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.
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