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456695 Optimal Experiment Design for Uncertain Biological Systems

This work considers OED for uncertain biological systems. Two different approaches to OED are comparatively investigated - OED based on deterministic descriptions of uncertainties and OED based on probabilistic descriptions of uncertainties. Deterministic experiment design with respect to all uncertainty realizations, including worst-case uncertainties, can potentially lead to overly conservative designs that underperform when maximizing information content of experiments. On the other hand, the probabilistic OED approach allows for taking into consideration the probability associated with each uncertainty realization. In this case, hence, the uncertainty realizations that have a very small probability of occurrence can be accounted for in the design of experiments, thus resulting in possibly less conservative designs. In the Probabilistic OED approach, the generalized polynomial chaos framework [11] is employed along with Galerkin projection [3] to propagate the uncertainty of the parameter estimates through the system dynamics, resulting in the full probability distribution of system states and outputs at each time point [9]. The deterministic OED approach is formulated through a max-min optimization problem [5]. The D-optimality criterion [10], which corresponds to the determinant of the Fisher Information (FI) Matrix, is used as a quantitative measure for information content of experiments in both approaches.

The efficacy of both approaches is compared for the canonical Wnt (Wnt/β-catenin) signaling pathway, which is ubiquitous in mammalian cell systems and orchestrates critical processes such as cell growth and differentiation [2,7]. A previously reported compartmentalized stochastic model of the Wnt/β-catenin pathway is adopted in this work [8]. The OED approaches are intended to design an optimal Wnt stimulation profile (input), which aims at maximizing the information content of the response of the signaling pathway.

The use of a stochastic model that quantitatively accounts for all sources of uncertainty enables obtaining a more representative description of the naturally occurring behavior in a given cell population, thus resulting in a superior performance of experimental design. Stimulation of the signaling pathway with the optimal Wnt profiles show that the probabilistic OED approach yields more accurate parameter estimates, as compared to the deterministic OED. However, the higher accuracy obtained with the probabilistic approach comes at the expense of allowing for a pre-specified probability of constraint violation.

References:

[1] Asprey, S.P. and Macchietto, S. (2002). Designing robust optimal dynamic experiments. *Journal of Process Control*, 12, 545-556.

[2] Clevers, H. and Nusse, R. (2012) Wnt/β-Catenin signaling and disease. *Cell*, 149, 1192-1205.

[3] Ghanem, R.G. and Spanos, P.D. (1991) Stochastic Fine Elements: A Spectral Approach. Springer-Verlag, New York.

[4] Kaern, M., Elston, T.C., Blake, W.J., Collins, J.J (2005). Stochasticity in gene expression: From theories to phenotypes. *Nature Reviews Genetics*, 6 , 451- 463.

[5] Korkel, S., Kostina, E., Bock, H.G., Schloder, J.P. (2004). Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes. *Optimization Methods and Software Journal*, 19, 327-338.

[6] Kreutz, C. and Timmer, J. (2009). Systems biology: Experimental design. *FEBS Journal*, 276, 923-942

[7] MacDonald, B., Tamai, K., and He, X. (2009). Wnt/beta-catenin signaling: Components, mechanisms, and diseases. *Developmental Cell*, 17, 9-26.

[8] Martin-Casas, M. and Mesbah, A. (2016). A Stochastic model for the canonical Wnt Signaling pathway. *Conference in Foundations of Systems Biology and Engineering*, Submitted.

[9] Mesbah, A. and Streif, S. (2015). A probabilistic approach to robust optimal experiment design with chance constraints. In *Proceedings of the 9th International Symposium on Advanced Control of Chemical Processes*, 100 -105.

[10] Pronzato, L. (2008). Optimal experiment design and related control problems. *Automatica*, 44, 303-325.

[11] Xiu, D. and Karniadakis, G. (2002). The Wiener-Askey polynomial chaos for stochastic differential equations. *SIAM Journal on Scientic Computing*, 24, 619-644.

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