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.
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