Using the effective Flory polymer model

Wayne K. Dawson

Summary

The purpose of this model is to approximate the features of the fundimental parameter $\nu$ as a function of the sequence dependence in the Flory polymer swelling model. The details of these parameters and how they are obtained (theoretically) is explained in the flory model.

For denatured polymers, the root-mean-square (rms) end-to-end separation distance can be obtained experimentally from the radius of gyration. The parameter $\nu$ expresses the dependence of the rms end-to-end separation-distance as a function of the number of residues in the sequence. Because this property can be found for each measured length, if follows that every point of the sequence exhibits a rms-separation distance that only depends on the number of residues that lie between a given pair of residues $i$ and $j$. The separation distance for each order pair of indices $(i,j)$ in a given polymer chain is $r_{ij}$; where, at the extreme ends, $i \equiv 1$ and $j \equiv N$. The properties of $\nu$ dependent on the specific conditions of the polymer and solvent which in turn influence the second and third virial coefficients.

The dependence of the end-to-end separation distance on sequence length is expressed by the parameter $\alpha_{ij}$ where

$$r_{ij} = \alpha_{ij} r_{{ij}_o} = \left( \frac{N_{ij}}{\xi}... ... \alpha_{ij} = \left( \frac{N_{ij}}{\xi} \right)^{\nu - 1/2}$$ (1)

where $r_{{ij}_o}$ ( $=\sqrt{(\xi N)} b$) is the rms distance for a Gaussian polymer chain structure (with $b$ the mer-to-mer separation distance), $N_{ij}$ is the number of residues separation index $i$ and $j$, and $\xi$ is the persistence length.

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

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

where $\nu_{ij}$ expresses the weight of this parameter as a function of length.

The behavior of $\nu_{ij}$ in typical RNA conditions is shown in Fig 1. In the figure, the RNA tends to swell slightly at short lengths (due to the exposure of RNA to solvent) and it become more globular at very long lengths where it can encounter more of itself if it becomes globular. The region labeled $N_c$ indicates the critical length where this transition occurs. The parameters $R_c$ indication the coexistence region that lies between $\nu _1$ and $\nu _2$.

The advantage of this method is that you can obtain and control the properties of this polymer swelling effect without having to solve for the second and third virial coefficients ($B_2$ and $B_3$). This model does not replace the Flory model as much as it circumvents the difficulties of solving non-integral sixth order polymomial equations. The virial coefficients are very strongly dependent on the persistence length ($\xi$), and therefore, finding good parameterizations depends on both $\xi$ and the $N_c$.

Figure 1: A conceptual model for understanding how the parameter $\nu$ changes as a function of sequence length. Here, $N_c$ is the critical length where the transition between the solvent expanded structure and the globular state occurs. $R_c$ expresses the range of the coexistence region: a property of Van der Waals like models such as this. The end-to-end separation weights $\nu _1$ and $\nu _2$ correspond to the short range and long range weights respectively. Solved using the standard Flory model, these weights are determined by the virial coefficients $B_2$ and $B_3$.