Causality and independence in perfectly adapted dynamical systems

Perfect adaptation in a dynamical system is the phenomenon that one or more variables have an initial transient response to a persistent change in an external stimulus but revert to their original value as the system converges to equilibrium. The causal ordering algorithm can be used to construct an equilibrium causal ordering graph that represents causal relations and a Markov ordering graph that implies conditional independences from a set of equilibrium equations. Based on this, we formulate sufficient graphical conditions to identify perfect adaptation from a set of first-order differential equations. Furthermore, we give sufficient conditions to test for the presence of perfect adaptation in experimental equilibrium data. We apply our ideas to a simple model for a protein signalling pathway and test its predictions both in simulations and on real-world protein expression data. We demonstrate that perfect adaptation in this model can explain why the presence and orientation of edges in the output of causal discovery algorithms does not always appear to agree with the direction of edges in biological consensus networks.