Abstract
The unfolding transition from compact globule to stretched coil in bio-polymers
was already predicted 15 years ago $1$, based on heuristic
arguments. Nowadays, the theoretical study of this transition is dominated by
numerical simulations. The self-avoiding walk in two dimensions is often
used to model the unfolding of polymers $2,3$. The
benefit of self-avoiding walks is that local interactions like monomer-monomer
attraction and the excluded volume effect are taken into account. The
disadvantage is that one is limited to short walks due to the computational cost
(up to chain length $55$ $2$). The present work uses random walks
without self-avoiding constraints in an attempt to obtain analytical instead of
numerical results.
In $4$ we use a one-dimensional random walk with Markovian increments
to compare two polymer stretching experiments. This model can also be used to
calculate analytically a force-temperature state diagram. The obtained state
diagram is similar to the diagrams of the numerical simulations. The most
remarkable property of these diagrams, the re-entrance behavior due to
residual entropy, is also observed in our calculations $5$.
The disadvantages of the present model are that its application is restricted to
flexible molecules in a poor solvent, and that the excluded volume effect is not
taken into account. The benefit of this simplified model is, that it can be
solved completely in closed form. It is also possible to generalize the model to
higher dimensions (see $6$ for some general results about
chains with Markovian increments).\\
1) A. Halperin, E. B. Zhulina, On the Deformation Behaviour
of Collapsed Polymers, Europhys. Lett. 15, 417 (1991).
\\
2) S. Kumar, I. Jensen, J. L. Jacobsen and A. J. Guttmann, Role
of conformatinal entropy in force-induced bio-polymer unfolding,
cond-mat/0702436 (Accepted in Phys. Rev. Lett.)
\\
3) D. Marenduzzo, A. Maritan, A. Rosa and F. Seno, Stretching
of a Polymer below the $þeta$ Point, Phys. Rev. Lett. 90, 088301 (2003)
\\
4) E. Van der Straeten, J. Naudts,
A one-dimensional model for theoretical analysis of single molecule
experiments,
J. Phys. A: Math. Gen. 39, 5715 (2006).
\\
5) E. Van der Straeten and J. Naudts, Residual entropy
in
theoretical model for single molecules, in preparation
\\
6) J. Naudts and E. Van der Straeten,
Transition records of stationary Markov chains,
Phys. Rev. E. 74, 040103 (2006).
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