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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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 2 / 13
Generalized Shadow Hybrid Monte Carlo (GSHMC)
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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 3 / 13
Generalized Shadow Hybrid Monte Carlo (GSHMC)
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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 3 / 13
Generalized Shadow Hybrid Monte Carlo (GSHMC)
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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 3 / 13
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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 4 / 13
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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 4 / 13
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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 4 / 13
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.
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 5 / 13
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, H.C. (1980). “Molecular dynamics simulations at constant pressure and/or temperature“. J. Chem. Phys. 72:2384.
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 5 / 13
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
Andersen, H.C. (1980). “Molecular dynamics simulations at constant pressure and/or temperature“. J. Chem. Phys. 72:2384.
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 5 / 13
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.
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 6 / 13
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.
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 6 / 13
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]
}
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 7 / 13
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]
}
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 7 / 13
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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 8 / 13
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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 8 / 13
Results
Spider toxin in membrane/water environmentCoarse grained system with 7810 elements
Figure: Toxin movement towards water/membrane interface
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 9 / 13
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)
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 10 / 13
Results
0 500 1000 1500 2000Time (ps)
290
295
300
305
310
315(K
)
NPT-GSHMCNVT-GSHMCNPT-GSHMC averageNVT-GSHMC average
Temperature
(E. Akhmatskaya, B. Escribano, M. F.-P., unpublished, 2013)
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 11 / 13
Results
0 5000 10000 15000 20000Time (ps)
0
1
2
3
4
Dis
tanc
e (n
m)
NPT-GSHMCNVT-GSHMC
Distance
(E. Akhmatskaya, B. Escribano, M. F.-P., unpublished, 2013)
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 12 / 13
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
Mario Fernandez Pendas (BCAM) 18 October 2013 - BCAM (Bilbao) 13 / 13