The theory of transition between $α$-helix, $β$-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 $β$-strand, the long-range bonding parameter that accounts for the interaction between a pair of bonded $β$-strands, and a correction factor for the initiation of a $β$-sheet. Either increasing local constraint factor or long-range bonding parameter can cause a transition from $α$-helix or random coil conformation to $β$-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 $β$-strands into different $β$-sheets gives rise to significant contribution to partition function, and makes major differences between beta structure and helical structure.
Spectral analysis of protein backbone geometry reveals abrupt helix--coil boundaries
College of Mathematics and System Science, Xinjiang University（新疆大学数学与系统科学学院）；Xinjiang Key Laboratory of Biological Resources and Genetic Engineering, College of Life Science and Technology, Xinjiang University（新疆大学生命科学与技术学院生物资源与基因工程重点实验室）；Shenzhen X-Institute（深圳X研究院）
