A Quantitative Design Guideline for Biomolecular Positive Feedback Systems

Feedback is at the core of biological systems found in medicine and in biotechnology. While design metaphors from control engineering are widely used to understand negative feedback in such systems, they are relatively uncommon for positive feedback, especially for biomolecular circuits. Here, we extended a block diagram modelling framework for the design of positive feedback. We found a quantitative design guideline for the strength of the positive feedback, which when wrapped around a saturation function can give a threshold and a hysteretic response. The critical feedback strength was inversely proportional to the saturation value and directly proportional to the input scale where saturation starts. We found that this saturation-threshold-hysteresis hierarchy persisted in a realistic model of a positively autoregulated gene. We showed how this classical model fitted well in a block diagram framework with multiplicative feedback and derived expressions for the critical feedback strength in terms of the saturation parameters. The dependence of the critical feedback on the parameters matched with the obtained design guideline. To complete a rigorous workflow, we discussed how Groebner Bases computations and an Interval Newton algorithm can be used to provide validated numerical solutions in biological positive feedback systems. These results should be helpful in the analysis and design of biomolecular systems with applications to the control of biomedical systems and in biotechnology.