  Chain molecules play important roles in industry and in living cells. Our
focus here is on distinct ways of modeling the stiffness inherent in a chain
molecule. We consider three types of stiffnesses -- one yielding an energy
penalty for local bends (energetic stiffness) and the other two forbidding
certain classes of chain conformations (entropic stiffness). Using detailed
Wang-Landau microcanonical Monte Carlo simulations, we study the interplay
between the nature of the stiffness and the ground state conformation of a
self-attracting chain. We find a wide range of ground state conformations
including a coil, a globule, a toroid, rods, helices, zig-zag strands
resembling $\beta$-sheets, as well as knotted conformations allowing us to
bridge conventional polymer phases and biomolecular phases. An analytical
mapping is derived between the persistence lengths stemming from energetic and
entropic stiffness. Our study shows unambiguously that different stiffness play
different physical roles and have very distinct effects on the nature of the
ground state of the conformation of a chain, even if they lead to identical
persistence lengths.
