AICHE presentation on multistate reweighting methods

7/27/2019 AICHE presentation on multistate reweighting methods

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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.

AICHE presentation on multistate reweighting methods

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