  We consider nonlinear reaction systems satisfying mass-action kinetics with
slow and fast reactions. It is known that the fast-reaction-rate limit can be
described by an ODE with Lagrange multipliers and a set of nonlinear
constraints that ask the fast reactions to be in equilibrium. Our aim is to
study the limiting gradient structure which is available if the reaction system
satisfies the detailed-balance condition.
  The gradient structure on the set of concentration vectors is given in terms
of the relative Boltzmann entropy and a cosh-type dissipation potential. We
show that a limiting or effective gradient structure can be rigorously derived
via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation
Principle for gradient flows. In general, the effective entropy will no longer
be of Boltzmann type and the reactions will no longer satisfy mass-action
kinetics.
