  We discuss the relation between the emergence of new phases with broken
symmetry within the framework of simple models of biopolymers. We start with a
classic model for a chain molecule of spherical beads tethered together, with
the steric constraint that non-consecutive beads cannot overlap, and with a
pairwise attractive square well potential accounting for the hydrophobic effect
and promoting compaction. We then discuss the consequences of the successive
breaking of spurious symmetries. First, we allow the partial interpenetration
of consecutive beads. In addition to the standard high temperature coil phase
and the low temperature collapsed phase, this results in a new class of
marginally compact ground states comprising conformations reminiscent of
$\alpha$-helices and $\beta$-sheets, the building blocks of the native states
of globular proteins. We then discuss the effect of a further symmetry breaking
of the cylindrical symmetry on attaching a side-sphere to the backbone beads
along the negative normal of the chain, to mimic the presence of side chains in
real proteins. This leads to the emergence of a novel phase within the
previously obtained marginally compact phase, with the appearance of more
complex secondary structure assemblies. The potential importance of this new
phase in the \textit{de novo} design of self-assembled peptides is highlighted.
