Principal component analysis of molecular dynamics simulations is a popular method to account for the essential dynamics of the system on a low-dimensional free energy landscape. Using Cartesian coordinates, first the translation and overall rotation need to be removed from the trajectory. Since the rotation depends via the moment of inertia on the molecule's structure, this separation is only straightforward for relatively rigid systems. Adopting millisecond molecular dynamics simulations of the folding of villin headpiece and the functional dynamics of BPTI provided by D. E. Shaw Research, it is demonstrated via a comparison of local and global rotational fitting that the structural dynamics of flexible molecules necessarily results in a mixing of overall and internal motion. Even for the small-amplitude functional motion of BPTI, the conformational distribution obtained from a Cartesian principal component analysis therefore reflects to some extend the dominant overall motion rather than the much smaller internal motion of the protein. Internal coordinates such as backbone dihedral angles, on the other hand, are found to yield correct and well-resolved energy landscapes for both examples. The virtues and shortcomings of the choice of various fitting schemes and coordinate sets as well as the generality of these results are discussed in some detail.

1.
F.
Rao
and
A.
Caflisch
,
J. Mol. Biol.
342
,
299
(
2004
).
2.
N.-V.
Buchete
and
G.
Hummer
,
J. Phys. Chem. B
112
,
6057
(
2008
).
3.
F.
Noe
,
C.
Schütte
,
E.
Vanden-Eijnden
,
L.
Reich
, and
T.
Weikl
,
Proc. Natl. Acad. Sci. U.S.A.
106
,
19011
(
2009
).
4.
G. R.
Bowman
,
K. A.
Beauchamp
,
G.
Boxer
, and
V. S.
Pande
,
J. Chem. Phys.
131
,
124101
(
2009
).
5.
J.-H.
Prinz
 et al,
J. Chem. Phys.
134
,
174105
(
2011
).
6.
G.
Hummer
,
New J. Phys.
7
,
34
(
2005
).
7.
O. F.
Lange
and
H.
Grubmüller
,
J. Chem. Phys.
124
,
214903
(
2006
).
8.
C.
Micheletti
,
G.
Bussi
, and
A.
Laio
,
J. Chem. Phys.
129
,
074105
(
2008
).
9.
R.
Hegger
and
G.
Stock
,
J. Chem. Phys.
130
,
034106
(
2009
).
10.
M. A.
Rohrdanz
,
W.
Zheng
, and
C.
Clementi
,
Annu. Rev. Phys. Chem.
64
,
295
(
2013
).
11.
A.
Amadei
,
A. B. M.
Linssen
, and
H. J. C.
Berendsen
,
Proteins
17
,
412
(
1993
).
12.
P.
Das
,
M.
Moll
,
H.
Stamati
,
L. E.
Kavraki
, and
C.
Clementi
,
Proc. Natl. Acad. Sci. U.S.A.
103
,
9885
(
2006
).
13.
O. F.
Lange
and
H.
Grubmüller
,
Proteins
62
,
1053
(
2006
).
14.
S. V.
Krivov
and
M.
Karplus
,
Proc. Natl. Acad. Sci. U.S.A.
105
,
13841
(
2008
).
15.
J. S.
Hub
and
B. L.
de Groot
,
PLoS Comput. Biol.
5
,
e1000480
(
2009
).
16.
S. V.
Krivov
,
J. Chem. Theory Comput.
9
,
135
(
2013
).
17.
G.
Perez-Hernandez
,
F.
Paul
,
T.
Giorgino
,
G.
De Fabritiis
, and
F.
Noe
,
J. Chem. Phys.
139
, (
2013
).
18.
I. T.
Jolliffe
,
Principal Component Analysis
(
Springer
,
New York
,
2002
).
19.
T.
Ichiye
and
M.
Karplus
,
Proteins
11
,
205
(
1991
).
20.
A. E.
Garcia
,
Phys. Rev. Lett.
68
,
2696
(
1992
).
21.
A.
Kitao
and
N.
,
Curr. Opin. Struct. Biol.
9
,
164
(
1999
).
22.
B. L.
de Groot
,
X.
Daura
,
A. E.
Mark
, and
H.
Grubmüller
,
J. Mol. Biol.
309
,
299
(
2001
).
23.
Y.
Mu
,
P. H.
Nguyen
, and
G.
Stock
,
Proteins
58
,
45
(
2005
).
24.
G. G.
Maisuradze
,
A.
Liwo
, and
H. A.
Scheraga
,
Phys. Rev. Lett.
102
,
238102
(
2009
).
25.
J. N.
Onuchic
,
Z. L.
Schulten
, and
P. G.
Wolynes
,
Annu. Rev. Phys. Chem.
48
,
545
(
1997
).
26.
K. A.
Dill
and
H. S.
Chan
,
Nat. Struct. Biol.
4
,
10
(
1997
).
27.
D. J.
Wales
,
Energy Landscapes
(
Cambridge University Press
,
Cambridge
,
2003
).
28.
C.
Eckart
,
Phys. Rev.
47
,
550
(
1935
).
29.
E. B.
Wilson
,
J. J. C.
Decius
, and
P. C.
Cross
,
Molecular Vibrations
(
McGraw-Hill
,
New York
,
1955
).
30.
A. D.
McLachlan
,
Acta Cryst. A
28
,
656
(
1972
).
31.
J. D.
Louck
and
H. W.
Galbraith
,
Rev. Mod. Phys.
48
,
69
(
1976
).
32.
L. R.
Allen
,
S. V.
Krivov
, and
E.
Paci
,
PLoS Comput. Biol.
5
,
e1000428
(
2009
).
33.
N.
Hori
,
G.
Chikenji
,
R. S.
Berry
, and
S.
Takada
,
Proc. Natl. Acad. Sci. U.S.A.
106
,
73
(
2009
).
34.
I. V.
Kalgin
,
A.
Caflisch
,
S. F.
Chekmarev
, and
M.
Karplus
,
J. Phys. Chem. B
117
,
6092
(
2013
).
35.
D. M. D.
van Aalten
,
B. L.
de Groot
,
J. B. C.
Finday
,
H. J. C.
Berendsen
, and
A.
Amadei
,
J. Comput. Chem.
18
,
169
(
1997
).
36.
N.
Elmaci
and
R. S.
Berry
,
J. Chem. Phys.
110
,
10606
(
1999
).
37.
A.
Altis
,
P. H.
Nguyen
,
R.
Hegger
, and
G.
Stock
,
J. Chem. Phys.
126
,
244111
(
2007
).
38.
A.
Altis
,
M.
Otten
,
P. H.
Nguyen
,
R.
Hegger
, and
G.
Stock
,
J. Chem. Phys.
128
,
245102
(
2008
).
39.
L.
Riccardi
,
P. H.
Nguyen
, and
G.
Stock
,
J. Phys. Chem. B
113
,
16660
(
2009
).
40.
A.
Jain
,
R.
Hegger
, and
G.
Stock
,
J. Phys. Chem. Lett.
1
,
2769
(
2010
).
41.
S.
Omori
,
S.
Fuchigami
,
M.
Ikeguchi
, and
A.
Kidera
,
J. Chem. Phys.
132
,
115103
(
2010
).
42.
S.
Piana
,
K.
Lindorff-Larsen
, and
D. E.
Shaw
,
Proc. Natl. Acad. Sci. U.S.A.
109
,
17845
(
2012
).
43.
D. E.
Shaw
 et al,
Science
330
,
341
(
2010
).
44.
K. N.
Kudin
and
A. Y.
Dymarsky
,
J. Chem. Phys.
122
,
224105
(
2002
).
45.
Y.
Zhou
,
M.
Cook
, and
M.
Karplus
,
Biophys. J.
79
,
2902
(
2000
).
46.
C.
Frohlich
,
Sci. Am.
242
,
155
(
1980
).
47.
W.
Kabsch
,
Acta Cryst. A
32
,
922
(
1976
).
48.
P. H.
Hünenberger
,
A. E.
Mark
, and
W. F.
van Gunsteren
,
J. Mol. Biol.
252
,
492
(
1995
).
49.
R.
Abseher
and
M.
Nilges
,
J. Mol. Biol.
279
,
911
(
1998
).
50.
A.
Amadei
,
G.
Chillemi
,
M. A.
Ceruso
,
A.
Grottesi
, and
A.
Di Nola
,
J. Chem. Phys.
112
,
9
(
2000
).
51.
E. A.
Coutsias
,
C.
Seok
,
M. P.
Jacobson
, and
K. A.
Dill
,
J. Comput. Chem.
25
,
510
(
2004
).
52.
D. L.
Theobald
and
D. S.
Wuttke
,
Proc. Natl. Acad. Sci. U.S.A.
103
,
18521
(
2006
).
53.
S.
Fuchigami
,
S.
Omori
,
M.
Ikeguchi
, and
A.
Kidera
,
J. Chem. Phys.
132
,
104109
(
2010
).
54.
Reference 55 includes numerous references on the superpositioning problem.
55.
V.
Gapsys
and
B. L.
de Groot
,
Biophys. J.
104
,
196
(
2013
).
56.
C.
Micheletti
,
Phys. Life Rev.
10
,
1
(
2013
).
57.
J.
Jellinek
and
D. H.
Li
,
Phys. Rev. Lett.
62
,
241
(
1989
).
58.
J. J.
Prompers
and
R.
Brüschweiler
,
Proteins
46
,
177
(
2002
).
59.
E.
Johnson
,
Proteins
80
,
2645
(
2012
).
60.
V.
Hornak
 et al,
Proteins
65
,
712
(
2006
).
61.
R. B.
Best
and
G.
Hummer
,
J. Phys. Chem. B
113
,
9004
(
2009
).
62.
K.
Lindorff-Larsen
 et al,
Proteins
78
,
1950
(
2010
).
63.
W. L.
Jorgensen
,
J.
Chandrasekhar
,
J. D.
Madura
,
R. W.
Impey
, and
M.
Klein
,
J. Chem. Phys.
79
,
926
(
1983
).
64.
H. W.
Horn
 et al,
J. Chem. Phys.
120
,
9665
(
2004
).
65.
S.
Pronk
 et al,
Bioinformatics
29
,
845
(
2013
).
66.
C. J.
McKnight
,
P. T.
Matsudaira
, and
P. S.
Kim
,
Nat. Struct. Biol.
4
,
180
(
1997
).
67.
A.
Jain
and
G.
Stock
,
J. Phys. Chem. B
(published online).
68.
A.
Wlodawer
,
J.
Walter
,
R.
Huber
, and
L.
Sjölin
,
J. Mol. Biol.
180
,
301
(
1984
).
69.
P. H.
Nguyen
and
G.
Stock
,
J. Chem. Phys.
119
,
11350
(
2003
).
70.
Y.
Mu
,
P. H.
Nguyen
, and
G.
Stock
,
Proteins
64
,
798
(
2006
).
71.
G. G.
Maisuradze
and
D. M.
Leitner
,
Proteins
67
,
569
(
2007
).
72.
A.
Jain
and
G.
Stock
,
J. Chem. Theory Comput.
8
,
3810
(
2012
).
73.
J. A.
Hartigan
and
M. A.
Wong
,
Appl. Stat.
28
,
100
(
1979
).
74.
S. V.
Krivov
and
M.
Karplus
,
Proc. Natl. Acad. Sci. U.S.A.
101
,
14766
(
2004
).
75.
M. J.
Grey
,
C.
Wang
, and
A. G.
Palmer
 III
,
J. Am. Chem. Soc.
125
,
14324
(
2003
).
76.
See the supplementary material at http://dx.doi.org/10.1063/1.4885338 for details on various fitting procedures for HP35 (Fig. S1), several PCAs for BPTI (Figs. S2-S6), and the clustering of BPTI (Figs. S7 and S8).

Supplementary Material

You do not currently have access to this content.