  The two-vibron dynamics associated to amide-I vibrations in a 3D
$\alpha$-helix is described according to a generalized Davydov model. The helix
is modeled by three spines of hydrogen-bonded peptide units linked via covalent
bonds. It is shown that the two-vibron energy spectrum supports both a
two-vibron free states continuum and two kinds of bound states, called TVBS-I
and TVBS-II, connected to the trapping of two vibrons onto the same amide-I
mode and onto two nearest neighbor amide-I modes belonging to the same spine,
respectively. At low temperature, non vanishing interspine hopping constants
yield a three dimensional nature of both TVBS-I and TVBS-II which the wave
functions extend over the three spines of the helix. At biological temperature,
the pairs are confined in a given spine and exhibit the same features as the
bound states described within a one-dimensional model. The interplay between
the temperature and the 3D nature of the helix is also responsible for the
occurrence of a third bound state called TVBS-III which refers to the trapping
of two vibrons onto two different spines. The experimental signature of the
existence of bound states is discussed through the simulation of their infrared
pump-probe spectroscopic response. Finally, the fundamental question of the
breather-like behavior of two-vibron bound states is addressed.
