Models for the 310-helix/coil and a-helix/coil equilibria have been derived. The theory is based on classifying residues into helical or nonhelical (coil) conformations. Statistical weights are assigned to residues in a helical conformation with an associated helical hydrogen bond, a helical conformation with no hydrogen bond, an N-cap position, a C-cap position, or the reference coil conformation. The models for a-helix formation and 3,,-helix formation have also been combined to describe a three-state equilibrium in which a-helical, 3,,-helical, and coil conformations are populated. The results are compared with the modified Lifson-Roig theory for the a-helix/coil equilibrium. The comparison accounts for the experimental observations that 310-helices tend to be short and a-helices are not favored for any length. This work may provide a framework for quantitatively rationalizing experimental work on isolated 310-helices and mixed 310-/a-helices.Keywords: a-helix; helix-coil transition theory; N-cap; a-helix; 3,,-helix Polypeptide helices are stabilized by periodic hydrogen bonds between backbone CO and NH groups. They can be described by the number of residues per helical turn and the number of atoms contained in the ring made by each hydrogen bond (Bragg et al., 1950). The ideal a-helix has a periodicity of 3.6 residues per turn, encloses 13 atoms in a ring by formation of an i , i + 4 C=O.. .H-N hydrogen bond and is thus a 3.6,,-helix.The 3,,-helix is a more tightly wound helix, stabilized by i, i + 3 C=O. . .H-N hydrogen bonds, whereas the s-(4.4,,-)helix is wound less tightly with i, i + 5 C=O. . .H-N hydrogen bonds.The 3,,-helix was proposed and discussed (Huggins, 1943; Bragg et al., 1950;Donohue, 1953) earlier than the a-helix (Pauling et al., 1951). Although appreciably less common than a-helices, 3-4'70 of residues in crystal structures are in 3,,-helices (Barlow & Thornton, 1988; Karpen et al., 1992), making the 3,,-helix the fourth most common type of secondary structure in proteins after a-helices, @-sheets, and reverse turns. Most 3,,-helices are short, only three or four residues long, compared with a mean of ten residues in a-helices (Barlow & Thornton, 1988 Hubbard, 1984;Barlow & Thornton, 1988): 32% of helices in crystal structures were found to have a 3,,-type hydrogen bond at their N-termini and 34% at their C-termini (Baker & Hubbard, 1984). Strong amino acid preferences have been observed for different locations within a 3,,-helix in a recent survey of crystal structures (Karpen et al., 1992).3,,-Helices have also been studied in isolated peptides. It has been proposed that peptides made of the standard 20 L-amino acids can form 3,0-helices (Miick et al., 1992), or at least populate a significant amount of 3,,-helix at their N-and C-termini. The fraction of 3,,-helix present in a helical peptide is certainly strongly sequence dependent (Fiori et al., 1994;Zhou et al., 1994). 3,,-Helix formation can be induced by the introduction of a c,,,-disubstituted a-amino acid, of which a-ami...