A new theory and an exact computer algorithm for calculating kinetics and thermodynamic properties of a particle system are described. The algorithm avoids trapping in metastable states, which are typical challenges for Molecular Dynamics (MD) simulations on rough energy landscapes. It is based on the division of the full space into Voronoi cells. Prior knowledge or coarse sampling of space points provides the centers of the Voronoi cells. Short time trajectories are computed between the boundaries of the cells that we call milestones and are used to determine fluxes at the milestones. The flux function, an essential component of the new theory, provides a complete description of the statistical mechanics of the system at the resolution of the milestones. We illustrate the accuracy and efficiency of the exact Milestoning approach by comparing numerical results obtained on a model system using exact Milestoning with the results of long trajectories and with a solution of the corresponding Fokker-Planck equation. The theory uses an equation that resembles the approximate Milestoning method that was introduced in 2004 [A. K. Faradjian and R. Elber, J. Chem. Phys. 120(23), 10880-10889 (2004)]. However, the current formulation is exact and is still significantly more efficient than straightforward MD simulations on the system studied.

1.
D. E.
Shaw
,
M. M.
Deneroff
,
R. O.
Dror
,
J. S.
Kuskin
,
R. H.
Larson
,
J. K.
Salmon
,
C.
Young
,
B.
Batson
,
K. J.
Bowers
,
J. C.
Chao
,
M. P.
Eastwood
,
J.
Gagliardo
,
J. P.
Grossman
,
C. R.
Ho
,
D. J.
Ierardi
,
I.
Kolossvary
,
J. L.
Klepeis
,
T.
Layman
,
C.
McLeavey
,
M. A.
Moraes
,
R.
Mueller
,
E. C.
Priest
,
Y. B.
Shan
,
J.
Spengler
,
M.
Theobald
,
B.
Towles
, and
S. C.
Wang
,
Commun. ACM
51
(
7
),
91
-
97
(
2008
).
2.
R. H.
Austin
,
K. W.
Beeson
,
L.
Eisenstein
,
H.
Frauenfelder
, and
I. C.
Gunsalus
,
Biochem.
14
(
24
),
5355
-
5373
(
1975
).
3.
R.
Olender
and
R.
Elber
,
J. Chem. Phys.
105
(
20
),
9299
-
9315
(
1996
).
4.
P. G.
Bolhuis
,
D.
Chandler
,
C.
Dellago
, and
P. L.
Geissler
,
Annu. Rev. Phys. Chem.
53
,
291
-
318
(
2002
).
5.
D. W. H.
Swenson
and
P. G.
Bolhuis
,
J. Chem. Phys.
141
(
4
),
044101
(
2014
).
6.
R.
Elber
and
M.
Karplus
,
Science
235
(
4786
),
318
-
321
(
1987
).
7.
D.
Moroni
,
P. G.
Bolhuis
, and
T. S.
van Erp
,
J. Chem. Phys.
120
(
9
),
4055
-
4065
(
2004
).
8.
B. W.
Zhang
,
D.
Jasnow
, and
D. M.
Zuckerman
,
J. Chem. Phys.
132
(
5
),
054107
(
2010
).
9.
A. K.
Faradjian
and
R.
Elber
,
J. Chem. Phys.
120
(
23
),
10880
-
10889
(
2004
).
10.
R.
Czerminski
and
R.
Elber
,
J. Chem. Phys.
92
(
9
),
5580
-
5601
(
1990
).
11.
K. A.
Fichthorn
and
Y. Z.
Lin
,
J. Chem. Phys.
138
(
16
),
164104
(
2013
).
12.
S.
Kirmizialtin
and
R.
Elber
,
J. Phys. Chem. A
115
(
23
),
6137
-
6148
(
2011
).
13.
S. M.
Kreuzer
,
R.
Elber
, and
T. J.
Moon
,
J. Phys. Chem. B
116
(
28
),
8662
-
8691
(
2012
).
14.
S.
Kirmizialtin
,
V.
Nguyen
,
K. A.
Johnson
, and
R.
Elber
,
Structure
20
(
4
),
618
-
627
(
2012
).
15.
E.
Vanden-Eijnden
and
M.
Venturoli
,
J. Chem. Phys.
130
(
19
),
194101
(
2009
).
16.
P.
Majek
and
R.
Elber
,
J. Chem. Theory Comput.
6
(
6
),
1805
-
1817
(
2010
).
17.
A.
Dickson
,
A.
Warmflash
, and
A. R.
Dinner
,
J. Chem. Phys.
130
(
7
),
074104
(
2009
).
18.
A.
Warmflash
,
P.
Bhimalapuram
, and
A. R.
Dinner
,
J. Chem. Phys.
127
(
15
),
154112
(
2007
).
19.
E.
Vanden-Eijnden
and
M.
Venturoli
,
J. Chem. Phys.
131
(
4
),
044120
(
2009
).
20.
R.
Kubo
,
M.
Toda
, and
N.
Hashitsume
,
Statistical Physics II: Nonequilibrium Statistical Mechanics
(
Springer Verlag
,
Berlin
,
1978
).
21.
R.
Czerminski
and
R.
Elber
,
Int. J. Quantum Chem.
38
,
167
-
186
(
1990
).
22.
R.
Olender
and
R.
Elber
,
J. Mol. Struct.: THEOCHEM
398
,
63
-
71
(
1997
).
24.
R.
Elber
and
A.
West
,
Proc. Natl. Acad. Sci. U. S. A.
107
,
5001
-
5005
(
2010
).
25.
S.
Kreuzer
and
R.
Elber
,
Biophys. J.
105
(
4
),
951
-
961
(
2013
).
26.
G. S.
Jas
,
W. A.
Hegefeld
,
P.
Majek
,
K.
Kuczera
, and
R.
Elber
,
J. Phys. Chem. B
116
(
23
),
6598
-
6610
(
2012
).
27.
A. E.
Cardenas
and
R.
Elber
,
J. Chem. Phys.
141
(
5
),
054101
(
2014
).
28.
A.
Cardenas
and
R.
Elber
, Mol. Phys. (to be published).
29.
S. M.
Kreuzer
,
T. J.
Moon
, and
R.
Elber
,
J. Chem. Phys.
139
(
12
),
121902
(
2013
).
30.
A. E.
Cardenas
and
R.
Elber
,
Mol. Phys.
111
(
22-23
),
3565
-
3578
(
2013
).
31.
G. H.
Golub
and
C.
Van Loan
,
Matrix Computation
, 4th ed. (
Johns Hopkins University Press
,
Baltimore
,
2012
).
32.
A. T.
Hawk
and
D. E.
Makarov
,
J. Chem. Phys.
135
(
22
),
224109
(
2011
).
33.
S.
Viswanath
,
S. M.
Kreuzer
,
A. E.
Cardenas
, and
R.
Elber
,
J. Chem. Phys.
139
(
17
),
174105
(
2013
).
34.
A. M. A.
West
,
R.
Elber
, and
D.
Shalloway
,
J. Chem. Phys.
126
(
14
),
145104
(
2007
).
35.
A. T.
Hawk
,
J. Chem. Phys.
138
(
15
),
154105
(
2013
).
36.
F.
Nuske
,
B. G.
Keller
,
G.
Perez-Hernandez
,
A.
Mey
, and
F.
Noe
,
J. Chem. Theory Comput.
10
(
4
),
1739
-
1752
(
2014
).
37.
E.
Vanden Eijnden
,
M.
Venturoli
,
G.
Ciccotti
, and
R.
Elber
,
J. Chem. Phys.
129
(
17
),
174102
(
2008
).
38.
P.
Reimann
,
G. J.
Schmid
, and
P.
Hanggi
,
Phys. Rev. E
60
(
1
),
R1
-
R4
(
1999
).
39.
B.
Leimkuhler
and
C.
Matthews
,
J. Chem. Phys.
138
(
17
),
174102
(
2013
).
40.
R. J.
Allen
,
D.
Frenkel
, and
P. R.
ten Wolde
,
J. Chem. Phys.
124
(
2
),
024102
(
2006
).
41.
C.
Schutte
,
F.
Noe
,
J. F.
Lu
,
M.
Sarich
, and
E.
Vanden-Eijnden
,
J. Chem. Phys.
134
(
20
),
204105
(
2011
).
42.
D.
Shalloway
and
A. K.
Faradjian
,
J. Chem. Phys.
124
(
5
),
054112
(
2006
).
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