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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 $\gamma = 1.75$ 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 $i$ and $j$ with an end-to-end separation distance of $r_{ij}$ and contained within a volume element of $4 \pi
r_{ij}^2 dr$ is

p(r_{ij})\Delta (r/b)
= C_{ij}^{\gamma \delta}
\left( \f...
...eft( \frac{r_{ij}}{b} \right)^{\delta} \right\}
\Delta (r/b)
\end{displaymath} (1)

where $\gamma$ is a weight that accounts for the self-avoiding character of the polymer and contributes to the the excluded volume, $\delta $ is the weight on the exponential function, and $C_{ij}^{\gamma \delta}$ is a normalization constant for the distribution function

C_{ij}^{\gamma \delta} =
\vartheta_{ij}^{\gamma + 1/\delta}}
{4 \pi \Gamma (\gamma + 1/\delta)}.
\end{displaymath} (2)

where $\Gamma (x)$ is a Gamma-function. The constant $\vartheta_{ij}$ 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,

\vartheta_{ij} = \left(
\frac{\Gamma (\gamma + 3/\delta)}{\...
\frac{b^2}{\langle r_{ij}^2 \rangle }
\right)^{\delta / 2}
\end{displaymath} (3)

In its most general form,

\langle r_{ij}^2 \rangle^{1/2} = \varsigma \xi^{1-\nu}N_{ij}^\nu b
\end{displaymath} (4)

where $\varsigma$ is an unspecified weight factor, $\xi$ is the persistence length (a measure of the correlation) and $\nu$ is the weight on $N$. It seems best to say that one should generally assume that $\varsigma \equiv 1$ in these problems without serious evidence to the contrary. The central limit theorem predicts that $\nu \equiv
1/2$ 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 $\gamma \equiv 1$ and $\delta \equiv 2$, $p(r_{ij})$ 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 $p(r_{ij})$ resembles a Gaussian polymer chain (GPC). When $p(r_{ij})$ is Gaussian, the parameters in (1) become $\gamma \equiv 1$ and $\delta \equiv 2$. The strict model of the GPC maintains $\gamma \equiv 1$; however, since a Gaussian function is a special case of a more general family known as $\Gamma$-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 $\nu = 1/2$.

It is important to note that in the McKenzie-Domb (McD) equations, eq (1) is altered slightly to the form

p(r_{ij}) = C_{ij}^{t \delta}
\left( \frac{r_{ij}}{b} \rig...
...eft( \frac{r_{ij}}{b} \right)^{\delta} \right\}
\Delta (r/b)
\end{displaymath} (5)

hence, $\gamma = (t + 2)/\delta$. 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.

next up previous
Next: Reasonable values for the Up: Using the McKenzie-Moore-Domb-Fisher parameters Previous: Using the McKenzie-Moore-Domb-Fisher parameters
Wayne Dawson 2006-10-31