Steady state is an essential concept in reaction networks. Its stability reflects fundamental characteristics of several biological phenomena such as cellular signal transduction and gene expression. Because biochemical reactions occur at the cellular level, they are affected by unavoidable fluctuations. Although several methods have been proposed to detect and analyze the stability of steady states for deterministic models, these methods cannot be applied to stochastic reaction networks. In this paper, we propose an algorithm based on algebraic computations to calculate parameter regions for constrained steady-state distribution of stochastic reaction networks, in which the means and variances satisfy some given inequality constraints. To evaluate our proposed method, we perform computer simulations for three typical chemical reactions and demonstrate that the results obtained with our method are consistent with the simulation results.

1.
M.
Thattai
and
A.
van Oudenaarden
, “
Intrinsic noise in gene regulatory networks
,”
Proc. Natl. Acad. Sci. U.S.A.
98
,
8614
8619
(
2001
).
2.
T.
Shibata
and
K.
Fujimoto
, “
Noisy signal amplification in ultrasensitive signal transduction
,”
Proc. Natl. Acad. Sci. U.S.A.
102
,
331
336
(
2005
).
3.
G.
Fichera
,
M. A.
Sneider
, and
J.
Wyman
, “
On the existence of a steady state in a biological system
,”
Proc. Natl. Acad. Sci. U.S.A.
74
,
4182
4184
(
1977
).
4.
D.
Angeli
,
J. E.
Ferrell
, and
E. D.
Sontag
, “
Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems
,”
Proc. Natl. Acad. Sci. U.S.A.
101
,
1822
1827
(
2004
).
5.
K.
Gatermann
,
M.
Eiswirth
, and
A.
Sensse
, “
Toric ideals and graph theory to analyze hopf bifurcations in mass action systems
,”
J. Symb. Comput.
40
,
1361
1382
(
2005
).
6.
G.
Craciun
,
A.
Dickenstein
,
A.
Shiu
, and
B.
Sturmfels
, “
Toric dynamical systems
,”
J. Symb. Comput.
44
,
1551
1565
(
2009
).
7.
M.
Mincheva
, “
Oscillations in non-mass action kinetics models of biochemical reaction networks arising from pairs of subnetworks
,”
J. Math. Chem.
50
,
1111
1125
(
2012
).
8.
M.
Domijan
and
M.
Kirkilionis
, “
Bistability and oscillations in chemical reaction networks
,”
J. Math. Biol.
59
,
467
501
(
2009
).
9.
L. S.
Tsimring
, “
Noise in biology
,”
Rep. Prog. Phys.
77
,
026601
(
2014
).
10.
Y.
Hasegawa
and
M.
Arita
, “
Optimal implementations for reliable circadian clocks
,”
Phys. Rev. Lett.
113
,
108101
(
2014
).
11.
D. A.
McQuarrie
, “
Stochastic approach to chemical kinetics
,”
J. Appl. Probab.
4
,
413
478
(
1967
).
12.
D. T.
Gillespie
, “
A rigorous derivation of the chemical master equation
,”
Physica A
188
,
404
425
(
1992
).
13.
D. T.
Gillespie
, “
Exact stochastic simulation of coupled chemical reactions
,”
J. Phys. Chem.
81
,
2340
2361
(
1977
).
14.
M. A.
Gibson
and
J.
Bruck
, “
Efficient exact stochastic simulation of chemical systems with many species and many channels
,”
J. Phys. Chem. A
104
,
1876
1889
(
2000
).
15.
D. T.
Gillespie
, “
Approximate accelerated stochastic simulation of chemically reacting systems
,”
J. Chem. Phys.
115
,
1716
1733
(
2001
).
16.
B.
Munsky
and
M.
Khammash
, “
The finite state projection algorithm for the solution of the chemical master equation
,”
J. Chem. Phys.
124
,
044104
(
2006
).
17.
A.
Gupta
,
J.
Mikelson
, and
M.
Khammash
, “
A finite state projection algorithm for the stationary solution of the chemical master equation
,”
J. Chem. Phys.
147
,
154101
(
2017
).
18.
V.
Kazeev
,
M.
Khammash
,
M.
Nip
, and
C.
Schwab
, “
Direct solution of the chemical master equation using quantized tensor trains
,”
PLoS Comput. Biol.
10
,
1
19
(
2014
).
19.
D.
Wang
and
B.
Xia
, “Stability analysis of biological systems with real solution classification,” in Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation (ACM, 2005), pp. 354–361.
20.
A. K.
Manrai
and
J.
Gunawardena
, “
The geometry of multisite phosphorylation
,”
Biophys. J.
95
,
5533
5543
(
2008
).
21.
M.
Thomson
and
J.
Gunawardena
, “
The rational parameterisation theorem for multisite post-translational modification systems
,”
J. Theor. Biol.
261
,
626
636
(
2009
).
22.
I.
Martínez-Forero
,
A.
Peláez-López
, and
P.
Villoslada
, “
Steady state detection of chemical reaction networks using a simplified analytical method
,”
PLoS One
5
,
e10823
(
2010
).
23.
P. M.
Loriaux
,
G.
Tesler
, and
A.
Hoffmann
, “
Characterizing the relationship between steady state and response using analytical expressions for the steady states of mass action models
,”
PLoS Comput. Biol.
9
,
e1002901
(
2013
).
24.
D.
Siegal-Gaskins
,
E.
Franco
,
T.
Zhou
, and
R. M.
Murray
, “
An analytical approach to bistable biological circuit discrimination using real algebraic geometry
,”
J. R. Soc. Interface
12
,
20150288
(
2015
).
25.
I.
Otero-Muras
,
P.
Yordanov
, and
J.
Stelling
, “
Chemical reaction network theory elucidates sources of multistability in interferon signaling
,”
PLoS Comput. Biol.
13
,
1
28
(
2017
).
26.
J. M. M.
González
, “
Revealing regions of multiple steady states in heterogeneous catalytic chemical reaction networks using Gröbner basis
,”
J. Symb. Comput.
80
,
521
537
(
2017
).
27.
D. A.
Cox
,
J.
Little
, and
D.
O’Shea
,
Ideals, Varieties, and Algorithms
(
Springer-Verlag
,
New York
,
2007
).
28.
A.
Gupta
,
C.
Briat
, and
M.
Khammash
, “
A scalable computational framework for establishing long-term behavior of stochastic reaction networks
,”
PLoS Comput. Biol.
10
,
1
16
(
2014
).
29.
A.
Gupta
and
M.
Khammash
, “
Computational identification of irreducible state-spaces for stochastic reaction networks
,”
SIAM J. Appl. Dyn. Syst.
17
,
1213
1266
(
2018
).
30.
E. M.
Ozbudak
,
M.
Thattai
,
I.
Kurtser
,
A. D.
Grossman
, and
A.
van Oudenaarden
, “
Regulation of noise in the expression of a single gene
,”
Nat. Genet.
31
,
69
(
2002
).
31.
M.
Kæn
,
W. J.
Blake
, and
J.
Collins
, “
The engineering of gene regulatory networks
,”
Annu. Rev. Biomed. Eng.
5
,
179
206
(
2003
).
32.
J.
Kuntz
,
P.
Thomas
,
G.-B.
Stan
, and
M.
Barahona
, “Rigorous bounds on the stationary distributions of the chemical master equation via mathematical programming,” e-print arXiv:1702.05468 (2017).
33.
Y.
Sakurai
and
Y.
Hori
, “
Optimization-based synthesis of stochastic biocircuits with statistical specifications
,”
J. R. Soc. Interface
15
,
20170709
(2018
).
34.
J.
Elf
and
E.
Måns
, “
Fast evaluation of fluctuations in biochemical networks with the linear noise approximation
,”
Genome Res.
13
,
2475
2484
(
2003
).
35.
M.
Scott
,
B.
Ingalls
, and
M.
Kærn
, “
Estimations of intrinsic and extrinsic noise in models of nonlinear genetic networks
,”
Chaos
16
,
026107
(
2006
).
36.
N. V.
Kampen
,
Stochastic Processes in Physics and Chemistry
(
Elsevier
,
Amsterdam
,
2007
).
37.
E. W. J.
Wallace
,
D. T.
Gillespie
,
K. R.
Sanft
, and
L. R.
Petzold
, “
Linear noise approximation is valid over limited times for any chemical system that is sufficiently large
,”
IET Syst. Biol.
6
,
102
115
(
2012
).
38.
C. A.
Gómez-Uribe
and
G. C.
Verghese
, “
Mass fluctuation kinetics: Capturing stochastic effects in systems of chemical reactions through coupled mean-variance computations
,”
J. Chem. Phys.
126
,
024109
(
2007
).
39.
L.
Ferm
,
P.
Lötstedt
, and
A.
Hellander
, “
A hierarchy of approximations of the master equation scaled by a size parameter
,”
J. Sci. Comput.
34
,
127
151
(
2008
).
40.
C. H.
Lee
,
K. H.
Kim
, and
P.
Kim
, “
A moment closure method for stochastic reaction networks
,”
J. Chem. Phys.
130
,
134107
(
2009
).
41.
C. S.
Gillespie
, “
Moment-closure approximations for mass-action models
,”
IET Syst. Biol.
3
,
52
58
(
2009
).
42.
R.
Grima
, “
A study of the accuracy of moment-closure approximations for stochastic chemical kinetics
,”
J. Chem. Phys.
136
,
154105
(
2012
).
43.
A.
Ale
,
P.
Kirk
, and
M. P. H.
Stumpf
, “
A general moment expansion method for stochastic kinetic models
,”
J. Chem. Phys.
138
,
174101
(
2013
).
44.
D. T.
Gillespie
, “
The chemical Langevin equation
,”
J. Chem. Phys.
113
,
297
306
(
2000
).
45.
D.
Schnoerr
,
G.
Sanguinetti
, and
R.
Grima
, “
Approximation and inference methods for stochastic biochemical kinetics—A tutorial review
,”
J. Phys. A Math. Theor.
50
,
093001
(
2017
).
46.
R.
Grima
, “
An effective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions
,”
J. Chem. Phys.
133
,
035101
(
2010
).
47.
P.
Thomas
,
H.
Matuschek
, and
R.
Grima
, “
How reliable is the linear noise approximation of gene regulatory networks?
,”
BMC Genomics
14
,
S5
(
2013
).
48.
Z.
Cao
and
R.
Grima
, “
Linear mapping approximation of gene regulatory networks with stochastic dynamics
,”
Nat. Commun.
9
,
3305
(
2018
).
49.
B.
Mishra
,
Algorithmic Algebra
(
Springer-Verlag
,
New York
,
1993
).
50.
D.
Wang
, “
Decomposing polynomial systems into simple systems
,”
J. Symb. Comput.
25
,
295
314
(
1998
).
51.
R.
Grima
, “
Linear-noise approximation and the chemical master equation agree up to second-order moments for a class of chemical systems
,”
Phys. Rev. E
92
,
042124
(
2015
).
52.
M. J.
Keeling
, “
Multiplicative moments and measures of persistence in ecology
,”
J. Theor. Biol.
205
,
269
281
(
2000
).
53.
I.
Nåsell
, “
An extension of the moment closure method
,”
Theor. Popul. Biol.
64
,
233
239
(
2003
).
54.
D.
Schnoerr
,
G.
Sanguinetti
, and
R.
Grima
, “
Comparison of different moment-closure approximations for stochastic chemical kinetics
,”
J. Chem. Phys.
143
,
185101
(
2015
).
55.
D.
Wang
, “
Computing triangular systems and regular systems
,”
J. Symb. Comput.
30
,
221
236
(
2000
).
56.
L.
Yang
,
X.
Hou
, and
B.
Xia
, “
A complete algorithm for automated discovering of a class of inequality-type theorems
,”
Sci. China Ser. F
44
,
33
49
(
2001
).
57.
L.
Yang
and
B.
Xia
, “Real solution classifications of parametric semi-algebraic systems,” in Algorithmic Algebra and Logic – Proceedings of the A3L 2005 (Verlag, 2005), pp. 281–289.
58.
D. S.
Arnon
,
G. E.
Collins
, and
S.
McCallum
, “
Cylindrical algebraic decomposition I: The basic algorithm
,”
SIAM J. Comput.
13
,
865
877
(
1984
).
59.
D. S.
Arnon
,
G. E.
Collins
, and
S.
McCallum
, “
Cylindrical algebraic decomposition II: An adjacency algorithm for the plane
,”
SIAM J. Comput.
13
,
878
889
(
1984
).
60.
S. K.
Hortsch
and
A.
Kremling
, “
Characterization of noise in multistable genetic circuits reveals ways to modulate heterogeneity
,”
PLoS One
13
,
1
20
(
2018
).
61.
R.
Grima
,
N. G.
Walter
, and
S.
Schnell
, “
Single-molecule enzymology à la michaelismenten
,”
FEBS J.
281
,
518
530
(
2014
).
62.
I.
Prigogine
and
R.
Lefever
, “
Symmetry breaking instabilities in dissipative systems. II
,”
J. Chem. Phys.
48
,
1695
1700
(
1968
).
63.
M. B.
Elowitz
,
A. J.
Levine
,
E. D.
Siggia
, and
P. S.
Swain
, “
Stochastic gene expression in a single cell
,”
Science
297
,
1183
1186
(
2002
).
64.
Y.
Togashi
and
K.
Kaneko
, “
Transitions induced by the discreteness of molecules in a small autocatalytic system
,”
Phys. Rev. Lett.
86
,
2459
2462
(
2001
).
65.
D. F.
Anderson
,
G. A.
Enciso
, and
M. D.
Johnston
, “
Stochastic analysis of biochemical reaction networks with absolute concentration robustness
,”
J. R. Soc. Interface
11
,
20130943
(
2014
).
66.
D.
Anderson
and
D.
Cappelletti
, “Discrepancies between extinction events and boundary equilibria in reaction networks,” e-print arXiv:1809.04613 (2018).
67.
D.
Schnoerr
,
G.
Sanguinetti
, and
R.
Grima
, “
Validity conditions for moment closure approximations in stochastic chemical kinetics
,”
J. Chem. Phys.
141
,
084103
(
2014
).
68.
P.
Thomas
,
N.
Popović
, and
R.
Grima
, “
Phenotypic switching in gene regulatory networks
,”
Proc. Natl. Acad. Sci. U.S.A.
111
,
6994
6999
(
2014
).
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