Planning identification experiments for cell signaling pathways: An NFκB case study
Mathematical modeling of cell signaling pathways has become a very important and challenging problem in recent years. The importance comes from possible applications of obtained models. It may help us to understand phenomena appearing in single cells and cell populations on a molecular level. Furthermore, it may help us with the discovery of new drug therapies. Mathematical models of cell signaling pathways take different forms. The most popular way of mathematical modeling is to use a set of nonlinear ordinary differential equations (ODEs). It is very difficult to obtain a proper model. There are many hypotheses about the structure of the model (sets of variables and phenomena) that should be verified. The next step, fitting the parameters of the model, is also very complicated because of the nature of measurements. The blotting technique usually gives only semi-quantitative observations, which are very noisy and collected only at a limited number of time moments. The accuracy of parameter estimation may be significantly improved by a proper experiment design. Recently, we have proposed a gradient-based algorithm for the optimization of a sampling schedule. In this paper we use the algorithm in order to optimize a sampling schedule for the identification of the mathematical model of the NFκB regulatory module, known from the literature. We propose a two-stage optimization approach: a gradient-based procedure to find all stationary points and then pair-wise replacement for finding optimal numbers of replicates of measurements. Convergence properties of the presented algorithm are examined.
Box, G. E. P. and Lucas, H. L. (1959). Design of experiments in nonlinear situations, Biometrika46(1/2): 77-90.
Box, M. J. (1968). The occurrence of replications in optimal designs of experiments to estimate parameters in non-linear models, Journal of the Royal Statistical Society. Series B30(2): 290-302.
Chernoff, H. (1972). Sequential Analysis and Optimal Design, SIAM, Philadelphia, PA.
D'Argenio, D. Z. (1981). Optimal sampling times for pharmacokinetic experiments, Journal of Pharmacokinetics and Biopharmaceutics9(6): 739-756.
de Jong, H. (2002). Modeling and simulation of genetic regulatory systems: A literature review, Journal of Computational Biology9(1): 67-103.
DiStefano, J. J. (1981). Optimized blood sampling protocols and sequential design of kinetic experiments, American Journal of Physiology9(240): R259-R265.
Fedorov, V. V. (1972). Theory of Optimal Experiments, Academic Press, New York, NY.
Fujarewicz, K. (2007). Planning identification experiments for cell signaling pathways using sensitivity analysis, Proceedings of the 23rd IFIP Conference on System Modelling and Optimization, Cracow, Poland, pp. 262-263.
Fujarewicz, K. (2008). Optimal scheduling for parameter estimation of cell signaling pathway models—A gradient approach, Proceedings of the 25th IASTED International Multi-Conference on Biomedical Engineering, Innsbruck, Austria, pp. 232-236.
Fujarewicz, K. (2009). Planning identification experiments for nfkb signaling pathway, Proceedings of the 15th National Conference on Application of Mathematics in Biology and Medicine, Szczyrk, Poland, pp. 64-53.
Fujarewicz, K. and Galuszka, A. (2004). Generalized backpropagation through time for continuous time neural networks and discrete time measurements, in L. Rutkowski, J. Siekmann, R. Tadeusiewicz and L. A. Zadeh (Eds.) Artificial Intelligence and Soft Computing—ICAISC 2004, Lecture Notes in Computer Science, Vol. 3070, Springer-Verlag, Berlin, pp. 190-196.
Fujarewicz, K., Kimmel, M., Lipniacki, T. and Swierniak, A. (2007). Adjoint systems for models of cell signalling pathways and their application to parameter fitting, IEEE/ACM Transactions on Computational Biology and Bioinformatics4(3): 322-335.
Goodwin, G. C. and Payne, R. L. (1977). Dynamic System Identification: Experiment Design and Data Analysis, Academic Press, New York, NY.
Hoffman, A., Levchenko, A., Scott, M. L. and Baltimore, D. (2002). The iκb-nf-κb signaling module: Temporal control and selective gene activation, Science298: 1241-1245.
Jacquez, J. (1998). Designs of experiments, Journal of Franklin Institute335(2).
Jacquez, J. and Greif, P. (1985). Numerical parameter identifiability and estimability: Integrating identifiability, estimability, and optimal sampling design, Mathematical Biosciences77: 201-227.
Kiefer, J. (1961). Optimum designs in regression problems, II. The Annals of Mathematical Statistics32(1): 298-325.
Kutalik, Z., Cho, K. and Wolkenhauer, O. (2004). Optimal sampling time selection for parameter estimation in dynamic pathway modeling, BioSystems75(1-3): 43-55.
Lee, E., Boone, D., Chai, S., Libby, S., Chien, M., Lodolce, J. and Ma, A. (2000). Failure to regulate tnf-induced nf-κb and cell death responses in a20-deficient mice, Science289(5488): 2350-2354.
Lipniacki, T., Paszek, P., Brasier, A. R., Luxon, B. and Kimmel, M. (2004). Mathematical model of nf-κb regulatory module, Journal of Theoretical Biology228(2): 195-215.
Pronzato, L. and Walter, E. (1985). Robust experiment design via stochastic approximation, Mathematical Biosciences75: 103-120.
Tod, M. and Rocchisani, J. M. (1997). Comparison of ed, eid and api criteria for the robust optimization of sampling times in pharmacokinetics, Journal of Pharmacokinetics and Biopharmaceutics25(4): 515-537.