Dynamical Dimension Flat dimension : by Karhunen-Loève expansion (PCA)

1. Context Model reduction techniques produce small but still accurate models from larger ones. There are many reasons for simplifying systems biology models: a) discard unnecessary complexity; b) reveal critical parameters and regulation loops; c) compare models and find common patterns; d) study robustness. The transcription factor NFB has been discovered 20 years ago and is still in the spotlights. It is involved in a wide diversity of domains such as immune and inflammatory responses, cell survival and apoptosis, cellular stress and neuro-degenerative diseases, cancer and development. Starting with a rather complex model of NFB signaling pathway, we obtain its “limit simplifications”. These are compared among them and with existing models. • Limit Simplifications • A modular technique : • define modules : set of intermediate species • compute simple submechanisms (SS) for modules • use idempotent algebra for the rates of SS : important complexity reduction! • Example: transcriptional module of p50, one of the two proteins forming NFB. Definition of simple submechanisms Limit simplifications Order conditions on non-critical parameters SS rates depend on critical parameters P50 is upregulated by IB downregulated by NF B P50 is downregulated by IB (X11) upregulated by NF B (X7) Model Comparison Models are distinguished by three complexity indices (n,m,p) : numbers of species, reactions, parameters. M(14,27,36) and M(14,27,29) have the same reaction graph, kinetic laws are different; M(14,27,29) was proposed by Lipniacki M(6,9,13) a simplification of M(14,27,29) M(39,67,88) most complex model Model 80% 95% 99% 99.9% 100% M(39,67,88) 2 4 6 8 15 M(14,27,36) 3 4 6 7 11 M(14,27,29) 2 3 4 5 9 M(6,9,13) 2 3 4 4 5 Visualization of principal components : grouped species oscillate in phase Conclusions Limit simplifications are robust reductions of reaction mechanisms. They allow to identify critical and non-critical parameters. They also allow to compare models, by reducing them to the same reaction graph and by comparing their kinetic laws. Finding equivalent sets of parameter values for different models is made easy. Dynamical dimension determination and PCA analysis are other general tools for model comparison. All models shown here have rather similar behaviour : they oscillate with similar phase relations between main species. Nevertheless, our most complex model M(39,67,88) and the one reduced from it M(14,27,36) have more complex oscillations than Lipniacki model. This is shown by the larger dynamical dimension. These two models are also less robust (result not illustrated here) in the sense that random variations of their parameters produce relatively larger variations of the characteristic times of the oscillations (period, dumping time). Model Nonlin.dim. M(39,67,88) 5 M(14,27,36) 3 M(14,27,29) 2 M(6,9,13) 2 Model Reduction and Model Comparison for NFB Signaling Ovidiu Radulescu (IRMAR and IRISA, Rennes, France), Andrei Zinovyev (Institut Curie, Paris, France), Alain Lilienbaum (CNRS URA 2115, Paris, France) contact: ovidiu.radulescu@univ-rennes1.fr Intermediate species Terminal species Reversible reactions Irreversible reactions Dynamics All models oscillate after application of a signal. Depending on parameters, oscillations can be damped, or sustained. Non-conservative models: the total NFkB quantity has small amplitude oscillations Conservative models: the total NFkB quantity is constant Dynamical Dimension Flat dimension : by Karhunen-Loève expansion (PCA) Dimension of linear manifold embedding the limiting trajectories for various ‘‘explained’’ variance threshold Invariant manifold of M(14,27,29) Singular value curves are used to estimate the nonlinear dynamical dimension