Polymer Physics & DNA

Model Definition

The end-to-end vector is the sum of all segment vectors:

Each segment vector $\mathbf{l}_i$ is isotropically distributed on a sphere of radius $l$. The segments are uncorrelated:

The contour length (maximum extension) is $L = Nl$, while the typical end-to-end distance is much smaller, scaling as $\sqrt{N}\, l$.

Mean-Square End-to-End Distance

We compute $\langle R^2 \rangle = \langle \mathbf{R} \cdot \mathbf{R} \rangle$:

Splitting into diagonal ($i = j$) and off-diagonal ($i \neq j$) terms:

For the FJC, segments are uncorrelated, so $\langle \mathbf{l}_i \cdot \mathbf{l}_j \rangle = 0$for $i \neq j$, and $\langle l_i^2 \rangle = l^2$. Therefore:

This is the fundamental result of random walk polymer physics. The root-mean-square end-to-end distance scales as $R_{\text{rms}} = l\sqrt{N}$, just like a diffusing particle after $N$ steps.

Distribution of End-to-End Distance

By the central limit theorem, for large $N$, each component of$\mathbf{R}$ is Gaussian distributed. The probability distribution of $\mathbf{R}$ is:

The radial distribution (probability of $|\mathbf{R}| = R$) is:

The most probable distance is $R^* = \sqrt{2Nl^2/3}$, while the average is $\langle R \rangle = (8Nl^2/3\pi)^{1/2}$. Both scale as $\sqrt{N}$.

Persistence Length and Tangent Correlations

The worm-like chain (WLC), or Kratky-Porod model, treats the polymer as a continuous elastic rod with bending stiffness $\kappa$. The bending energy is:

where $\hat{\mathbf{t}}(s) = \partial \mathbf{r}/\partial s$ is the unit tangent vector and $s$ is the arc length. The persistence length is defined as:

The tangent-tangent correlation function decays exponentially:

For $s \ll l_p$, the chain appears straight (rod-like). For$s \gg l_p$, it appears flexible (random-coil-like).

DNA as a Worm-Like Chain

Double-stranded DNA is the best-characterized semiflexible polymer. Its mechanical properties have been measured with extraordinary precision using optical tweezers, magnetic tweezers, and atomic force microscopy.

• • Persistence length: $l_p \approx 50$ nm (150 bp) in physiological salt
• • Rise per base pair: 0.34 nm (B-form)
• • Bending modulus: $\kappa = l_p k_BT \approx 205$ pN·nm$^2$
• • Stretch modulus: $S \approx 1000$ pN
• • Torsional stiffness: $C \approx 440$ pN·nm$^2$

Stretching Experiments and the Overstretching Transition

The force-extension curve of DNA shows three distinct regimes:

• • Entropic regime (0–5 pN): WLC behavior, force increases gradually
• • Enthalpic stretching (5–65 pN): backbone stretching beyond contour length, described by extensible WLC: $x/L = 1 - (k_BT/(4l_pF))^{1/2} + F/S$
• • Overstretching transition (~65 pN): cooperative transition where DNA extends to ~1.7 times its B-form contour length

The overstretching transition at 65 pN is a thermodynamic phase transition between B-form DNA and an extended state. The work per base pair at the transition is:

This is comparable to the base-pairing free energy, supporting the model that overstretching involves disruption of Watson-Crick base pairs.

DNA Looping and Cyclization

The probability of DNA cyclization (end-to-end closure) is the Jacobson-Stockmayer$J$-factor:

For DNA shorter than the persistence length, cyclization is strongly suppressed by the bending energy penalty. For DNA much longer than $l_p$,$J \propto L^{-3/2}$ (Gaussian chain scaling). There is an optimal length for cyclization at $L \approx 3\text{-}4 \times l_p \approx 500$ bp.

The Excluded Volume Effect

Real polymer chains cannot overlap themselves. This excluded volumeeffect causes the chain to swell beyond the ideal (Gaussian) prediction. Flory's mean-field theory estimates the free energy as the sum of elastic and interaction terms:

where $v$ is the excluded volume parameter (effective volume per monomer). The first term penalizes stretching (entropic elasticity), and the second penalizes compression (two-body repulsion in a volume $R^3$).

Minimizing $F(R)$ with respect to $R$:

The Flory exponent is $\nu = 3/5 = 0.6$, compared to $\nu = 1/2$for ideal chains. The exact value from renormalization group theory is$\nu \approx 0.588$ in 3D, remarkably close to Flory's result.

The scaling $R \propto N^{3/5}$ means swollen chains are more extended than ideal chains. This is relevant for unfolded proteins and flexible polymers in good solvents. In a theta solvent ($v = 0$), the ideal chain scaling $R \propto N^{1/2}$ is recovered.

Radius of Gyration

The radius of gyration $R_g$ is a more experimentally accessible measure of chain size than the end-to-end distance. It is defined as:

For an ideal chain, the relationship between $R_g$ and $R$ is:

The radius of gyration can be measured by static light scattering, small-angle X-ray scattering (SAXS), or small-angle neutron scattering (SANS). For unfolded proteins, the empirical scaling is $R_g \approx 2.0 \cdot N^{0.59}$ Angstrom in denaturing conditions.

Polymer Dynamics: The Rouse Model

The dynamics of an ideal polymer chain are described by the Rouse model, where each monomer is connected by harmonic springs and experiences Stokes drag. The relaxation time of the $p$-th Rouse mode is:

where $\tau_R$ is the longest (Rouse) relaxation time. The diffusion coefficient of the center of mass is:

For DNA in solution, hydrodynamic interactions are important, leading to the Zimm model where $\tau_Z \propto N^{3\nu} \approx N^{1.76}$ and$D \propto 1/R \propto N^{-\nu}$.