---
title: "Stretch a Single DNA Molecule Interactive Lab | Sandbox Physics"
description: "How much of a pull stretches DNA, and how much moves the bead?"
canonical_url: "https://sandboxphysics.com/labs/dna-tether-elasticity/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/dna-tether-elasticity/"
---

FR19-A · A molecular spring

# Stretch a Single DNA Molecule

**How much of a pull stretches DNA, and how much moves the bead?** The imposed distance is shared between DNA extension and the bead’s displacement from the trap center. A softer trap takes up more of that distance, so the same apparatus separation produces less molecular tension.

Drag an optical trap away from a fixed anchor. Watch the DNA straighten and the bead fall behind the focus, then compare the force with the filtered detector record.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The Marko–Siggia interpolation for an inextensible worm-like chain sets the mean tension. A locally harmonic, one-dimensional bead model produces thermal motion and a first-order detector response. The strand is schematic; bead size and thermal motion are magnified. DNA internal relaxation, walls, salt changes, torsion, intrinsic stretch and overstretching are omitted. All records are synthetic.

Synthetic detector record

## The instrument shares the pull

Fluctuations, bandwidth and uncertainty

$\sigma_{\rm record}=10\,\mathrm{nm}$

The recorded variance is computed from the saved calibrated detector samples. The reference variance comes from integrating the bead spectrum through a first-order low-pass detector. A finite record has sampling scatter; the reference is an ensemble expectation.

$\sigma_{\rm det}^2=\frac{k_{\rm B}T}{k_{\rm t}+k_{\rm m}}\frac{2\pi f_{\rm d}}{2\pi f_{\rm d}+1/\tau},\qquad \tau=\frac{\gamma}{k_{\rm t}+k_{\rm m}}$

The formula uses seconds and hertz. Exported times use milliseconds. The seed fixes stochastic scatter; calibration gain introduces systematic scale error. Temperature, drag and the distance zero are held known. The model does not include instrumental drift or independent electronic noise.

The saved file includes physical parameters, seed, model version, raw model positions, filtered positions, recorded forces and the estimates shown here.

[Continue with single-molecule biophysics](https://sandboxphysics.com/stories/single-molecule/)

Physics tutorial

## A bead and a molecule share the displacement

**Background** One end of a DNA molecule is fixed. The other end pulls on a bead held by light. Moving the focus stretches the DNA and displaces the bead from the focus.

**Why it matters** The separation of the apparatus is easy to set. The molecule’s extension must be found from the bead position, and the force from its calibrated displacement.

Start with the essentials

- **Focus question**: Why does a softer trap make the same DNA look softer?
- **One-sentence intuition**: DNA extension and bead displacement add to the imposed separation, so the apparatus and molecule act as springs in series.

### Core mathematical model

#### Force balance along the tether

$$
F_{\rm WLC}(x)=k_{\rm t}(D-x)
$$

The anchor is at zero and pulling to the right is positive. The extension is the distance to the bead attachment; the focus coordinate uses the same bead-radius correction. Force is in piconewtons and distance in nanometers.

#### The worm-like chain interpolation

$$
F_{\rm WLC}(x)=\frac{k_{\rm B}T}{L_{\rm p}}\left[\frac{1}{4(1-x/L)^2}-\frac14+\frac{x}{L}\right]
$$

Contour length is the length along the chain; persistence length measures how far its direction remains correlated. Pulling first removes bends. Near the contour length, removing the remaining bends costs much more force.

#### Two different stiffnesses

$$
k_{\rm series}=\frac{k_{\rm t}k_{\rm m}}{k_{\rm t}+k_{\rm m}},\qquad \langle\delta x^2\rangle=\frac{k_{\rm B}T}{k_{\rm t}+k_{\rm m}},\quad k_{\rm m}=\frac{\mathrm dF_{\rm WLC}}{\mathrm dx}
$$

Slowly moving the focus probes series stiffness. At fixed focus, the bead feels both restoring slopes, so its local thermal variance depends on their sum. The fluctuation formula is a local harmonic approximation.

### Common difficulties

#### A quieter bead can mean a slower detector

**Typical misconception** A smooth detector trace proves that the DNA has become stiffer.

**Better mental model** Compare the bandwidth with the bead relaxation time. The detector can remove high-frequency motion before the samples are stored.

### Run the experiment

1. 01

   #### Watch the distance split

   Drag the bright trap ring to the right. Compare A stretched chain with The trap gives way.

   **What to observe:** The bead follows the focus incompletely. A softer trap absorbs more of the imposed displacement, so the same apparatus separation produces less DNA tension.

2. 02

   #### Find the steep part

   Choose Hard to straighten. Reduce separation, then bring it back up while watching the reference curve below.

   **What to observe:** Extension approaches the fixed contour length. Equal increments of extension require rapidly increasing force.

3. 03

   #### Slow the detector

   Compare A stretched chain and Motion disappears. Keep the physical parameters fixed and vary only Detector bandwidth.

   **What to observe:** The saved detector trace loses fast fluctuations, but the underlying bead record and equilibrium force stay the same. A variance estimate that ignores filtering overestimates the total restoring stiffness.

Physical conclusion

Measure bead position and trap displacement separately, and keep detector response in the force analysis.

## Related experiments and articles

Single-molecule forces and motors · Reading path

### Thermal motion, molecular elasticity, unfolding and motors

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why DNA gets harder to stretch**](https://sandboxphysics.com/stories/single-molecule/dna-elasticity/)
2. [Article **How much of a trap movement stretches DNA?**](https://sandboxphysics.com/stories/single-molecule/series-compliance/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)