Protein structural fluctuation, typically measured by Debye-Waller factors, or B-factors, is a manifestation of protein flexibility, which strongly correlates to protein function. The flexibility-rigidity index (FRI) is a newly proposed method for the construction of atomic rigidity functions required in the theory of continuum elasticity with atomic rigidity, which is a new multiscale formalism for describing excessively large biomolecular systems. The FRI method analyzes protein rigidity and flexibility and is capable of predicting protein B-factors without resorting to matrix diagonalization. A fundamental assumption used in the FRI is that protein structures are uniquely determined by various internal and external interactions, while the protein functions, such as stability and flexibility, are solely determined by the structure. As such, one can predict protein flexibility without resorting to the protein interaction Hamiltonian. Consequently, bypassing the matrix diagonalization, the original FRI has a computational complexity of

${\cal O}(N^2)$
O(N2)⁠. This work introduces a fast FRI (fFRI) algorithm for the flexibility analysis of large macromolecules. The proposed fFRI further reduces the computational complexity to
${\cal O}(N)$
O(N)
. Additionally, we propose anisotropic FRI (aFRI) algorithms for the analysis of protein collective dynamics. The aFRI algorithms permit adaptive Hessian matrices, from a completely global 3N × 3N matrix to completely local 3 × 3 matrices. These 3 × 3 matrices, despite being calculated locally, also contain non-local correlation information. Eigenvectors obtained from the proposed aFRI algorithms are able to demonstrate collective motions. Moreover, we investigate the performance of FRI by employing four families of radial basis correlation functions. Both parameter optimized and parameter-free FRI methods are explored. Furthermore, we compare the accuracy and efficiency of FRI with some established approaches to flexibility analysis, namely, normal mode analysis and Gaussian network model (GNM). The accuracy of the FRI method is tested using four sets of proteins, three sets of relatively small-, medium-, and large-sized structures and an extended set of 365 proteins. A fifth set of proteins is used to compare the efficiency of the FRI, fFRI, aFRI, and GNM methods. Intensive validation and comparison indicate that the FRI, particularly the fFRI, is orders of magnitude more efficient and about 10% more accurate overall than some of the most popular methods in the field. The proposed fFRI is able to predict B-factors for α-carbons of the HIV virus capsid (313 236 residues) in less than 30 seconds on a single processor using only one core. Finally, we demonstrate the application of FRI and aFRI to protein domain analysis.

1.
M. P.
Allen
and
D. J.
Tildesley
,
Computer Simulation of Liquids
(
Clarendon Press
,
Oxford
,
1987
).
2.
C. B.
Anfinsen
, “
Einfluss der configuration auf die wirkung den
,”
Science
181
,
223
230
(
1973
).
3.
A. R.
Atilgan
,
S. R.
Durrell
,
R. L.
Jernigan
,
M. C.
Demirel
,
O.
Keskin
, and
I.
Bahar
, “
Anisotropy of fluctuation dynamics of proteins with an elastic network model
,”
Biophys. J.
80
,
505
515
(
2001
).
4.
I.
Bahar
,
A. R.
Atilgan
,
M. C.
Demirel
, and
B.
Erman
, “
Vibrational dynamics of proteins: Significance of slow and fast modes in relation to function and stability
,”
Phys. Rev. Lett.
80
,
2733
2736
(
1998
).
5.
I.
Bahar
,
A. R.
Atilgan
, and
B.
Erman
, “
Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential
,”
Folding Des.
2
,
173
181
(
1997
).
6.
A.
Bakan
,
L. M.
Meireles
, and
I.
Bahar
, “
Prody: Protein dynamics inferred from theory and experiments
,”
Bioinformatics
27
,
1575
1577
(
2011
).
7.
B. R.
Brooks
,
R. E.
Bruccoleri
,
B. D.
Olafson
,
D.
States
,
S.
Swaminathan
, and
M.
Karplus
, “
Charmm: A program for macromolecular energy, minimization, and dynamics calculations
,”
J. Comput. Chem.
4
,
187
217
(
1983
).
8.
F.
Chiti
and
C. M.
Dobson
, “
Protein misfolding, functional amyloid, and human disease
,”
Annu. Rev. Biochem.
75
,
333
366
(
2006
).
9.
Q.
Cui
and
I.
Bahar
,
Normal Mode Analysis: Theory and Applications to Biological and Chemical Systems
(
Chapman and Hall/CRC
,
2010
).
10.
Q.
Cui
,
G. J.
Li
,
J.
Ma
, and
M.
Karplus
, “
A normal mode analysis of structural plasticity in the biomolecular motor f(1)-atpase
,”
J. Mol. Biol.
340
(
2
),
345
372
(
2004
).
11.
O. N. A.
Demerdash
and
J. C.
Mitchell
, “
Density-cluster NMA: A new protein decomposition technique for coarse-grained normal mode analysis
,”
Proteins
80
(
7
),
1766
1779
(
2012
).
12.
P. J.
Flory
, “
Statistical thermodynamics of random networks
,”
Proc. R. Soc. London, Ser. A
351
,
351
378
(
1976
).
13.
H.
Frauenfelder
,
S. G.
Slihar
, and
P. G.
Wolynes
, “
The energy landscapes and motion of proteins
,”
Science
254
(
5038
),
1598
1603
(
1991
).
14.
Z. N.
Gerek
and
S. B.
Ozkan
, “
A flexible docking scheme to explore the binding selectivity of PDZ domains
,”
Protein Sci.
19
,
914
928
(
2010
).
15.
N.
Go
,
T.
Noguti
, and
T.
Nishikawa
, “
Dynamics of a small globular protein in terms of low-frequency vibrational modes
,”
Proc. Natl. Acad. Sci. U.S.A.
80
,
3696
3700
(
1983
).
16.
K.
Hinsen
, “
Analysis of domain motions by approximate normal mode calculations
,”
Proteins
33
,
417
429
(
1998
).
17.
K.
Hinsen
, “
Structural flexibility in proteins: Impact of the crystal environment
,”
Bioinformatics
24
,
521
528
(
2008
).
18.
W.
Humphrey
,
A.
Dalke
, and
K.
Schulten
, “
VMD – Visual molecular dynamics
,”
J. Mol. Graphics
14
(
1
),
33
38
(
1996
).
19.
D. J.
Jacobs
,
A. J.
Rader
,
L. A.
Kuhn
, and
M. F.
Thorpe
, “
Protein flexibility predictions using graph theory
,”
Proteins: Struct., Funct., Genet.
44
(
2
),
150
165
(
2001
).
20.
O.
Keskin
,
I.
Bahar
,
D.
Flatow
,
D. G.
Covell
, and
R. L.
Jernigan
, “
Molecular mechanisms of chaperonin GroEL-GroES function
,”
Biochemistry
41
,
491
501
(
2002
).
21.
D. A.
Kondrashov
,
A. W.
Van Wynsberghe
,
R. M.
Bannen
,
Q.
Cui
, and
J. G. N.
Phillips
, “
Protein structural variation in computational models and crystallographic data
,”
Structure
15
,
169
177
(
2007
).
22.
S.
Kundu
,
J. S.
Melton
,
D. C.
Sorensen
, and
J. G. N.
Phillips
, “
Dynamics of proteins in crystals: Comparison of experiment with simple models
,”
Biophys. J.
83
,
723
732
(
2002
).
23.
M.
Levitt
,
C.
Sander
, and
P. S.
Stern
, “
Protein normal-mode dynamics: Trypsin inhibitor, crambin, ribonuclease and lysozyme
,”
J. Mol. Biol.
181
(
3
),
423
447
(
1985
).
24.
G. H.
Li
and
Q.
Cui
, “
A coarse-grained normal mode approach for macromolecules: An efficient implementation and application to Ca(2+)-ATPase
,”
Biophys. J.
83
,
2457
2474
(
2002
).
25.
D. R.
Livesay
,
S.
Dallakyan
,
G. G.
Wood
, and
D. J.
Jacobs
, “
A flexible approach for understanding protein stability
,”
FEBS Lett.
576
,
468
476
(
2004
).
26.
J. P.
Ma
, “
Usefulness and limitations of normal mode analysis in modeling dynamics of biomolecular complexes
,”
Structure
13
,
373
180
(
2005
).
27.
J. N.
Onuchic
,
Z.
Luthey-Schulten
, and
P. G.
Wolynes
, “
Theory of protein folding: The energy landscape perspective
,”
Annu. Rev. Phys. Chem.
48
,
545
600
(
1997
).
28.
J. K.
Park
,
R.
Jernigan
, and
Z.
Wu
, “
Coarse grained normal mode analysis vs. refined gaussian network model for protein residue-level structural fluctuations
,”
Bull. Math. Biol.
75
,
124
160
(
2013
).
29.
A. J.
Rader
,
D. H.
Vlad
, and
I.
Bahar
, “
Maturation dynamics of bacteriophage hk97 capsid
,”
Structure
13
,
413
421
(
2005
).
30.
R. J.
Renka
, “
Multivariate interpolation of large sets of scattered data
,”
ACM Trans. Math. Software
14
(
2
),
139
148
(
1988
).
31.
M. G.
Rossman
and
A.
Liljas
, “
Recognition of structural domains in globular proteins
,”
J. Mol. Biol.
85
,
177
181
(
1974
).
32.
M.
Schroder
and
R. J.
Kaufman
, “
The mammalian unfolded protein response
,”
Annu. Rev. Biochem.
74
,
739
789
(
2005
).
33.
D.
Sept
and
F. C.
MacKintosh
, “
Microtubule elasticity: Connecting all-atom simulations with continuum mechanics
,”
Phys. Rev. Lett.
104
(
1
),
018101
(
2010
).
34.
L.
Skjaerven
,
S. M.
Hollup
, and
N.
Reuter
, “
Normal mode analysis for proteins
,”
J. Mol. Struct.: THEOCHEM
898
,
42
48
(
2009
).
35.
G.
Song
and
R. L.
Jernigan
, “
vGNM: A better model for understanding the dynamics of proteins in crystals
,”
J. Mol. Biol.
369
(
3
),
880
893
(
2007
).
36.
F.
Tama
and
C. K.
Brooks
 III
, “
Diversity and identity of mechanical properties of icosahedral viral capsids studied with elastic network normal mode analysis
,”
J. Mol. Biol.
345
,
299
314
(
2005
).
37.
F.
Tama
and
Y. H.
Sanejouand
, “
Conformational change of proteins arising from normal mode calculations
,”
Protein Eng.
14
,
1
6
(
2001
).
38.
F.
Tama
,
M.
Valle
,
J.
Frank
, and
C. K.
Brooks
 III
, “
Dynamic reorganization of the functionally active ribosome explored by normal mode analysis and cryo-electron microscopy
,”
Proc. Natl. Acad. Sci. U.S.A.
100
(
16
),
9319
9323
(
2003
).
39.
M.
Tasumi
,
H.
Takenchi
,
S.
Ataka
,
A. M.
Dwidedi
, and
S.
Krimm
, “
Normal vibrations of proteins: Glucagon
,”
Biopolymers
21
,
711
714
(
1982
).
40.
W. I.
Thacker
,
J.
Zhang
,
L. T.
Watson
,
J. B.
Birch
,
M. A.
Iyer
, and
M. W.
Berry
, “
Algorithm 905: SHEPPACK: Modified Shepard algorithm for interpolation of scattered multivariate data
,”
ACM Trans. Math. Software
37
(
3
),
1
20
(
2010
).
41.
M. M.
Tirion
, “
Large amplitude elastic motions in proteins from a single-parameter, atomic analysis
,”
Phys. Rev. Lett.
77
,
1905
1908
(
1996
).
42.
C. W.
von der Lieth
,
K.
Stumpf-Nothof
, and
U.
Prior
, “
A bond flexibility index derived from the constitution of molecules
,”
J. Chem. Inf. Comput. Sci.
36
,
711
716
(
1996
).
43.
Y.
Wang
,
A. J.
Rader
,
I.
Bahar
, and
R. L.
Jernigan
, “
Global ribosome motions revealed with elastic network model
,”
J. Struct. Biol.
147
,
302
314
(
2004
).
44.
G. W.
Wei
, “
Wavelets generated by using discrete singular convolution kernels
,”
J. Phys. A: Math. Gen.
33
,
8577
8596
(
2000
).
45.
S. H.
White
and
W. C.
Wimley
, “
Membrane protein folding and stability: Physical principles
,”
Annu. Rev. Biophys. Biomol. Struct.
28
,
319
365
(
1999
).
46.
K. L.
Xia
,
K.
Opron
, and
G. W.
Wei
, “
Multiscale multiphysics and multidomain models: Flexibility and rigidity
,”
J. Chem. Phys.
139
,
194109
(
2013
).
47.
C.
Xu
,
D.
Tobi
, and
I.
Bahar
, “
Allosteric changes in protein structure computed by a simple mechanical model: Hemoglobin t ↔ r2 transition
,”
J. Mol. Biol.
333
,
153
168
(
2003
).
48.
L. W.
Yang
and
C. P.
Chng
, “
Coarse-grained models reveal functional dynamics–I. Elastic network models–Theories, comparisons and perspectives
,”
Bioinf. Biol. Insights
2
,
25
45
(
2008
).
49.
W.
Zheng
,
B. R.
Brooks
, and
D.
Thirumalai
, “
Allosteric transitions in the chaperonin groel are captured by a dominant normal mode that is most robust to sequence variations
,”
Biophys. J.
93
,
2289
2299
(
2007
).
50.
W. J.
Zheng
and
S.
Doniach
, “
A comparative study of motor-protein motions by using a simple elastic-network model
,”
Proc. Natl. Acad. Sci. U.S.A.
100
(
23
),
13253
13258
(
2003
).
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