  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.
