---
title: "Why a molecule unfolds at different forces"
description: "Derive survival and first-passage distributions under a force clamp and a ramp, and retain the information in censored records."
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language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/single-molecule/unfolding-hazard/"
last_modified: "2026-10-28"
---

[Single-molecule physics from bead motion to motors](https://sandboxphysics.com/stories/single-molecule/)

L2 · Chapter 4

# Why a molecule unfolds at different forces

Sandbox Physics · 2026-10-28

Crossing an unfolding barrier is random. Force lowers the forward barrier and increases the chance of escape, but it does not assign a unique unfolding force. Repeated pulls give a distribution. Explaining that distribution requires both molecular escape kinetics and the applied force history.

## Before you begin

Use chapter three on force and loading rate. Probabilities lie between zero and one. An integral adds risks over short time intervals; an exponential describes loss proportional to the surviving population.

[Review the preceding chapter](https://sandboxphysics.com/stories/single-molecule/series-compliance/)

- Distinguish rate, waiting time and unfolding force.
- Derive survival under a constant force and a linear ramp.
- Use the information in records that end before unfolding.

## What an instantaneous rate means

Track one folded state and one absorbing unfolded state. Conditional on remaining folded, the probability of unfolding during a short interval $dt$ is approximately $k(F)dt$. The rate $k(F)$ has units of inverse seconds. It is not the cumulative unfolding probability.

Let $S(t)$ be the probability of remaining folded at time $t$. Only a molecule that survives to this time can unfold in the next interval. The loss rate therefore includes $S(t)$. Integrating this conditional rule gives the survival law.

$$
\frac{dS}{dt}=-k(F(t))S(t),\qquad S(t)=\exp\!\left[-\int_0^t k(F(s))\,ds\right]
$$

This is a memoryless single-barrier model conditional on the force. It omits refolding, molecular heterogeneity and hidden folded states.

Reading: Evans & Ritchie

## How force changes a barrier

The distance from the folded state to the transition state along extension is $x^\ddagger$. In the Bell approximation, force lowers the barrier by $Fx^\ddagger$. Escape grows exponentially with this energy relative to thermal energy. The zero-force rate is $k_0$.

Picture the transition state as the narrow passage required to leave a stable shape. One effective distance describes how force changes access to that passage. At high force, barrier-shape changes can invalidate the approximation. The Lab uses specified teaching parameters.

$$
k(F)=k_0\exp\!\left(\frac{Fx^\ddagger}{k_BT}\right)
$$

Force times distance is an energy, so the exponent is dimensionless. A larger transition distance gives a stronger force dependence.

Reading: G. I. Bell · Evans & Ritchie

## A constant force produces a spread of waiting times

With ideal force control, the rate is constant. Survival is exponential and the mean waiting time is the inverse rate. A molecule that has already waited a long time has the same conditional chance of escape during the next short interval.

Take $k_0=0.10\;\mathrm{s^{-1}}$, $x^\ddagger=0.50\;\mathrm{nm}$ and $k_BT=4.11\;\mathrm{pN\,nm}$. At $10.0\;\mathrm{pN}$, the rate is about $0.338\;\mathrm{s^{-1}}$ and the mean wait is $2.96\;\mathrm{s}$. The chance of still being folded after three seconds is about $0.363$. Such records contain useful information.

$$
S(t)=e^{-kt},\qquad p(t)=ke^{-kt},\qquad \langle t\rangle=\frac1k
$$

A density multiplied by a small interval gives a probability. The mean wait is not an appointment for each individual molecule.

Folded fraction Time (s)

Analytic survival reference for the worked example at constant 10 pN. A smooth probability curve describes an ensemble; an individual molecule unfolds once at a random time.

Reading: Evans & Ritchie

## A ramp converts waiting into an unfolding force

Increase force from $F_0$ at a constant loading rate $r$, so $F=F_0+rt$. Escape becomes faster as force grows. Replacing time by force in the accumulated hazard gives the unfolding-force distribution.

To sample an event, draw a uniform number $U$ between zero and one and wait until cumulative hazard reaches $-\ln U$. This can be inverted analytically for a Bell ramp. It avoids guessing event times with a coarse time step. A new seed changes the realization but preserves the distribution.

$$
\alpha=\frac{x^\ddagger}{k_BT},\qquad S(F)=\exp\!\left[-\frac{k_0}{\alpha r}(e^{\alpha F}-e^{\alpha F_0})\right],\qquad p(F)=\frac{k(F)}rS(F)
$$

This holds for $r>0$ and $F\ge F_0$. The Lab starts at $F_0=2\;\mathrm{pN}$; a zero starting force sets the second exponential to one. Constant endpoint speed generally changes loading rate through series compliance; integrate its actual force history.

Reading: Evans & Ritchie

## Read an event in the frozen default

Keep the Lab default separate from the preceding clamp example. Its synthetic parameters are $k_0=0.100\;\mathrm{s^{-1}}$, $x^\ddagger=0.450\;\mathrm{nm}$ and $k_BT=4.114\;\mathrm{pN\,nm}$. A six-second ramp starts at $2\;\mathrm{pN}$ and rises at $12\;\mathrm{pN\,s^{-1}}$. The detector pole is $8\;\mathrm{Hz}$ with $0.020\;\mathrm{s}$ sampling. Seed $49$ generates $128$ molecules.

For the first molecule, the continuous event occurs before the sampled threshold crossing. Across the ensemble, random waiting and detection latency both affect the fitted rate. Every default molecule is detected within six seconds. At a smaller clamp force, retaining censored records becomes especially important.

Synthetic unfolding default: first-passage truth, detected events and ensemble estimates.

| Quantity | Saved value |
| --- | --- |
| First molecule, latent event time | $1.85230\;\mathrm{s}$ |
| First molecule, detector event time | $1.86\;\mathrm{s}$ |
| Specified zero-force rate | $0.100\;\mathrm{s^{-1}}$ |
| Zero-force rate from all latent events | $0.0932\;\mathrm{s^{-1}}$ |
| Zero-force rate from all detected events | $0.0918\;\mathrm{s^{-1}}$ |

Reading: Sandbox Physics

## Try the prediction

### [Compare one molecule with an ensemble](https://sandboxphysics.com/labs/stochastic-unfolding-force/)

1. Choose Hold the force. Predict whether some molecules remain folded after three seconds. Compare one record with ensemble survival.
2. Change the seed and see which features change. Keep records without a detected event at the observation endpoint as right censored.
3. Compare Give it time with Pull quickly, separating applied force, detector delay and the ensemble distribution.

## Exercises and explanations

1. Does surviving two seconds make unfolding more likely during the next second?

   Read the explanation

   Not in this constant-force, single-state memoryless model. If measured conditional risks change with waiting time, investigate hidden states, drift or heterogeneity.

2. How does discarding records without an observed event affect the mean wait?

   Read the explanation

   It biases the mean downward because long waits are preferentially removed. Keep the observation endpoint and mark right censoring to retain the information that the molecule waited at least that long.

[Previous: How much of a trap movement stretches DNA?](https://sandboxphysics.com/stories/single-molecule/series-compliance/) [Next: Inferring a barrier from many unfolding events](https://sandboxphysics.com/stories/single-molecule/loading-rate-inference/)

## Sources and calculation scope

1. [Sandbox Physics, FR19 synthetic default records and SHA-256 manifest (2026-10-28)](https://sandboxphysics.com/data/fr19/manifest.json)

   Frozen default parameters, seeds and saved observation channels for the three teaching Labs. All records are synthetic; the manifest lists each JSON dataset and its checksum.

2. [Evans & Ritchie, Dynamic Strength of Molecular Adhesion Bonds, Biophysical Journal 72, 1541–1555 (1997)](https://pmc.ncbi.nlm.nih.gov/articles/PMC1184350/)

   Force-dependent escape and loading-rate-dependent rupture. Our Bell first-passage calculation uses a single irreversible barrier and specified teaching rates. It is not a protein-specific fit or a reconstruction of the paper’s data.

3. [G. I. Bell, Models for the Specific Adhesion of Cells to Cells, Science 200(4342), 618–627 (1978)](https://doi.org/10.1126/science.347575)

   Printed page 622, equation (16), and page 623, Figure 6: a phenomenological force-dependent bond lifetime and a tilted energy landscape. We use its reciprocal-rate form for one irreversible teaching barrier, with no rebinding or protein-specific parameters.