Why DNA gets harder to stretch

Two different ways to measure a chain

The contour length $L$ follows the DNA backbone. The extension $z$ measures the separation of its ends along the pulling direction. A long chain can adopt many bent shapes, so a small extension can contain a much longer backbone.

The persistence length $L_p$ describes how far the local tangent retains its direction. A larger value means greater bending stiffness. We model DNA as a continuous inextensible chain in three-dimensional thermal equilibrium. The illustrated curve suggests bending; it is not a simulated microscopic conformation.

Reading: Marko & Siggia · Wang et al.

Straightening removes available shapes

Without an applied force, the chain can bend in many directions. Holding its ends farther apart reduces the number of compatible shapes. A force is required to maintain this less probable state. This entropic elasticity depends on temperature and bending stiffness.

Write the fractional extension as $q=z/L$ . The Marko–Siggia interpolation below joins the weak-stretching response to the strong increase in force near full extension. It is a practical approximation with a defined domain.

F(z)=\frac{k_BT}{L_p}\left[\frac{1}{4(1-z/L)^2}-\frac14+\frac zL\right],\qquad 0\le z<L

Use nanometers for lengths and piconewton nanometers for thermal energy. The interpolation approximates the statistical-mechanical worm-like chain; it is not an exact solution at every extension.

Tension (pN)Extension (nm)

Analytic interpolation reference: a 1000 nm inextensible chain with 50 nm persistence length and thermal energy 4.11 pN nm. The increasing slope is local molecular stiffness.

Reading: Marko & Siggia

Limits provide an independent check

Expand the force near zero extension. The constant terms cancel, leaving a linear spring with stiffness $3k_BT/(2L_pL)$ . Doubling contour length doubles the absolute extension under the same small force.

Close to contour length, the first term dominates. The remaining fractional slack decreases as the inverse square root of force. The diverging force follows from inextensibility. Real DNA can stretch its backbone and undergo structural changes at larger forces.

F\simeq\frac{3k_BT}{2L_pL}z\quad(z/L\ll1),\qquad 1-\frac zL\simeq\frac12\sqrt{\frac{k_BT}{FL_p}}\quad(FL_p\gg k_BT)

These limits catch missing factors, unit errors and confusion between extension and contour length. The strong-force limit still requires the inextensible approximation to be appropriate.

Reading: Marko & Siggia · Wang et al.

Worked example from eighty to ninety percent

Choose teaching parameters $L=1000\;\mathrm{nm}$ , $L_p=50\;\mathrm{nm}$ and $k_BT=4.11\;\mathrm{pN\,nm}$ . The prefactor is $0.0822\;\mathrm{pN}$ . At eighty percent extension, the bracket is $6.80$ , giving $0.559\;\mathrm{pN}$ . At ninety percent, it is $25.65$ , giving $2.11\;\mathrm{pN}$ .

The extra extension is only one tenth of the contour length, yet the force nearly quadruples. Local stiffness is the slope of this curve. Its steep increase changes both bead displacement and instrument compliance once we attach a trap.

Reading: Marko & Siggia

Try the prediction

Straighten the tether gradually

Predict the force change between eighty and ninety percent extension. Drag the trap and compare molecular extension with tension.
Keep chain parameters fixed and compare soft and stiff traps. Separate movement of the focus, movement of the bead and extension of the molecule.
Read the local-fluctuation assumption. A large drag recomputes equilibrium; the drawn chain is a geometric illustration.

Exercises and explanations

At fixed fractional extension, what happens to the interpolated tension when contour length doubles?

Read the explanation

It remains unchanged. Fractional extension and persistence length are the same. Absolute extension doubles and local stiffness halves.

Can an inextensible worm-like-chain curve continue through its contour length?

Read the explanation

No. Its extension must stay below contour length. Higher-force modeling needs an additional stretch elasticity and a stated range before structural transitions matter.

Sources and calculation scope

Wang et al., Stretching DNA with Optical Tweezers, Biophysical Journal 72, 1335–1346 (1997)

Figure 1 defines the surface–DNA–bead geometry; the elasticity analysis separates entropic and stretch compliance. Our collinear, inextensible teaching model uses its own parameters and synthetic records. No experimental points are digitized.

Marko & Siggia, Stretching DNA, Macromolecules 28, 8759–8770 (1995)

Worm-like-chain elasticity and the interpolation between weak and strong stretching. The displayed interpolation is an approximation to the statistical-mechanical model; enthalpic stretch, twist and overstretching are outside our Lab.