A Unified Model of $\alpha$-Helix/$\beta$-Sheet/Random-Coil Transition in Proteins

The theory of transition between $\alpha$-helix, $\beta$-sheet and random coil conformation of a protein is discussed through a simple model, that includes both short and long-range interactions. Besides the bonding parameter and helical initiation factor in Zimm-Bragg model, three new parameters are introduced to describe beta structure: the local constraint factor for a single residue to be contained in a $\beta$-strand, the long-range bonding parameter that accounts for the interaction between a pair of bonded $\beta$-strands, and a correction factor for the initiation of a $\beta$-sheet. Either increasing local constraint factor or long-range bonding parameter can cause a transition from $\alpha$-helix or random coil conformation to $\beta$-sheet conformation. The sharpness of transition depends on the competition between short and long-range interactions. Other effective factors, such as the chain length and temperature, are also discussed. In this model, the entropy due to different ways to group $\beta$-strands into different $\beta$-sheets gives rise to significant contribution to partition function, and makes major differences between beta structure and helical structure.