Balanced truncation for model reduction of biological oscillators

Alberto Padoan,Fulvio Forni,Rodolphe Sepulchre

doi:10.1007/s00422-021-00888-4

Abstract

Model reduction is a central problem in mathematical biology. Reduced order models enable modeling of a biological system at different levels of complexity and the quantitative analysis of its properties, like sensitivity to parameter variations and resilience to exogenous perturbations. However, available model reduction methods often fail to capture a diverse range of nonlinear behaviors observed in biology, such as multistability and limit cycle oscillations. The paper addresses this need using differential analysis. This approach leads to a nonlinear enhancement of classical balanced truncation for biological systems whose behavior is not restricted to the stability of a single equilibrium. Numerical results suggest that the proposed framework may be relevant to the approximation of classical models of biological systems.

Highlights

Model reduction is a central problem in mathematical biology (Fall et al 2005; Murray 2007; Alon 2007; Keener and Sneyd 2008; Del Vecchio and Murray 2015)
Biological Cybernetics (2021) 115:383–395 that evolve on different timescale, which enables the separation of their dynamics into “slow” and “fast.” This property is exploited by methods based on timescale separation to reduce the complexity of a system by approximating the fast variables with their steady-state values
Contributions This paper proposes a model reduction framework for biological systems whose behavior is not restricted to the stability of a single equilibrium

Summary

Introduction

Model reduction is a central problem in mathematical biology (Fall et al 2005; Murray 2007; Alon 2007; Keener and Sneyd 2008; Del Vecchio and Murray 2015). Biological Cybernetics (2021) 115:383–395 that evolve on different timescale, which enables the separation of their dynamics into “slow” and “fast.” This property is exploited by methods based on timescale separation to reduce the complexity of a system by approximating the fast variables with their steady-state values This approach preserves the biological meaning of the state space variables and favors modularity between processes with widely separated timescales (Grunberg and Del Vecchio 2019). While in some cases this maintains biological interpretability and compatibility with open systems modeling (Rao et al 2014), lumping often requires expert knowledge, which makes it unappealing for constructive design and quantitative verification Another common approach to model reduction of biochemical systems is based on sensitivity analysis and optimization (Danø et al 2006; Zhang and Goutsias 2010; Prescott and Papachristodoulou 2012; Hangos 2013), which build reduced order models by minimizing an error function (such as the sensitivity of a variable) within a given range of candidate models. Motivated and inspired by this line of research, the present paper extends the applicability of balanced truncation to behaviors that are not restricted to the stability of a single equilibrium

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Conclusion