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| In [[polymer science]], the '''Lifson–Roig model''' is a [[helix-coil transition model]] applied to the [[alpha helix]]-[[random coil]] transition of [[polypeptide]]s;<ref name="Lifson">{{cite journal |author=Lifson S, Roig A |year=1961 |title=On the theory of helix-coil transition in polypeptides |journal=J Chem Phys |volume=34 |pages=1963–1974 |doi=10.1063/1.1731802 |issue=6}}</ref> 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.
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| 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 acid]]s and branched polymers. The transfer-matrix approach is especially elegant for homopolymers, since the statistical mechanics may be solved exactly using a simple [[eigenanalysis]].
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| ==Parameterization==
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| The Lifson–Roig model is characterized by three parameters: the [[statistical weight]] for [[nucleation|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.<ref name="Doig">{{cite journal |doi=10.1002/pro.5560040708 |author=Doig AJ, Baldwin RL |year=1995 |title=N- and C-capping preferences for all 20 amino acids in alpha-helical peptides |journal=Protein Sci |volume=4 |issue=7 |pages=1325–1336 |pmid=7670375 |pmc=2143170}}</ref> 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''.
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| {| class="wikitable" style="text-align:center" align="right"
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| |-
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| ! ''M'' !! hh !! hc !! ch !! cc
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| |-
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| ! hh
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| | w || v || 0 || 0
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| |-
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| ! hc
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| | 0 || 0 || <math>\sqrt{nc}</math> || c
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| |-
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| ! ch
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| | v || v || 0 || 0
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| |-
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| ! cc
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| | 0 || 0 || n || 1
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| |}
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| 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 1.) The [[partition function (statistical mechanics)|partition function]] for the helix-coil transition equilibrium is
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| :<math> | |
| Z = V \left(\prod_{i=0}^{N+1} M(i)\right)\tilde{V}
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| </math>
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| where ''V'' is the end [[coordinate vector|vector]] <math>V=[0 0 0 1]</math>, arranged to ensure the coil state of the first and last residues in the polymer.
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| This strategy for parameterizing helix-coil transitions was originally developed for [[alpha helix|alpha helices]], whose [[hydrogen bond]]s occur between residues ''i'' and ''i+4''; however, it is straightforward to extend the model to [[310 helix|3<sub>10</sub> helices]] and [[pi helix|pi helices]], with ''i+3'' and ''i+5'' hydrogen bonding patterns respectively. The complete alpha/3<sub>10</sub>/pi transfer matrix includes weights for transitions between helix types as well as between helix and coil states. However, because 3<sub>10</sub> helices are much more common in the [[tertiary structure]]s of proteins than pi helices, extension of the Lifson–Roig model to accommodate 3<sub>10</sub> helices - resulting in a 9x9 transfer matrix when capping is included - has found a greater range of application.<ref name="Rohl">{{cite journal |author=Rohl CA, Doig AJ |year=1996 |title=Models for the 3(10)-helix/coil, pi-helix/coil, and alpha-helix/3(10)-helix/coil transitions in isolated peptides |journal=Protein Sci |pmid=8844857 |volume=5 |issue=8 |pmc=2143481 |pages=1687–1696 |doi=10.1002/pro.5560050822}}</ref> Analogous extensions of the Zimm-Bragg model have been put forth but have not accommodated mixed helical conformations.<ref name="Sheinerman">{{cite journal |doi=10.1021/ja00145a022 |author=Sheinerman FB, Brooks CL |year=1995 |title=310 helices in peptides and proteins as studied by modified Zimm-Bragg theory |journal=J Am Chem Soc |volume=117 |issue=40 |pages=10098–10103}}</ref>
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Lifson-Roig model}}
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| [[Category:Polymer physics]]
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| [[Category:Protein structure]]
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| [[Category:Statistical mechanics]]
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