Ramp approximations of sigmoid control functions in gene networks

In models for networks of regulatory interactions of biological molecules, the sigmoid relationship between concentration of regulating bodies and the production rates they control has led to the use of continuous-time ‘switching’ systems, sometimes referred to as Glass networks, which result from a simplifying assumption that the switching behaviour occurs instantaneously at particular threshold values. Though this assumption produces highly tractable models, it also causes analytic difficulties, such as non-uniqueness, in certain cases, due to the discontinuities of the system. Here, the use of ramp functions is explored as an alternative approximation to the sigmoid, which restores continuity to the vector field and removes the assumption of infinitely steep switching by linearly interpolating the focal point values used in a corresponding Glass network. A general framework for describing a ramp system using the ‘focal points’ of the corresponding Glass network is given. Solutions of two-dimensional networks are explored, and then higher-dimensional networks under certain restrictions. Periodic behaviour is explored using mappings between threshold boundaries. Limitations in these methods are explored, and a general proof of the existence of periodic solutions in negative feedback loops with ramp interactions is given.