Approximate solutions of the chemical master equation and the chemical Fokker-Planck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the moment-closure approach used to obtain such approximations, calling it an ad hoc method. In this article, we give a new variational derivation of moment-closure equations which provides us with an intuitive understanding of their properties and failure modes and allows us to correct some of these problems. We use mixtures of product-Poisson distributions to obtain a flexible parametric family which solves the commonly observed problem of divergences at low system sizes. We also extend the recently introduced entropic matching approach to arbitrary ansatz distributions and Markov processes, demonstrating that it is a special case of variational moment closure. This provides us with a particularly principled approximation method. Finally, we extend the above approaches to cover the approximation of multi-time joint distributions, resulting in a viable alternative to process-level approximations which are often intractable.

1.
N. G.
van Kampen
,
Stochastic Processes in Physics and Chemistry
(
Elsevier
,
2007
).
2.
C.
Kuehn
,
Control of Self-Organizing Nonlinear Systems
(
Springer
,
2016
), pp.
253
271
.
3.
D.
Schnoerr
,
G.
Sanguinetti
, and
R.
Grima
,
J. Chem. Phys.
141
,
084103
(
2014
).
4.
D.
Schnoerr
,
G.
Sanguinetti
, and
R.
Grima
,
J. Chem. Phys.
143
,
185101
(
2015
).
5.
T.
Ramalho
,
M.
Selig
,
U.
Gerland
, and
T. A.
Enßlin
,
Phys. Rev. E
87
,
022719
(
2013
).
6.
D. T.
Gillespie
,
J. Chem. Phys.
113
,
297
(
2000
).
7.
G. L.
Eyink
,
Phys. Rev. E
54
,
3419
(
1996
).
8.
R. H.
Leike
and
T. A.
Enßlin
,
Entropy
19
,
402
(
2017
).
9.
D.
Brigo
,
B.
Hanzon
, and
F.
Le Gland
,
Bernoulli
5
,
495
(
1999
).
10.
D.
Brigo
and
G.
Pistone
,
Computational Information Geometry
(
Springer
,
2017
), pp.
217
265
.
11.
C.
Archambeau
,
D.
Cornford
,
M.
Opper
, and
J.
Shawe-Taylor
, in
Gaussian Processes in Practice
(PMLR,
2007
), pp.
1
16
.
12.
M.
Opper
and
G.
Sanguinetti
,
Advances in Neural Information Processing Systems 20
(The MIT Press,
2008
), pp.
1105
1112
.
13.
T.
Sutter
,
A.
Ganguly
, and
H.
Koeppl
,
J. Mach. Learn. Res.
17
,
6544
6580
(
2016
).
14.
B.
Bravi
and
P.
Sollich
,
Phys. Biol.
14
,
045010
(
2017
).
15.
J.
Trȩbicki
and
K.
Sobczyk
,
Probab. Eng. Mech.
11
,
169
(
1996
).
16.
P.
Smadbeck
and
Y. N.
Kaznessis
,
Proc. Natl. Acad. Sci. U. S. A.
110
,
14261
(
2013
).
17.
A.
Andreychenko
,
L.
Mikeev
, and
V.
Wolf
,
ACM Trans. Model. Comput. Simul.
25
(
2
), (
2015
).
18.
D. F.
Anderson
,
G.
Craciun
, and
T. G.
Kurtz
,
Bull. Math. Biol.
72
,
1947
(
2010
).
19.
D. F.
Anderson
and
S. L.
Cotter
,
Bull. Math. Biol.
78
,
2390
(
2016
).
20.
B.
Munsky
and
M.
Khammash
,
J. Chem. Phys.
124
,
044104
(
2006
).
21.
P.
Deuflhard
,
W.
Huisinga
,
T.
Jahnke
, and
M.
Wulkow
,
SIAM J. Sci. Comput.
30
,
2990
(
2008
).
22.
E.
Lakatos
,
A.
Ale
,
P. D.
Kirk
, and
M. P.
Stumpf
,
J. Chem. Phys.
143
,
094107
(
2015
).
23.
C.
Gardiner
and
S.
Chaturvedi
,
J. Stat. Phys.
17
,
429
(
1977
).
24.
J.
Ohkubo
,
J. Chem. Phys.
129
,
044108
(
2008
).
25.
M.
Infusino
,
T.
Kuna
,
J. L.
Lebowitz
, and
E. R.
Speer
,
J. Math. Anal. Appl.
452
,
443
(
2017
).
You do not currently have access to this content.