We present a multiscale method for the determination of collective reaction coordinates for macromolecular dynamics based on two recently developed mathematical techniques: diffusion map and the determination of local intrinsic dimensionality of large datasets. Our method accounts for the local variation of molecular configuration space, and the resulting global coordinates are correlated with the time scales of the molecular motion. To illustrate the approach, we present results for two model systems: all-atom alanine dipeptide and coarse-grained src homology 3 protein domain. We provide clear physical interpretation for the emerging coordinates and use them to calculate transition rates. The technique is general enough to be applied to any system for which a Boltzmann-sampled set of molecular configurations is available.

1.
B.
Qi
,
S.
Muff
,
A.
Caflisch
, and
A. R.
Dinner
,
J. Phys. Chem. B
114
,
6979
(
2010
).
2.
R. B.
Best
and
G.
Hummer
,
Proc. Natl. Acad. Sci. U.S.A.
107
,
1088
(
2010
).
3.
S.
Yang
,
J. N.
Onuchic
, and
H.
Levine
,
J. Chem. Phys.
125
,
054910
(
2006
).
4.
A.
Berezhkovskii
and
A.
Szabo
,
J. Chem. Phys.
122
,
014503
(
2005
).
5.
R.
Du
,
V. S.
Pande
,
A. Y.
Grosberg
,
T.
Tanaka
, and
E. S.
Shakhnovich
,
J. Chem. Phys.
108
,
334
(
1998
).
6.
A.
Ma
and
A. R.
Dinner
,
J. Phys. Chem. B
109
,
6769
(
2005
).
7.
R. B.
Best
and
G.
Hummer
,
Proc. Natl. Acad. Sci. U.S.A.
102
,
6732
(
2005
).
8.
W.
E
,
W.
Ren
and
E.
Vanden-Eijnden
,
J. Chem. Phys.
126
,
164103
(
2007
).
9.
L.
Maragliano
and
E.
Vanden-Eijnden
,
Chem. Phys. Lett.
446
,
182
(
2007
).
10.
C.
Dellago
,
P. G.
Bolhuis
,
F. S.
Csajka
, and
D.
Chandler
,
J. Chem. Phys.
108
,
1964
(
1998
).
11.
A. K.
Faradjian
and
R.
Elber
,
J. Chem. Phys.
120
,
10880
(
2004
).
12.
I. T.
Jolliffe
,
Principal Components Analysis
(
Springer
,
Berlin
,
1986
).
13.
P. H.
Nguyen
,
Proteins
65
,
898
(
2006
).
14.
Y.
Mu
,
P. H.
Nguyen
, and
G.
Stock
,
Proteins
58
,
45
(
2005
).
15.
S. T.
Roweis
and
L. K.
Saul
,
Science
290
,
2323
(
2000
).
16.
J. B.
Tenenbaum
,
V.
de Silva
, and
J. C.
Langford
,
Science
290
,
2319
(
2000
).
17.
P.
Das
,
M.
Moll
,
H.
Stamati
,
L. E.
Kavraki
, and
C.
Clementi
,
Proc. Natl. Acad. Sci. U.S.A.
103
,
9885
(
2006
).
18.
Y.
Yao
,
J.
Sun
,
X.
Huang
,
G. R.
Bowman
,
G.
Singh
,
M.
Lesnick
,
L. J.
Guibas
,
V. S.
Pande
,
G.
Carlsson
,
J. Chem. Phys.
130
,
144115
(
2009
).
19.
R. R.
Coifman
and
S.
Lafon
,
Appl. Comput. Harmon. Anal.
21
,
5
(
2006
).
20.
R. R.
Coifman
and
M.
Maggioni
,
Appl. Comput. Harmon. Anal.
21
,
53
(
2006
).
21.
R. R.
Coifman
 et al,
Proc. Natl. Acad. Sci. U.S.A.
102
,
7426
(
2005
).
22.
R. R.
Coifman
 et al,
Proc. Natl. Acad. Sci. U.S.A.
102
,
7432
(
2005
).
23.
R. R.
Coifman
,
I. G.
Kevrekidis
,
S.
Lafon
,
M.
Maggioni
, and
B.
Nadler
,
Multiscale Model. Simul.
7
,
842
(
2008
).
24.
A. D.
Szlam
,
M.
Maggioni
, and
R. R.
Coifman
,
J. Mach. Learn. Res.
9
,
1711
(
2008
).
25.
S.
Mahadevan
and
M.
Maggioni
,
J. Mach. Learn. Res.
8
,
2169
(
2007
).
26.
P. W.
Jones
,
M.
Maggioni
, and
R.
Schul
,
Proc. Nat. Acad. Sci. U.S.A.
105
,
1803
(
2008
).
27.
A.
Little
,
Y.-M.
Jung
, and
M.
Maggioni
,
Multiscale Estimation of Intrinsic Dimensionality of Data Sets
,
AAAI Fall Symposium Series
(November 5-7,
2009
,
Arlington, Virginia, USA
). Available online: http://aaai.org/ocs/index.php/FSS/FSS09/paper/view/950.
28.
The functions
$\phi _i(\protect \boldsymbol{x})/\phi _0(\protect \boldsymbol{x})$
φi(x)/φ0(x)
are eigenfunctions of the backward Fokker–Plank operator. The forward and backward Fokker–Planck operators are adjoint to one another, share the same set of eigenvalues, and their eigenfunctions differ by a factor of
$\phi _0(\protect \boldsymbol{x})$
φ0(x)
.
29.
See supplementary material at http://dx.doi.org/10.1063/1.3569857 for more details.
30.
H. A.
Kramers
,
Physica
7
,
284
(
1940
).
31.
L.
Huang
and
D. E.
Makarov
,
J. Chem. Phys.
128
,
114903
(
2008
).
32.
G.
Hummer
,
New J. Phys.
7
,
34
(
2005
).
33.
D. A.
Case
,
T. A.
Darden
,
T. E.
Cheatham
 III
,
C. L.
Simmerling
,
J.
Wang
,
R. E.
Duke
,
R.
Luo
,
K. M.
Merz
,
D. A.
Pearlman
,
M.
Crowley
,
R. C.
Walker
,
W.
Zhang
,
B.
Wang
,
S.
Hayik
,
A.
Roitberg
,
G.
Seabra
,
K. F.
Wong
,
F.
Paesani
,
X.
Wu
,
S.
Brozell
,
V.
Tsui
,
H.
Gohlke
,
L.
Yang
,
C.
Tan
,
J.
Mongan
,
V.
Hornak
,
G.
Cui
,
P.
Beroza
,
D. H.
Mathews
,
C.
Schafmeister
,
W. S.
Ross
, and
P. A.
Kollman
, AMBER9 University of California, San Francisco,
2006
.
34.
C.
Clementi
,
H.
Nymeyer
, and
J. N.
Onuchic
,
J. Mol. Biol.
298
,
937
(
2000
).
35.
C.
Clementi
and
S. S.
Plotkin
,
Protein Sci.
13
,
1750
(
2004
).
36.
L.
Li
,
L. A.
Mirny
, and
E. I.
Shakhnovich
,
Nat. Struct. Biol.
7
,
336
(
2000
).
37.
V. P.
Grantcharova
,
D. S.
Riddle
,
J. V.
Santiago
, and
D.
Baker
,
Nat. Struct. Biol.
5
,
714
(
1998
).
38.
S. S.
Cho
,
Y.
Levy
, and
P. G.
Wolynes
,
Proc. Natl. Acad. Sci. U.S.A.
103
,
586
(
2006
).
39.
E.
Plaku
,
H.
Stamati
,
C.
Clementi
, and
L. E.
Kavraki
,
Proteins
67
,
897
(
2007
).
40.
P.
Das
,
S.
Matysiak
, and
C.
Clementi
,
Proc. Natl. Acad. Sci. U.S.A.
102
,
10141
(
2005
).
41.
D. van der
Spoel
,
E.
Lindahl
,
B.
Hess
,
G.
Groenhof
,
A. E.
Mark
, and
H. J. C.
Berendsen
,
J. Comput. Chem.
26
,
1701
(
2005
).
42.
A. R.
Viguera
,
C.
Vega
, and
L.
Serrano
,
Proc. Natl. Acad. Sci. U.S.A.
99
,
5349
(
2002
).
43.
As in Hummer's work, we also perform Metropolis Monte Carlo on Pi, the probability of being in the ith cell. The elements of R and P are related through detailed balance: Ri + 1, i/Ri, i + 1 = Pi + 1/Pi. While the Bayesian analysis method provides estimates of the free-energy, we use the more finely sampled free-energy directly from our simulation for the evaluation of the Kramers' integrals.

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