Before revision:  Since both properties have been tuned through natural selection, their adaptation can be formalized in the framework of mathematical optimization. By using a succinct model , we found that simultaneous optimization of regularity and entrainability entails  inherent features of the circadian mechanism irrespective of model details .
After revision:  We found by using a phase model with multiple inputs that achieving the maximal limit of regularity and entrainability entails inherent features of the circadian mechanism irrespective of model details .