  We study the two-body problem of the ultracold fermionic alkaline-earth
(like) atoms in the electronic $^1$S$_0$ state ($g$-state) and $^3$P$_0$ state
($e$-state), which are confined in a quasi-one-dimensional (quasi-1D) tube. In
addition, in the axial direction, the $g$-atom experience a 1D optical lattice
and the $e$-atom is localized by a harmonic potential. In this work, we propose
two appropriate tight-binding models, which are applicable for the cases that
the odd-wave scattering between the $g$- and $e$-atom is negligible or not,
respectively. We further give a microscopic derivation for the inter-atomic
interaction parameters of these tight-binding models, by exactly calculating
the low-energy inter-atomic scattering amplitude. Our results show that, as one
can predict, these interaction parameters can be efficiently controlled by the
confinement potentials. We further exam the simple "projection approximation"
with which one derives the interaction parameters by directly projecting the 3D
Huang-Yang pseudopotential on the ground state of the confinement and the
lowest band of the optical lattice. We find that one should be very careful
about determining the interaction parameters in the tight-binding models.
Furthermore, we calculate the spin-exchanging rate, which dependents on the
incident quasi-momentum $k$ of the $g$-atom, for the recent experiment (L.
Riegger, {\it et. al.,} Phys. Rev. Lett. {\bf 120}, 143601 (2018)) of
$^{173}$Yb atoms in this quasi-(1+0)D system, and study finite-momentum effect
in this experiment. Our results show that in this system the finite-momentum
effect of the $g$-atom is very significant, and the momentum of the $g$-atoms
in this experiment may be pretty high (already in the second Brillouin zone of
the optical lattice)
