The vibrational subsystem analysis is a useful approach that allows for evaluating the spectrum of modes of a given system by integrating out the degrees of freedom accessible to the environment. The approach could be utilized for exploring the collective dynamics of a membrane protein (system) coupled to the lipid bilayer (environment). However, the application to membrane proteins is limited due to high computational costs of modeling a sufficiently large membrane environment unbiased by end effects, which drastically increases the size of the investigated system. We derived a recursive formula for calculating the reduced Hessian of a membrane protein embedded in a lipid bilayer by decomposing the membrane into concentric cylindrical domains with the protein located at the center. The approach allows for the design of a time- and memory-efficient algorithm and a mathematical understanding of the convergence of the reduced Hessian with respect to increasing membrane sizes. The application to the archaeal aspartate transporter GltPh illustrates its utility and efficiency in capturing the transporter’s elevator-like movement during its transition between outward-facing and inward-facing states.

1.
Z.
Cournia
 et al,
J. Membr. Biol.
248
,
611
(
2015
).
2.
F. J.
Alenghat
and
D. E.
Golan
,
Curr. Top. Membr.
72
,
89
(
2013
).
3.
D. P.
Ng
,
B. E.
Poulsen
, and
C. M.
Deber
,
Biochim. Biophys. Acta, Biomembr.
1818
,
1115
(
2012
).
4.
H.
Yin
and
A. D.
Flynn
,
Annu. Rev. Biomed. Eng.
18
,
51
(
2016
).
5.
F.
Cymer
and
D.
Schneider
,
Cell Adhes. Migr.
4
,
299
(
2010
).
6.
J. C.
Cheung
and
C. M.
Deber
,
Biochemistry
47
,
1465
(
2008
).
7.
A.-N.
Bondar
,
C.
del Val
, and
S. H.
White
,
Structure
17
,
395
(
2009
).
8.
T. R.
Lezon
and
I.
Bahar
,
Biophys. J.
102
,
1331
(
2012
).
9.
I. H.
Shrivastava
and
I.
Bahar
,
Biophys. J.
90
,
3929
(
2006
).
10.
B.
Isin
 et al,
Biophys. J.
95
,
789
(
2008
).
11.
J. Y.
Lee
 et al,
Biophys. J.
113
,
2634
(
2017
).
12.
H.
Li
 et al,
Nucleic Acids Res.
45
,
W374
(
2017
).
13.
X.
Wu
,
M.
Han
, and
D.
Ming
,
J. Chem. Phys.
143
,
134113
(
2015
).
14.
K.
Hinsen
 et al,
Chem. Phys.
261
,
25
(
2000
).
15.
H. L.
Woodcock
 et al,
J. Chem. Phys.
129
,
214109
(
2008
).
16.
W.
Zheng
and
B. R.
Brooks
,
Biophys. J.
89
,
167
(
2005
).
17.
I.
Bahar
 et al,
Chem. Rev.
110
,
1463
(
2010
).
18.
D.
Ming
and
M. E.
Wall
,
Phys. Rev. Lett.
95
,
198103
(
2005
).
19.
S.
Luber
and
M.
Reiher
,
ChemPhysChem
10
,
2049
(
2009
).
20.
S.
Yamamoto
 et al,
J. Chem. Theory Comput.
8
,
977
(
2012
).
21.
J.
Kessler
,
J.
Kapitán
, and
P.
Bouř
,
J. Phys. Chem. Lett.
6
,
3314
(
2015
).
22.
M.
Lu
,
B.
Poon
, and
J.
Ma
,
J. Chem. Theory Comput.
2
,
464
(
2006
).
23.
M. A.
Lomize
 et al,
Nucleic Acids Res.
40
,
D370
(
2012
).
24.
I.
Bahar
,
B.
Erman
, and
L.
Monnerie
,
Macromolecules
24
,
3618
(
1991
).
25.
P. J.
Flory
,
Statistical Mechanics of Chain Molecules
  (
Hanser & Gardner
,
Cincinnati, Ohio
,
1989
).
26.
R. A.
Horn
and
C. R.
Johnson
,
Matrix Analysis
, 2nd ed. (
Cambridge University Press
,
Cambridge; New York
,
2012
).
27.
B.
Hess
,
Phys. Rev. E
65
,
031910
(
2002
).
28.
R.
Li
,
Y.
Xi
, and
Y.
Saad
,
Numer. Linear Algebra Appl.
23
,
706
(
2016
).
29.
D.
Yernool
 et al,
Nature
431
,
811
(
2004
).
30.
O.
Boudker
 et al,
Nature
445
,
387
(
2007
).
31.
W.
Helfrich
,
Z. Naturforsch. C
28
,
693
(
1973
).
32.
L.
Lamparter
and
M.
Galic
,
Front. Cell Dev. Biol.
8
,
684
(
2020
).
33.
R.
Phillips
 et al,
Nature
459
,
379
(
2009
).
34.
A.
Bakan
,
L. M.
Meireles
, and
I.
Bahar
,
Bioinformatics
27
,
1575
(
2011
).
35.
S.
Zhang
 et al, “
ProDy 2.0: Increased scale and scope after 10 years of protein dynamics modelling with Python
,”
Bioinformatics
(to be published,
2021
).
36.
H.
Li
 et al,
Nucleic Acids Res.
44
,
D415
(
2016
).
37.
K.
Goossens
and
H.
De Winter
,
J. Chem. Inf. Model.
58
,
2193
(
2018
).
38.
D. E.
Chandler
 et al,
Biophys. J.
106
,
2503
(
2014
).
39.
A.
Truszkowski
 et al,
J. Cheminf.
6
,
45
(
2014
).
40.
N.
Leioatts
,
T. D.
Romo
, and
A.
Grossfield
,
J. Chem. Theory Comput.
8
,
2424
(
2012
).
41.
T. D.
Romo
and
A.
Grossfield
,
Proteins: Struct., Funct., Bioinf.
79
,
23
(
2011
).
42.
M.
Gur
,
E.
Zomot
, and
I.
Bahar
,
J. Chem. Phys.
139
,
121912
(
2013
).
43.
M. M.
Tirion
,
Phys. Rev. Lett.
77
,
1905
(
1996
).
44.
K.
Hinsen
,
Proteins: Struct., Funct., Bioinf.
33
,
417
(
1998
).
45.
A. R.
Atilgan
 et al,
Biophys. J.
80
,
505
(
2001
).
46.
B.
Isin
,
P.
Doruker
, and
I.
Bahar
,
Biophys. J.
82
,
569
(
2002
).
47.
F.
Tama
 et al,
Proteins: Struct., Funct., Bioinf.
41
,
1
(
2000
).
48.
P.
Doruker
and
R. L.
Jernigan
,
Proteins: Struct., Funct., Bioinf.
53
,
174
(
2003
).
49.
G.
Song
,
PLoS Comput. Biol.
16
,
e1007855
(
2020
).

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