(27cj) Constraining the Predictions of Conserved SMAD Signaling Pathway through Parameter Identifiability Informed Experimental Design.

The SMAD signaling pathway is a conserved pathway that regulates growth, development, and adult homeostasis from worms to humans. The transforming growth factor - β (TGF-β) is a ligand from the super-family of TGF-β which binds to the receptors and activates the SMAD signaling pathway by phosphorylating SMAD2 (pSMAD2). pSMAD2 forms homo-dimeric, hetero-dimeric, and hereto-trimeric complexes with the transcriptional activator SMAD4. The active signaling complex, (pSMAD2)₂SMAD4, translocate to the nucleus and regulates gene expression. When the signaling pathway is turned on by the activation of the receptors, there is a preferential localization of SMAD2 species in the nucleus. Whereas, when an inhibitor that deactivates the active receptors is added to the system, the accumulation of SMAD2 species in the nucleus falls off. (Schmierer and Hill, 2007)

We developed a mathematical model (adapted from Schmierer et al., 2008) to infer the dynamics of the SMAD signaling pathway. Mathematical models, such as ODE-based models, are widely applied in Systems Biology to build hypotheses, guide experimental design, and make predictions about the system behavior. These models have numerous free parameters, which are estimated by fitting the model to experimental data, such as fluorescence intensity over a time course, using global optimization algorithms, such as evolutionary algorithms. From a geometric perspective, these models form a hyper-surface of all possible predictions in high-dimensional data space called the model manifold (Quinn et al., 2022).

The structure of these model manifolds can make it difficult to estimate model parameters. Data-fitting results in a large number of parameter sets with a high goodness-of-fit but large uncertainties associated with them. The eigenvalues of the Fisher Information Matrix (FIM) (Ashyraliyev et al., 2009; Transtrum and Qiu, 2012), which is a measure of the amount of information an observable random variable carries about an unknown parameter, for such a model span over many decades. Such models are deemed “,” a characteristic of systems biology models (Gutenkunst et al., 2007). In other words, the fitness landscape near a local minimum has an enormous aspect ratio, with small eigenvalues leading to sloppy (wide) directions, and large eigenvalues leading to stiff (narrow) directions (hierarchy of widths).

We found that the SMAD signaling model also has large uncertainties associated with the parameters (O(θ) ∼ 10¹⁵) and the eigenvalues of the FIM span over 15 decades. The predictions for the total nuclear active signaling complex resulting from these parameter sets were severely under-constrained. We reconstructed a simplified model manifold based on the hierarchy of widths and observed that the model predictions vary markedly as we move along the sloppy (wider) axis, even though the model fits remain excellent. Fixing some of the parameters to experimentally measured values reported in literature reduced variation between the parameter sets but did not reduce the variability in model predictions.

To make the model less sloppy and constrain the predictions, we then employed profile-likelihood (Raue et al., 2009) to discern all the parameters i.e., parameters that cannot be learned from data, and to design experiments based on their likelihood to be maximally informative in constraining model predictions. This allowed us to constrain the predictions and reduce the parameter uncertainty of the model parameters significantly. Such a parameter identifiability informed experimental design reduced variation in parameter sets obtained through global optimization techniques and the uncertainty associated with them and their predictions.

Through this analysis, we aim to highlight the idea that systems biology models are sloppy, and even with a high goodness-of-fit, parameter sets can result in highly under-constrained predictions. A careful analysis of the model structure can provide insight into making models less sloppy and lead to predictions with a narrower confidence interval.

References:

Ashyraliyev, Maksat, et al. "Systems biology: parameter estimation for biochemical models." The FEBS journal 276.4 (2009): 886-902.

Gutenkunst, Ryan N., et al. "Universally sloppy parameter sensitivities in systems biology models." PLoS computational biology 3.10 (2007): e189.

Quinn, Katherine N., et al. "Information geometry for multiparameter models: New perspectives on the origin of simplicity." Reports on Progress in Physics (2022).

Raue, Andreas, et al. "Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood." Bioinformatics 25.15 (2009): 1923-1929.

Schmierer, Bernhard, and Caroline S. Hill. "TGFβ–SMAD signal transduction: molecular specificity and functional flexibility." Nature reviews Molecular cell biology 8.12 (2007): 970-982.

Schmierer, Bernhard, et al. "Mathematical modeling identifies Smad nucleocytoplasmic shuttling as a dynamic signal-interpreting system." Proceedings of the National Academy of Sciences 105.18 (2008): 6608-6613.

Transtrum, Mark K., and Peng Qiu. "Optimal experiment selection for parameter estimation in biological differential equation models." BMC bioinformatics 13.1 (2012): 1-12.