results = BAR(w_F, w_R)
df_bar = results['Delta_f']
ddf_bar = results['dDelta_f']
bar_analytical = (f_k_analytical[k1] - f_k_analytical[k])
bar_error = bar_analytical - df_bar
print(
"BAR estimator for reduced free energy from states %d to %d is %f +/- %f"
% (k, k1, df_bar, ddf_bar))
stddev_away("BAR estimator", bar_error, ddf_bar)
print("==============================================")
print(" Testing computeEXP ")
print("==============================================")
print("EXP forward free energy")
for k in range(K - 1):
for k in range(K):
stdevs[k, k] = 0
print stdevs
print "=============================================="
print " Testing computeBAR "
print "=============================================="
nonzero_indices = numpy.array(range(K))[Nk_ne_zero]
Knon = len(nonzero_indices)
for i in range(Knon - 1):
k = nonzero_indices[i]
k1 = nonzero_indices[i + 1]
w_F = u_kln[k, k1, 0:N_k[k]] - u_kln[k, k, 0:N_k[k]] # forward work
w_R = u_kln[k1, k, 0:N_k[k1]] - u_kln[k1, k1, 0:N_k[k1]] # reverse work
(df_bar, ddf_bar) = BAR(w_F, w_R)
bar_analytical = (f_k_analytical[k1] - f_k_analytical[k])
bar_error = bar_analytical - df_bar
print "BAR estimator for reduced free energy from states %d to %d is %f +/- %f" % (
k, k1, df_bar, ddf_bar)
print "BAR estimator differs by %f standard deviations" % numpy.abs(
bar_error / ddf_bar)
print "=============================================="
print " Testing computeEXP "
print "=============================================="
print "EXP forward free energy"
for k in range(K - 1):
if N_k[k] != 0:
w_F = u_kln[k, k + 1, 0:N_k[k]] - u_kln[k, k, 0:N_k[k]] # forward work
def check_hybrid_round_trip_elimination(topology_proposal,
positions,
ncmc_nsteps=50,
NSIGMA_MAX=6.0):
"""
Test the hybrid system by switching between lambda = 1 and lambda = 0, then using BAR to compute the free energy
difference. As the test is designed so that both endpoints are the same, the free energy difference should be zero.
Parameters
----------
topology_proposal : TopologyProposal
The topology proposal to test.
This must be a null transformation, where topology_proposal.old_system == topology_proposal.new_system
ncmc_steps : int, optional, default=50
Number of NCMC switching steps, or 0 for instantaneous switching.
NSIGMA_MAX : float, optional, default=6.0
"""
functions = {
'lambda_sterics': 'lambda',
'lambda_electrostatics': 'lambda',
'lambda_bonds': 'lambda',
'lambda_angles': 'lambda',
'lambda_torsions': 'lambda'
}
# Initialize engine
from perses.annihilation import NCMCGHMCAlchemicalIntegrator
from perses.annihilation.new_relative import HybridTopologyFactory
#The current and "proposed" positions are the same, since the molecule is not changed.
factory = HybridTopologyFactory(topology_proposal, positions, positions)
forward_integrator = NCMCGHMCAlchemicalIntegrator(temperature,
factory.hybrid_system,
functions,
nsteps=ncmc_nsteps,
direction='insert')
reverse_integrator = NCMCGHMCAlchemicalIntegrator(temperature,
factory.hybrid_system,
functions,
nsteps=ncmc_nsteps,
direction='delete')
platform = openmm.Platform.getPlatformByName("Reference")
forward_context = openmm.Context(factory.hybrid_system, forward_integrator,
platform)
reverse_context = openmm.Context(factory.hybrid_system, reverse_integrator,
platform)
# Make sure that old system and new system are identical.
if not (topology_proposal.old_system == topology_proposal.new_system):
raise Exception(
"topology_proposal must be a null transformation for this test (old_system == new_system)"
)
for (k, v) in topology_proposal.new_to_old_atom_map.items():
if k != v:
raise Exception(
"topology_proposal must be a null transformation for this test (retailed atoms must map onto themselves)"
)
nequil = 5 # number of equilibration iterations
niterations = 50 # number of round-trip switching trials
logP_work_n_f = np.zeros([niterations], np.float64)
for iteration in range(nequil):
positions = simulate_hybrid(factory.hybrid_system, functions, 0.0,
factory.hybrid_positions)
#do forward switching:
for iteration in range(niterations):
# Equilibrate
positions = simulate_hybrid(factory.hybrid_system, functions, 0.0,
factory.hybrid_positions)
# Check that positions are not NaN
if (np.any(np.isnan(positions / unit.angstroms))):
raise Exception("Positions became NaN during equilibration")
# Hybrid NCMC
forward_integrator.reset()
forward_context.setPositions(positions)
forward_integrator.step(ncmc_nsteps)
logP_work = forward_integrator.getTotalWork(forward_context)
# Check that positions are not NaN
if (np.any(np.isnan(positions / unit.angstroms))):
raise Exception("Positions became NaN on Hybrid NCMC switch")
# Store log probability associated with work
logP_work_n_f[iteration] = logP_work
logP_work_n_r = np.zeros([niterations], np.float64)
for iteration in range(nequil):
positions = simulate_hybrid(factory.hybrid_system, functions, 1.0,
factory.hybrid_positions)
#do forward switching:
for iteration in range(niterations):
# Equilibrate
positions = simulate_hybrid(factory.hybrid_system, functions, 1.0,
factory.hybrid_positions)
# Check that positions are not NaN
if (np.any(np.isnan(positions / unit.angstroms))):
raise Exception("Positions became NaN during equilibration")
# Hybrid NCMC
reverse_integrator.reset()
reverse_context.setPositions(positions)
reverse_integrator.step(ncmc_nsteps)
logP_work = reverse_integrator.getTotalWork(forward_context)
# Check that positions are not NaN
if (np.any(np.isnan(positions / unit.angstroms))):
raise Exception("Positions became NaN on Hybrid NCMC switch")
# Store log probability associated with work
logP_work_n_r[iteration] = logP_work
work_f = -logP_work_n_f
work_r = -logP_work_n_r
from pymbar import BAR
[df, ddf] = BAR(work_f, work_r)
print("df = %12.6f +- %12.5f kT" % (df, ddf))
if (abs(df) > NSIGMA_MAX * ddf):
msg = 'Delta F (%d steps switching) = %f +- %f kT; should be within %f sigma of 0\n' % (
ncmc_nsteps, df, ddf, NSIGMA_MAX)
msg += 'logP_work_n:\n'
msg += str(work_f) + '\n'
msg += str(work_r) + '\n'
raise Exception(msg)
def check_alchemical_hybrid_elimination_bar(topology_proposal,
positions,
ncmc_nsteps=50,
NSIGMA_MAX=6.0,
geometry=False):
"""
Check that the hybrid topology, where both endpoints are identical, returns a free energy within NSIGMA_MAX of 0.
Parameters
----------
topology_proposal
positions
ncmc_nsteps
NSIGMA_MAX
Returns
-------
"""
from perses.annihilation import NCMCGHMCAlchemicalIntegrator
from perses.annihilation.new_relative import HybridTopologyFactory
#make the hybrid topology factory:
factory = HybridTopologyFactory(topology_proposal, positions, positions)
platform = openmm.Platform.getPlatformByName("Reference")
hybrid_system = factory.hybrid_system
hybrid_topology = factory.hybrid_topology
initial_hybrid_positions = factory.hybrid_positions
n_iterations = 50 #number of times to do NCMC protocol
#alchemical functions
functions = {
'lambda_sterics':
'2*lambda * step(0.5 - lambda) + (1.0 - step(0.5 - lambda))',
'lambda_electrostatics': '2*(lambda - 0.5) * step(lambda - 0.5)',
'lambda_bonds': 'lambda',
'lambda_angles': 'lambda',
'lambda_torsions': 'lambda'
}
w_f = np.zeros(n_iterations)
w_r = np.zeros(n_iterations)
#make the alchemical integrators:
#forward_integrator = NCMCGHMCAlchemicalIntegrator(temperature, hybrid_system, functions, nsteps=ncmc_nsteps, direction='insert')
#reverse_integrator = NCMCGHMCAlchemicalIntegrator(temperature, hybrid_system, functions, nsteps=ncmc_nsteps, direction='delete')
#first, do forward protocol (lambda=0 -> 1)
for i in range(n_iterations):
forward_integrator = NCMCGHMCAlchemicalIntegrator(temperature,
hybrid_system,
functions,
nsteps=ncmc_nsteps,
direction='insert')
equil_positions = simulate_hybrid(hybrid_system, functions, 0.0,
initial_hybrid_positions)
context = openmm.Context(hybrid_system, forward_integrator, platform)
context.setPositions(equil_positions)
forward_integrator.step(ncmc_nsteps)
w_f[i] = -1.0 * forward_integrator.getLogAcceptanceProbability(context)
print(i)
del context, forward_integrator
#now, reverse protocol
for i in range(n_iterations):
reverse_integrator = NCMCGHMCAlchemicalIntegrator(temperature,
hybrid_system,
functions,
nsteps=ncmc_nsteps,
direction='delete')
equil_positions = simulate_hybrid(hybrid_system, functions, 1.0,
initial_hybrid_positions)
context = openmm.Context(hybrid_system, reverse_integrator, platform)
context.setPositions(equil_positions)
reverse_integrator.step(ncmc_nsteps)
w_r[i] = -1.0 * reverse_integrator.getLogAcceptanceProbability(context)
print(i)
del context, reverse_integrator
from pymbar import BAR
[df, ddf] = BAR(w_f, w_r)
print("df = %12.6f +- %12.5f kT" % (df, ddf))
if (abs(df) > NSIGMA_MAX * ddf):
msg = 'Delta F (%d steps switching) = %f +- %f kT; should be within %f sigma of 0\n' % (
ncmc_nsteps, df, ddf, NSIGMA_MAX)
msg += 'logP_work_n:\n'
msg += str(w_f) + '\n'
msg += str(w_r) + '\n'
raise Exception(msg)
def DoBAR(fwds, revs, label, verbose):
"""
BAR to combine fwd and rev data of dGs.
Here, don't multiply dGs_R by -1 since BAR calls for reverse work value.
Parameters
----------
fwds: dictionary of forward work values for each window
revs: dictionary of reverse work values for each window
label: string label of what it is (only for printing output)
Returns
-------
dgs: 1D list of accumulated list of energy values. Ex. if each step was 2,
then dgs would be [0,2,4...]
gsdlist: 1D list of accompanying stdevs to the dgs list
"""
fwd_ss = {} # subsampled version of fwds
rev_ss = {} # subsampled version of revs
dg_bar = np.zeros([len(fwds)], np.float64) # allocate storage: dG steps
gsd_bar = np.zeros([len(fwds)],
np.float64) # allocate storage: dG stdev steps
dgs = np.zeros([len(fwds)], np.float64) # allocate storage: dG accumulated
gsdlist = np.zeros([len(fwds)],
np.float64) # allocate storage: dG stdev accum
#corr_time = np.zeros([len(fwds)], np.float64)
corr_time = {}
for key, value in fwds.items(
): # this notation changes in python3: http://tinyurl.com/j3uq3me
# compute correlation time
g = timeseries.statisticalInefficiency(value)
corr_time[key] = [g]
# compute indices of UNcorrelated timeseries, then extract those samples
indices = timeseries.subsampleCorrelatedData(value, g)
fwd_ss[key] = value[indices]
for key, value in revs.items(
): # this notation changes in python3: http://tinyurl.com/j3uq3me
# compute correlation time
g = timeseries.statisticalInefficiency(value)
corr_time[key].append(g)
# compute indices of UNcorrelated timeseries, then extract those samples
indices = timeseries.subsampleCorrelatedData(value, g)
rev_ss[key] = value[indices]
bar = {}
# then apply BAR estimator to get dG for each step
for kF, kR in zip(sorted(fwd_ss.keys()),
sorted(list(rev_ss.keys()), reverse=True)):
dg_bar[kF], gsd_bar[kF] = BAR(fwd_ss[kF], rev_ss[kR])
bar[kF] = [np.sum(dg_bar), dg_bar[kF], gsd_bar[kF]]
# calculate the net dG standard deviation = sqrt[ sum(s_i^2) ]
gsd = (np.sum(np.power(gsd_bar, 2)))**0.5
net = 0.
netsd = 0.
for i, g in enumerate(dg_bar):
# accumulate net dGs into running sums (plot this)
dgs[i] = dg_bar[i] + net
net = dgs[i]
# combine the stdevs: s = sqrt(s1^2 + s2^2 + ...)
gsdlist[i] = ((gsd_bar[i])**2. + (netsd)**2.)**0.5
netsd = gsdlist[i]
if verbose == True:
print('\n\n#####---Correlation Times for dG_{}--#####'.format(label))
print('Window'.rjust(3), 'F'.rjust(5), 'R'.rjust(9))
for k, v in corr_time.items():
print("{:3d} {:10.3f} {:10.3f}".format(k, v[0], v[1]))
print("\n\n#####---BAR estimator for dG_{}---#####".format(label))
print('Window'.rjust(3), 'dG'.rjust(5), 'ddG'.rjust(11),
"Uncert.".rjust(11))
print("---------------------------------------------------------")
for k, v in bar.items():
str = '{:3d} {:10.4f} {:10.4f} +- {:3.4f}'.format(
k, v[0], v[1], v[2])
print(str)
print(("\nNet dG_{} energy difference = {:.4f} +- {:.4f} kcal/mol".format(
label, np.sum(dg_bar), gsd)))
return dgs, gsdlist
for k in range(K):
stdevs[k,k] = 0
print(stdevs)
print("==============================================")
print(" Testing computeBAR ")
print("==============================================")
nonzero_indices = numpy.array(list(range(K)))[Nk_ne_zero]
Knon = len(nonzero_indices)
for i in range(Knon-1):
k = nonzero_indices[i]
k1 = nonzero_indices[i+1]
w_F = u_kln[k, k1, 0:N_k[k]] - u_kln[k, k, 0:N_k[k]] # forward work
w_R = u_kln[k1, k, 0:N_k[k1]] - u_kln[k1, k1, 0:N_k[k1]] # reverse work
results = BAR(w_F, w_R, return_dict=True)
df_bar = results['Delta_f']
ddf_bar = results['dDelta_f']
bar_analytical = (f_k_analytical[k1]-f_k_analytical[k])
bar_error = bar_analytical - df_bar
print("BAR estimator for reduced free energy from states %d to %d is %f +/- %f" % (k,k1,df_bar,ddf_bar))
stddev_away("BAR estimator",bar_error,ddf_bar)
print("==============================================")
print(" Testing computeEXP ")
print("==============================================")
print("EXP forward free energy")
for k in range(K-1):
if N_k[k] != 0:
w_F = u_kln[k, k+1, 0:N_k[k]] - u_kln[k, k, 0:N_k[k]] # forward work