  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. With the help of the causal ordering algorithm, one can construct
graphical representations of dynamical systems that represent the causal
relations between the variables and the conditional independences in the
equilibrium distribution. We apply these tools to formulate sufficient
graphical conditions for identifying 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 this method to a simple model for a protein signalling pathway
and test its predictions both in simulations and using real-world protein
expression data. We demonstrate that perfect adaptation can lead to misleading
orientation of edges in the output of causal discovery algorithms.
