Holomorphic families of long C^2's

We construct a holomorphically varying family
of complex surfaces X_s, parametrized by
the points s in any Stein manifold, such that
every X_s is a long C^2 (a complex manifold exhausted
by an increasing sequence of domains biholomorphic to C^2)
such that X_s is biholomorphic to C^2 for some but not
all values of s. This extends the recent work of
E.F.Wold and answers a question posed by L.Meersseman.