  In the framework of a lattice-model study of protein folding, we investigate
the interplay between designability, thermodynamic stability, and kinetics. To
be ``protein-like'', heteropolymers must be thermodynamically stable, stable
against mutating the amino-acid sequence, and must be fast folders. We find two
criteria which, together, guarantee that a sequence will be ``protein like'':
i) the ground state is a highly designable stucture, i. e. the native structure
is the ground state of a large number of sequences, and ii) the sequence has a
large $\Delta/\Gamma$ ratio, $\Delta$ being the average energy separation
between the ground state and the excited compact conformations, and $\Gamma$
the dispersion in energy of excited compact conformations. These two criteria
are not incompatible since, on average, sequences whose ground states are
highly designable structures have large $\Delta/\Gamma$ values. These two
criteria require knowledge only of the compact-state spectrum. These claims are
substantiated by the study of 45 sequences, with various values of
$\Delta/\Gamma$ and various degrees of designability, by means of a
Borst-Kalos-Lebowitz algorithm, and the Ferrenberg-Swendsen histogram
optimization method. Finally, we report on the reasons for slow folding. A
comparison between a very slow folding sequence, an average folding one and a
fast folding one suggests that slow folding originates from a proliferation of
nearly compact low-energy conformations, not present for fast folders.
