  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.
