Statics, metastable states, and barriers in protein folding: A replica variational approach

Protein folding is analyzed using a replica variational formalism to investigate some free energy landscape characteristics relevant for dynamics. A random contact interaction model that satisfies the minimum frustration principle is used to describe the coil-globule transition (characterized by ${\mathrm{T}}_{\mathrm{CG}}$), glass transitions (by ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{K}}$), and folding transition (by ${\mathrm{T}}_{\mathrm{F}}$). Trapping on the free energy landscape is characterized by two characteristic temperatures, one dynamic (${\mathrm{T}}_{\mathrm{A}}$) and the other static [${\mathrm{T}}_{\mathrm{K}}$ (${\mathrm{T}}_{\mathrm{A}}$>${\mathrm{T}}_{\mathrm{K}}$)], which are similar to those found in mean field theories of the Potts glass. (i) Above ${\mathrm{T}}_{\mathrm{A}}$, the free energy landscape is monotonous and the polymer is melted both dynamically and statically. (ii) Between ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{K}}$, the melted phase is still dominant thermodynamically, but frozen metastable states, exponentially large in number, appear. (iii) A few lowest minima become thermodynamically dominant below ${\mathrm{T}}_{\mathrm{K}}$, where the polymer is totally frozen. In the temperature range between ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{K}}$, barriers between metastable states are shown to grow with decreasing temperature, suggesting super-Arrhenius behavior in a sufficiently large system. Due to evolutionary constraints on fast folding, the folding temperature ${\mathrm{T}}_{\mathrm{F}}$ is expected to be higher than ${\mathrm{T}}_{\mathrm{K}}$, but may or may not be higher than ${\mathrm{T}}_{\mathrm{A}}$. Diverse scenarios of the folding kinetics are discussed based on phase diagrams that take into account the dynamical transition, as well as the static ones.