The reaction-diffusion master equation (RDME) is a lattice stochastic reaction-diffusion model that has been used to study spatially distributed cellular processes. The RDME is often interpreted as an approximation to spatially continuous models in which molecules move by Brownian motion and react by one of several mechanisms when sufficiently close. In the limit that the lattice spacing approaches zero, in two or more dimensions, the RDME has been shown to lose bimolecular reactions. The RDME is therefore not a convergent approximation to any spatially continuous model that incorporates bimolecular reactions. In this work we derive a new convergent RDME (CRDME) by finite volume discretization of a spatially continuous stochastic reaction-diffusion model popularized by Doi. We demonstrate the numerical convergence of reaction time statistics associated with the CRDME. For sufficiently large lattice spacings or slow bimolecular reaction rates, we also show that the reaction time statistics of the CRDME may be approximated by those from the RDME. The original RDME may therefore be interpreted as an approximation to the CRDME in several asymptotic limits.

1.
S.
Neves
,
P.
Tsokas
,
A.
Sarkar
,
E.
Grace
,
P.
Rangamani
,
S.
Taubenfeld
,
C.
Alberini
,
J.
Schaff
,
R.
Blitzer
,
I.
Moraru
, and
R.
Iyengar
,
Cell
133
,
666
(
2008
).
2.
B.
Hendriks
,
L.
Opresko
,
H.
Wiley
, and
D.
Lauffenburger
,
J. Biol. Chem.
278
,
23343
(
2003
).
3.
G.
Von Dassow
,
E.
Meir
,
E. M.
Munro
, and
G. M.
Odell
,
Nature
406
,
188
(
2000
).
4.
L.
Ma
,
J.
Wagner
,
J. J.
Rice
,
W.
Hu
,
A. J.
Levine
, and
G. A.
Stolovitzky
,
Proc. Natl. Acad. Sci. U.S.A.
102
,
14266
(
2005
).
5.
J.
Muñoz-García
,
Z.
Neufeld
,
B. N.
Kholodenko
, and
H. M.
Sauro
,
PLoS Comput. Biol.
5
,
e1000330
(
2009
).
6.
S. A.
Isaacson
,
D. M.
McQueen
, and
C. S.
Peskin
,
Proc. Natl. Acad. Sci. U.S.A.
108
,
3815
(
2011
).
7.
J.
Elf
and
M.
Ehrenberg
,
IEE Proc. Sys. Biol.
1
,
230
(
2004
).
8.
A.
Arkin
and
H. H.
McAdams
,
Proc. Natl. Acad. Sci. U.S.A.
94
,
814
(
1997
).
9.
W. J.
Blake
,
M.
Kaern
,
C. R.
Cantor
, and
J. J.
Collins
,
Nature
422
,
633
(
2003
).
10.
J.
Raser
and
E.
O'Shea
,
Science
304
,
1811
(
2004
).
11.
D. E.
Shaw
,
R. O.
Dror
,
J. K.
Salmon
,
J. P.
Grossman
,
K. M.
Mackenzie
,
J. A.
Bank
,
C.
Young
,
M. M.
Deneroff
,
B.
Batson
,
K. J.
Bowers
,
E.
Chow
,
M. P.
Eastwood
,
D. J.
Ierardi
,
J. L.
Klepeis
,
J. S.
Kuskin
,
R. H.
Larson
,
K.
Lindorff-Larsen
,
P.
Maragakis
,
M. A.
Moraes
,
S.
Piana
,
Y.
Shan
, and
B.
Towles
, “
Millisecond-scale molecular dynamics simulations on anton
,”
Proceedings of the Conference on High Performance Computing, Networking, Storage and Analysis (SC09)
(
ACM
,
New York
,
2009
).
12.
E.
Teramoto
and
N.
Shigesada
,
Prog. Theor. Phys.
37
,
29
(
1967
).
13.
M.
Doi
,
J. Phys. A
9
,
1465
(
1976
).
14.
M.
Doi
,
J. Phys. A
9
,
1479
(
1976
).
15.
M. V.
Smoluchowski
,
Z. Phys. Chem.
92
,
129
(
1917
).
16.
J.
Keizer
,
J. Phys. Chem.
86
,
5052
(
1982
).
17.
C. W.
Gardiner
,
K. J.
McNeil
,
D. F.
Walls
, and
I. S.
Matheson
,
J. Stat. Phys.
14
,
307
(
1976
).
18.
C. W.
Gardiner
,
Handbook of Stochastic Methods: For Physics, Chemistry, and the Natural Sciences
, 2nd ed.,
Springer Series in Synergetics
Vol.
13
(
Springer Verlag
,
New York
,
1996
).
19.
N. G.
Van Kampen
,
Stochastic Processes in Physics and Chemistry
(
North-Holland
,
Amsterdam
,
2001
).
20.
R.
Erban
,
S. J.
Chapman
, and
P. K.
Maini
, “
A practical guide to stochastic simulations of reaction-diffusion processes
,” eprint arXiv:0704.1908 (
2007
).
21.
S. A.
Isaacson
and
C. S.
Peskin
,
SIAM J. Sci. Comput.
28
,
47
(
2006
).
22.
S. A.
Isaacson
,
SIAM J. Appl. Math.
70
,
77
(
2009
).
23.
D. A.
McQuarrie
,
J. Appl. Probab.
4
,
413
(
1967
).
24.
S. A.
Isaacson
,
J. Phys. A: Math. Theor.
41
,
065003
(
2008
).
25.
S.
Engblom
,
L.
Ferm
,
A.
Hellander
, and
P.
Lötstedt
,
SIAM J. Sci. Comput.
31
,
1774
(
2009
).
26.
R.
Erban
and
S. J.
Chapman
,
Phys. Biol.
6
,
046001
(
2009
).
27.
R. A.
Kerr
,
T. M.
Bartol
,
B.
Kaminsky
,
M.
Dittrich
,
J.
Chang
,
S. B.
Baden
,
T. J.
Sejnowski
, and
J. R.
Stiles
,
SIAM J. Sci. Comput.
30
,
3126
(
2008
).
28.
S. S.
Andrews
and
D.
Bray
,
Phys. Biol.
1
,
137
(
2004
).
29.
A.
Donev
,
V. V.
Bulatov
,
T.
Oppelstrup
,
G. H.
Gilmer
,
B.
Sadigh
, and
M. H.
Kalos
,
J. Comput. Phys.
229
,
3214
(
2010
).
30.
K.
Takahashi
,
S.
Tanase-Nicola
, and
P. R.
ten Wolde
,
Proc. Natl. Acad. Sci. U.S.A.
107
,
2473
(
2010
).
31.
D. T.
Gillespie
,
J. Phys. Chem.
81
,
2340
(
1977
).
32.
A. B.
Bortz
,
M. H.
Kalos
, and
J. L.
Lebowitz
,
J. Comput. Phys.
17
,
10
(
1975
).
33.
S.
Arjunan
and
M.
Tomita
,
Syst. Synth. Biol.
4
,
35
(
2010
).
34.
S.
Hellander
,
A.
Hellander
, and
L.
Petzold
,
Phys. Rev. E
85
(
1
),
042901
(
2012
).
35.
D.
Fange
,
O. G.
Berg
,
P.
Sjöberg
, and
J.
Elf
,
Proc. Natl. Acad. Sci. U.S.A.
107
,
19820
(
2010
).
36.
D. T.
Gillespie
, “
Voxel size restrictions in the reaction-diffusion master equation
,” private communication (
2013
).
37.
W.
Naumann
,
J. Chem. Phys.
98
,
2353
(
1993
).
38.
V.
Kuzovkov
and
E.
Kotomin
,
Rep. Prog. Phys.
51
,
1479
(
1988
).
39.
K.
Seki
,
A. I.
Shushin
,
M.
Wojcik
, and
M.
Tachiya
,
J. Phys.: Condens. Matter
19
,
065117
(
2007
).
40.
M.
Klann
,
A.
Ganguly
, and
H.
Koeppl
,
Bioinformatics
28
,
i549
(
2012
).
41.
J.
Lipkova
,
K. C.
Zygalakis
,
S. J.
Chapman
, and
R.
Erban
,
SIAM J. Appl. Math.
71
,
714
(
2011
).
42.
M. B.
Flegg
,
S.
Rüdiger
, and
R.
Erban
,
J. Chem. Phys.
138
,
154103
(
2013
).
43.
I. C.
Agbanusi
and
S. A.
Isaacson
, “
A comparison of bimolecular reaction models for stochastic reaction-diffusion systems
,”
Bull. Math. Biol.
(in press).
44.
H.
Wang
,
C. S.
Peskin
, and
T. C.
Elston
,
J. Theor. Biol.
221
,
491
(
2003
).
45.
A.
Hellander
and
P.
Lötstedt
,
Multiscale Model. Simul.
8
,
1691
(
2010
).
46.
B.
Bayati
,
P.
Chatelain
, and
P.
Koumoutsakos
,
J. Comput. Phys.
230
,
13
(
2011
).
47.
S. A.
Isaacson
and
D.
Isaacson
,
Phys. Rev. E
80
,
066106
(
2009
).
48.
C. A.
Yates
,
R. E.
Baker
,
R.
Erban
, and
P. K.
Maini
,
Phys. Rev. E
86
,
021921
(
2012
).
49.
M.
Vigelius
and
B.
Meyer
,
PloS ONE
7
,
e33384
(
2012
).
50.
L.
Ferm
,
A.
Hellander
, and
P.
Lötstedt
,
J. Comput. Phys.
229
,
343
(
2010
).
51.
Z.
Cai
,
J.
Douglas
 Jr.
, and
M.
Park
,
Adv. Comput. Math.
19
,
3
(
2003
).
52.
F.
Kühner
,
L. T.
Costa
,
P. M.
Bisch
,
S.
Thalhammer
,
W. M.
Heckl
, and
H. E.
Gaub
,
Biophys. J.
87
,
2683
(
2004
).
53.
O.
Dushek
,
P. A.
van der Merwe
, and
V.
Shahrezaei
,
Biophys. J.
100
,
1189
(
2011
).
54.
T.
Hahn
,
Comput. Phys. Commun.
168
,
78
(
2005
).
55.
S. G.
Johnson
, Cubature integration library, available at: http://ab-initio.mit.edu/wiki/index.php/Cubature.
You do not currently have access to this content.