The Chemical Langevin Equation (CLE), which is a stochastic differential equation driven by a multidimensional Wiener process, acts as a bridge between the discrete stochastic simulation algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions there is another, simple formulation of the CLE with only m1+m2 Wiener processes, whereas the standard approach uses 2m1+m2. We demonstrate that there are considerable computational savings when using this latter formulation. Such transformations of the CLE do not cause a loss of accuracy and are therefore distinct from model reduction techniques. We illustrate our findings by considering alternative formulations of the CLE for a human ether a-go-go related gene ion channel model and the Goldbeter–Koshland switch.

1.
T. A.
Turner
,
S.
Schnell
, and
K.
Burrage
,
Comput. Biol. Chem.
28
,
165
(
2004
).
2.
E.
Klipp
,
R.
Herwig
,
A.
Kowald
,
C.
Wierling
, and
H.
Lehrach
,
Systems Biology in Practice: Concepts, Implementation and Application
(
Wiley-VCH
,
Weinheim
,
2005
).
3.
H. H.
McAdams
and
A.
Arkin
,
Proc. Natl. Acad. Sci. U.S.A.
94
,
814
(
1997
).
4.
J.
Hasty
,
J.
Pradines
,
M.
Dolnik
, and
J. J.
Collins
,
Proc. Natl. Acad. Sci. U.S.A.
97
,
2075
(
2000
).
5.
M. B.
Elowitz
,
A. J.
Levine
,
E. D.
Siggia
, and
P. S.
Swain
,
Science
297
,
1183
(
2002
).
6.
F. J.
Isaacs
,
J.
Hasty
,
C. R.
Cantor
, and
J. J.
Collins
,
Proc. Natl. Acad. Sci. U.S.A.
100
,
7714
(
2003
).
7.
T.
Tian
and
K.
Burrage
,
J. Theor. Biol.
227
,
229
(
2004
).
8.
J. G.
Restrepo
,
J. N.
Weiss
, and
A.
Karma
,
Biophys. J.
95
,
3767
(
2008
).
9.
D. T.
Gillespie
,
J. Phys. Chem.
81
,
2340
(
1977
).
10.
11.
N. G.
Van Kampen
,
Stochastic Processes in Physics and Chemistry
, 3rd ed., (
Elsevier
,
Amsterdam
,
2007
).
12.
T. G.
Kurtz
,
J. Chem. Phys.
57
,
2976
(
1972
).
13.
K.
Ball
,
T. G.
Kurtz
,
L.
Popovic
, and
G.
Rempala
,
Ann. Appl. Probab.
16
,
1925
(
2006
).
14.
D. F.
Anderson
,
J. Chem. Phys.
127
,
214107
(
2007
).
15.
D. T.
Gillespie
,
J. Chem. Phys.
115
,
1716
(
2001
).
16.
T.
Tian
and
K.
Burrage
,
J. Chem. Phys.
121
,
10356
(
2004
).
17.
A.
Auger
,
P.
Chatelain
, and
P.
Koumoutsakos
,
J. Chem. Phys.
125
,
084103
(
2006
).
18.
D. T.
Gillespie
,
J. Chem. Phys.
113
,
297
(
2000
).
19.
B.
Øksendal
,
Stochastic Differential Equations: An Introduction with Applications
, 6th ed. (
Springer
,
New York
,
2007
).
20.
D. J.
Wilkinson
,
Stochastic Modelling for Systems Biology, Mathematical and Computational Biology Series
(
CRC Press
,
Boca Raton
,
2006
).
21.
E. J.
Allen
,
L. J. S.
Allen
,
A.
Arciniega
, and
P. E.
Greenwood
,
Stochastic Anal. Appl.
26
,
274
(
2008
).
22.
M.
Ullah
and
O.
Wolkenhauer
,
J. Theor. Biol.
260
,
340
(
2009
).
23.
D.
Adalsteinsson
,
D.
McMillen
, and
T.
Elston
,
BMC Bioinf.
5
,
24
(
2004
), http://www.biomedcentral.com/1471-2105/5/24.
24.
A.
Singer
,
R.
Erban
,
I. G.
Kevrekidis
, and
R. R.
Coifman
,
Proc. Natl. Acad. Sci. U.S.A.
106
,
16090
(
2009
), http://www.pnas.org/content/106/38/16090 (abstract).
25.
P. E.
Kloeden
and
E.
Platen
,
Numerical Solution of Stochastic Differential Equations
(
Springer
,
Berlin
,
1992
).
26.
T.
Brennan
,
M.
Fink
, and
B.
Rodriguez
,
Eur. J. Pharm. Sci.
36
,
62
(
2009
).
27.
A.
Goldbeter
and
D. E.
Koshland
,
Proc. Natl. Acad. Sci. U.S.A.
78
,
6840
(
1981
).
28.
B.
Munsky
and
M.
Khammash
,
J. Chem. Phys.
124
,
044104
(
2006
).
29.
S.
MacNamara
,
K.
Burrage
, and
R. B.
Sidje
,
Multiscale Model. Simul.
6
,
1146
(
2008
), .
30.
S.
MacNamara
,
A. M.
Bersani
,
K.
Burrage
, and
R. B.
Sidje
,
J. Chem. Phys.
129
,
095105
(
2008
).
31.
W.
E
,
D.
Liu
, and
E.
Vanden-Eijnden
,
J. Chem. Phys.
123
,
194107
(
2005
).
32.
E. L.
Haseltine
and
J. B.
Rawlings
,
J. Chem. Phys.
117
,
6959
(
2002
).
33.
R.
Tomioka
,
H.
Kimura
,
T. J.
Kobayashi
, and
K.
Aihara
,
J. Theor. Biol.
229
,
501
(
2004
).
34.
C. A.
Gómez-Uribe
and
G. C.
Verghese
,
J. Chem. Phys.
126
,
024109
(
2007
).
35.
A.
Singh
and
J. P.
Hespanha
, in
Decision and Control, 2006 45th IEEE Conference
, pp.
2063
2068
, available online at .
36.
D. J.
Higham
and
R.
Khanin
,
The Open Applied Mathematics Journal
2
,
59
(
2008
).
37.
D. T.
Gillespie
,
Am. J. Phys.
64
,
1246
(
1996
).
38.
C. -Y. F.
Huang
and
J. E.
Ferrell
,
Proc. Natl. Acad. Sci. U.S.A.
93
,
10078
(
1996
), http://www.pnas.org/content/93/19/10078 (abstract).
39.
V.
Sotiropoulos
,
M. -N.
Contou-Carrere
,
P.
Daoutidis
, and
Y. N.
Kaznessis
,
IEEE/ACM Trans. Comput. Biol. Bioinf.
6
,
470
(
2009
).
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