Topologies needed for perfect adaptation

It is now being greatly studied for the relationship between the topologies of biological networks and their functions. However, previous studies on network topologies underlie certain biological function are all based on specific mechanisms or dynamics. Here, using a very general mathematical description, we investigate the topologies needed for perfect adaptation-a core function for living creatures in nature, just based on the interactions relationship among nodes: promotion, inhibition, and no relation. We find a necessary interactions relationship among nodes for the network to achieve perfect adaptation. And using this necessary condition, we can conclude that the coherent feedforward loop is incapable of performing perfect adaptation function and show that the saturation condition for the key protein CheR in the network of E. coli chemotaxis is essential.