  We study nuclear symmetry energy and the thermodynamic instabilities of
asymmetric nuclear matter in a self-consistent manner by using a modified
quark-meson coupling model where the confining interaction for quarks inside a
nucleon is represented by a phenomenologically averaged potential in an equally
mixed scalar-vector harmonic form. The nucleon-nucleon interaction in nuclear
matter is then realized by introducing additional quark couplings to $\sigma$,
$\omega$, and $\rho$ mesons through mean-field approximations. We find an
analytic expression for the symmetry energy ${\cal E}_{sym}$ as a function of
its slope $L$. Our result establishes a linear correlation between $L$ and
${\cal E}_{sym}$. We also analyze the constraint on neutron star radii in
$(pn)$ matter with $\beta$ equilibrium.
