Lifson-Roig model

In polymer science, the Lifson-Roig model is a helix-coil transition model applied to the alpha helix-random coil transition of polypeptides;Lifson S, Roig A. (1961). On the theory of helix-coil transition in polypeptides. "J Chem Phys" 34: 1963-74.] 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.

Parameterization

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.Doig AJ, Baldwin RL. (1995). N- and C-capping preferences for all 20 amino acids in alpha-helical peptides. "Protein Sci" 4: 1325-36.] 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 :$Z\; =\; V\; left(prod\_\{i=0\}^\{N+1\}\; M(i)\; ight)\; ilde\{V\}$

where "V" is the end vector $V=\; [0\; 0\; 0\; 1]$, arranged to ensure the coil state of the first and last residues in the polymer.

This strategy for parameterizing helix-coil transitions was originally developed for alpha helices, whose hydrogen bonds occur between residues "i" and "i+4"; however, it is straightforward to extend the model to 3₁₀ helices and pi helices, with "i+3" and "i+5" hydrogen bonding patterns respectively. The complete alpha/3₁₀/pi transfer matrix includes weights for transitions between helix types as well as between helix and coil states. However, because 3₁₀ helices are much more common in the tertiary structures of proteins than pi helices, extension of the Lifson-Roig model to accommodate 3₁₀ helices - resulting in a 9x9 transfer matrix when capping is included - has found a greater range of application.Rohl CA, Doig AJ. (1996). Models for the 310-helix/coil, pi-helix/coil, and alpha-helix/310-helix/coil transitions in isolated peptides. "Protein Sci" 5: 1687-96.] Analogous extensions of the Zimm-Bragg model have been put forth but have not accommodated mixed helical conformations.Sheinerman FB, Brooks CL. (1995). 310 helices in peptides and proteins as studied by modified Zimm-Bragg theory. "J Am Chem Soc" 117:10098-10103.]

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