---
title: "Stochastic Unfolding Force Spectroscopy Interactive Lab | Sandbox Physics"
description: "Why does the same molecule unfold at different forces? Unfolding is a random escape over a molecular energy barrier."
canonical_url: "https://sandboxphysics.com/labs/stochastic-unfolding-force/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/stochastic-unfolding-force/"
---

FR19-B · Waiting for a molecule to open

# Stochastic Unfolding Force Spectroscopy

**Why does the same molecule unfold at different forces?** Unfolding is a random escape over a molecular energy barrier. A stronger pull raises the escape rate, yet each molecule waits a different time. Pulling faster leaves less time to escape at low force and shifts the unfolding distribution upward.

Pull a folded domain with an ideal force clamp. Compare individual extension jumps with a population, then narrow the detector bandwidth to see how a jump arrives late.

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

Model assumptions

A single irreversible Bell transition releases a fixed extension in a linear handle. The force is imposed by ideal instantaneous feedback; bead drag, feedback delay, refolding and atomistic protein structure are excluded. Rates and geometry are teaching parameters. All records are synthetic.

128 synthetic molecules · fixed seed

## One jump, a whole distribution

### Extension of the selected molecule

Raw detector Filtered detector Model truth

### How many remain folded?

Detected population Ideal Bell survival

The default record contains an unfolding event. The gold curve follows the detector response.

Rate estimate, censoring and export

$\widehat k_0=\frac{N_{\rm events}}{\sum_i\int_0^{t_i}e^{F(s)x^\ddagger/(k_{\rm B}T)}\,ds}$

The barrier distance is supplied. For each saved filtered trace, subtract the filtered folded baseline and find the first half-jump crossing. The likelihood includes every trajectory, with the full six-second exposure for censored records. A slow detector shifts crossing times and biases this estimate. The export includes the seed, units, model version, imposed force protocol, exact event times, all raw and filtered positions, detected event times and both rate estimates.

The quoted standard error uses the event-count approximation under the assumed Bell model. It excludes barrier-distance uncertainty, force calibration error, baseline misspecification and detector bias.

[Continue the single-molecule course](https://sandboxphysics.com/stories/single-molecule/)

Physics tutorial

## Why unfolding force has a distribution

**Background** A folded domain can open after a random wait. Pulling lowers the activation barrier, so an event becomes more likely as force rises.

**Why it matters** A single jump is easy to see. Learning a molecular rate requires many molecules and a record of how long each one was observed.

Start with the essentials

- **Focus question**: Why does the same molecule unfold at different forces?
- **One-sentence intuition**: An unfolding force records both a random waiting time and the force applied during that wait.

### Core mathematical model

#### Force changes the opening rate

$$
k(F)=k_0\exp\!\left(\frac{Fx^\ddagger}{k_{\rm B}T}\right)
$$

The zero-force rate has units of inverse seconds. The barrier distance is measured along the pulling direction in nanometers. Positive tension lowers the barrier in this single-path Bell approximation.

#### Survival depends on the whole protocol

$$
S(t)=\exp\!\left[-\int_0^t k(F(s))\,ds\right],\qquad F(t)=F_0+rt
$$

Survival means the domain remains folded. During a force ramp, a molecule can open at any time; the cumulative rate determines the distribution of its opening force.

#### Ideal force balance locates the bead

$$
x_{\rm mol}=x_{\rm folded}+\frac{F}{\kappa_{\rm handle}}+u\Delta x,\qquad x_{\rm trap}-x_{\rm bead}=\frac{F}{\kappa_{\rm trap}}
$$

The molecular extension ends at the bead attachment. The indicator is zero before opening and one afterwards. Ideal feedback moves the trap with the extending molecule and keeps the prescribed tension.

### Common difficulties

#### A peak is not a breaking threshold

**Typical misconception** The peak of the unfolding-force distribution is the force required to open every molecule.

**Better mental model** The peak is the most frequent result under one loading protocol. Changing the loading rate changes the time available to cross the barrier and shifts that peak.

#### Keep the unfinished trajectories

**Typical misconception** A molecule that stays folded contributes no useful information.

**Better mental model** It survived the entire observation interval. Leaving it out removes long waits and usually makes opening appear faster than it was.

### Run the experiment

1. 01

   #### Predict what a slower pull changes

   Compare Give it time and Pull quickly. Watch the population curve against tension, then choose Next molecule several times.

   **What to observe:** A slower ramp leaves more time to open at lower forces. Individual molecules still open at different points because their waiting times are random.

2. 02

   #### Separate waiting from increasing force

   Choose Hold the force. Predict the shape of the folded fraction before watching the full replay.

   **What to observe:** The force is constant, yet different molecules open at different times. The ideal population follows an exponential survival curve.

3. 03

   #### Make the detector slow

   Choose A slow detector, then increase Detector bandwidth without changing the force protocol. Compare the two curves and the inferred rate.

   **What to observe:** The domain opens at the same model time, but the filtered position reaches its half-jump threshold later at low bandwidth. Treating that delay as molecular waiting biases the estimated rate downward.

Physical conclusion

Compare populations at known loading rates, keep censored waits, and account for the detector response.

## 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 a molecule unfolds at different forces**](https://sandboxphysics.com/stories/single-molecule/unfolding-hazard/)
2. [Article **Inferring a barrier from many unfolding events**](https://sandboxphysics.com/stories/single-molecule/loading-rate-inference/)
3. [Article **Can a motor model predict a new load protocol?**](https://sandboxphysics.com/stories/single-molecule/predicting-protocols/)
4. [Lab **ATP Molecular Motor** Reserve the load step for prediction](https://sandboxphysics.com/labs/atp-molecular-motor/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

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