In polymer science, the Lifson-Roig model is a helix-coil transition model applied to the alpha helix-random coil transition of polypeptides; it is a refinement of the Zimm-Bragg model that recognizes that a polypeptide alpha helix is only stabilized by a hydrogen bond only once three consecutive residues have adopted the helical conformation. To consider three consecutive residues each with two states (helix and coil), the Lifson-Roig model uses a 4x4 transfer matrix instead of the 2x2 transfer matrix of the Zimm-Bragg model, which considers only two consecutive residues. However, the simple nature of the coil state allows this to be reduced to a 3x3 matrix for most applications.

The Zimm-Bragg and Lifson-Roig models are but the first two in a series of analogous transfer-matrix methods in polymer science that have also been applied to nucleic acids and branched polymers. The transfer-matrix approach is especially elegant for homopolymers, since the statistical mechanics may be solved exactly using a simple eigenanalysis. The Lifson-Roig model is characterized by three parameters: the statistical weight for nucleating a helix, the weight for propagating a helix and the weight for forming a hydrogen bond, which is granted only if three consecutive residues are in a helical state.

Weights are assigned at each position in a polymer as a function of the conformation of the residue in that position and as a function of its two neighbors. A statistical weight of 1 is assigned to the "reference state" of a coil unit whose neighbors are both coils, and a "nucleation" unit is defined (somewhat arbitrarily) as two consecutive helical units neighbored by a coil. A major modification of the original Lifson-Roig model introduces "capping" parameters for the helical termini, in which the N- and C-terminal capping weights may vary independently.

The correlation matrix for this modification can be represented as a matrix M, reflecting the statistical weights of the helix state h and coil state c. The Lifson-Roig model may be solved by the transfer-matrix method using the transfer matrix M shown at the right, where w is the statistical weight for helix propagation, v for initiation, n for N-terminal capping, and c for C-terminal capping. (In the traditional model n and c are equal to 0.) The partition function for the helix-coil transition equilibrium is

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