Abstract
Within the recently developed Hamiltonian formulation of the Zimm and Bragg model we re-evaluate several size dependent approximations of model partition function. Our size analysis is based on the comparison of chain length N with the maximal correlation (persistence) length of helical conformation. For the first time we re-derive the partition function of zipper model by taking the limits of the Zimm–Bragg eigenvalues. The critical consideration of applicability boundaries for the single-sequence (zipper) and the long chain approximations has shown a gap in description for the range of experimentally relevant chain lengths of 5–10 persistence lengths . Correction to the helicity degree expression is reported. For the exact partition function we have additionally found, that: at the transition temperature reaches its asymptotic behavior of infinite N; the transition interval needs about a thousand persistence lengths to saturate at its asymptotic, infinite length value. Obtained results not only contribute to the development of the Zimm–Bragg model, but are also relevant for a wide range of Biotechnologies, including the Biosensing applications.
Highlights
The model suggested by Zimm and Bragg [1,2] in 1960s contains two size-related relevant scales: the degree of polymerization (DP) N which is related to the polypeptide chain length and the temperature-dependent correlation length ξ describing the scale of correlations in the secondary structure
While the Zimm–Bragg model is mostly used in its limiting, infinite size limit, recent advances in BioNanotechnologies trigger the need for a detailed consideration of chain length related effects
While in the current study there is no intention to apply the Zimm–Bragg model to describe the adsorption of long single-strand DNA (ssDNA) to CNT, better understanding of different limits of the model is a first step towards possible future developments in this direction
Summary
The model suggested by Zimm and Bragg [1,2] in 1960s contains two size-related relevant scales: the degree of polymerization (DP) N which is related to the polypeptide chain length and the temperature-dependent correlation length ξ describing the scale of correlations in the secondary structure. While in the current study there is no intention to apply the Zimm–Bragg model to describe the adsorption of long ssDNA to CNT, better understanding of different limits of the model is a first step towards possible future developments in this direction Another recent study [8] has reported on Zimm–Bragg model application to describe the helix-coil transition in the classical experimental system of synthetic polypeptide homopolymers of different lengths. Suggested Hamiltonian formulation of the Zimm–Bragg model [9] offers straightforward computation of helicity degree from the exact partition function at any degree of polymerization N It allows to derive different approximations of partition function in a straightforward and controllable manner and makes it easier to justify the limits of their applicability. We re-derive the well-known approximations of the Zimm–Bragg model, thoroughly discuss the limits of their applicability and report the dependence of transition temperature and interval on degree of polymerization N
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