---
title: "Inferring a barrier from many unfolding events"
description: "Derive the loading-rate shift of rupture forces, fit censored records, and diagnose bias from compliance and finite detector response."
canonical_url: "https://sandboxphysics.com/stories/single-molecule/loading-rate-inference/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/single-molecule/loading-rate-inference/"
last_modified: "2026-10-28"
---

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

L3 · Chapter 5

# Inferring a barrier from many unfolding events

Sandbox Physics · 2026-10-28

A faster pull leaves less time for escape at low force, shifting the unfolding distribution upward. Comparing loading rates reveals how force changes escape kinetics. The useful input is each record’s force history, event and observation endpoint, including records without an event.

## Before you begin

Chapter four on survival and accumulated hazard. A logarithm turns exponential dependence into a line. Likelihood compares how readily different parameters could have produced the observed records.

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

- Derive the most probable unfolding force.
- Estimate a zero-force rate with censoring at a known transition distance.
- Identify instrumental bias in an inferred rate.

## Read the shift before fitting it

Take the logarithm of the first-passage density and differentiate with respect to force. At an internal maximum, the escape rate equals loading rate times $x^\ddagger/(k_BT)$. Substituting the Bell rate gives a peak force linear in log loading rate.

This suggests a useful graph: peak force against log loading rate. Its slope is inversely proportional to transition distance. Histogram bins and record length affect an estimated peak, so a quantitative fit should use individual records.

$$
F_* = \frac{k_BT}{x^\ddagger}\ln\!\left(\frac{r x^\ddagger}{k_0k_BT}\right)
$$

The peak is interior only when $F_*>F_0$. Otherwise the density is largest at $F_0$, which is two piconewtons in the Lab.

Reading: Evans & Ritchie

## Worked example for a tenfold faster ramp

Use $k_BT=4.11\;\mathrm{pN\,nm}$, $x^\ddagger=0.50\;\mathrm{nm}$ and $k_0=0.10\;\mathrm{s^{-1}}$. At $10\;\mathrm{pN\,s^{-1}}$ the internal peak is about $20.5\;\mathrm{pN}$. At $100\;\mathrm{pN\,s^{-1}}$ it is about $39.5\;\mathrm{pN}$. The shift is $18.9\;\mathrm{pN}$, equal to thermal energy divided by transition distance times the natural logarithm of ten.

These are results of one teaching model. Different experimental slopes can motivate checks of barrier shape, multiple pathways and force calibration. A segmented line alone cannot identify an atomic structure.

Reading: Evans & Ritchie

## Censored records contribute to likelihood

Record $i$ ends at $t_i$. Set $\delta_i=1$ for an observed unfolding event and zero otherwise. With transition distance fixed, integrate the exponential force factor along the actual force history to obtain $A_i$. Every event contributes a rate factor; every record contributes survival.

Weighted exposures have units of seconds. Multiplying event factors and survival probabilities gives the likelihood. Its logarithm makes the zero-force-rate dependence explicit. Differentiation gives event count divided by weighted exposure. At a common constant force, events divided by total time estimates the rate at that force; divide by the force-enhancement factor to recover the zero-force rate. With no events, exposure still constrains an upper bound.

$$
A_i=\int_0^{t_i}e^{F_i(t)x^\ddagger/(k_BT)}dt,\qquad \ln L(k_0)=N_{\rm event}\ln k_0-k_0\sum_i A_i+\mathrm{constant}
$$

Transition distance is fixed, so terms independent of the zero-force rate enter the constant. Censored records still contribute exposure.

$$
\frac{d\ln L}{dk_0}=\frac{N_{\rm event}}{k_0}-\sum_i A_i=0,\qquad \widehat k_0=\frac{\sum_i\delta_i}{\sum_i A_i}
$$

This also assumes exact event times. Estimating both distance and zero-force rate needs multiple loading conditions; their estimates may be strongly correlated.

Reading: Evans & Ritchie

## A threshold can lengthen the apparent wait

In a real assay, both bead and detector take time to follow unfolding. First consider an ideal reference with detector lag alone: a noiseless isolated displacement step reaches half height after a delay through a continuous first-order filter.

The unfolding Lab assumes instantaneous bead force balance. It adds independent Gaussian noise to position samples and then applies a discrete single-pole filter, without separate bead relaxation or thermal motion. Detection subtracts the known folded baseline after the same filter, then finds the first half-jump crossing. Latent and detected events are saved separately; records without a crossing at the endpoint remain right censored.

$$
y(t)=\Delta z\left(1-e^{-(t-t_0)/\tau_d}\right),\qquad t_{1/2}-t_0=\tau_d\ln2
$$

This continuous noiseless reference uses $\tau_d=(2\pi f_d)^{-1}$. A sampled crossing is localized only within a sample interval; noise can also shift its first occurrence.

$$
y_j=ay_{j-1}+(1-a)x_j,\qquad a=e^{-2\pi f_d\Delta t}
$$

This recurrence generates the saved unfolding detector data. Its pole parameter comes from the continuous response; near Nyquist, calculate the response from the recurrence itself.

Reading: Lang et al.

## Reserve a new protocol for prediction

Fit several ramps, freeze the parameters, and predict a constant-force wait or a segmented ramp. Comparing only the histograms used in fitting can conceal an omitted process.

If the new protocol fails systematically, first check force history, series compliance and detection. Then consider additional barriers or refolding. Chapter eight applies this prediction test to a motor network.

Reading: Evans & Ritchie · Lang et al.

## Try the prediction

### [Recompute an estimate from saved events](https://sandboxphysics.com/labs/stochastic-unfolding-force/)

1. Predict the peak shift, then compare Give it time with Pull quickly. These presets retain molecular parameters.
2. Export the ensemble and inspect event and censoring fields. Recompute zero-force rate from weighted exposure.
3. Reduce bandwidth within one protocol and compare latent-event inference with threshold-event inference. A slow detector also switches to a force clamp, so that preset does not isolate bandwidth alone.

## Exercises and explanations

1. Twenty constant-force events occur during forty seconds of total exposure. What is the rate estimate at that force?

   Read the explanation

   The force-dependent rate estimate is $0.500\;\mathrm{s^{-1}}$. Exposure includes records ending without an event. Divide by the force-enhancement factor to estimate the zero-force rate.

2. Where is the maximum when a slow ramp puts the internal stationary point below the starting force?

   Read the explanation

   At the starting-force boundary. For a ramp beginning at two piconewtons, search only at or above two piconewtons. An interior stationary point matters only when it lies in the supported interval.

[Previous: Why a molecule unfolds at different forces](https://sandboxphysics.com/stories/single-molecule/unfolding-hazard/) [Next: How ATP gives random transitions a direction](https://sandboxphysics.com/stories/single-molecule/chemical-motor/)

## Sources and calculation scope

1. [Lang et al., An Automated Two-Dimensional Optical Force Clamp for Single Molecule Studies, Biophysical Journal 83, 491–501 (2002)](https://pmc.ncbi.nlm.nih.gov/articles/PMC1302163/)

   The apparatus and feedback discussion motivate separating applied load, bead response and detector response. The unfolding and motor experiments assume ideal prescribed-force control; they do not simulate this complete feedback apparatus.

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.