Assembly of protein complexes like virus shells, the centriole, the nuclear pore complex, or the actin cytoskeleton is strongly determined by their spatial structure. Moreover, it is becoming increasingly clear that the reversible nature of protein assembly is also an essential element for their biological function. Here we introduce a computational approach for the Brownian dynamics of patchy particles with anisotropic assemblies and fully reversible reactions. Different particles stochastically associate and dissociate with microscopic reaction rates depending on their relative spatial positions. The translational and rotational diffusive properties of all protein complexes are evaluated on-the-fly. Because we focus on reversible assembly, we introduce a scheme which ensures detailed balance for patchy particles. We then show how the macroscopic rates follow from the microscopic ones. As an instructive example, we study the assembly of a pentameric ring structure, for which we find excellent agreement between simulation results and a macroscopic kinetic description without any adjustable parameters. This demonstrates that our approach correctly accounts for both the diffusive and reactive processes involved in protein assembly.

1.
R. R.
Gabdoulline
and
R. C.
Wade
,
Biophys. J.
72
,
1917
(
1997
).
2.
W.
Roos
,
R.
Bruinsma
, and
G.
Wuite
,
Nat. Phys.
6
,
733
(
2010
).
3.
A.
Zlotnick
and
S.
Mukhopadhyay
,
Trends Microbiol.
19
,
14
(
2011
).
4.
5.
E.
Karsenti
,
F.
Nédélec
, and
T.
Surrey
,
Nat. Cell Biol.
8
,
1204
(
2006
).
6.
P.
Gönczy
,
Nat. Rev. Mol. Cell Biol.
13
,
425
(
2012
).
7.
F.
Alber
,
S.
Dokudovskaya
,
L. M.
Veenhoff
,
W.
Zhang
,
J.
Kipper
,
D.
Devos
,
A.
Suprapto
,
O.
Karni-Schmidt
,
R.
Williams
,
B. T.
Chait
 et al.,
Nature (London)
450
,
683
(
2007
).
8.
M. A.
D’Angelo
and
M. W.
Hetzer
,
Trends Cell Biol.
18
,
456
(
2008
).
9.
E. J.
Tran
and
S. R.
Wente
,
Cell
125
,
1041
(
2006
).
10.
B.
Geiger
and
A.
Bershadsky
,
Curr. Opin. Cell Biol.
13
,
584
(
2001
).
11.
A. D.
Bershadsky
,
C.
Ballestrem
,
L.
Carramusa
,
Y.
Zilberman
,
B.
Gilquin
,
S.
Khochbin
,
A. Y.
Alexandrova
,
A. B.
Verkhovsky
,
T.
Shemesh
, and
M. M.
Kozlov
,
Eur. J. Cell Biol.
85
,
165
(
2006
).
12.
H.
Wolfenson
,
Y. I.
Henis
,
B.
Geiger
, and
A. D.
Bershadsky
,
Cell Motil. Cytoskeleton
66
,
1017
(
2009
).
13.
C. C.
Beltzner
and
T. D.
Pollard
,
J. Biol. Chem.
283
,
7135
(
2007
).
14.
K.
Guo
,
J.
Shillcock
, and
R.
Lipowsky
,
J. Chem. Phys.
131
,
015102
(
2009
).
15.
T. D.
Pollard
,
Annu. Rev. Biophys. Biomol. Struct.
36
,
451
(
2007
).
16.
K.
Guo
,
J.
Shillcock
, and
R.
Lipowsky
,
J. Chem. Phys.
133
,
155105
(
2010
).
17.
G.
Schreiber
,
G.
Haran
, and
H.-X.
Zhou
,
Chem. Rev.
109
,
839
(
2009
).
18.
C. A.
Mirkin
,
R. L.
Letsinger
,
R. C.
Mucic
, and
J. J.
Storhoff
,
Nature (London)
382
,
607
(
1996
).
19.
L.
Di Michele
and
E.
Eiser
,
Phys. Chem. Chem. Phys.
15
,
3115
(
2013
).
20.
Y.
Wang
,
Y.
Wang
,
D. R.
Breed
,
V. N.
Manoharan
,
L.
Feng
,
A. D.
Hollingsworth
,
M.
Weck
, and
D. J.
Pine
,
Nature (London)
491
,
51
(
2012
).
21.
C.
Knorowski
and
A.
Travesset
,
Curr. Opin. Solid State Mater. Sci.
15
,
262
(
2011
).
22.
S.
Sacanna
,
W.
Irvine
,
P. M.
Chaikin
, and
D. J.
Pine
,
Nature (London)
464
,
575
(
2010
).
23.
S.
Sacanna
,
M.
Korpics
,
K.
Rodriguez
,
L.
Colón-Meléndez
,
S.-H.
Kim
,
D. J.
Pine
, and
G.-R.
Yi
,
Nat. Commun.
4
,
1688
(
2013
).
24.
M. E.
Leunissen
,
R.
Dreyfus
,
F. C.
Cheong
,
D. G.
Grier
,
R.
Sha
,
N. C.
Seeman
, and
P. M.
Chaikin
,
Nat. Mater.
8
,
590
(
2009
).
25.
L.
Di Michele
,
F.
Varrato
,
J.
Kotar
,
S. H.
Nathan
,
G.
Foffi
, and
E.
Eiser
,
Nat. Commun.
4
,
2007
(
2013
).
26.
J. E.
Baschek
,
H. C.
Klein
, and
U. S.
Schwarz
,
BMC Biophys.
5
,
22
(
2012
).
27.
M. v.
Smoluchowski
,
Zeitschrift für physikalische Chemie
92
,
9
(
1917
).
28.
F. C.
Collins
and
G. E.
Kimball
,
J. Colloid Sci.
4
,
425
(
1949
).
29.
K.
Šolc
and
W.
Stockmayer
,
Intl. J. Chem. Kinet.
5
,
733
(
1973
).
30.
D.
Shoup
,
G.
Lipari
, and
A.
Szabo
,
Biophys. J.
36
,
697
(
1981
).
32.
A.
Shushin
,
J. Chem. Phys.
110
,
12044
(
1999
).
33.
M.
Schlosshauer
and
D.
Baker
,
J. Phys. Chem. B
106
,
12079
(
2002
).
34.
M.
Schlosshauer
and
D.
Baker
,
Protein Sci.
13
,
1660
(
2004
).
35.
N.
Agmon
,
J. Chem. Phys.
81
,
2811
(
1984
).
36.
N.
Agmon
and
A.
Szabo
,
J. Chem. Phys.
92
,
5270
(
1990
).
37.
S. H.
Northrup
,
S. A.
Allison
, and
J. A.
McCammon
,
J. Chem. Phys.
80
,
1517
(
1984
).
38.
S. H.
Northrup
and
H. P.
Erickson
,
Proc. Natl. Acad. Sci.
89
,
3338
(
1992
).
39.
H. X.
Zhou
,
J. Phys. Chem.
94
,
8794
(
1990
).
41.
R. R.
Gabdoulline
and
R. C.
Wade
,
J. Mol. Biol.
306
,
1139
(
2001
).
42.
G.
Zou
and
R. D.
Skeel
,
Biophys. J.
85
,
2147
(
2003
).
43.
R.
Alsallaq
and
H.-X.
Zhou
,
Structure
15
,
215
(
2007
).
44.
R.
Alsallaq
and
H.-X.
Zhou
,
Biophys. J.
92
,
1486
(
2007
).
45.
S.
Qin
,
X.
Pang
, and
H.-X.
Zhou
,
Structure
19
,
1744
(
2011
).
46.
M.
Klann
and
H.
Koeppl
,
Intl J. Mol. Sci.
13
,
7798
(
2012
).
47.
S. S.
Andrews
and
D.
Bray
,
Phys. Biol.
1
,
137
(
2004
).
49.
J. S.
van Zon
and
P. R.
Ten Wolde
,
J. Chem. Phys.
123
,
234910
(
2005
).
50.
M. J.
Morelli
and
P. R.
Ten Wolde
,
J. Chem. Phys.
129
,
054112
(
2008
).
51.
K.
Takahashi
,
S.
Tănase-Nicola
, and
P. R.
ten Wolde
,
Proc. Natl Acad. Sci.
107
,
2473
(
2010
).
52.
D. T.
Gillespie
,
J. Phys. Chem.
81
,
2340
(
1977
).
53.
J.
Hattne
,
D.
Fange
, and
J.
Elf
,
Bioinformatics
21
,
2923
(
2005
).
54.
D.
Fange
,
O. G.
Berg
,
P.
Sjöberg
, and
J.
Elf
,
Proc. Natl. Acad. Sci.
107
,
19820
(
2010
).
55.
D.
Fange
,
A.
Mahmutovic
, and
J.
Elf
,
Bioinformatics
28
,
3155
(
2012
).
56.
57.
M. F.
Hagan
and
D.
Chandler
,
Biophys. J.
91
,
42
(
2006
).
58.
H. D.
Nguyen
,
V. S.
Reddy
, and
C. L.
Brooks
 III
,
Nano Lett.
7
,
338
(
2007
).
59.
D.
Rapaport
,
Phys. Rev. E
86
,
051917
(
2012
).
60.
I. G.
Johnston
,
A. A.
Louis
, and
J. P.
Doye
,
J. Phys.: Condens. Matter
22
,
104101
(
2010
).
61.
C.
Horejs
,
M. K.
Mitra
,
D.
Pum
,
U. B.
Sleytr
, and
M.
Muthukumar
,
J. Chem. Phys.
134
,
125103
(
2011
).
62.
A. W.
Wilber
,
J. P.
Doye
,
A. A.
Louis
,
E. G.
Noya
,
M. A.
Miller
, and
P.
Wong
,
J. Chem. Phys.
127
,
085106
(
2007
).
63.
A. W.
Wilber
,
J. P.
Doye
, and
A. A.
Louis
,
J. Chem. Phys.
131
,
175101
(
2009
).
64.
J.
Largo
,
F. W.
Starr
, and
F.
Sciortino
,
Langmuir
23
,
5896
(
2007
).
65.
P. F.
Damasceno
,
M.
Engel
, and
S. C.
Glotzer
,
Science
337
,
453
(
2012
).
66.
X.
Liu
,
J. C.
Crocker
, and
T.
Sinno
,
J. Chem. Phys.
138
,
244111
(
2013
).
67.
F.
Sciortino
,
C.
De Michele
, and
J. F.
Douglas
,
J. Phys.: Condens. Matter
20
,
155101
(
2008
).
68.
J. D.
Halverson
and
A. V.
Tkachenko
,
Phys. Rev. E
87
,
062310
(
2013
).
69.
M.
Eigen
, “
Diffusion control in biochemical reactions
,” in
Quantum Statistcal Mechanics in the Natural Sciences
,
Studies in the Natural Sciences
Vol.
4
, edited by
S. L.
Mintz
, and
S. M.
Widmayer
(
Plenum Press
,
New York
,
1974
), pp.
37
61
.
70.
D.
Shoup
and
A.
Szabo
,
Biophys. J.
40
,
33
(
1982
).
71.
C.
Korn
and
U.
Schwarz
,
J. Chem. Phys.
126
,
095103
(
2007
).
72.
J.
Schluttig
,
D.
Alamanova
,
V.
Helms
, and
U. S.
Schwarz
,
J. Chem. Phys.
129
,
155106
(
2008
).
73.
J.
Schluttig
,
C. B.
Korn
, and
U. S.
Schwarz
,
Phys. Rev. E
81
,
030902
(
2010
).
74.
J.
García de la Torre
,
M. L.
Huertas
, and
B.
Carrasco
,
Biophys. J.
78
,
719
(
2000
).
75.
B.
Berger
,
P. W.
Shor
,
L.
Tucker-Kellogg
, and
J.
King
,
Proc. Natl. Acad. Sci.
91
,
7732
(
1994
).
76.
T. L.
Hill
,
An Introduction to Statistical Thermodynamics
(
Dover Publications
,
New York
,
1986
) (Unabridged, corr. republ. of the 2. print., Reading, MA, 1962).
77.
M.
Pogson
,
R.
Smallwood
,
E.
Qwarnstrom
, and
M.
Holcombe
,
Biosystems
85
,
37
(
2006
).
78.
M. T.
Klann
,
A.
Lapin
, and
M.
Reuss
,
BMC Syst. Biol.
5
,
71
(
2011
).
79.
J.
Paijmans
, “
The fundamental lower bound of the noise in transcriptional regulation
,” Master's thesis,
University of Amsterdam
, Amsterdam (July
2012
).
80.
N.
Kern
and
D.
Frenkel
,
J. Chem. Phys.
118
,
9882
(
2003
).
81.
A.
Arkhipov
,
Y.
Yin
, and
K.
Schulten
,
Biophys. J.
95
,
2806
(
2008
).
82.
A.
Frost
,
V. M.
Unger
, and
P. D.
Camilli
,
Cell
137
,
191
(
2009
).
83.
H. D.
Nguyen
and
C. L.
Brooks
 III
,
Nano Lett.
8
,
4574
(
2008
).
84.
O. M.
Elrad
and
M. F.
Hagan
,
Nano Lett.
8
,
3850
(
2008
).
85.
H. D.
Nguyen
,
V. S.
Reddy
, and
C. L.
Brooks
 III
,
J. Am. Chem. Soc.
131
,
2606
(
2009
).
86.
L.
Blanchoin
,
R.
Boujemaa-Paterski
,
C.
Sykes
, and
J.
Plastino
,
Physiol. Rev.
94
,
235
(
2014
).
87.
D. L. D.
Caspar
and
A.
Klug
,
Cold Spring Harbor Symposia on Quantitative Biology
(
Cold Spring Harbor Laboratory Press
,
1962
), Vol.
27
, pp.
49
50
.
88.
D.
Kitagawa
,
I.
Vakonakis
,
N.
Olieric
,
M.
Hilbert
,
D.
Keller
,
V.
Olieric
,
M.
Bortfeld
,
M. C.
Erat
,
I.
Flückiger
,
P.
Gönczy
, and
M. O.
Steinmetz
,
Cell
144
,
364
(
2011
).
89.
M.
van Breugel
,
M.
Hirono
,
A.
Andreeva
,
H.-a.
Yanagisawa
,
S.
Yamaguchi
,
Y.
Nakazawa
,
N.
Morgner
,
M.
Petrovich
,
I.-O.
Ebong
,
C. V.
Robinson
,
C. M.
Johnson
,
D.
Veprintsev
, and
B.
Zuber
,
Science
331
,
1196
(
2011
).
90.
P.
Guichard
,
V.
Hachet
,
N.
Majubu
,
A.
Neves
,
D.
Demurtas
,
N.
Olieric
,
I.
Fluckiger
,
A.
Yamada
,
K.
Kihara
,
Y.
Nishida
,
S.
Moriya
,
M. O.
Steinmetz
,
Y.
Hongoh
, and
P.
Gönczy
,
Curr. Biol.
23
,
1620
(
2013
).
91.
C.
Erlenkämper
and
K.
Kruse
,
J. Chem. Phys.
139
,
164907
(
2013
).
92.
R.
Kapral
,
J. Chem. Phys.
138
,
020901
(
2013
).
You do not currently have access to this content.