Homeostasis in Networks with Multiple Input Nodes and Robustness in Bacterial Chemotaxis

A biological system achieve homeostasis when there is a regulated quantity that is maintained within a narrow range of values. Here we consider homeostasis as a phenomenon of network dynamics. In this context, we improve a general theory for the analysis of homeostasis in network dynamical systems with distinguished input and output nodes, called `input-output networks'. The theory allows one to define `homeostasis types' of a given network in a `model independent' fashion, in the sense that the classification depends on the network topology rather than on the specific model equations. Each `homeostasis type' represents a possible mechanism for generating homeostasis and is associated with a suitable `subnetwork motif' of the original network. Our contribution is an extension of the theory to the case of networks with multiple input nodes. To showcase our theory, we apply it to bacterial chemotaxis, a paradigm for homeostasis in biochemical systems. By considering a representative model of Escherichia coli chemotaxis, we verify that the corresponding abstract network has multiple input nodes. Thus showing that our extension of the theory allows for the inclusion of an important class of models that were previously out of reach. Moreover, from our abstract point of view, the occurrence of homeostasis in the studied model is caused by a new mechanism, called input counterweight homeostasis. This new homeostasis mechanism was discovered in the course of our investigation and is generated by a balancing between the several input nodes of the network -- therefore, it requires the existence of at least two input nodes to occur. Finally, the framework developed here allows one to formalize a notion of `robustness' of homeostasis based on the concept of `genericity' from the theory dynamical systems. We discuss how this kind of robustness of homeostasis appears in the chemotaxis model.