[1]The helix-coil transition is used to describe the conformation equilibrium between helix and coil states for proteins or nucleic acids. Only helix (h) or coil (c) state is allowed for each residue in a polymer chain. The formation of the first helix residue is governed by the equilibrium equation:
[1]
The equilibrium constant Kch1 does not apply to the helix-coil transition between ch and hh because of the cooperative of α helix formation. Cooperative means the formation of one helical residue will help the helix formation of its neighboring residues, which is essentially the 1D Ising model. A second constant Kch2 is introduced in the equilibrium equation:
[2]
The quantities s and s are introduced now for simpler statistic calculation. s represents the nucleation constant and s is the propagation constant. Since s is the ratio of two equilibrium constants, the smaller the s value the larger the cooperative. The statistical weight of the fully coil state is set to unity usually. The any other conformation state within the polymer is represented the two parameters s and s if it is a homopolymer.
[3]
[4]
For example the fully coil state of a pentamer peptide, ccccc, has a statistical weight of unity 1; chccc, s s; cchhc, ss2; hhhhh, ss5; hcchh, s2s3;... The total number of conformation state will be 2n with n as the number of residues. The writing and calculation of the partition function is tedious as the polymer growing. The matrix equation allows calculation of the partition function Qn for a peptide with n resdiues . W(ih) represents the statistical weight of conformations with i of the number residues in h state.
[5]
The nucleation constant s is set to the value of 0.004 as determined from the reference J. Am. Chem. Soc. (1998) 120:10646 by Yang J., Zhao, K., Gong, Y., Vologodskii A. and Kalllenbach NR. It is assumed all residues share the same s. But the propagation constant s is residue specific since the identity of the side chain makes different helix propensities. The value of residue specific s is taken from the reference Protein Sci. (1996) 5:2623 by Rohl CA., Chakrabartty A. and Baldwin RL.
The reasonable assumptions used are that the helix propagation is from N terminal to C terminal and only one helix is allowed in a chain. The latter assumption is not appropriate when the chain length increases infinitely. The helical quantity H of a polypeptide is the summation of the statistical weight of each conformation times its number of residues in h state. In equation [6] n is the number of chain residues, j is the helical length. si indicates the propagation constant of residue at the ith position.
[6]
The total quantity W is given by equation [7].
[7]
The final helical content fH is just the fraction of H over W.
[8]
The CD absorbance at 222nm θ222 is calculated using equation published in Protein Sci. (1996) 5:2623 by Rohl CA., Chakrabartty A. and Baldwin RL. x is 2 for correction. θH is -42500 deg.·cm2·dmol-1 and θC is 640 deg.·cm2·dmol-1.
[9]