Structural properties and robustness of dynamical networks in nature

Despite their large scale, complexity and pervasive uncertainty, biological systems often preserve fundamental properties and qualitative behaviours. Their remarkable robustness suggests that some of their properties may depend on the system architecture rather than on precise parameter values.
This seminar introduces parameter-free structural approaches to determine whether a property is preserved for a whole family of uncertain systems exclusively due to its structure (i.e., the underlying graph topology along with qualitative information), regardless of parameter values. Conversely, robust approaches assess whether a property is preserved for a whole family of uncertain systems, when the uncertain parameters are known to belong to prescribed uncertainty sets. Examples from systems biology illustrate how structural and robust analysis can reveal the mechanisms underlying biological function and robustness, even when quantitative information is absent or scarce.
As a paradigmatic example, we focus on structural mechanisms underlying the emergence of sustained oscillations in biological systems. Oscillatory behaviour is central to systems biology, from biomolecular oscillators governing the cell cycle and metabolism to circadian regulation of hormone secretion, body temperature and metabolic activity. For a class of strongly 2-cooperative nonlinear dynamical systems, combining methods from the theory of cones, the spectral theory of totally positive matrices and Perron-Frobenius theory, we show that every solution emanating from an explicit set of initial conditions of positive measure converges to a periodic orbit. We illustrate how this result provides a structural explanation for robust oscillatory behaviour in well-known biological systems, including the Goodwin oscillator.

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