Enabling constant pressure hybrid Monte Carlo simulations ... · PDF fileEnabling constant pressure hybrid Monte Carlo simulations using the GROMACS molecular simulation package Mario

Enabling constant pressure hybrid Monte Carlosimulations using the GROMACS molecular

simulation package

Mario Fernandez Pendas

MSBMS GroupSupervised by Bruno Escribano and Elena Akhmatskaya

BCAM

18 October 2013

Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 1 / 13

Outline

Generalized Shadow Hybrid Monte Carlo (GSHMC)

Statistical ensembles

Andersen barostat

GSHMC with constant pressure and constant temperature

GROMACS software

Results

Conclusions



A Generalized Hybrid Monte Carlo method with respect to a modifiedHamiltonian was introduced by E. Akhmatskaya and S. Reich. It is called theGeneralized Shadow Hybrid Monte CarloAkhmatskaya, E., Reich, S. (2008). “GSHMC: An efficient method for molecular simulations”, J. Comput. Phys. 227: 4934-4954.

IdeaCombines Hamiltonian dynamics with Monte Carlo

Samples with respect to modified Hamiltonians

Partially updates momentum

AdvantagesEfficient sampling. Reduced discretization error. Improved acceptance ratefor large systems sizeRigorous temperature controlSampling complex systems while retaining the dynamical information
















Statistical Ensembles

In statistical mechanics, average values are defined as ensemble averageAn ensemble is a collection of all possible systems with different microscopicstates but identical macroscopic states

NVEMolecular dynamics is naturally performed in this ensemble

NVT (canonical)Hybrid Monte Carlo (HMC) is naturally performed in this ensemble

NPT (isothermal-isobaric)

To extend MD/HMC to NVT/NPT ensembles thermostats/barostats arerequired
















Andersen Barostat

Main IdeaThe coordinate vector r ∈ R3N is replaced by a scaled vectord := r/V1/3 ∈ R3N , where V is the volume of the simulation box

We consider an extended Lagrangian, where we use the dynamic value qof V (the piston degree of freedom), the external pressure α and the massof the piston µ:

L =

{12

q2/3d · [Md]− V(q1/3d) +µ

2q2 − αq

}From this Lagrangian we can get the Hamiltonian and new equations of

motion:dddt

=∂H∂pd

,dpd

dt= −∂H

∂d

dqdt

=∂H∂pq

,dpq

dt= −∂H

∂q

Andersen, H.C. (1980). “Molecular dynamics simulations at constant pressure and/or temperature“. J. Chem. Phys. 72:2384.


Andersen Barostat

Main IdeaThe coordinate vector r ∈ R3N is replaced by a scaled vectord := r/V1/3 ∈ R3N , where V is the volume of the simulation boxWe consider an extended Lagrangian, where we use the dynamic value qof V (the piston degree of freedom), the external pressure α and the massof the piston µ:

L =

{12

q2/3d · [Md]− V(q1/3d) +µ

2q2 − αq

}

From this Lagrangian we can get the Hamiltonian and new equations ofmotion:

dddt

=∂H∂pd

,dpd

dt= −∂H

∂d

dqdt

=∂H∂pq

,dpq

dt= −∂H

∂q



Andersen Barostat

Main IdeaThe coordinate vector r ∈ R3N is replaced by a scaled vectord := r/V1/3 ∈ R3N , where V is the volume of the simulation boxWe consider an extended Lagrangian, where we use the dynamic value qof V (the piston degree of freedom), the external pressure α and the massof the piston µ:

L =

{12

q2/3d · [Md]− V(q1/3d) +µ

2q2 − αq

}From this Lagrangian we can get the Hamiltonian and new equations of

motion:dddt

=∂H∂pd

,dpd

dt= −∂H

∂d

dqdt

=∂H∂pq

,dpq

dt= −∂H

∂q



GSHMC in NPT

MotivationFor some applications, like unfolding of peptides or crystallization process forpolymorphism in drugs, keeping constant pressure during the simulation isneeded

ObjectiveTo enable GSHMC simulations in NPT ensembles, GSHMC has beencombined with Andersen barostat

To implement NPT-GSHMC in the molecular software packageGROMACS

Akhmatskaya, E., Reich, S. (2008). “GSHMC: An efficient method for molecular simulations”, J. Comput. Phys. 227: 4934-4954.


GSHMC in NPT

MotivationFor some applications, like unfolding of peptides or crystallization process forpolymorphism in drugs, keeping constant pressure during the simulation isneeded

ObjectiveTo enable GSHMC simulations in NPT ensembles, GSHMC has beencombined with Andersen barostat

To implement NPT-GSHMC in the molecular software packageGROMACS

Akhmatskaya, E., Reich, S. (2008). “GSHMC: An efficient method for molecular simulations”, J. Comput. Phys. 227: 4934-4954.


NPT GSHMC

Main IdeaFor the momenta and positions (including the extended variable q), atime-reversible and symplectic method, the modified Velocity Verletintegrator, has been derived:

δ+t

{12

[(qn)2/3 + (qn−1)2/3]Mδ−t dn}

= −∇dV((qn)1/3dn)

Hairer, E., Lubich, C., Wanner, G. (2002). “Geometric Numerical Integration“, Springer-Verlag, Berlin, Heidelberg.

The new shadow Hamiltonian has been formulated:

H[4]∆t = H+

∆t2

24{2µQQ(3) − µQ2 + 2Q2/3D · [MD(3)]− Q2/3D · [MD]}

+∆t2

12

{(4Q

3Q1/3 −4Q2

9Q4/3

)D · [MD]− 2

3Q1/3 QD · [MD]

}


NPT GSHMC

Main IdeaFor the momenta and positions (including the extended variable q), atime-reversible and symplectic method, the modified Velocity Verletintegrator, has been derived:

δ+t

{12

[(qn)2/3 + (qn−1)2/3]Mδ−t dn}

= −∇dV((qn)1/3dn)

Hairer, E., Lubich, C., Wanner, G. (2002). “Geometric Numerical Integration“, Springer-Verlag, Berlin, Heidelberg.

The new shadow Hamiltonian has been formulated:

H[4]∆t = H+

∆t2

24{2µQQ(3) − µQ2 + 2Q2/3D · [MD(3)]− Q2/3D · [MD]}

+∆t2

12

{(4Q

3Q1/3 −4Q2

9Q4/3

)D · [MD]− 2

3Q1/3 QD · [MD]

}


GROMACS

GROMACSOpen-source software for molecular dynamics [14,000+ citations since1995]

Written in C

Supports all important algorithms expected from a modern moleculardynamics implementation

MPI, multithredding, GPU acceleration

GSHMC has been implemented in GROMACS by Bruno Escribano andit is not a part of the released version of GROMACS

The released version of GROMACS does not contain the Andersenbarostat algorithm

NPT-GSHMC has been implemented in the GROMACS package


GROMACS

GROMACSOpen-source software for molecular dynamics [14,000+ citations since1995]

Written in C

Supports all important algorithms expected from a modern moleculardynamics implementation

MPI, multithredding, GPU acceleration

GSHMC has been implemented in GROMACS by Bruno Escribano andit is not a part of the released version of GROMACS

The released version of GROMACS does not contain the Andersenbarostat algorithm

NPT-GSHMC has been implemented in the GROMACS package


Results

Spider toxin in membrane/water environmentCoarse grained system with 7810 elements

Figure: Toxin movement towards water/membrane interface


Results

0 500 1000Time (ps)

-200

-100

0

100

(bar

)

NPT-GSHMC average = 9.35NPT-GSHMCMD + Andersen average = 9.79MD + AndersenNVT-GSHMCNVT-GSHMC average = -140.186

Pressure10 bar

(E. Akhmatskaya, B. Escribano, M. F.-P., unpublished, 2013)


Results

0 500 1000 1500 2000Time (ps)

290

295

300

305

310

315(K

)

NPT-GSHMCNVT-GSHMCNPT-GSHMC averageNVT-GSHMC average

Temperature



Results

0 5000 10000 15000 20000Time (ps)

0

1

2

3

4

Dis

tanc

e (n

m)

NPT-GSHMCNVT-GSHMC

Distance



Conclusions

GSHMC is running in NPT ensembles, that is constant pressure andconstant temperature

Reference pressure can be achieved without losing efficiency

NPT-GSHMC has comparable sampling efficiency to NVT-GSHMC

Energies are comparable between NPT-GSHMC and NVT-GSHMC


Documents

Enabling constant pressure hybrid Monte Carlo simulations ... · PDF fileEnabling constant pressure hybrid Monte Carlo simulations using the GROMACS molecular simulation package Mario