The origin of this model is largely the pioneering work of these individuals. Fisher was the first to recognize and find a parameterization for the non-Gaussian character of the distribution function. The constant in the Gaussian-like function used in Mfold and the Vienna package has its origins in Fisher's work. Domb and Fisher both did a fair amount of study on the exponent, and McKenzie and Moore established further justification for working on the weight for the self-avoiding walk.
In the most general expression for the distribution function, the likelihood of finding mers and with an end-to-end separation distance of and contained within a volume element of is
where is a weight that accounts for the self-avoiding character of the polymer and contributes to the the excluded volume, is the weight on the exponential function, and is a normalization constant for the distribution function
where is a Gamma-function. The constant is is a dimensionless scaling parameter that is a function of the root-mean-square (rms) end-to-end separation-distance of the polymer chain,
In its most general form,
where is an unspecified weight factor, is the persistence length (a measure of the correlation) and is the weight on . It seems best to say that one should generally assume that in these problems without serious evidence to the contrary. The central limit theorem predicts that for a GPC; however, some aspects of RNA behavior tend to deviate from this ideal situation. The persistence length is a measure of the correlation between segments in a polymer.
This function assumes that the characteristics of the polymer can be reduced to nearly identical entities to a reasonable approximation. For the most part, the four bases in RNA can broadly meet these attributes. When and , becomes a Gaussian function that is the origin of its name Gaussian polymer chain (GPC).
A central theme in approaches that use this kind of model is resembles a Gaussian polymer chain (GPC). When is Gaussian, the parameters in (1) become and . The strict model of the GPC maintains ; however, since a Gaussian function is a special case of a more general family known as -functions, altering this parameter in (1) has for the most part the effect of changing the weight on equation but otherwise not affecting the general properties. Without invoking the ``-flory'' option, it is assume that .
It is important to note that in the McKenzie-Domb (McD) equations, eq (1) is altered slightly to the form
hence, . However, we have chosen to use the parameters in Eq (4) to maintain the same notation throughout.
The model strongly built around a continuum renormalization theory to get the correct polymer distribution function. Because of the difficulty of such models, they generally are restricted to the polymer in good solvent where strong self-(attractive)-interactions can be sufficiently ignored. Exactly what happens when you allow for strong self-interactions is a lot murkier. This model is provided as is. There are many problems with continuum renormalization theory that are difficult to address particularly when self-interaction is strong. The strong point of the continuum renormalization method is that reveals critical constants.