Stochastic effects dominate many chemical and biochemical processes. Their analysis, however, can be computationally prohibitively expensive and a range of approximation schemes have been proposed to lighten the computational burden. These, notably the increasingly popular linear noise approximation and the more general moment expansion methods, perform well for many dynamical regimes, especially linear systems. At higher levels of nonlinearity, it comes to an interplay between the nonlinearities and the stochastic dynamics, which is much harder to capture correctly by such approximations to the true stochastic processes. Moment-closure approaches promise to address this problem by capturing higher-order terms of the temporally evolving probability distribution. Here, we develop a set of multivariate moment-closures that allows us to describe the stochastic dynamics of nonlinear systems. Multivariate closure captures the way that correlations between different molecular species, induced by the reaction dynamics, interact with stochastic effects. We use multivariate Gaussian, gamma, and lognormal closure and illustrate their use in the context of two models that have proved challenging to the previous attempts at approximating stochastic dynamics: oscillations in p53 and Hes1. In addition, we consider a larger system, Erk-mediated mitogen-activated protein kinases signalling, where conventional stochastic simulation approaches incur unacceptably high computational costs.
REFERENCES
1.
D. J.
Wilkinson
, “Stochastic modelling for quantitative description of heterogeneous biological systems
,” Nat. Rev. Genet.
10
(2
), 122
–133
(2009
).2.
D. T.
Gillespie
, “Exact stochastic simulation of coupled chemical reactions
,” J. Phys. Chem.
81
(25
), 2340
–2361
(1977
).3.
T. R.
Kiehl
, R. M.
Mattheyses
, and M. K.
Simmons
, “Hybrid simulation of cellular behavior
,” Bioinformatics
20
(3
), 316
–322
(2004
).4.
J.
Puchalka
and A. M.
Kierzek
, “Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks
,” Biophys. J.
86
(3
), 1357
–1372
(2004
).5.
M.
Komorowski
, J.
Mikisz
, and M. P. H.
Stumpf
, “Decomposing noise in biochemical signaling systems highlights the role of protein degradation
,” Biophys. J.
104
(8
), 1783
–1793
(2013
).6.
M.
Komorowski
, B.
Finkenstadt
, C.
Harper
, and D.
Rand
, “Bayesian inference of biochemical kinetic parameters using the linear noise approximation
,” BMC Bioinf.
10
(1
), 343
(2009
).7.
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
(4
), 102
–115
(2012
).8.
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
(2
), 024109
(2007
).9.
M.
Ullah
and O.
Wolkenhauer
, “Investigating the two-moment characterisation of subcellular biochemical networks
,” J. Theor. Biol.
260
(3
), 340
–352
(2009
).10.
A.
Ale
, P.
Kirk
, and M. P. H.
Stumpf
, “A general moment expansion method for stochastic kinetic models
,” J. Chem. Phys.
138
(17
), 174101
(2013
).11.
L. A.
Goodman
, “Population growth of the sexes
,” in Mathematical Demography
, Biomathematics
(Springer
, Berlin, Heidelberg
, 1977
), Vol. 6
, pp. 469
–480
.12.
P.
Whittle
, “On the use of the normal approximation in the treatment of stochastic processes
,” J. R. Stat. Soc., Ser. B
19
(2
), 268
–281
(1957
).13.
T. I.
Matis
and I. G.
Guardiola
, “Achieving moment closure through cumulant neglect
,” Math. J.
12
(1
), 2:1
–2:18
(2010
).14.
C. S.
Gillespie
, “Moment-closure approximations for mass-action models
,” IET Syst. Biol.
3
(1
), 52
–58
(2009
).15.
P.
Milner
, C. S.
Gillespie
, and D. J.
Wilkinson
, “Moment closure approximations for stochastic kinetic models with rational rate laws
,” Math. Biosci.
231
(2
), 99
–104
(2011
).16.
I.
Krishnarajah
, A.
Cook
, G.
Marion
, and G.
Gibson
, “Novel moment closure approximations in stochastic epidemics
,” Bull. Math. Biol.
67
(4
), 855
–873
(2005
).17.
I.
Krishnarajah
, G.
Marion
, and G.
Gibson
, “Novel bivariate moment-closure approximations
,” Math. Biosci.
208
(2
), 621
–643
(2007
).18.
C. H.
Lee
, K.-H.
Kim
, and P.
Kim
, “A moment closure method for stochastic reaction networks
,” J. Chem. Phys.
130
(13
), 134107
(2009
).19.
M. J.
Keeling
, “Metapopulation moments: Coupling, stochasticity and persistence
,” J. Anim. Ecol.
69
(5
), 725
–736
(2000
).20.
A.
Singh
and J. P.
Hespanha
, “Lognormal moment closures for biochemical reactions
,” in Proceedings of the 45th IEEE Conference on Decision and Control
(IEEE
, 2006
), pp. 2063
–2068
.21.
A.
Singh
and J. P.
Hespanha
, “A derivative matching approach to moment closure for the stochastic logistic model
,” Bull. Math. Biol.
69
(6
), 1909
–1925
(2007
).22.
A.
Singh
and J. P.
Hespanha
, “Approximate moment dynamics for chemically reacting systems
,” IEEE Trans. Autom. Control
56
(2
), 414
–418
(2011
).23.
K.
Hausken
and J. F.
Moxnes
, “A closure approximation technique for epidemic models
,” Math. Comput. Modell. Dyn. Syst.
16
, 555
–574
(2010
).24.
P.
Smadbeck
and Y. N.
Kaznessis
, “A closure scheme for chemical master equations
,” Proc. Natl. Acad. Sci. U. S. A.
110
(35
), 14261
–14265
(2013
).25.
R.
Grima
, “A study of the accuracy of moment-closure approximations for stochastic chemical kinetics
,” J. Chem. Phys.
136
(15
), 154105
(2012
).26.
J. P.
Hespanha
, StochDynTools— a MATLAB toolbox to compute moment dynamics for stochastic networks of bio-chemical reactions. Available at http://www.ece.ucsb.edu/~hespanha, 2007.27.
P.
Milner
, C. S.
Gillespie
, and D. J.
Wilkinson
, “Moment closure based parameter inference of stochastic kinetic models
,” Stat. Comput.
23
(2
), 287
–295
(2013
).28.
T.
Toni
, D.
Welch
, N.
Strelkowa
, A.
Ipsen
, and M. P. H.
Stumpf
, “Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems
,” J. R. Soc., Interface
6
(31
), 187
–202
(2009
).29.
I. G.
Johnston
, “Efficient parametric inference for stochastic biological systems with measured variability
,” Stat. Appl. Genet. Mol. Biol.
13
(3
), 379
–390
(2014
).30.
J.
Owen
, D. J.
Wilkinson
, and C. S.
Gillespie
, “Likelihood free inference for Markov processes: A comparison
,” Stat. Appl. Genet. Mol. Biol.
14
(2
), 189
–209
(2015
).31.
P.
Kirk
, T. W.
Thorne
, and M. P. H.
Stumpf
, “Model selection in systems and synthetic biology
,” Curr. Opin. Biotechnol.
24
(4
), 767
–774
(2013
).32.
P.
Fearnhead
, D.
Prangle
, M. P.
Cox
, P. J.
Biggs
, and N. P.
French
, “Semi-automatic selection of summary statistics for ABC model choice
,” Stat. Appl. Genet. Mol. Biol.
13
(1
), 67
–82
(2014
).33.
L. R.
Mead
and N.
Papanicolaou
, “Maximum-entropy in the problem of moments
,” J. Math. Phys.
25
, 2404
–2417
(1984
).34.
M.
Komorowski
, M. J.
Costa
, D. A.
Rand
, and M. P. H.
Stumpf
, “Sensitivity, robustness, and identifiability in stochastic chemical kinetics models
,” Proc. Natl. Acad. Sci. U. S. A.
108
(21
), 8645
–8650
(2011
).35.
J.
Liepe
, S.
Filippi
, M.
Komorowski
, and M. P. H.
Stumpf
, “Maximizing the information content of experiments in systems biology
,” PLoS Comput. Biol.
9
(1
), e1002888
(2013
).36.
D.
Silk
, P.
Kirk
, C. P.
Barnes
, T.
Toni
, and M. P. H.
Stumpf
, “Model selection in systems biology depends on experimental design
,” PLoS Comput. Biol.
10
(6
), e1003650
(2014
).37.
D.
Silk
, P. D. W.
Kirk
, C. P.
Barnes
, T.
Toni
, A.
Rose
, S.
Moon
, M. J.
Dallman
, and M. P. H.
Stumpf
, “Designing attractive models via automated identification of chaotic and oscillatory dynamical regimes
,” Nat. Commun.
2
, 489
(2011
).38.
C. P.
Barnes
, D.
Silk
, X.
Sheng
, and M. P. H.
Stumpf
, “Bayesian design of synthetic biological systems
,” Proc. Natl. Acad. Sci. U. S. A.
108
(37
), 15190
–15195
(2011
).39.
P.
Smadbeck
and Y. N.
Kaznessis
, “Chemical master equation closure for computer-aided synthetic biology
,” Methods Mol. Biol. (Clifton, N.J.)
1244
, 179
–191
(2015
).40.
Y.
Taniguchi
, P. J.
Choi
, G.-W.
Li
, H.
Chen
, M.
Babu
, J.
Hearn
, A.
Emili
, and X. S.
Xie
, “Quantifying e. coli proteome and transcriptome with single-molecule sensitivity in single cells
,” Science
329
(5991
), 533
–538
(2010
).41.
N.
Friedman
, L.
Cai
, and X. S.
Xie
, “Linking stochastic dynamics to population distribution: An analytical framework of gene expression
,” Phys. Rev. Lett.
97
(16
), 168302
(2006
).42.
L.
Isserlis
, “On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables
,” Biometrika
12
(1-2
), 134
–139
(1918
).43.
A.
Stefanek
, M. C.
Guenther
, and J. T.
Bradley
, “Normal and inhomogeneous moment closures for stochastic process algebras
,” in 10th Workshop on Process Algebra and Stochastically Timed Activities 2011 (PASTA’11)
, September 2011
.44.
E. L.
Crow
and K.
Shimizu
, Lognormal Distributions: Theory and Applications
, Statistics, Textbooks and Monographs
(M. Dekker
, New York
, 1988
), Vol. 88
.45.
A. M.
Mathai
and P. G.
Moschopoulos
, “On a multivariate-gamma
,” J. Multivar. Anal.
39
, 135
–153
(1991
).46.
E.
Furman
, “On a multivariate gamma distribution
,” Stat. Probab. Lett.
78
, 2353
–2360
(2008
).47.
N.
Geva-Zatorsky
, N.
Rosenfeld
, S.
Itzkovitz
, R.
Milo
, A.
Sigal
, E.
Dekel
, T.
Yarnitzky
, Y.
Liron
, P.
Polak
, G.
Lahav
, and U.
Alon
, “Oscillations and variability in the p53 system
,” Mol. Syst. Biol.
2
, 2006.0033
(2006
).48.
H. A.
Harrington
, M.
Komorowski
, M.
Beguerisse-Díaz
, G. M.
Ratto
, and M. P. H.
Stumpf
, “Mathematical modeling reveals the functional implications of the different nuclear shuttling rates of erk1 and erk2
,” Phys. Biol.
9
(3
), 036001
(2012
).49.
P.
Kügler
, “Moment fitting for parameter inference in repeatedly and partially observed stochastic biological models
,” PLoS One
7
(8
), e43001
(2012
).50.
B.
Munsky
, B.
Trinh
, and M.
Khammash
, “Listening to the noise: Random fluctuations reveal gene network parameters
,” Mol. Syst. Biol.
5
(1
), 318
(2009
).51.
E. G.
Gilbert
, “The decoupling of multivariable systems by state feedback
,” SIAM J. Control
7
(1
), 50
–63
(1969
).52.
J. F.
Apgar
, J. E.
Toettcher
, D.
Endy
, F. M.
White
, and B.
Tidor
, “Stimulus design for model selection and validation in cell signaling
,” PLoS Comput. Biol.
4
(2
), e30
(2008
).53.
V.
John
, I.
Angelov
, A. A.
Oencuel
, and D.
Thevenin
, “Techniques for the reconstruction of a distribution from a finite number of its moments
,” Chem. Eng. Sci.
62
, 2890
–2904
(2007
).54.
J.
Hasenauer
, V.
Wolf
, A.
Kazeroonian
, and F. J.
Theis
, “Method of conditional moments (mcm) for the chemical master equation : A unified framework for the method of moments and hybrid stochastic-deterministic models
,” J. Math. Biol.
69
(3
), 687
–735
(2013
).55.
C. H.
Waddington
, The Strategy of the Genes: A Discussion of Some Aspects of Theoretical Biology
(Allen & Unwin
, London
, 1957
).56.
C.
Li
and J.
Wang
, “Quantifying Waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation
,” J. R. Soc., Interface
10
(89
), 20130787
(2013
).57.
P.
Smadbeck
and Y. N.
Kaznessis
, “On a theory of stability for nonlinear stochastic chemical reaction networks
,” J. Chem. Phys.
142
(18
), 184101
(2015
).58.
D.
Schnoerr
, G.
Sanguinetti
, and R.
Grima
, “Validity conditions for moment closure approximations in stochastic chemical kinetics
,” J. Chem. Phys.
141
(8
), 084103
(2014
).59.
K.
Erguler
and M. P. H.
Stumpf
, “Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map
,” Math. Biosci.
216
, 90
–99
(2008
).60.
A.
Golightly
and D. J.
Wilkinson
, “Bayesian inference for stochastic kinetic models using a diffusion approximation
,” Biometrics
61
(3
), 781
–788
(2005
).61.
A.
Golightly
and D. J.
Wilkinson
, “Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo
,” Interface Focus
1
(6
), 807
–820
(2011
).62.
A Python package that implements the method described here is available from our web page http://www.theosysbio.bio.ic.ac.uk.
© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
You do not currently have access to this content.
