We present an effective method for estimating the motion of proteins from the motion of attached probe particles in single-molecule experiments. The framework naturally incorporates Langevin dynamics to compute the most probable trajectory of the protein. By using a perturbation expansion technique, we achieve computational costs more than 3 orders of magnitude smaller than the conventional gradient descent method without loss of simplicity in the computation algorithm. We present illustrative applications of the method using simple models of single-molecule experiments and confirm that the proposed method yields reasonable and stable estimates of the hidden motion in a highly efficient manner.
REFERENCES
1.
K.
Svoboda
, C. F.
Schmidt
, B. J.
Schnapp
, and S. M.
Block
, Nature (London)
365
, 721
(1993
).2.
H.
Noji
, R.
Yasuda
, M.
Yoshida
, and K.
Kinosita
, Jr., Nature (London)
386
, 299
(1997
).3.
M.
Rief
, R. S.
Rock
, A. D.
Mehta
, M. S.
Mooseker
, R. E.
Cheney
, and J. A.
Spudich
, Proc. Natl. Acad. Sci. U.S.A.
97
, 9482
(2000
).4.
R.
Yasuda
, H.
Noji
, M.
Yoshida
, K.
Kinosita
, Jr., and H.
Itoh
, Nature (London)
410
, 898
(2001
).5.
W. J.
Greenleaf
, M. T.
Woodside
, and S. M.
Block
, Annu. Rev. Biophys. Biomol. Struct.
36
, 171
(2007
).6.
K.
Shiroguchi
and K.
Kinosita
, Jr., Science
316
, 1208
(2007
).7.
K.
Shiroguchi
, H. F.
Chin
, D. E.
Hannemann
, E.
Muneyuki
, E. M.
De La Cruz
, and K.
Kinosita
, Jr., PLoS Biol.
9
, e1001031
(2011
).8.
H.
Ueno
, S.
Nishikawa
, R.
Iino
, K. V.
Tabata
, S.
Sakakihara
, T.
Yanagida
, and H.
Noji
, Biophys. J.
98
, 2014
(2010
).9.
L.
Nugent-Glandorf
and T. T.
Perkins
, Opt. Lett.
29
, 2611
(2004
).10.
D. E.
Smith
, S. J.
Tans
, S. B.
Smith
, S.
Grimes
, D. L.
Anderson
, and C.
Bustamante
, Nature (London)
413
, 748
(2001
).11.
S.
Uemura
and S.
Ishiwata
, Nat. Struct. Biol.
4
, 308
(2003
).12.
H.
Itoh
, A.
Takahashi
, K.
Adachi
, H.
Noji
, R.
Yasuda
, M.
Yoshida
, and K.
Kinosita
, Jr., Nature (London)
427
, 465
(2004
).13.
T.
Watanabe-Nakayama
, S.
Toyabe
, S.
Kudo
, S.
Sugiyama
, M.
Yoshida
, and E.
Muneyuki
, Biochem. Biophys. Res. Commun.
366
, 951
(2008
).14.
J.
Liphardt
, S.
Dumont
, S. B.
Smith
, I.
Tinoco
, Jr., and C.
Bustamante
, Science
296
, 1832
(2002
).15.
S.
Toyabe
, T.
Okamoto
, T.
Watanabe-Nakayama
, H.
Taketani
, S.
Kudo
, and E.
Muneyuki
, Phys. Rev. Lett.
104
, 198103
(2010
).16.
E.
Geva
and J. L.
Skinner
, Chem. Phys. Lett.
288
, 225
(1998
).17.
A. M.
Berezhkovskii
, A.
Szabo
, and G. H.
Weiss
, J. Phys. Chem. B
104
, 3776
(2000
).18.
J.
Cao
, Chem. Phys. Lett.
327
, 38
(2000
).19.
J. B.
Witkoskie
and J.
Cao
, J. Chem. Phys.
121
, 6361
(2004
a).20.
O.
Flomenbom
, J.
Klafter
, and A.
Szabo
, Biophys. J.
88
, 3780
(2005
).21.
I. V.
Gopich
and A.
Szabo
, J. Chem. Phys.
124
, 154712
(2006
).22.
J. B.
Witkoskie
and J.
Cao
, J. Chem. Phys.
121
, 6373
(2004
b).23.
J. B.
Witkoskie
and J.
Cao
, J. Phys. Chem. B
112
, 5988
(2008
).24.
A.
Baba
and T.
Komatsuzaki
, Proc. Natl. Acad. Sci. U.S.A.
104
, 19297
(2007
).25.
A.
Baba
and T.
Komatsuzaki
, Phys. Chem. Chem. Phys.
13
, 1395
(2011
).26.
S.
Chung
and R.
Kennedy
, J. Neurosci. Methods
40
, 71
(1991
).27.
X.
Nan
, P. A.
Sims
, P.
Chen
, and X. S.
Xie
, J. Phys. Chem. B
109
, 24220
(2005
).28.
J. W. J.
Kerssemakers
, E. L.
Munteanu
, L.
Laan
, T. L.
Noetzel
, M. E.
Janson
, and M.
Dogterom
, Nature (London)
442
, 709
(2006
).29.
B. C.
Carter
, M.
Vershinin
, and S. P.
Gross
, Biophys. J.
94
, 306
(2008
).30.
B.
Bozorgui
, K.
Shundyak
, S.
Cox
, and D.
Frenkel
, Eur. Phys. J. E
31
, 411
(2010
).31.
C.
Dellago
, P. G.
Bolhuis
, F. S.
Csajka
, and D.
Chandler
, J. Chem. Phys.
108
, 1964
(1998
).32.
P. G.
Bolhuis
, D.
Chandler
, C.
Dellago
, and P. L.
Geissler
, Annu. Rev. Phys. Chem.
53
, 291
(2002
).33.
E.
Autieri
, P.
Faccioli
, M.
Sega
, F.
Pederiva
, and H.
Orland
, J. Chem. Phys.
130
, 064106
(2009
).34.
P.
Faccioli
, M.
Sega
, F.
Pederiva
, and H.
Orland
, Phys. Rev. Lett.
97
, 108101
(2006
).35.
M.
Sega
, P.
Faccioli
, F.
Pederiva
, G.
Garberoglio
, and H.
Orland
, Phys. Rev. Lett.
99
, 118102
(2007
).36.
M.
Miyazaki
and T.
Harada
, J. Chem. Phys.
134
, 085108
(2011
).37.
L.
Onsager
and S.
Machlup
, Phys. Rev.
91
, 1505
(1953
).38.
S.
Machlup
and L.
Onsager
, Phys. Rev.
91
, 1512
(1953
).39.
K. L.C.
Hunt
and J.
Ross
, J. Chem. Phys.
75
, 976
(1981
).40.
K.
Adachi
, K.
Oiwa
, T.
Nishizaka
, S.
Furuike
, H.
Noji
, H.
Itoh
, M.
Yoshida
, and K.
Kinosita
, Jr., Cell
130
, 309
(2007
).41.
K.
Hayashi
, H.
Ueno
, R.
Iino
, and H.
Noji
, Phys. Rev. Lett.
104
, 218103
(2010
).42.
Q.
Cui
, G.
Li
, J.
Ma
, and M.
Karplus
, J. Mol. Biol
340
, 345
(2004
).43.
Normal Mode Analysis: Theory and Applications to Biological and Chemical Systems
, 1st ed., edited by Q.
Cui
and I.
Bahar
(Chapman & Hall/CRC
, Boca Raton, FL
, 2006
).44.
Y.
Togashi
and A. S.
Mikhailov
, Proc. Natl. Acad. Sci. U.S.A.
104
, 8697
(2007
).45.
Y.
Togashi
, T.
Yanagida
, and A. S.
Mikhailov
, PLoS Comput. Biol.
6
, e1000814
(2010
).46.
F.
Jülicher
, A.
Ajdari
, and J.
Prost
, Rev. Mod. Phys.
69
, 1269
(1997
).47.
P.
Reimann
, Phys. Rep.
361
, 57
(2002
).48.
H. A.
Kramers
, Physica
, VII
, 284
(1940
).49.
If the system is locally equilibrated, we can adopt the Boltzmann distribution.
50.
J. D.
Gunton
, M. S.
Miguel
, and P. S.
Sahni
, Phase Transitions and Critical Phenomena
, edited by C.
Domb
and J. L.
Lebowitz
(Academic
, New York
, 1983
), Vol. 8
.51.
K.
Kawasaki
and T.
Ohta
, Physica
116A
, 573
(1982
).52.
J.
Carr
and R.
Pego
, Commun. Pure. Appl. Math.
XLII
, 523
(1989
).53.
R.
Mortensen
, J. Stat. Phys.
1
, 271
(1969
).54.
When x is close to the bottom of the effective potential,
$-G_x(x, y) \simeq - G_{xx}(x_{\protect \rm b}, y) (x - x_{\protect \rm b})$
, where $x_{\protect \rm b}$
represents the bottom position. Since $G_{xx}(x_{\protect \rm b}, y)$
is positive, Eq. (11) will be destabilized if the equation is integrated in the reverse-time direction.55.
ε is a nondimensional parameter, which guarantees the smallness of the second term in the right-hand side of Eq. (12). After the calculation, we substitute ε = 1 into the result.
56.
$\tau _{\protect \rm dim} \sim \gamma / G_{xx}(x_{\protect \rm b}) = \gamma /(a_1 + k)$
where $x_{\protect \rm b}$
represents the bottom position of the effective potential.57.
See supplementary material at http://dx.doi.org/10.1063/1.3574396 for the supporting figures.
58.
The standard deviation of the thermal fluctuation of x(t) is
$\protect \sqrt{2k_{\protect \mathrm{B}}T\Delta t / \gamma }$
in the case of discrete t. For model A, SD = 0.2.59.
W. H.
Press
, S. A.
Teukolsky
, W. T.
Vetterling
, and B. P.
Flannery
, Numerical Recipes: The Art of Scientific Computing
, 3rd ed. (Cambridge University Press
, New York
, 2007
), Chap. 20.60.
The Go-and-Back method is not an algorithm to search the global minimum point of
$S([x,y] ; \hbox{\protect \bm {\Pi} })$
along [x]-space, but systematically calculates the higher order approximation of the solution of the Euler-Lagrange equation [Eq. (8)]. Therefore, a monotonic decrease of $S([x,y] ; \hbox{\protect \bm {\Pi} })$
does not mean that the estimate gets stuck into the local minimum.61.
G.
Hummer
, New J. Phys.
7
, 516
(2005
).62.
R. B.
Best
and G.
Hummer
, Phys. Rev. Lett.
96
, 228104
(2006
).63.
S.
Yang
, J. N.
Onuchic
, and H.
Levine
, J. Chem. Phys.
125
, 054910
(2006
).64.
R. B.
Best
and G.
Hummer
, Proc. Natl. Acad. Sci. U.S.A.
107
, 1088
(2010
).65.
H. P.
Lu
, L.
Xun
, and X. S.
Xie
, Science
282
, 1877
(1998
).66.
B. P.
English
, W.
Min
, A. M.
van Oijen
, K. T.
Lee
, G.
Luo
, H.
Sun
, B. J.
Cherayil
, S. C.
Kou
, and X. S.
Xie
, Nat. Chem. Biol.
2
, 87
(2005
).67.
T.
Harada
and S.-i.
Sasa
, Phys. Rev. Lett.
95
, 130602
(2005
).68.
T.
Harada
, Europhys. Lett.
70
, 49
(2005
).69.
© 2011 American Institute of Physics.
2011
American Institute of Physics
You do not currently have access to this content.
