Oscillations in strongly 2-cooperative systems and their applications in systems biology

The emergence of sustained oscillations (via convergence to periodic orbits) in high-dimensional nonlinear dynamical systems is a highly non-trivial question with important applications in systems biology, including the understanding of bio-molecular oscillators ruling cell life-cycle and metabolism, as well as circadian rhythms in hormone secretion, body temperature and metabolic functions. In systems biology, the mechanism underlying such widespread oscillatory biological motifs is still not fully understood. From a mathematical perspective, the study of sustained oscillations is comprised of two parts: (i) showing that at least one periodic orbit exists and (ii) studying the stability of periodic orbits and/or characterising the initial conditions which yield solutions that converge to periodic trajectories. In this talk we will focus on a specific class of nonlinear dynamical systems that are strongly 2-cooperative. Employing results from the theory of cones of rank k, the spectral theory of totally positive matrices and Perron-Frobenius theory, we will show that strongly 2-cooperative systems admit an explicit set of initial conditions of positive measure, such that every solution emanating from this set converges to a periodic orbit. We will further demonstrate our results using the n-dimensional Goodwin oscillator and a 4-dimensional biological oscillator based on RNA–mediated regulation.