In statistical mechanics, the Zimm–Bragg model is a helix-coil transition model that describes helix-coil transitions of macromolecules, usually polymer chains. Most models provide a reasonable approximation of the fractional helicity of a given polypeptide; the Zimm–Bragg model differs by incorporating the ease of propagation (self-replication) with respect to nucleation. It is named for co-discoverers Bruno H. Zimm and J. K. Bragg.
Helix-coil transition models Helix-coil transition models assume that polypeptides are linear chains composed of interconnected segments. Further, models group these sections into two broad categories: coils, random conglomerations of disparate unbound pieces, are represented by the letter 'C', and helices, ordered states where the chain has assumed a structure stabilized by hydrogen bonding, are represented by the letter 'H'. Thus, it is possible to loosely represent a macromolecule as a string such as CCCCHCCHCHHHHHCHCCC and so forth. The number of coils and helices factors into the calculation of fractional helicity, θ {\displaystyle \theta \ } , defined as
θ = ⟨ i ⟩ N {\displaystyle \theta ={\frac {\left\langle i\right\rangle }{N}}}
where
⟨ i ⟩ {\displaystyle \left\langle i\right\rangle \ } is the average helicity and
N {\displaystyle N\ } is the number of helix or coil units.
Zimm–Bragg
The Zimm–Bragg model takes the cooperativity of each segment into consideration when calculating fractional helicity. The probability of any given monomer being a helix or coil is affected by which the previous monomer is; that is, whether the new site is a nucleation or propagation. By convention, a coil unit ('C') is always of statistical weight 1. Addition of a helix state ('H') to a previously coiled state (nucleation) is assigned a statistical weight σ s {\displaystyle \sigma s\ } , where σ {\displaystyle \sigma \ } is the nucleation parameter and s {\displaystyle s\ } is the equilibrium constant
s = [ H ] [ C ] {\displaystyle s={\frac {[H]}{[C]}}}
Adding a helix state to a site that is already a helix (propagation) has a statistical weight of s {\displaystyle s\ } . For most proteins,
σ ≪ 1 < s {\displaystyle \sigma \ll 1<s\ }
which makes the propagation of a helix more favorable than nucleation of a helix from coil state. From these parameters, it is possible to compute the fractional helicity θ {\displaystyle \theta \ } . The average helicity ⟨ i ⟩ {\displaystyle \left\langle i\right\rangle \ } is given by
⟨ i ⟩ = ( s q ) d q d s {\displaystyle \left\langle i\right\rangle =\left({\frac {s}{q}}\right){\frac {dq}{ds}}}
where q {\displaystyle q\ } is the partition function given by the sum of the probabilities of each site on the polypeptide. The fractional helicity is thus given by the equation
θ = 1 N ( s q ) d q d s {\displaystyle \theta ={\frac {1}{N}}\left({\frac {s}{q}}\right){\frac {dq}{ds}}}
Statistical mechanics The Zimm–Bragg model is equivalent to a one-dimensional Ising model and has no long-range interactions, i.e., interactions between residues well separated along the backbone; therefore, by the famous argument of Rudolf Peierls, it cannot undergo a phase transition. The statistical mechanics of the Zimm–Bragg model may be solved exactly using the transfer-matrix method. The two parameters of the Zimm–Bragg model are σ, the statistical weight for nucleating a helix and s, the statistical weight for propagating a helix. These parameters may depend on the residue j; for example, a proline residue may easily nucleate a helix but not propagate one; a leucine residue may nucleate and propagate a helix easily; whereas glycine may disfavor both the nucleation and propagation of a helix. Since only nearest-neighbour interactions are considered in the Zimm–Bragg model, the full partition function for a chain of N residues can be written as follows
Z = ( 0 , 1 ) ⋅ { ∏ j = 1 N W j } ⋅ ( 1 , 1 ) {\displaystyle {\mathcal {Z}}=\left(0,1\right)\cdot \left\{\prod _{j=1}^{N}\mathbf {W} _{j}\right\}\cdot \left(1,1\right)}
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