Universal functions on Stein manifolds

We study universal holomorphic functions on a Stein manifold M with projective compactification. Let {phi(n)} be a sequence of holomorphic automorphisms of M. We prove that if {phi(n)(-1)} is A run-away, then the set of all universal functions with respect to {phi(n)} in A(K) for all compact subsets K with a certain property is the intersection of countable number of open dense subsets in the space of all holomorphic functions on M. We also note that there is a close connection between the direction of run-awayness and a family of compact sets for which there exists a universal function.