Identification of alterations in the Jacobian of biochemical reaction networks from steady state covariance data at two conditions. (English) Zbl 1300.92029

Kügler, Philipp; Yang, Wei

J. Math. Biol. 68, No. 7, 1757-1783 (2014).

Summary: Model building of biochemical reaction networks typically involves experiments in which changes in the behavior due to natural or experimental perturbations are observed. Computational models of reaction networks are also used in a systems biology approach to study how transitions from a healthy to a diseased state result from changes in genetic or environmental conditions. In this paper we consider the nonlinear inverse problem of inferring information about the Jacobian of a Langevin type network model from covariance data of steady state concentrations associated to two different experimental conditions. Under idealized assumptions on the Langevin fluctuation matrices we prove that relative alterations in the network Jacobian can be uniquely identified when comparing the two data sets. Based on this result and the premise that alteration is locally confined to separable parts due to network modularity we suggest a computational approach using hybrid stochastic-deterministic optimization for the detection of perturbations in the network Jacobian using the sparsity promoting effect of \(\ell_p\)-penalization. Our approach is illustrated by means of published metabolomic and signaling reaction networks.

MSC:

92C40	Biochemistry, molecular biology
62P10	Applications of statistics to biology and medical sciences; meta analysis
92-08	Computational methods for problems pertaining to biology
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

biochemical reaction networks; Langevin type network; Langevin fluctuation matrices

Software:

KEGGtranslator; COVAIN; SDE Toolbox; Algorithm 432; KEGG; Sbmltoolbox

References:

[1]	Aderem, A, Systems biology: its practice and challenges, Cell, 121, 511-513, (2005)
[2]	Aldridge, BB; Burke, JM; Lauffenburger, DA; Sorger, PK, Physicochemical modelling of cell signalling pathways, Nat Cell Biol, 8, 1195-1203, (2006)
[3]	Bartels, RH; Stewart, GW, Solution of the matrix equation \(ax + xb = c\), Commun ACM, 15, 820-826, (1972) · Zbl 1372.65121
[4]	Bruckstein, AM; Donoho, DL; Elad, M, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Rev, 51, 34-81, (2009) · Zbl 1178.68619
[5]	Butcher, EC; Berg, EL; Kunkel, EJ, Systems biology in drug discovery, Nat Biotechnol, 22, 1253-1259, (2004) · Zbl 1058.35129
[6]	Chen, WW; Niepel, M; Sorger, PK, Classic and contemporary approaches to modeling biochemical reactions, Genes Develop, 24, 1861-1875, (2010)
[7]	Daubechies, I; Defrise, M; De-Mol, C, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun Pure Appl Math, 57, 1413-1457, (2004) · Zbl 1077.65055
[8]	Sol, A; Balling, R; Hood, L; Galas, D, Diseases as network perturbations, Cell Curr Opin Biotechnol, 21, 566-571, (2010)
[9]	Eissing T, Kuepfer L, Becker C, Block M, Coboeken K, Gaub T, Goerlitz L, Jaeger J, Loosen R, Ludewig B, Meyer M, Niederalt C, Sevestre M, Siegmund HU, Solodenko J, Thelen K, Telle U, Weiss W, Wendl T, Willmann S, Lippert J (2011) A computational systems biology software platform for multiscale modeling and simulation: integrating whole-body physiology, disease biology, and molecular reaction networks. Front Comput Physiol Med 2:4 · Zbl 1194.65061
[10]	Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems. Kluwer, Dordrecht · Zbl 0859.65054
[11]	Engl, HW; Flamm, C; Kügler, P; Lu, J; Müller, S; Schuster, P, Inverse problems in systems biology, Inverse Probl, 25, 123014, (2009) · Zbl 1193.34001
[12]	Franklin GF, Powell JD, Emami-Naeini A (2002) Feedback Control of Dynamical Systems. Prentice Hall, Englewood Cliffs
[13]	Golub GH, van Loan CF (1996) Matrix computations. The Johns Hopkins University Press, Baltimore and London · Zbl 0865.65009
[14]	Golub, GH; Nash, S; Loan, CF, A Hessenberg-Schur method for the problem \(ax + xb = c\), IEEE Trans Auto Contr, 24, 909-913, (1979) · Zbl 0421.65022
[15]	Golub, GH; Hansen, PC; O’Leary, DP, Tikhonov regularization and total least squares, SIAM J Matrix Anal Appl, 21, 185-194, (1999) · Zbl 0945.65042
[16]	Grasmair, M; Haltmeier, M; Scherzer, O, Sparse regularization with \(ℓ _q\) penalty term, Inverse Probl, 24, 055020, (2008) · Zbl 1157.65033
[17]	Hammarling, SJ, Numerical solution of the stable, non-negative definite Lyapunov equation, IMA J Num Anal, 2, 303-325, (1982) · Zbl 0492.65017
[18]	Hood, L; Perlmutter, RM, The impact of systems approaches on biological problems in drug discovery, Nat Biotechnol, 22, 1215-1217, (2004)
[19]	Hood, L; Heath, JR; Phelps, ME; Lin, B, Systems biology and new technologies enable predictive and preventive medicine, Science, 306, 640-643, (2004)
[20]	Iglesias PA, Ingalls BP (eds) (2010) Control theory and systems biology. MIT Press, Cambridge · Zbl 0882.35134
[21]	Kanehisa, M; Goto, S; Furumichi, M; Tanabe, M; Hirakawa, M, Kegg for representation and analysis of molecular networks involving diseases and drugs, Nucleic Acids Res, 38, d355-d360, (2010)
[22]	Kärkkäinen, T, An equation error method to recover diffusion from the distributed observation, Inverse Probl, 13, 1033, (1997) · Zbl 0882.35134
[23]	Keating, SM; Bornstein, BJ; Finney, A; Hucka, M, Sbmltoolbox: an sbml toolbox for Matlab users, Bioinformatics, 22, 1275-1277, (2006)
[24]	Kitano, H, Systems biology: a brief overview, Science, 295, 1662-1664, (2002) · Zbl 1033.37007
[25]	Klipp E, Liebermeister W, Wierling C, Kowald A, Lehrach H, Herwig R (2009) Systems biology: a textbook. Wiley-VCH, Weinheim
[26]	Koide, T; Lee Pang, W; Baliga, NS, The role of predictive modelling in rationally re-engineering biological systems, Nat Rev Micro, 7, 297-305, (2009)
[27]	Kuepfer L, Lippert J, Eissing T (2012) Multiscale mechanistic modeling in pharmaceutical research and development. In: Goryanin II, Goryachev AB (eds) Advances in systems biology, Advances in Experimental Medicine and Biology, vol 736. Springer, New York, pp 543-561
[28]	Kügler, P, Moment Fitting for parameter inference in repeatedly and partially observed stochastic biological models, PLoS ONE, 7, e43001, (2012)
[29]	Lai, MJ, On sparse solution of underdetermined linear systems, J Concr Appl Math, 8, 296-327, (2010) · Zbl 1194.65061
[30]	Ljung L (1998) System Identification: Theory for the User. Pearson Education, New York · Zbl 0615.93004
[31]	Lu, S; Pereverzev, SV; Tautenhahn, U, Regularized total least squares: computational aspects and error bounds, SIAM J Matrix Anal Appl, 31, 918-941, (2009) · Zbl 1198.65094
[32]	Maslov S, Ispolatov I (2007) Propagation of large concentration changes in reversible protein-binding networks. Proc Natl Acad Sci 104(34):13655-13660
[33]	Maslov, S; Sneppen, K; Ispolatov, I, Spreading out of perturbations in reversible reaction networks, New J Phys, 9, 273, (2007)
[34]	Moutselos, K; Kanaris, I; Chatziioannou, A; Maglogiannis, I; Kolisis, F, Keggconverter: a tool for the in-silico modelling of metabolic networks of the kegg pathways database, BMC Bioinf, 10, 324, (2009)
[35]	Orton RJ, Adriaens ME, Gormand A, Sturm OE, Kolch W, Gilbert DR (2009) Computational modelling of cancerous mutations in the egfr/erk signalling pathway. BMC Syst Biol 3:100
[36]	Picchini U (2007) SDE toolbox: simulation and estimation of stochastic differential equations with MATLAB. http://sdetoolbox.sourceforge.net
[37]	Ramlau R, Teschke G (2010) Sparse recovery in inverse problems. In: Fornasier M (ed) Theoretical foundations and numerical methods for sparse recovery, radon series on Computational and Applied Mathematics, vol 9. deGruyter, New York, pp 1-63 · Zbl 1210.65107
[38]	Scott M (2011) Applied stochastic processes in science and engineering. Free e-book, http://www.math.uwaterloo.ca · Zbl 1365.62208
[39]	Slotine JE, Li W (1991) Applied nonlinear control. Prentice-Hall, Englewood Cliffs · Zbl 0753.93036
[40]	Steuer, R; Kurths, J; Fiehn, O; Weckwerth, W, Observing and interpreting correlations in metabolomic networks, Bioinformatics, 19, 1019-1026, (2003)
[41]	Steuer R, Gross T, Selbig J, Blasius B (2006) Structural kinetic modeling of metabolic networks. Proc Natl Acad Sci 103(32):11868-11873
[42]	Sun, X; Weckwerth, W, Covain: a toolbox for uni- and multivariate statistics, time- series and correlation network analysis and inverse estimation of the differential Jacobian from metabolomics covariance data, Metabolomics, 306, 640-643, (2012)
[43]	Szallasi Z, Stelling J, Periwal V (eds) (2006) System modeling in cellular biology: from concepts to nuts and bolts. MIT Press, Cambridge · Zbl 1274.92012
[44]	Van Kampen NG (2007) Stochastic processes in physics and chemistry. North Holland, Amsterdam
[45]	Vershynin, R, How close is the sample covariance matrix totheactual covariance matrix?, J Theor Probab, 25, 655-686, (2012) · Zbl 1365.62208
[46]	Wilkinson JD (2012) Stochastic modelling for systems biology, 2nd edn. Chapman & Hall/CRC, London · Zbl 1300.92004
[47]	Wrzodek, C; Dräger, A; Zell, A, Keggtranslator: visualizing and converting the kegg pathway database to various formats, Bioinformatics, 27, 2314-2315, (2011)
[48]	Zarzer, CA, On Tikhonov regularization with non-convex sparsity constraints, Inverse Probl, 25, 025006, (2009) · Zbl 1161.65044