2.2 The Double Helix

Step 1: Beer-Lambert Law for individual nucleotides

For a solution of isolated nucleotides at concentration c with path length l:

$$A = \varepsilon c l$$

where $\varepsilon$ is the molar extinction coefficient. At 260 nm, individual nucleotides have $\varepsilon \approx 8,000$--$15,000$ M$^{-1}$cm$^{-1}$.

Step 2: Electronic coupling between stacked bases

When bases are stacked in a duplex, the $\pi \to \pi^*$ transition dipoles of adjacent bases interact through exciton coupling. The interaction Hamiltonian between two stacked chromophores is:

$$H_{12} = \frac{\vec{\mu}_1 \cdot \vec{\mu}_2 - 3(\vec{\mu}_1 \cdot \hat{r})(\vec{\mu}_2 \cdot \hat{r})}{4\pi\varepsilon_0 r^3}$$

where $\vec{\mu}_1, \vec{\mu}_2$ are the transition dipole moments and r is their separation (~3.4 Angstrom).

Step 3: Exciton splitting creates two states

The coupling splits the excited state into two exciton states with energies $E_{\pm} = E_0 \pm H_{12}$. For parallel stacked bases, the oscillator strength redistributes: the lower-energy transition gains intensity while the higher-energy transition (at 260 nm) loses intensity.

Step 4: The effective extinction coefficient of duplex DNA

The observed extinction coefficient of duplex DNA is reduced (hypochromic) compared to the sum of individual bases:

$$\varepsilon_{duplex} = \varepsilon_{ss} \cdot (1 - h)$$

where h is the hypochromicity factor ($h \approx 0.27$ for B-DNA at 260 nm).

Step 5: Hyperchromicity upon denaturation

When the duplex denatures, stacking is disrupted and the full extinction coefficient is recovered. The relative increase is:

$$\text{Hyperchromicity} = \frac{A_{ss} - A_{ds}}{A_{ds}} = \frac{\varepsilon_{ss} - \varepsilon_{duplex}}{\varepsilon_{duplex}} = \frac{h}{1-h}$$

$$= \frac{0.27}{0.73} \approx 0.37 = 37\%$$

Step 6: Monitoring the melting transition

The fraction of denatured DNA at temperature T can be expressed as:

$$f_d(T) = \frac{A_{260}(T) - A_{260}^{ds}}{A_{260}^{ss} - A_{260}^{ds}}$$

At $T = T_m$, $f_d = 0.5$. The sharpness of the transition (cooperativity) depends on the enthalpy: a two-state transition with $\Delta H° = -400$ kJ/mol produces a transition width of only ~10 C, making Tm measurement precise.

Step 1: Define the tangent-tangent correlation

For a worm-like chain, the correlation between tangent vectors $\hat{t}(s)$ at positions s and s' along the contour decays exponentially:

$$\langle \hat{t}(s) \cdot \hat{t}(s') \rangle = e^{-|s - s'|/L_p}$$

$L_p$ is the persistence length: the characteristic distance over which the chain "remembers" its direction. For B-DNA, $L_p \approx 50$ nm (~150 bp).

Step 2: Write the end-to-end vector as an integral

The end-to-end vector $\vec{R}$ for a chain of contour length L is:

$$\vec{R} = \int_0^L \hat{t}(s) \, ds$$

Step 3: Compute the mean-square end-to-end distance

$$\langle R^2 \rangle = \langle \vec{R} \cdot \vec{R} \rangle = \int_0^L \int_0^L \langle \hat{t}(s) \cdot \hat{t}(s') \rangle \, ds \, ds' = \int_0^L \int_0^L e^{-|s-s'|/L_p} \, ds \, ds'$$

Step 4: Evaluate the double integral

Let $u = s - s'$. Using the symmetry of the integrand and computing:

$$\langle R^2 \rangle = 2L_p \int_0^L \left(1 - e^{-(L-s)/L_p}\right) ds$$

Evaluating this integral yields:

$$\boxed{\langle R^2 \rangle = 2L_p L\left[1 - \frac{L_p}{L}\left(1 - e^{-L/L_p}\right)\right]}$$

Step 5: Verify limiting cases

Rigid rod ($L \ll L_p$): Taylor expanding $e^{-L/L_p} \approx 1 - L/L_p + L^2/(2L_p^2)$ gives $\langle R^2 \rangle \approx L^2$ (as expected for a straight rod).

Flexible coil ($L \gg L_p$): The exponential term vanishes, giving $\langle R^2 \rangle \approx 2L_p L$. This is the random walk result with step size (Kuhn length) $b = 2L_p \approx 100$ nm.

Step 6: Biological significance -- DNA bending and nucleosome wrapping

The bending energy for curving DNA to radius R over length L is:

$$E_{bend} = \frac{L_p \cdot k_BT}{2} \cdot \frac{L}{R^2}$$

For nucleosome wrapping (R = 4.2 nm, L = 50 nm, $L_p = 50$ nm): $E_{bend} \approx 70 \, k_BT$. This enormous bending penalty is overcome by ~14 histone-DNA contact points and electrostatic interactions, explaining why nucleosome assembly requires histone chaperones and ATP-dependent remodeling.