aiche presentation on multistate reweighting methods
TRANSCRIPT
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Using multistate reweighting to rapidly explore
molecular parameter space
Himanshu Paliwal and Michael R. Shirts
AICHE Annual meeting
Session: Recent advances in molecular simulation III
Oct 29th 2012
Shirts Research Group
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Molecular design in chemical and parameter space
requires high throughput thermodynamic calculations
Observables like free energies of solvation, binding etc. are used for
Computational drug design
Design of new chromatographic surfaces
Design of new solvents
Parameterizing force fields
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Two problems in computing large number of
thermodynamic estimates
Computational cost to generate samples is very high.
Finding accurate and inexpensive simulation parameters for
calculating thermodynamic properties of interest
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Every sampled thermodynamic state has some information
about the neighboring unsampled thermodynamic state
UA XA
UA (XA)
UB
UCUD
UB (XA)
UC (XA)
UD (XA)
Sampled state
Unsampled
states
Sampled energies
Reevaluated
energies
OA
OB
OC
OD
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How do we rapidly scan the vast simulation parameter
space of nonbonded interaction parameters?
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Different choices of simulation parameters translate to
different accuracies in thermodynamic estimates
Accuracy of thermodynamic estimates depend how accurately
potential energies are estimated. For example
Thermodynamic observables
Free energy differences
We dont know how accurate the potential energy should be to get desired
accuracy in thermodynamic estimates.
00110
)(expln1
01 UUGGG
dxxU
dxxUA
A
))(exp(
))(exp(
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Coulomb interaction is estimated as a sum of real space
and Fourier space contributions
Parameters for Coulomb using PME Short range cutoff (rc,coul)
(Gaussian width) or Etol
Fourier spacing (Fsp)
Order of spline (Order)
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How do we choose the cutoff between
short and long range Lennard-Jones contributions?
Parameters for LJ
Short range cutoff (rc,LJ)
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Choosing a switching distance
Tradeoff between minimizing force discontinuities and error in
thermodynamic estimates. (Only important for MD)
Parameters for Coulomb and LJ switch
Coulomb switch distance (rswi,coul)
LJ switch distance (rswi,LJ)
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How accurate does the potential energy need to be to get
desired accuracy in thermodynamic estimates?
O1
O2
Observables space (O)Potential energy space (E)
P1
P2
Parameter space (P)E1
E2
O1, O2 could be free
energy and enthalpy of
phase change
E1, E2 could be LJ and
Coulomb potentialP1, P2 could be LJ cutoff
and Coulomb cutoff
Studied Not Studied
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Parameter space for the search is combinatorially
large, hence we split the search
Total number of combinations before split is 2,592,000.
After split number of combinations is 5184.
Paramters Choices # of choices
Order of beta spline [3, 4, 5, 6] 4
Ewald tolerance [10-2, 10-4, 10-6, 10-8, 10-10] 5
Fourier spacing (nm) [0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16 , 0.18, 0.20] 9
Coulomb cutoff (nm) [0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5 ] 10
Width of Coulomb switch (nm) [0.2, 0.18, 0.16, 0.14, 0.12, 0.10, 0.08, 0.06, 0.04, 0.02, 0.01, 0.001] 12
LJ cutoff (nm) [0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5 ] 10
Width of LJ switch (nm) [0.2, 0.18, 0.16, 0.14, 0.12, 0.10, 0.08, 0.06, 0.04, 0.02, 0.01, 0.001] 12
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We search for converged parameters for three
benchmark systems
The three systems in the benchmark set which were used tovalidate free energy methods are used in this study also.
Methane solvation
Dipole Inversion
Anthracene solvation
We calculate the following observables: Free energy of transformation
Enthalpy of transformation
Heat of vaporization of TIP3P water
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We want a computationally inexpensive parameter set
which gives accurate thermodynamic estimates
Parameter set
Definitions:
Expensive parameters give converged potential energies and hence
converged thermodynamic estimates.
Optimized parameters give potential energies and thermodynamic
estimates statistically indistinguishable w.r.t expensive but will be
computationally cheap.
Benchmark parameters: the set we started with.
LJswiLJccoulswicoulc rrrrFspEtolOrder ,,,, ,,,,,,
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Different parameter sets represent different
thermodynamic states
Multistate Bennett Acceptance Ratio (MBAR) can reweight the data fromsampled states to predict thermodynamic estimates for the unsampled
states.
Initial state sampling UB (XB)
UB,0 (XB,0)
0
1
UB,1 (XB,0)0
1
UB,1 (XB,1)UB,0 (XB,1)
GB
Reevaluate UE (XB)
UE,0 (XB,0) UE,1 (XB,0)
UE,1 (XB,1)UE,0 (XB,1)
Ui,0 (XB,0) Ui,1 (XB,0)
Ui,1 (XB,1)Ui,0 (XB,1)
Reevaluate Ui (XB)
GE Gi
GBE GEi
GBi
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Samples generated only using the benchmark set are
used to predict GEi for 5184 parameter sets
Takes only a minute to reevaluate one set of parameters
Reduction in time consumption : 540 CPU years 1 CPU month
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Our search results in optimized parameters that are
statistically indistinguishable from expensive
Predicted using only
benchmark set
Direct
differences
Calculated using
both sets
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Efficiency achieved using reweighting formalism is
very promising.
We search through 5200 parameter combinations whichinvolves calculation of:
3 million observables for 60,000 thermodynamic states.
Using MD for all the sets the same analysis would have taken
540 CPU years.
We used MD for a single set with re-evaluation of energies for
rest of the sets and the whole exercise took
One CPU month.
Re-evaluation formalism is faster than raw sample generation
by roughly
a factor of 4000
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How do we deal with changes in molecular geometry
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Parameter scans and molecular transformations not only
involve alchemical transformations in LJ and Coulomb
but also changes in geometry
Alchemical changes (growing and disappearing of atoms):
Changes in sigma, epsilon and charges
Geometry changes: Change in bond length, bond angle, dihedral angle
Present free energy methods lack the capacity to do freeenergy analysis for transformations involving changes in
geometry.
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A perturbed molecular geometry is never seen in the
simulation of original geometry.
In the FEP calculation
We need to calculate UTIP3P(XTIP4P) .
But we will never see a TIP3P water geometry in a TIP4P water simulation.
We can introduce a linear map Tij which maps TIP4P to TIP3P geometryand then we can evaluate UTIP3P(Tij(XTIP4P))
PTIPPTIPPTIPPTIPPTIP
UUPTIPGPTIPGG44334
)(expln1
43
TIP4P TIP3P
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We use linear maps between molecular geometries to
estimate thermodynamic property differences
The linear map Tij introduces a Jacobian Jij term in the partition function
integral.
Jij can be analytically calculated and can be included in the free energy
estimating algorithm.
This way we dont have to change anything in the MD code.
K
j
N
nK
kjn
w
kkk
jn
w
i
i
j
xufN
xuf
1 1
1
))(exp(
))(exp(ln
)ln(
1))(()( ijjnijijn
w
i JxTUxu
MBAR Eq.
unchanged !!Jij is included in
effective potential
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We test our new algorithm by estimating free energies
and enthalpies for four transformations
A set of truncated harmonic oscillators
Force constant and Equilibrium distance is changed(2)
TIP4P TIP3P molecule in liq. Phase
Charge, sigma, epsilon is changed and additional
site is introduced (4)
SPC-E TIP3P molecule in liq. Phase
Charge, sigma, epsilon, OH bond length and
HOH bond angle is changed (5)
TIP4P SPC-E molecule in liq. phase
Charge, sigma, epsilon, OH bond length, HOH bond
angle is changed and extra site is introduced(6)
TIP4P TIP3P
SPC-E TIP3P
TIP4P SPCE
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Mapping makes configurations of different geometries
visible in simulations of all other intermediate states
ki
Okj Oki
kj
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We validate water transformations using
thermodynamic cycles.
Analytical free energy for truncated harmonic oscillators with spread
Water transformations
(Ghyd)TIP4Pand (Ghyd)
TIP3Pare evaluated without mapping, G2a is
evaluated using mapping and G1a can be evaluated analytically.
(Ghyd)TIP3P
(Ghyd)TIP4P
(G1a) (G2a)
N TIP3Pmolecules
in vacuum
N TIP4P
molecules
in vacuum
N TIP4P
molecules
in TIP4P water
N TIP3Pmolecules
in TIP3P water
N TIP4P
molecules
in TIP4P water
N SPC-E
molecules
in SPC-E water
N TIP3P
molecules
in TIP3P water
(G2a)
(G2b)
(G2c)
aa
PTIP
hyd
PTIP
hyd GGGG 1234 )()( 0222 abc GGG
)2ln()( ii analyticalf
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Multistate reweighting with mapping drastically reduces
uncertainty of the calculation
Direct calculations
Linear removal of charge, soft core removal of LJ interactions
One molecule solvated using 21 intermediate states, each state simulated for 10 ns.
Mapped calculations
Linearly variation of charge, LJ and geometry over 21 states , each state simulated for 10
ns.
TransformationDirect calculation
Ghyd (kJ/mol)
Mapped
Ghyd (kJ/mol)
TIP3P TIP4P -0.0205 0.0633 0.0410 0.0002
SPC-E TIP3P 3.9479 0.0632 4.0503 0.0001
TIP4P SPC-E -3.9274 0.0656 - 4.0912 0.0002
300 X
Lower uncertainty
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Mapped calculations require less number of
intermediate states compared to direct calculations
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Mapped calculations require less number of samples
compared to direct calculations
MBAR without mapping requires 20,000 samples/state and 21 intermediate states to reach a
statistical uncertainty of 0.06 kJ/mol in free energy estimate.
MBAR with mapping requires just 50 samples/state and 11 intermediate state to reach the
same precision. Speed up in this particular system is by a factor of800
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Conclusion
It is possible to estimate thermodynamic properties for large number of
unsampled states using samples from just a few states.
It is possible to estimate free energies for states which do not share similargeometries using a linear transformation which maps the two different
geometries.
Reweighting technique along with mapping algorithm can substantially
speed up the parameter scans.
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Acknowledgement
I thank ..
you all for your time and patience
the organizers for giving me a platform to discuss my research
NSF CHE-1152786 for funding.
the entire Shirts group for all the encouragement and support.