sampling the free energy surfaces of collective variables · free energy estimators for eabf ak is...
TRANSCRIPT
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Sampling the free energy surfacesSampling the free energy surfaces
of collective variablesof collective variables
Enhanced Samplingand Free-Energy Calculations Urbana, 12 September 2018
Jérôme Hénin
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Please interrupt!
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structure
model systems
struct biol
struct bioinform
interactions(force fields)
phys chem theoretical chem
physics
algorithms
CS
maths
structure(refinement)
molecularinteractions
dynamics
thermodynamics
struct biol
biophysics
pharmacology
biomolecularsimulation
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Biology and the curse of dimensionality
φ
ψ
A(φ,ψ)
we need reduced representations
made of few selected coordinates● for human intuition● for importance sampling
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Outline
● Free energy
● Collective variables
● Free energy landscapes
● Methods to compute (estimate) FE landscapes
– from probability distribution (histograms)
– from forces (thermodynamic integration)
– from adapted biasing potential (metadynamics)
● Methods to sample FE landscapes
– umbrella sampling
– metadynamics : adaptive biasing potential
– adaptive biasing force
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Tetramethylammonium – acetone binding
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Free energy
● free energy differences ↔ probability ratios
● macrostates (A, B) are collections of microstates (atom coordinates x)
● →probabilities of macrostates are sums (integrals) over microstates
● probabilities of microstates follow Boltzmann distribution
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Collective variables
● geometric variables that depend on the positions of several atoms(hence “collective”)
● mathematically: functions of atomic coordinates
● example: distance between two atoms
● distance between the centers of mass of groups of atoms G1, G2
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Probability distribution of a collective variable
● we know the 3N-dimensional probability distribution of atom coordinates x:
● what is the probability distribution of
● theory: sum (integral) over all the values of x corresponding to a value of z
● in simulations: sample and calculate a histogram of coordinate z
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Probability distribution of a collective variable
(1) from unbiased simulation
5 10
distance (Å)
50000
1e+05
1.5e+05
2e+05
prob
abili
ty d
ensi
ty ρ
(lo
g sc
ale)
Probability distributionTMA-acetone pair in vacuum, 1 ns unbiased MD
ρ = 0
5 10
distance (Å)
1
10
100
1000
10000
1e+05
prob
abili
ty d
ensi
ty ρ
(lo
g sc
ale)
Probability distributionTMA-acetone pair in vacuum, 1 ns unbiased MD
ρ = 0
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Probability distribution of a collective variable
(2) with enhanced sampling
4 6 8 10 12
distance (Å)
0
5e+07
1e+08
1.5e+08
2e+08
2.5e+08
3e+08
pro
ba
bili
ty d
en
sity
ρ
Probability distributionTMA-acetone pair in vacuum
ρ = 1
5 10
distance (Å)
1
10
100
1000
10000
1e+05
1e+06
1e+07
1e+08
1e+09
prob
abili
ty d
ensi
ty ρ
(lo
g sc
ale)
enhanced sampling (ABF)
unbiased
Probability distributionTMA-acetone pair in vacuum
ρ = 1
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From probability to free energy
4 6 8 10 12
distance (Å)
1
10
100
1000
10000
1e+05
1e+06
1e+07
1e+08
prob
abili
ty d
ensi
ty ρ
(lo
g sc
ale)
Probability distributionTMA-acetone pair in vacuum
ρ = 1
4 6 8 10 12
distance (Å)
0
5
10
15
20
fre
e e
ne
rgy
(kca
l/mo
l)
Free energy profile for TMA - acetone pair
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Ways to calculate the free energy
● from unbiased histogram
● from biased histogram (importance sampling) with bias Vbias(z)
– in Umbrella Sampling, need to find values of C!
● estimate and integrate free energy derivative (gradient):Thermodynamic Integration
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Umbrella sampling
4 6 8 10 12
distance (Å)
0
500
1000
1500
2000
2500
3000
num
ber
of s
ampl
es
Histograms from Umbrella Sampling
● distribute (stratify) sampling using multiple confinement restraints
● combine partial information of each histogram by computing relative free energies
– WHAM (weighted histogram analysis method)
– MBAR (multistate Bennett’s acceptance ratio)
● requires overlap between sampling in adjacent windows
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Multi-channel free energy landscape
hiddenbarrier
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Multi-channel free energy landscape
hiddenbarrier
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Umbrella Sampling: stratification
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Umbrella Sampling or Not Sampling?
benefit of adaptive sampling methods: no stratification needed
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Orthogonal relaxation in ABF
Hénin, Tajkhorshid, Schulten & Chipot, Biophys J. 2008
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Adaptive sampling 1: adaptive biasing potential
where At converges to A
Free energy profile A(z) is linked to distribution of transition coordinate:
ABP: time-dependent biased potential
Long-time biased distribution:
that is, a uniform distribution.
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Adaptive Biasing Potential : Metadynamics
● adaptive bias is sum of Gaussian functions created at current position
● pushes coordinate away from visited regions
● convergence requires careful tuning of time dependence of the bias(“well-tempered” metadynamics)
Illustration: Parrinello group, ETH Zürich
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Adaptive sampling 2: Adaptive Biasing Force (ABF)
where A’t converges to A’
● ABF: time-dependent biasing force
● long-time biased distribution is uniform, as in ABP
● how do we estimate A’?
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Free energy derivative is a mean force
is a projected force (defined by coordinate transform)
is a geometric (entropic) term
den Otter J. Chem. Phys. 2000
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Simpler estimator of free energy gradient
● for each variable ξi, force is measured along arbitrary vector field(Ciccotti et al. 2005)
● orthogonality condition:
● free energy gradient:
● there are other estimators:
– from constraint force (original ABF, Darve & Pohorille 2001)
– from time derivatives of coordinate (Darve & Pohorille 2008)
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1. Stretching deca-alanine
Hénin & Chipot JCP 2004
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2. Sampling deca-alanine?
Chipot & Hénin JCP 2005
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3. Sampling in higher dimension
Hénin et al. JCTC 2010
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4. More robust sampling for poor coordinates:
Multiple-Walker ABF
● good performance with hidden barriers (Minoukadeh, Chipot, Lelièvre 2010)
● can sample systems using incomplete set of collective variables?
ABF, 1 x 100 ns MW-ABF, 32 x 3 ns
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ABF: a tale of annoying geometry
Estimator of free energy gradient:
● for each variable ξi, force is “measured” along arbitrary vector field vi(Ciccotti et al. 2005)
● orthogonality conditions:
● free energy gradient:
● geometric calculations are sometimes intractable(e.g. second derivatives of elaborate coordinates)
● orthogonality conditions are additional constraints
● in practice, many cases where ABF is unavailable
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extended-system Adaptive Biasing Force (eABF)
● idea: Lelièvre, Rousset & Stoltz 2007
● implementation: Fiorin, Klein & Hénin 2013
Get rid of geometry by watching an unphysical variable λ ,harmonically coupled to our geometric coordinate:
λ undergoes Langevin dynamics with mass m.Mass and force constant based on desired fluctuation and period:
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eABF trajectories
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Tight vs. loose coupling
λ
z z
λ
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Free energy estimators for eABF
● Ak is an estimator of free energy A, asymptotically accurate for high k
● other estimators lift this “stiff spring” requirement:
– umbrella integration (Kästner & Thiel 2005, Zheng & Yang 2012,Fu, Shao, Chipot & Cai 2016)
– CZAR (Lesage, Lelièvre, Stoltz & Hénin 2017)
● using these estimators, eABF is a hybrid adaptive method(free energy estimate is separate from bias)
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Hybrid methods
● adaptive sampling combines free energy estimation and enhanced sampling
● hybrid methods: bias based on one estimator, use another estimator to compute final free energy
● examples:
– unbiased sampling with thermodynamic integration
– metadynamics with thermodynamic integration
– eABF dynamics with UI or CZAR estimator
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Different estimates at very short sampling times
4 6 8 10 12
distance (Å)
0
10
20
30
fre
e e
ne
rgy
(kca
l/mo
l)ABF
metadynamics
mtd-TI
SMD (single non-eq work)
SMD-TI
Free energy profile for TMA - acetone pairfrom 100 ps simulations
● same long-time results, but different short-time convergence!● caution: may be system-dependent● efficiency of sampling vs. biases in short-time estimates
→ benefit of hybrid methods
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Thank you!
Questions?