Abstract
An important aspect of systems biology research is the so-called “reverse engineering” of cellular metabolic dynamics from measured input-output data. This allows researchers to estimate and validate both the pathway’s structure as well as the kinetic constants. In this paper, the recently published ‘Proximate Parameter Tuning’ (PPT) method for the identification of biochemical networks is analysed. In particular, it is shown that the described PPT algorithm is essentially equivalent to a sequential linear programming implementation of a constrained optimization problem. The corresponding objective function consists of two parts, the first emphasises the data fitting where a residual 1-norm is used, and the second emphasises the proximity of the calculated parameters to the specified nominal values, using an ∞-norm. The optimality properties of PPT algorithm solution as well as its geometric interpretation are analyzed. The concept of optimal parameter locus is applied for the exploration of the entire family of optimal solutions. An efficient implementation of the parameter locus is also developed. Parallels are drawn with 1-norm parameter deviation regularization which attempt to fit the data with a minimal number of parameters. Finally, a small example is used to illustrate all of these properties.
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Banga, J.R., Moles, C.G., Alonso, A.A., 2003. Global optimization of bioprocesses using stochastic and hybrid methods. In: Frontiers in Global Optimization, Nonconvex Optimization and Its Applications, vol. 74, pp. 45–70. Kluwer Academic, Dordrecht.
Brown, M., He, F., Zhan, C., Yeung, L.F., 2008. Nonparametric collocation ODE parameter estimation: application in biochemical pathway modelling. In: UKACC International Conference on Control, September 2008, Manchester, UK.
Brown, M., He, F., Papadopoulos, G., 2009. Dynamic basis pursuit regularization for complex biochemical pathway identification. Accepted for IEEE Conference on Decision and Control, Shanghai, P.R. China, December.
Conrad, E.D., Tyson, J.J., 2006. Modeling molecular interaction networks with nonlinear ordinary differential equations. In: Szallasi, Z., Stelling, J., Periwal, V. (Eds.), Systems Modelling in Cellular Biology: From Concept to Nuts and Bolts. MIT Press, Cambridge.
Dimelow, R.J., Wilkinson, S.J., 2009. Control of translation initiation: a model-based analysis from limited experimental data. J. R. Soc. Interface 6, 51–62.
Golub, G.H., Hansen, P.C., Leary, D.P.O., 1999. Tikhonov regularization and total least squares. SIAM J. Matrix. Anal. Appl. 21, 185–194.
Gutenkunst, R.N., Waterfall, J.J., Casey, F.P., Brown, K.S., Myers, C.R., Sethna, J.P., 2007. Universally sloppy parameter sensitivities in systems biology models. PLoS Comput. Biol 3, 1871–1878.
Hansen, P.C., 1997. Rank-deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion. SIAM Monogr. Math. Model. Comput., vol. 4. SIAM, Philadelphia.
He, F., Brown, M., Yeung, L.F., 2008. On the complexity-sensitivity trade-off for the NF-κB pathway modelling. In: IEEE World Congress on Computational Intelligence, Hong Kong, pp. 3932–3939.
Jin, Y., Yue, H., Liang, Y., Kell, D.B., 2007. Improving data fitting of a signal transduction model by global sensitivity analysis. In: American Control Conference, New York City, USA, pp. 2708–2713.
Johansen, T.A., 1997. On Tikhonov regularization, bias and variance in nonlinear system identification. Automatica 33(3), 441–446.
Kozubowski, T.J., Podgorski, K., 2003. Log-Laplace distributions. Int. Math. J. 3, 467–495.
Lei, F., Jorgensen, S.B., 2001. Estimation of kinetic parameters in a structured yeast model using regularization. J. Biotechnol. 88, 223–237.
Liebermeister, W., Baur, U., Klipp, E., 2005. Biochemical network models simplified by balanced truncation. FEBS J. 272, 4034–4043.
Muller, S., Lu, J., Kugler, P., Engl, H.W., 2008. Parameter identification in systems biology: solving ill-posed inverse problem using regularization. Report of Johann Radon Institute for Computational and Applied Mathematics (RICAM), 2008-25. http://www.ricam.oeaw.ac.at/publications/reports/08/rep08-25.pdf.
Okino, M.S., Mavrovouniotis, M.L., 1998. Simplification of mathematical models of chemical reaction systems. Chem. Rev. 98, 391–408.
Papadopoulos, G., Brown, M., 2007. Feature sensitivity of biochemical signalling pathways. In: IEEE Symp. Comp. Intell. Bioinf. Comput. Biol., April 2007, Hawaii, USA, pp. 373–380.
Rodriguez, M., Mendes, P., Banga, J.R., 2006a. A hybrid approach for efficient and robust parameter estimation in biochemical pathways. Biosystems 83, 248–265.
Rodriguez, M., Egea, J.A., Banga, J.R., 2006b. Novel metaheuristic for parameter estimation in nonlinear dynamic biology systems. BMC Bioinformatics 7, 483–501.
Voit, E.O., 2000. Computational Analysis of Biochemical Systems. Cambridge University Press, Cambridge.
Wilkinson, S.J., Benson, N., Kell, D.B., 2008. Proximate Parameter Tuning for biochemical networks with uncertain kinetic parameters. Mol. Biosyst. 4, 74–97.
Yue, H., Brown, M., Kell, D.B., Knowles, J., Wang, H., Broomhead, D., 2006. Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of an NF-κB signalling pathway. Mol. Biosyst. 2(12), 640–649.
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Brown, M., He, F. & Wilkinson, S.J. Properties of the Proximate Parameter Tuning Regularization Algorithm. Bull. Math. Biol. 72, 697–718 (2010). https://doi.org/10.1007/s11538-009-9466-y
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DOI: https://doi.org/10.1007/s11538-009-9466-y