Fundamentals

In an linear (non-circular) RNA sequence of length $N$, we can express the position of each monomer (or mer) by an index. If we denote the 5$^\prime$ end by the index $1$ and the 3$^\prime$ end by the index $N$, then for we can let index $i$ corresponds to the index of monomer $i$, where $1\le i\le N$. We denote the correlation between two monomer by the ordered pair $(i,j)$, where $i\ne j$ and $1 \le i < j \le N$. For a true randomly oriented polymer, the average correlation between any indices $i$ and $j$ is zero. However, when there is bonding between two mers, the correlation is much greater. It is an expression for this correlation that we seek.

One important observable thermodynamic state variable of a polymer is the radius of gyration ($s_o$). The radius of gyration is a measure of the root-mean-square (rms) separation-distance between indices $1$ and $N$ of the non-circular polymer. If the radius of gyration is known, then we can find the rms end-to-end separation distance from the relationship $r_{rms} =\sqrt 6 s_o$. For monomers that lie somewhere within the sequence, it is convenient to express these subsequence lengths in terms of the difference of the indices $N_{ij} = j - i + 1$, where $N_{1N} = N$. It goes without saying that rules that apply to $N_{1N}$ must also apply to $N_{ij}$ because we can construct individual polymer fragments of length $N_{ij} \leftarrow N$ and measure them and see that their behavior is a function of their length:

$$\left\langle {r^2} \right\rangle^{1/2} = r_{rms} (N),\;\mbo... ...eft\langle {r_{ij}^2 } \right\rangle^{1/2} = r_{rms} (N_{ij})$$ (1)

The GPC condition represents a special case of the rms end-to-end separation-distance of a polymer chain where

$$\left\langle {r_{ij}^2 } \right\rangle _o =\xi N_{ij} b^2$$ (2)

where $b$ is the separation distance between monomers (in this case nucleic acids), and $\xi$ is the persistence length. The latter variable ($\xi )$ is known as the persistence length or the Kuhn length and expresses the correlation between neighboring monomers: $i.e.$, the number of monomers that tend to group together and behave as though they were one single unit. In the traditional model, $r_{ij}$ actually refers to the extreme ends of the polymer chain where $i \equiv 1$ and $j \equiv N$, with $N$ the total number of monomers in the polymer chain. When the length ($L$) of the polymer chain is stretched out from end-to-end in a linear fashion, $L = Nb$.

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

$$p(r_{ij} )\Delta (r/b)=4\pi C_{ij}^{\gamma \delta } \mathop ... ...{r_{ij} }{b}} \right)}\nolimits^\delta } \right\}\Delta (r/b)$$ (3)

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

$$C_{ij}^{\gamma \delta} = \frac{\delta \vartheta_{ij}^{\gamma + 1/\delta}} {4 \pi \Gamma (\gamma + 1/\delta)}.$$ (4)

where $\Gamma (x)$ is a Gamma-function. The constant $\vartheta_{ij}$ is a dimensionless scaling parameter that is a function of the root-mean-square (rms) end-to-end separation-distance of the polymer chain,

$$\vartheta_{ij} =\mathop {\left( {\frac{\Gamma (\gamma +3/\de... ...e {r_{ij}^2 } \right\rangle }} \right)}\nolimits^{\delta /2}$$ (5)

In its most general form,

$$\left\langle {r_{ij}^2} \right\rangle^{1/2}=\xi^{1-\nu} N_{ij}^\nu b$$ (6)

where $\nu$ is the weight on $N$. 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 as we will show. 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).