The multiscale coarse-graining (MS-CG) method is a method for determining the effective potential energy function for a coarse-grained (CG) model of a system using the data obtained from molecular dynamics simulation of the corresponding atomically detailed model. The MS-CG method, as originally formulated for systems at constant volume, has previously been given a rigorous statistical mechanical basis for the canonical ensemble. Here, we propose and test a version of the MS-CG method suitable for the isothermal-isobaric ensemble. The method shows how to construct an effective potential energy function for a CG system that generates the correct volume fluctuations as well as correct distribution functions in the configuration space of the CG sites. The formulation of the method requires introduction of an explicitly volume dependent term in the potential energy function of the CG system. The theory is applicable to simulations with isotropic volume fluctuations and cases where both the atomistic and CG models do not have any intramolecular constraints, but it is straightforward to extend the theory to more general cases. The present theory deals with systems that have short ranged interactions. (The extension to Coulombic forces using Ewald methods requires additional considerations.) We test the theory for constant pressure MS-CG simulations of a simple model of a solution. We show that both the volume dependent and the coordinate dependent parts of the potential are transferable to larger systems than the one used to obtain these potentials.
REFERENCES
1.
M. P.
Allen
and D. P.
Tildesley
, Computer Simulation of Liquids
(Oxford University Press
, Oxford
, 1987
).2.
M.
Frenkel
and B.
Smit
, Understanding Molecular Simulation: From Algorithms to Applications
(Academic
, New York
, 2002
).3.
S. O.
Nielsen
, C. F.
Lopez
, G.
Srinivas
, and M. L.
Klein
, J. Phys.: Condens. Matter
16
, R481
(2004
), and references therein.4.
T.
Head-Gordon
and S.
Brown
, Curr. Opin. Struct. Biol.
13
, 160
(2003
), and references therein.5.
V.
Tozzini
, Curr. Opin. Struct. Biol.
15
, 144
(2005
), and references therein.6.
I.
Bahar
and A.
Rader
, Curr. Opin. Struct. Biol.
15
, 586
(2005
), and references therein.7.
G. S.
Ayton
, W. G.
Noid
, and G. A.
Voth
, Curr. Opin. Struct. Biol.
17
, 192
(2007
).8.
M.
Müller
, K.
Katsov
, and M.
Schick
, Phys. Rep.
434
, 113
(2006
), and references therein.9.
J.
Baschnagel
, K.
Binder
, P.
Doruker
, A. A.
Gusev
, O.
Hahn
, K.
Kremer
, W. L.
Mattice
, F.
Müller-Plathe
, M.
Murat
, W.
Paul
, S.
Santos
, U. W.
Suter
, and V.
Tries
, Advances in Polymer Science
(Springer-Verlag
, Berlin/Heidelberg
, 2000
), Vol. 152
, p. 41
, and references therein.10.
Coarse-Graining of Condensed Phase and Biomolecular Systems
, edited by G. A.
Voth
(CRC
, Boca Raton, FL
, 2009
).11.
M.
Levitt
, J. Mol. Biol.
104
, 59
(1976
).12.
N.
Gō
and H.
Abe
, Biopolymers
20
, 991
(1981
).13.
H.
Abe
and N.
Gō
, Biopolymers
20
, 1013
(1981
).14.
S.
Miyazawa
and R. L.
Jernigan
, Macromolecules
18
, 534
(1985
).15.
A. P.
Lyubartsev
and A.
Laaksonen
, Phys. Rev. E
52
, 3730
(1995
).16.
J. C.
Shelley
, M. Y.
Shelley
, R. C.
Reeder
, S.
Bandyopadhyay
, and M. L.
Klein
, J. Phys. Chem. B
105
, 4464
(2001
).17.
S. J.
Marrink
, A. H.
de Vries
, and A. E.
Mark
, J. Phys. Chem. B
108
, 750
(2004
).18.
A.
Liwo
, S.
Oldziej
, C.
Czaplewski
, U.
Kozlowska
, and H. A.
Scheraga
, J. Phys. Chem. B
108
, 9421
(2004
).19.
A.
Liwo
, C.
Czaplewski
, J.
Pillardy
, and H. A.
Scheraga
, J. Chem. Phys.
115
, 2323
(2001
).20.
I. G.
Kevrekidis
, C. W.
Gear
, and G.
Hummer
, AIChE J.
50
, 1346
(2004
).21.
N. -V.
Buchete
, J. E.
Straub
, and D.
Thirumalai
, Protein Sci.
13
, 862
(2004
).22.
D.
Curcó
, R.
Nussinov
, and C.
Aleman
, J. Phys. Chem. B
111
, 14006
(2007
).23.
M.
Lu
and J.
Ma
, Proc. Natl. Acad. Sci. U.S.A.
105
, 15358
(2008
).24.
M. A.
Jonikas
, R. J.
Radmer
, A.
Laederach
, R.
Das
, S.
Pearlman
, D.
Herschlag
, and R. B.
Altman
, RNA
15
, 189
(2009
).25.
R. B.
Pandey
and B. L.
Farmer
, J. Chem. Phys.
130
, 044906
(2009
).26.
T.
Ha-Duong
, N.
Basdevant
, and D.
Borgis
, Chem. Phys. Lett.
468
, 79
(2009
).27.
Z.
Zhang
and W.
Wriggers
, J. Phys. Chem. B
112
, 14026
(2008
).28.
A. Y.
Shih
, A.
Arkhipov
, P. L.
Freddolino
, and K.
Schulten
, J. Phys. Chem. B
110
, 3674
(2006
).29.
P. J.
Bond
, J.
Holyoake
, A.
Ivetac
, S.
Khalid
, and M. S.
Sansom
, J. Struct. Biol.
157
, 593
(2007
).30.
D.
Reith
, M.
Pütz
, and F.
Müller-Plathe
, J. Comput. Chem.
24
, 1624
(2003
).31.
L.
Monticelli
, S. K.
Kandasamy
, X.
Periole
, R. G.
Larson
, D. P.
Tieleman
, and S. -J.
Marrink
, J. Chem. Theory Comput.
4
, 819
(2008
).32.
K.
Moritsugu
and J. C.
Smith
, Biophys. J.
95
, 1639
(2008
).33.
C. F.
Abrams
, L.
Delle Site
, and K.
Kremer
, Phys. Rev. E
67
, 021807
(2003
).34.
R. E.
Rudd
and J. Q.
Broughton
, Phys. Rev. B
58
, R5893
(1998
).35.
R. L. C.
Akkermans
and W. J.
Briels
, J. Chem. Phys.
114
, 1020
(2001
).36.
H.
Fukunaga
, J.
Takimoto
, and M.
Doi
, J. Chem. Phys.
116
, 8183
(2002
).37.
S. O.
Nielsen
, C. F.
Lopez
, G.
Srinivas
, and M. L.
Klein
, J. Chem. Phys.
119
, 7043
(2003
).38.
V.
Molinero
and W. A.
Goddard
, J. Phys. Chem. B
108
, 1414
(2004
).39.
G. S.
Ayton
, H. L.
Tepper
, D. T.
Mirijanian
, and G. A.
Voth
, J. Chem. Phys.
120
, 4074
(2004
).40.
S. D.
Chao
, J. D.
Kress
, and A.
Redondo
, J. Chem. Phys.
122
, 234912
(2005
).41.
V.
Tozzini
and J. A.
McCammon
, Chem. Phys. Lett.
413
, 123
(2005
).42.
J. -W.
Chu
and G. A.
Voth
, Proc. Natl. Acad. Sci. U.S.A.
102
, 13111
(2005
).43.
P. A.
Golubkov
and P.
Ren
, J. Chem. Phys.
125
, 064103
(2006
).44.
H.
Gohlke
and M.
Thorpe
, Biophys. J.
91
, 2115
(2006
).45.
D. A.
Kondrashov
, Q.
Cui
, and G. N.
Phillips
, Biophys. J.
91
, 2760
(2006
).46.
L. -J.
Chen
, H. -J.
Qian
, Z. -Y.
Lu
, Z. -S.
Li
, and C. -C.
Sun
, J. Phys. Chem. B
110
, 24093
(2006
).47.
S.
Izvekov
and G. A.
Voth
, J. Phys. Chem. B
109
, 2469
(2005
).48.
S.
Izvekov
and G. A.
Voth
, J. Chem. Phys.
123
, 134105
(2005
).49.
Y.
Wang
, S.
Izvekov
, T.
Yan
, and G. A.
Voth
, J. Phys. Chem. B
110
, 3564
(2006
).50.
S.
Izvekov
and G. A.
Voth
, J. Chem. Theory Comput.
2
, 637
(2006
).51.
J.
Zhou
, I. F.
Thorpe
, S.
Izvekov
, and G. A.
Voth
, Biophys. J.
92
, 4289
(2007
).52.
I. F.
Thorpe
, J.
Zhou
, and G. A.
Voth
, J. Phys. Chem. B
112
, 13079
(2008
).53.
S.
Izvekov
, A.
Violi
, and G. A.
Voth
, J. Phys. Chem. B
109
, 17019
(2005
).54.
Q.
Shi
, S.
Izvekov
, and G. A.
Voth
, J. Phys. Chem. B
110
, 15045
(2006
).55.
W. G.
Noid
, J. -W.
Chu
, G. S.
Ayton
, V.
Krishna
, S.
Izvekov
, G. A.
Voth
, A.
Das
, and H. C.
Andersen
, J. Chem. Phys.
128
, 244114
(2008
).56.
W. G.
Noid
, P.
Liu
, Y.
Wang
, J. -W.
Chu
, G. S.
Ayton
, S.
Izvekov
, H. C.
Andersen
, and G. A.
Voth
, J. Chem. Phys.
128
, 244115
(2008
).57.
L.
Lu
and G. A.
Voth
, J. Phys. Chem. B
113
, 1501
(2009
).58.
A.
Das
and H. C.
Andersen
, J. Chem. Phys.
131
, 034102
(2009
).59.
60.
I. R.
McDonald
, Mol. Phys.
23
, 41
(1972
).61.
H. C.
Andersen
, J. Chem. Phys.
72
, 2384
(1980
).62.
If we had attempted to carry out this derivation using unscaled coordinates, we would not be able to get simple results like Eqs. (16) and (17) for the following reason. The mapping operator for unscaled coordinates in a periodic system with fluctuating volume depends explicitly on volume in a complicated way and, consequently, gives rise to nonzero partial derivatives with respect to volume. This would have led to much more complicated results and might have made the calculations intractable. This problem is avoided by performing the calculation using the scaled coordinates, where the position mapping operator does not have any explicit volume dependence, leading to much simpler expressions.
63.
Here we are concerned only with the case of homogeneous isotropic fluid systems. For other systems, different considerations will, of course, apply.
64.
W. H.
Press
, S. A.
Teukolsky
, W. T.
Vetterling
, and B. P.
Flannery
, Numerical Recipes in C: The Art of Scientific Computing
(Cambridge University Press
, Cambridge
, 1992
).65.
Note, however, that constant volume coarse graining procedures do not usually provide a method for obtaining the contribution to the pressure, but the method for doing so could be derived from the current work. Attempts to calculate the virial pressure of a CG system without taking this term into account cannot generally get the correct answer. For a simple example, see Appendix A for a discussion of an ideal gas mixture.
66.
J. B.
Sturgeon
and B. B.
Laird
, J. Chem. Phys.
112
, 3474
(2000
).67.
J. M.
Sanz-Serna
and M. P.
Calvo
, Numerical Hamiltonian Problems
(Chapman and Hall
, London
, 1994
).68.
E.
Hairer
, C.
Lubich
, and G.
Wanner
, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations
(Springer
, Berlin
, 2006
).© 2010 American Institute of Physics.
2010
American Institute of Physics
You do not currently have access to this content.
