The effect of integral control in oscillatory and chaotic reaction kinetic networks

Integral control is ubiquitously used in industrial processes to keep variables robustly regulated at a given setpoint. Integral control is also present in many biological systems where it, implemented through reaction kinetic networks of genes, proteins and molecules, protects the organism against external variations. One difference between industrial control systems and organisms is that oscillatory behavior seems to be more common in biology. This is probably because engineers can choose to design systems that avoid oscillations. Looking at regulation from the viewpoint of biological systems, the prevalence of oscillations leads to a question which is not often asked in traditional control engineering: how can regulatory and adaptive mechanisms function and coexist with oscillations? And furthermore: does integral control provide some kind of robust regulation in oscillatory systems? Here we present an analysis of the effect of integral control in oscillatory systems. We study nonlinear reaction kinetic networks where integral control is internally present and how these systems behave for parameter values that produce periodic and chaotic oscillations. In addition, we also study how the behavior of an oscillatory reaction kinetic network, the Brusselator, changes when integral control is added to it. Our results show that integral control, when internally present, in an oscillatory system robustly defends the average level of a controlled variable. This is true for both periodic and chaotic oscillations. Although we use reaction kinetic networks in our study, the properties we find are applicable to all systems that contain integral control.