  We explore the dynamical large-deviations of a lattice heteropolymer model of
a protein by means of path sampling of trajectories. We uncover the existence
of non-equilibrium dynamical phase-transitions in ensembles of trajectories
between active and inactive dynamical phases, whose nature depends on
properties of the interaction potential. When the full heterogeneity of
interactions due to the amino-acid sequence is preserved, as in a fully
interacting model or in a heterogeneous version of the G\={o} model where only
native interactions are considered, the transition is between the equilibrium
native state and a highly native but kinetically trapped state. In contrast,
for the homogeneous G\={o} model, where there is a single native energy and the
sequence plays no role, the dynamical transition is a direct consequence of the
static bi-stability between unfolded and native states. In the heterogeneous
case the native-active and native-inactive states, despite their static
similarity, have widely varying dynamical properties, and the transition
between them occurs even in lattice proteins whose sequences are designed to
make them optimal folders.
