It has recently been suggested by Mu et al [Proteins58, 45 (2005)] to use backbone dihedral angles instead of Cartesian coordinates in a principal component analysis of molecular dynamics simulations. Dihedral angles may be advantageous because internal coordinates naturally provide a correct separation of internal and overall motion, which was found to be essential for the construction and interpretation of the free energy landscape of a biomolecule undergoing large structural rearrangements. To account for the circular statistics of angular variables, a transformation from the space of dihedral angles {φn} to the metric coordinate space {xn=cosφn,yn=sinφn} was employed. To study the validity and the applicability of the approach, in this work the theoretical foundations underlying the dihedral angle principal component analysis (dPCA) are discussed. It is shown that the dPCA amounts to a one-to-one representation of the original angle distribution and that its principal components can readily be characterized by the corresponding conformational changes of the peptide. Furthermore, a complex version of the dPCA is introduced, in which N angular variables naturally lead to N eigenvalues and eigenvectors. Applying the methodology to the construction of the free energy landscape of decaalanine from a 300ns molecular dynamics simulation, a critical comparison of the various methods is given.

1.
W. F.
van Gunsteren
,
D.
Bakowies
,
R.
Baron
 et al,
Angew. Chem., Int. Ed.
45
,
4064
(
2007
).
2.
J. N.
Onuchic
,
Z. L.
Schulten
, and
P. G.
Wolynes
,
Annu. Rev. Phys. Chem.
48
,
545
(
1997
).
3.
K. A.
Dill
and
H. S.
Chan
,
Nat. Struct. Biol.
4
,
10
(
1997
).
4.
D. J.
Wales
,
Energy Landscapes
(
Cambridge University Press
,
Cambridge
,
2003
).
5.
I. T.
Jolliffe
,
Principal Component Analysis
(
Springer
,
New York
,
2002
).
6.
T.
Ichiye
and
M.
Karplus
,
Proteins
11
,
205
(
1991
).
7.
8.
A.
Amadei
,
A. B. M.
Linssen
, and
H. J. C.
Berendsen
,
Proteins
17
,
412
(
1993
).
9.
S.
Hayward
,
A.
Kitao
,
F.
Hirata
, and
N.
Go
,
J. Mol. Biol.
234
,
1207
(
1993
).
11.
O. F.
Lange
and
H.
Grubmüller
,
J. Phys. Chem. B
110
,
22842
(
2006
).
12.
F.
Noe
,
D.
Krachtus
,
J. C.
Smith
, and
S.
Fischer
,
J. Chem. Theory Comput.
2
,
840
(
2006
).
13.
R.
Abseher
and
M.
Nilges
,
J. Mol. Biol.
279
,
911
(
1998
).
14.
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
).
15.
N.
Elmaci
and
R. S.
Berry
,
J. Chem. Phys.
110
,
10606
(
1999
).
16.
T. H.
Reijmers
,
R.
Wehrens
, and
L. M. C.
Buydens
,
Chemom. Intell. Lab. Syst.
56
,
61
(
2001
).
17.
Y.
Mu
,
P. H.
Nguyen
, and
G.
Stock
,
Proteins
58
,
45
(
2005
).
18.
G. E.
Sims
,
I.-G.
Choi
, and
S.-H.
Kim
,
Proc. Natl. Acad. Sci. U.S.A.
102
,
618
(
2005
).
19.
J.
Wang
and
R.
Brüschweiler
,
J. Chem. Theory Comput.
2
,
18
(
2006
).
20.
B.
Alakent
,
P.
Doruker
, and
M. C.
Camurdan
,
J. Chem. Phys.
121
,
4756
(
2004
).
21.
V.
Schultheis
,
T.
Hirschberger
,
H.
Carstens
, and
P.
Tavan
,
J. Chem. Theory Comput.
1
,
515
(
2005
).
22.
A.
Ma
and
A. R.
Dinner
,
J. Phys. Chem. B
109
,
6769
(
2005
).
23.
E.
Meerbach
,
E.
Dittmer
,
I.
Horenko
, and
C.
Schütte
,
Lect. Notes Phys.
703
,
475
(
2006
).
24.
J. D.
Chodera
,
W. C.
Swope
,
J. W.
Pitera
, and
K. A.
Dill
,
Multiscale Model. Simul.
5
,
1214
(
2006
).
25.
P.
Das
,
M.
Moll
,
H.
Stamati
,
L. E.
Kavraki
, and
C.
Clementi
,
Proc. Natl. Acad. Sci. U.S.A.
103
,
9885
(
2006
).
26.
O. F.
Lange
and
H.
Grubmüller
,
Proteins
62
,
1052
(
2006
).
27.
P. H.
Nguyen
,
Proteins
65
,
898
(
2006
).
28.
R.
Hegger
,
A.
Altis
,
P. H.
Nguyen
, and
G.
Stock
,
Phys. Rev. Lett.
98
,
028102
(
2007
).
29.
K.
Hinsen
,
Proteins
64
,
795
(
2006
).
30.
Y.
Mu
,
P. H.
Nguyen
, and
G.
Stock
,
Proteins
64
,
798
(
2006
).
31.
N. I.
Fisher
,
Statistical Analysis of Circular Data
(
Cambridge University Press
,
Cambridge
,
1996
).
32.
S.
Woutersen
and
P.
Hamm
,
J. Phys. Chem. B
104
,
11316
(
2000
).
33.
S.
Woutersen
,
R.
Pfister
,
P.
Hamm
,
Y.
Mu
,
D.
Kosov
, and
G.
Stock
,
J. Chem. Phys.
117
,
6833
(
2002
).
34.
R.
Schweitzer-Stenner
,
F.
Eker
,
Q.
Huang
, and
K.
Griebenow
,
J. Am. Chem. Soc.
123
,
9628
(
2001
).
35.
J.
Graf
,
P. H.
Nguyen
,
G.
Stock
, and
H.
Schwalbe
,
J. Am. Chem. Soc.
129
,
1179
(
2007
).
36.
Y.
Mu
and
G.
Stock
,
J. Phys. Chem. B
106
,
5294
(
2002
).
37.
Y.
Mu
,
D. S.
Kosov
, and
G.
Stock
,
J. Phys. Chem. B
107
,
5064
(
2003
).
38.
S.
Gnanakaran
and
A. E.
Garcia
,
J. Phys. Chem. B
107
,
12555
(
2003
).
39.
H. J. C.
Berendsen
,
D.
van der Spoel
, and
R.
van Drunen
,
Comput. Phys. Commun.
91
,
43
(
1995
).
40.
D.
van der Spoel
,
E.
Lindahl
,
B.
Hess
,
G.
Groenhof
,
A. E.
Mark
, and
H. J. C.
Berendsen
,
J. Comput. Chem.
26
,
1701
(
2005
).
41.
W. F.
van Gunsteren
,
S. R.
Billeter
,
A. A.
Eising
,
P. H.
Hünenberger
,
P.
Krüger
,
A. E.
Mark
,
W. R. P.
Scott
, and
I. G.
Tironi
,
Biomolecular Simulation: The GROMOS96 Manual and User Guide
(
Vdf Hochschulverlag AG an der ETH Zürich
,
Zürich
,
1996
).
42.
H. J. C.
Berendsen
,
J. P. M.
Postma
,
W. F.
van Gunsteren
, and
J.
Hermans
, in
Intermolecular Forces
, edited by
B.
Pullman
(
Reidel
,
Dordrecht
,
1981
), pp.
331
342
.
43.
T.
Darden
,
D.
York
, and
L.
Petersen
,
J. Chem. Phys.
98
,
10089
(
1993
).
44.

A direct back-calculation of the dihedral angles is not possible. But since the time indices of the original trajectory and the principal components are identical, we can use these indices to identify corresponding dihedral angles.

45.

Details of the identification of the metastable conformational states and their transition matrices are given in Ref. 17.

46.

As the complete analysis is performed in the space of dihedral angle principal components, there is no need to invoke the Jacobian transformation between these coordinates and the atomic Cartesian coordinates (Ref. 50).

47.
O. F.
Lange
and
H.
Grubmüller
,
J. Chem. Phys.
124
,
214903
(
2006
).
48.
S.
Yang
,
J. N.
Onuchic
, and
H.
Levine
,
J. Chem. Phys.
125
,
054910
(
2006
).
49.
S. V.
Krivov
and
M.
Karplus
,
Proc. Natl. Acad. Sci. U.S.A.
101
,
14766
(
2004
).
50.
S.
He
and
H. A.
Scheraga
,
J. Chem. Phys.
108
,
271
(
1998
).
You do not currently have access to this content.