Zeitschrift für Informatik und Systembiologie

Stability Constraints of Markov State Kinetic Models Based on Routh- Hurwitz Criterion

Abstract

Sarmis M, Orjuela R, Bouteiller JC, Ambert N, Legendre A, Bischoff S, Haeberlé O and Baudry M

In computational neuroscience, receptors, channels and more generally signaling pathways are often modeled with Markov state models to represent biochemical reactions, which are then implemented with bilinear equations. One of the goals of these models, once calibrated with experimental results is to predict the dynamics of the biological system they represent in response to molecular perturbations and therefore facilitate and enhance the success rate of drug discovery and development. To model receptors under both pathological and physiological conditions, modelers usually modify the ligand association and dissociation parameters in the kinetic model during the optimization phase of model development. However, some parameter values may lead to unstable models, making calibration very difficult, time-consuming and inefficient before performing predictive in silico studies. In order to guarantee model stability during the parameter optimization phase, we propose to linearize bilinear kinetic models around an operating point. Considering the model input as piecewise constant, we propose an algorithm based on the Routh-Hurwitz criterion to generate stability constraints on model parameters. As an example, we apply this algorithm to the gamma-aminobutyric acid (GABA) receptor subtype A (GABAA receptor) model, as developed by Pugh and Raman (2005). The results obtained with the Routh-Hurwitz criterion provide constraint equations. These equations, once integrated into the parameter optimization process, guarantee the stability of the model and thus the success of the optimization process. An additional benefit is that the constraint equations allow determining the boundaries of the stability domain of the model. In the example provided, the Routh-Hurwitz criterion indicates that the model with the chosen parameters becomes unstable if GABA concentration rises above 6.54 mM. The proposed algorithm has also the advantage of being fast and easy to implement.