Phase diagram of a semiflexible polymer chain in a θ solvent: Application to protein folding

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Abstract
We consider a lattice model of a semiflexible homopolymer chain in a bad solvent. Beside the temperature T, this model is described by (i) a curvature energy εh, representing the stiffness of the chain; (ii) a nearest‐neighbor attractive energy εv, representing the solvent; and (iii) the monomer density ρ=N/Ω, where N and Ω denote, respectively, the number of monomers and the number of lattice sites. This model is a simplified view of the protein folding problem, which encompasses the geometrical competition between secondary structures (the curvature term modelling helix formation) and the global compactness (modeled here by the attractive energy), but contains no side chain information. By allowing the monomer density ρ to depart from unity one has made a first (albeit naive) step to include the role of the water. In previous analytical studies, we considered only the (fully compact) case ρ=1, and found a first order freezing transition towards a crystalline ground state (also called the native state in the protein literature). In this paper, we extend this calculation to the description of both compact and noncompact phases. The analysis is done first at a mean‐field level. We then find that the transition from the high temperature swollen coil state to the crystalline ground state is a two‐step process for which (i) there is first a θ collapse transition towards a compact ‘‘liquid’’ globule, and (ii) at low temperature, this ‘‘liquid’’ globule undergoes a discontinuous freezing transition. The mean‐field value of the θ collapse temperature is found to be independent of the curvature energy εh. This mean‐field analysis is improved by a variational bound, which confirms the independence of the θ collapse temperature with respect to εh. This result is confirmed by a Monte Carlo simulation, although with a much lower value of the θ temperature. This lowering of the collapse transition allows the possibility (for large εh) of a direct first order freezing transition, from a swollen coil to the crystalline ground state. For small values of εh, the mean‐field two‐step mechanism remains valid. In the protein folding problem, the ‘‘liquid’’ compact phase is likely to be related to the ‘‘molten globule’’ phase. The properties of this model system thus suggest that, even though side chain disordering is not taken into account, disordering of the backbone of a protein may still be a sufficient mechanism to drive the system from the native state into the molten globule state.
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