Semi-algebraic optimization of temperature compensation in a general switch-type negative feedback model of circadian clocks.

Temperature compensation is an essential property of circadian oscillators which enables them to act as physiological clocks. We have analyzed the temperature compensating behavior of a generalized transcriptional-translational negative feedback oscillator with a hard hysteretic switch and rate constants with an Arrhenius-type temperature dependence. These oscillations can be considered as the result of a lowpass filtering operator acting on a train of rectangular pulses. Such a signal-processing viewpoint makes it possible to express, in a semi-algebraic manner, the period length, the oscillator's control (sensitivity) coefficients, and the first and second-order derivatives of the period-temperature relationship. We have used the semi-algebraic approach to investigate a 3-dimensional Goodwin-type representation of the oscillator, where local optimization for temperature compensation has been considered. In the local optimization, activation energies are found, which lead to a zero first order derivative and to a closest-to-zero second order derivative at a given reference temperature. We find that the major contribution to temperature compensation over an extended temperature range is given by the (local) zero first order derivative, while only minor contributions to temperature compensation are given by an optimized second order derivative. In biological terms this could be interpreted to relate to a circadian clock mechanism which during evolution is being optimized for a certain but relative narrow (habitat) temperature range.