Dynamical models play a crucial role in the systematic analysis of complex biological systems, which are characterized by nonlinearities and uncertainties. The usual existence of several hypotheses to describe the dynamics of biological systems, along with the consideration that models of such systems typically have several unknown parameters, have motivated the use of model-based optimal experiment design (OED) approaches for model discrimination (Fedorov, 1972) and parameter estimation (Pronzato, 2008). The primary notion of OED for model discrimination and parameter estimation is to systematically design input signals aiming at, respectively, separating predictions of the rival models and maximizing the information content of experiments for accurate parameter estimation. However, a major limitation of standard OED approaches is their reliance on point estimates of the system parameters, which can be largely different from their true values, leading to (possibly) ineffective input designs. In addition, an optimal input for model discrimination is unlikely to be adequate for parameter estimation and vice versa.
This work addresses the problem of OED for simultaneous model discrimination and parameter estimation in polynomial nonlinear systems with probabilistic time-invariant uncertainties. The proposed OED approach obviates the conservatism of the deterministic robust OED formulations by explicitly accounting for the probabilistic information of system uncertainties. Chance constraints are included to tolerate predetermined levels of risk in the designed experiments in terms of state constraint violations (Mesbah and Streif, 2015). This study uses the generalized polynomial chaos framework (Xiu and Karnadiakis, 2002) in conjunction with Galerkin projection (Ghanem et al. 1991, Streif et al., 2014) for efficient propagation of the probabilistic uncertainties. The similarity between probability distributions of states/outputs is quantified in terms of the Wasserstein distance (Dobrushin, 1970, Vallender, 1972). The D-optimality criterion is adopted as a quantitative measure for information content of the experiments. The to-be-minimized objective function used in the optimization is a geometric mean of the model discrimination and parameter estimation criteria.
The proposed optimal experiment design approach is demonstrated for the ubiquitous canonical Wnt signaling pathway, which regulates cell growth and fate determination at the molecular level (Li et al., 2010). Activation of the pathway by the wild type ligand Wnt results in accumulation of the transcriptional coactivator β-catenin in the cytoplasm. The proposed OED approach is used to discriminate between two alternative β-catenin phosphorylation mechanisms performed by two components of the pathway, i.e. CK1α and GSK3 (Li et al., 2010, Hernandez et al., 2012). In the so-called processive mechanism, β-catenin binds to the destruction complex and does not detach until it is fully phosphorylated and flagged for degradation. Conversely, in the distributive mechanism β-catenin binds and unbinds before and after each phosphorylation event, and thus not fully assembled destruction complexes (only with GSK3 or with CK1α) could also participate in the phosphorylation of β-catenin (Hernandez et al. 2012). Two dynamical models are developed to appropriately describe the distinctive events of each mechanism (Jensen et al., 2010, Hernandez et al., 2012). The input of the system is chosen to be the rate of degradation of Axin once it is sequestered by LRP on the cell membrane, the magnitude of which is related to the concentration of pathway activator Wnt (Jensen et al. 2010). Concentration of free β-catenin is assumed to be the noise-corrupted output of the system.
To evaluate the performance of the robust OED approach for simultaneous model discrimination and parameter estimation, Monte Carlo runs are performed using the optimized input profile and a nominal input profile. The simulation results reveal that the OED approach is able to effectively separate the probability distributions of model outputs over critical transient behavior of the biological system in the presence of uncertainties. In addition, the designed input leads to smaller average estimation errors for the kinetic parameters of the biological system. A key feature of the robust OED approach is the ability to ensure constraint satisfaction with a prespecified probability level in a stochastic setting.
Dobrushin, R.L. (1970). Prescribing a System of Random Variables by Conditional Distributions. SIAM Journal of Scientific Computation, 3, 458-486.
Fedorov, V. (1972). Theory of Optimal Experiments. Academic Press, New York.
Ghanem R.G. and Spanos P.D. (1991). Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York.
Hernandez, A., Klein, A., Kirschner, M. (2012). Kinetic Responses of β-catenin specify the sites of Wnt Control, Science, 338, 1337-1340.
Jensen, P.B., Pedersen, L., Krishna S., Jensen M. (2010). Wnt signaling in oncogenesis and embryogenesis: a look outside the nucleus. Biophysical Journal, 98, 943-950.
Li, V., Ng, S., Boersema, P., Low, T., Karthaus, W., Gerlach, J., Mohammed, S., Heck, A., Maurice, M. M., Mahmoudi, T., Clevers, S. (2010). Wnt Signaling through Inhibition of β-catenin Degradation in an Intact Axin1 Complex. Cell, 149, 1245-1256.
Mesbah A., Streif, S (2015). A Probabilistic Approach for Robust Optimal Experiment Designs with Chance Constraints. ADCHEM, In Press.
Pronzato, L. (2008). Optimal Experimental Design and Some Related Control Problems. Automatica, 44, 303-325.
Streif, S., Petzke, F., Mesbah, A., Findeisen, R., Braatz, R.D. (2014). Optimal Experiment Design for Probabilistic Model Discrimination using Polynomial Chaos. In Proceedings of the IFAC World Congress, pages 4103-4109. Cape Town, South Africa.
Vallender, S.S. (1972). Calculation of the Wasserstein Distance between Probability Distributions on the Line. Theory of Probability and its Applications, 18(4), 784–786.
Xiu, D., Karniadakis, G. (2002). The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal of Scientific Computation, 24, 619-644.
See more of this Group/Topical: Computing and Systems Technology Division