Ideal chain

The Freely - Jointed Chain Model ( German model of freely moving chain) in biophysics is the simplest model, whereby a polymer can be described. The model neglects interactions between the monomers, so that they can rotate around its two ends as desired, which corresponds mathematically to a random walk. As improvement, the wormlike chain model represents, in which the monomers restricted in their mobility, so that polymers can be described with stiffness.

Properties

A polymer represented in this model as a chain of rigid pieces of the length of the so-called Kuhn length - maximum length is thus defined by

Given. The segments are free to move, like a hinge ( here, however, three-dimensional). This results in a random walk with the step length and the step number. For large, the central limit theorem applies.

In this approach, no interactions between the monomers are adopted, the energy of the polymer is assumed to be independent of its shape. This means, in thermodynamic equilibrium all possible configurations are equally probable, the polymer it passes through all the passage of time - the fluctuations are described by the Maxwell - Boltzmann distribution.

Is the end-to -end vector of the ideal chain and the vectors to individual monomers. These randomly distributed vectors have three components in the x-, y -and z- direction. We assume that the number of monomers N is large, so that the central limit theorem applies. The figure below shows the sketch of an ideal short chain:

The ends of the chain do not coincide, but since they fluctuate freely of course applies to the mean ( expected value ):

There are statistically independent, it follows from the central limit theorem, that are normally distributed: more precisely follow in 3D and according to a normal distribution of the mean 0 with variance: