A model for polymers

In the previous sections, we have essentially ignored the dependence of the persistence length on the expressions for polymer-solvent interaction. The program does not assume that the structure of the polymer is at the convenience of our eyes and or reagents. Rather, the program is concerned with how nature has reduced the number of degrees of freedom on the biopolymer to produce the structure one sees in the journal or the textbook. Having a persistence length different from $\xi \equiv 1$ means that we must divide the sequence into $N/\xi$ ``effective'' mers each of which has a length $\xi b$. The total length of the stretched out sequence is therefore the same: $(N/\xi)(\xi b) = N b$, but the manipulation does require some modification.

Three parameters are needed to describe $\alpha$: $N$, $\xi$ and $\nu$. Using the definition $r_o^2 = \xi N b^2$ as the unperturbed rms end-to-end separation-distance, we write $\alpha$ as

$$r=\alpha r_o =\mathop {\left( {\frac{N}{\xi }} \right)}\noli... ...mathop {\left( {\frac{N}{\xi }} \right)}\nolimits^{\nu -1/2}$$ (86)

where $\xi$ is the persistence length, which is set at the very beginning of the calculation, and $N$ is specified by the particulars of the given RNA sequence. The variable $\nu$ is what must be found in (86).

To handle the general case of stacking in RNA, one must estimate the rms end-to-end separation-distance for each base-pair $(i,j)$ in the polymer chain rather than just the extreme ends ($i \equiv 1$ and $j\equiv N)$. Therefore, the model must be modified slightly because the $\alpha$ parameter depends on the length of the chain, yet all chain lengths of the same polymer must exhibit the same $r$ dependence on $N$. Since each order pair $(i,j)$ must exhibit this same property, one single $\alpha$ parameter is simply not appropriate. Then the result for $\alpha_{ij}$ follows from (86)

$$r_{ij} =\alpha _{ij} r_{\mathop {ij}\nolimits_o } =\mathop {... ...p {\left( {\frac{N_{ij} }{\xi }} \right)}\nolimits^{\nu -1/2}$$ (87)

The entropy of folding for a GPC solutions ( $\delta \equiv 2)$ is

$$\Delta S(N_{ij} )=\frac{k_B T}{\xi }\left\{ {\nu \delta \gam... ...N_{ij} }} \right)}\nolimits^{\nu \delta } } \right)} \right\}$$ (88)

where $\Psi_\nu = (\xi/\lambda)^{1/\nu}/\xi$, $\lambda$ is the separation distance between two base pairs when treated as a kind of ball shaped object (I'll try to add a figure to describe this more clearly in an updated version of this text), and $\zeta$ is related to $\vartheta$ in (5)

$$\zeta (\delta ,\gamma )=\mathop {\left( {\frac{\Gamma (\gamm... ...)}{\Gamma (\gamma +1/\delta )}} \right)}\nolimits^{\delta /2}$$ (89)

The value of $\nu$ in (88) must satisfy a modified version of (78) or (80), due to the fact that these problems involve effective mers and not the monomers themselves. Thus (78) becomes

$$\alpha^5 - \alpha^3 = \frac{B_2 (N/\xi)^{1/2}}{(\xi b)^3} + 2 \frac{B_3}{(\xi b)^6 \alpha^3}$$ (90)

and for $\delta \ne 2$ and $\gamma \ne 1$, (90) takes on the form in (83)

$$\alpha^{\delta + 3} - \alpha^3 = \frac{3 B_2 (N/\xi)^{1/2}}... ... b)^3} + \frac{6 B_3}{(\delta \gamma + 1) (\xi b)^6 \alpha^3}$$ (91)

Only positive roots of (90) and (91) can be used and the dominant term in (90) and (91) is $B_2$ when $N$ is large. If $B_2$ is negative and $B_3$ positive, then $\nu < 1/2$ for any legitimate value of $N$. Hence, specifying a negative value for $B_2$ will insure that the response of the polymer will resemble a globular system for $N$ larger than some length dependent on the balance between the two-body and three-body interaction terms. Values of $B_3$ must be positive to generate a positive real root in this case. If $B_2$ is positive, then, irrespective of $B_3$ (usually positive), $\nu > 1/2$ and $B_3$ can be neglected in most cases. Finally, when $B_2 = 0$, the equation reduces to the GPC for large $N$.

The limits on $\nu$ and the corresponding polymer-solvent conditions and the corresponding virial coefficients are summarized in Table 1.

The exponent $\nu _{ij}$ can be expressed as a function of $\alpha_{ij}$ through the following relationship

$$\nu _{ij} =\frac{1}{2}\left( {1+\frac{2\ln (\alpha _{ij} )}{\ln (N_{ij} /\xi )}} \right)$$ (92)

so for $0<\alpha _{ij} <1$, $0<\nu _{ij} <1/2$, and for $1\le \alpha _{ij}$, $1/2\le \nu _{ij}$. In general, physical values for $\nu _{ij}$ range from $0.3<\nu _{ij} <0.7$.

Table 1: Relationship between solvent conditions, the state of the polymer as expressed by the Flory parameter ($\nu$) and the virial coefficients.

$\nu$	solvent conditions	characteristics of the polymer	virial coefficients
0.33	poor	the globular state $r \propto N^{1/3}$	$B_2 < 0$, $B_3 > 0$
0.50	athermal	obeys Gaussian statistics $r \propto N^{1/2}$	$B_2 = 0$, $B_3 = 0$
0.60	good	polymer swells $r \propto N^{3/5}$	$B_2 > 0$