Probability distribution function used in JS

For those of you unfamiliar with what a Gaussian polymer chain, I have to refer you to explanations in any reputable polymer science textbook. I'll eventually try to write an introduction myself, but I have to apologize, I just haven't had the time.

First I just explain where the Jacobson-Stockmayer (JS) equation comes from. We start with the familiar Gaussian polymer chain equation.

$$W(r) \Delta r = \left( \frac{\beta^2}{\pi} \right)^{3/2} \exp \left\{ - \beta^2 r^2 \right\} 4 \pi r^2 \Delta r$$ (1)

where $\beta^2 = 3/2\xi N b^2$, $b$ is the monomer-to-monomer (or mer-to-mer) separation distance between individual monomer on a polymer chain and $\xi$ is the persistence length (a measure of the correlation between individual monomers). The persistence length is given in units of $b$ (the mer-to-mer distance) and is usually greater than $1$ mer because. Some may note that $\beta^2 = \vartheta$ if they have encountered other pages here. We then consider the root-mean-square (RMS) end-to-end separation distance $\langle r^2 \rangle$ in this expression

$$\langle r^2 \rangle = \left( \frac{\beta^2}{\pi} \right)^{3/... ...t\{ - \beta^2 r^2 \right\} 4 \pi r^4 dr} = \frac{3}{2 \beta^2}$$ (2)

where the RMS value itself is $\langle r^2 \rangle = \xi N b^2$.

It is important to remember that the probability function (1) is not expressing the length of the chain but the end-to-end separation distance of monomer 1 and monomer $N$ (where an RNA sequence is numbered as 1 at the 5' end and $N$ at the 3' end). This RMS end-to-end separation distance has a finite volume. Therefore, Jacobson and Stockmayer reasoned that the probability that the two ends of the polymer chain of length $N$ will localize within the same volume $v_s$ is

$$p_{v_s}(N) = \left( \frac{\beta^2}{\pi} \right)^{3/2} \int_{... ... \pi r^4 dr} \sim \left( \frac{\beta^2}{\pi} \right)^{3/2} v_s$$ (3)

and because $\langle r^2 \rangle = \xi N b^2$, (3) simplifies to

$$p_{v_s}(N) = \left( \frac{3}{2 \pi \xi N} \right)^{3/2} \left( \frac{v_s}{b^3} \right).$$ (4)

Given that one has chosen intelligent parameters for $v_s$ in relation to $b$, Eqn (4) is also a sensable probability function to go further with. So, with no explicit temperature dependence, the entropy is simply $\Delta S = k_B \ln ( p_{v_s}(N) )$. Therefore,

$$\Delta S (N) \sim - k_B \left\{ \frac{3}{2} \ln \left( \fra... ... \frac{v_s}{b^3} \right) \right\} - \frac{3}{2} k_B \ln ( N )$$

$$= - \left\{ A (\xi b, v_s) - \frac{3}{2} k_B \ln (N) \right\}$$ (5)

where $A(\xi b, v_s)$ is determined by estimates of $\xi$, $b$ and $v_s$. A similar expression was obtained by Poland and Scheraga (1965) by a somewhat different and perhaps more rigorous approach.

In addition to this, some consideration was given to the fact that real polymers actually occupy space (a novel idea). The Gaussian polymer chain model is a random walk and does not care whether it crosses part of the path of a previous step. In fact, one possible structure is one in which you can fold the polymer back and forth between the same two points. This is obviously not physical. From the theoretical work of Fisher, it was shown that one effective solution is to change weight on the second term in Eqn (5) from 3/2 to about 1.75.

The key issue here is what values of $v_s$ would make sense. In these models, one should perceive the supposed monomers in these models (including ours) as some sort of spherical balls or a blobish sort of collection of these balls whose individual center-of-mass corresponds to the position of the drawn object. Most likely, one should not (for typical blobs) expect that the center-of-mass of each blob would come closer than the respective mer-to-mer separation distance ($b$). In short, $v_s \geq b^3$. Indeed, a proper estimate for the distance between the center-of-mass of each nucleic acid in a base-pair is about $2 b$ or $v_s/b^3 \sim 8$. (One can see this is so by considering the arrangement and general distances between the nucleic acids in the base-pairs of a DNA helix and realize that it is at least on the order of $b$.) This renders $A(\xi b, v_s)$ far too small. Therefore, the estimates from this parameter have usually been empirical for RNA.

What we call the ``JS-model'' is actually only such in the sense that we honor the variability of the weight (3/2) in Eqn (5). In fact, we show in our work that the JS-model is actually an approximation solution that satisfies a subset of our more general approach.

Here, we only say that the JS-model can be found by simplifying the McKenzie-Moore-Domb-Fisher distribution function. This model has the general form

$$p(r_{ij}) \Delta (r/b) = A_{\delta \gamma} C_{ij}^{\gamma ... ...eft( \frac{r_{ij}}{b} \right)^{\delta} \right\} \Delta (r/b)$$ (6)

where $r_{ij}$ is the end-to-end separation distance between two residues of index $i$ and $j$, $p(r_{ij})$ is a probability distribution function used to express this end-to-end separation distance, $\gamma$ is in part a factor in the self-avoiding (or excluded volume) weight, $\delta$ is the weight on the exponential function, $\vartheta_{ij}$ is a dimensionless scaling parameter, $A_{\delta \gamma}$ is the spherically symmetric solid angle weight (not to be confused with $A(\xi b, v_s)$ in Eqn (5)!!), and $C_{ij}^{\gamma \delta}$ is a normalization constant for the distribution function.

When $\delta \equiv 2$, this equation becomes

$$p(r_{ij}) = A_{2 \gamma} C_{ij}^{2 \gamma } \left( \frac{r... ...} \left( \frac{r_{ij}}{b} \right)^2 \right\} \Delta (r/b).$$ (7)

Again, since there is no implicit temperature dependence, the entropy becomes the natural log of this expression,

$$S(r_{ij}) = k_B \ln \left( p(r_{ij}) \right) = k_B \left\{... ... - \vartheta_{ij} \left( \frac{r_{ij}}{b} \right)^2 \right\}.$$ (8)

In the JS equation, the change in entropy is measured from the denatured structure ( $r_{ij} = R_{ij}$) and the native state ( $r_{ij} = \lambda b$), where $\lambda$ is a scalar constant proportional to the separation between the two monomers when treated as beads on a string and $R_{ij}$ is the root-mean-square separation distance between $i$ and $j$ in the denatured state. The resulting loss in entropy as the structure folds from the denature state to the native state has the form

$$\Delta S (N_{ij}) = S ( \lambda b ) - S ( R_{ij} ) = - \fra... ... 1/2) \left( 1 - \frac{1}{ \Psi N_{ij} } \right) \right\}$$ (9)

where $\xi$ is the persistence length (a measure of the correlation of the nucleotides at nearest neighboring ends), $\Psi = \xi/\lambda^2$ is a constant with $\lambda$ the base-pair separation distance in units of $b$. Since the free energy is $\Delta G = \Delta H - T \Delta S$, for large $N_{ij}$, the leading contribution to the free energy due to entropy loss has the form $-T \Delta S = \gamma k_B T \ln ( N_{ij} )$, which resembles the JS equation used in the Turner energy rules.

It is for this reason that we honor the tradition of calling this the JS-model. However, it should be firmly planted in the reader's mind that this bears but a superficial resemblance to the JS-model traditionally used in the dynamic programming algorithm and only reduced to such under limiting conditions.