DNA Mechanics & Topology

The Worm-Like Chain (Kratky-Porod) Model

DNA is modeled as a continuously flexible rod with bending modulus $\kappa$. The bending energy of a configuration is:

where $\hat{t}(s)$ is the unit tangent vector at arc length $s$ and$R(s)$ is the local radius of curvature. The persistence length $L_p$ is defined as the characteristic decay length of tangent correlations:

For dsDNA: $L_p \approx 50$ nm $\approx 150$ bp. This means dsDNA is rigid on scales below 50 nm and flexible on longer scales. For comparison: ssDNA has$L_p \approx 1$ nm, RNA has $L_p \approx 2$ nm, and microtubules have$L_p \approx 5$ mm.

Tangent Correlation Function

The tangent-tangent correlation function decays exponentially. To derive this, consider the energy of bending between two points separated by $\Delta s$. In 2D, the angle $\theta$ between tangents is Gaussian-distributed with variance$\langle\theta^2\rangle = \Delta s / L_p$. Using$\langle\cos\theta\rangle = \exp(-\langle\theta^2\rangle/2)$ in 2D (or more precisely, averaging over all fluctuation modes):

This result is exact for the WLC in 3D. It means orientational memory is exponentially lost over a characteristic distance $L_p$.

End-to-End Distance

The mean square end-to-end distance is computed by integrating the tangent correlation:

Substituting the exponential correlation and performing the double integral:

This interpolates between two limits:

• • Rigid rod ($L \ll L_p$): $\langle R^2 \rangle \approx L^2$
• • Flexible coil ($L \gg L_p$): $\langle R^2 \rangle \approx 2L_pL$ (random walk with step size $2L_p$, the Kuhn length)

Marko-Siggia Force-Extension Formula

The WLC force-extension relation cannot be obtained in closed form, but Marko and Siggia (1995) provided an excellent interpolation formula. For a WLC under force $F$ with fractional extension $z = x/L$:

This has the correct asymptotic behavior:

• • Low force: $F \approx \frac{3k_BT}{2L_pL}x$ (linear, entropic spring)
• • High force: $F \approx \frac{k_BT}{4L_p(1-z)^2}$ (diverges as $x \to L$)

This formula fits single-molecule stretching data (optical tweezers, magnetic tweezers) for dsDNA with $L_p = 50$ nm to within 1% accuracy up to forces of ~10 pN, beyond which the extensible WLC (including backbone stretching) is needed.

Topological Invariant: Linking Number

For a closed circular DNA, the linking number $\text{Lk}$is a topological invariant that can only be changed by breaking and rejoining strands. It decomposes into:

where Tw (twist) counts the helical turns of one strand around the other, and Wr (writhe) measures the coiling of the helical axis in 3D space. For relaxed B-DNA, $\text{Lk}_0 = N/h$ where$N$ is the number of base pairs and $h = 10.5$ bp/turn.

Supercoiling Density and Free Energy

The supercoiling density is the fractional change in linking number:

In bacteria, DNA is maintained at $\sigma \approx -0.06$ by the balance of gyrase (introduces negative supercoils) and topoisomerase I (removes them). The free energy stored in supercoiling is:

where $C$ is the torsional modulus ($C \approx 75$ nm $\times k_BT$). The quadratic dependence on $\sigma$ means the energy is symmetric for positive and negative supercoiling. Topoisomerases maintain the cell at the optimal $\sigma$ that facilitates strand separation for replication and transcription.

Topoisomerases

Topoisomerases are the enzymes that manage DNA topology:

• • Type I: Single-strand break, relaxes one turn at a time ($\Delta\text{Lk} = \pm 1$). No ATP required
• • Type II (e.g., gyrase): Double-strand break, passes one segment through another ($\Delta\text{Lk} = \pm 2$). ATP-dependent

The distribution of topoisomers in a relaxed population follows a Gaussian:$P(\Delta\text{Lk}) \propto \exp(-\frac{2\pi^2 C}{Lk_BT}(\Delta\text{Lk})^2)$, which allows experimental measurement of $C$ from gel electrophoresis of topoisomer ladders.

Jacobson-Stockmayer Factor

DNA looping is essential for gene regulation. The Jacobson-Stockmayer factor $j(L)$ is the effective concentration of one end of a polymer at the position of the other end. For a Gaussian chain (long DNA, $L \gg L_p$):

This follows from the Gaussian probability distribution for the end-to-end vector$\mathbf{R}$ of a WLC:

Setting $\mathbf{R} = 0$ for loop closure: $j(L) = P(0)$. The looping probability decreases as $L^{-3/2}$ for long DNA.

Short DNA: Bending Energy Barrier

For DNA shorter than a few persistence lengths ($L \lesssim 3L_p$), the Gaussian approximation breaks down. The bending energy required to form a loop of radius$R_{\text{loop}} = L/(2\pi)$ is:

For a 100 bp loop ($L = 34$ nm): $G_{\text{bend}} \approx 29 k_BT$, making spontaneous looping extremely rare. The Shimada-Yamakawa theory provides the full WLC result including both entropic and energetic contributions:

where $f(L/L_p)$ is a correction factor that accounts for the non-Gaussian statistics of short chains. The looping probability has a maximum near $L \approx 500$ bp, balancing the bending energy penalty (short loops) against the entropic dilution (long loops).

Wrapping Free Energy

In the nucleosome, 147 bp of DNA ($\approx 50$ nm) are wrapped in 1.67 superhelical turns around the histone octamer (radius $R \approx 4.2$ nm). The bending energy cost is:

Substituting $L_p = 50$ nm, $L = 50$ nm, $R = 4.2$ nm:

This enormous bending energy (~175 kJ/mol) must be compensated by favorable histone-DNA interactions. Approximately 14 contact points between the histone octamer and the DNA minor groove contribute a total binding energy of ~28-40 $k_BT$ per contact site (electrostatic + hydrogen bonds), giving a total adhesion energy of ~100-140 $k_BT$. The net wrapping free energy is therefore $\Delta G \approx -30$ to $-70\,k_BT$, depending on DNA sequence (some sequences are easier to bend).

Unwrapping Equilibrium

DNA spontaneously unwraps from nucleosome ends ("breathing"). The equilibrium constant for unwrapping $n$ base pairs from one end is:

where $\Delta G_{\text{contact}}(n)$ is the lost histone-DNA contact energy and$\Delta G_{\text{bend}}(n)$ is the recovered bending energy. The competition between these terms determines the unwrapping profile. FRET and force spectroscopy experiments show:

• • Outer wraps (first ~20 bp from each end) unwrap transiently at equilibrium ($K \sim 0.01{-}0.1$)
• • Inner wraps are much more stable ($K < 10^{-4}$)
• • Applied force of ~3-5 pN is sufficient to unwrap the outer turns
• • Full unwrapping requires ~20 pN in single-molecule pulling experiments

Force-Extension-Torque Relationship

When DNA is simultaneously stretched and twisted (as in magnetic tweezers), the elastic response depends on three moduli: the bending persistence length $L_p$, the torsional persistence length $C$, and the stretch modulus $S$. The torque in the DNA is:

As the DNA is overwound or underwound at constant force, the extension initially remains constant (torque builds up). Above a critical torque $\tau_c$, the DNA buckles to form plectonemic supercoils:

Above $\tau_c$, each additional turn removes a fixed amount of extension as it is absorbed into plectonemes. The slope $dx/d(\text{turns})$ depends on force and is a key experimental observable.

Phase Diagram

The response of DNA to torsion depends on whether it is overwound or underwound:

• • Extended phase: $|\tau| < \tau_c$. DNA stores torsional energy as uniform twist. Extension unchanged
• • Plectonemic phase (positive supercoiling): $\tau > \tau_c$. Excess turns form right-handed plectonemes, shortening the molecule
• • Melted/denatured phase (negative supercoiling at low force): $\sigma < -0.02$ at $F < 0.5$ pN. Underwinding denatures the DNA locally, creating single-stranded bubbles

At high forces ($F > 6$ pN), the plectonemic phase is suppressed and the DNA undergoes a structural transition to an overwound form (P-DNA) at high positive torsion, or to denatured regions at high negative torsion. This rich phase behavior has been mapped in detail by magnetic tweezers experiments.