---
title: "Single-molecule physics from bead motion to motors"
description: "Learn DNA elasticity, stochastic unfolding and ATP-driven motors through eight connected chapters and three interactive experiments."
canonical_url: "https://sandboxphysics.com/stories/single-molecule/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/single-molecule/"
---

FR19

# Single-molecule physics from bead motion to motors

Sandbox Physics · 2026-10-28

A bead displaced by a molecule measures a tiny force. Follow that force from thermal motion to a stretching DNA chain, a random unfolding event and an ATP-driven motor. Learn what a trajectory reveals, how the instrument changes it, and how to test a model with a new load.

## The force meter is part of the experiment

The bead is large enough to locate with light, yet small enough to respond to a molecular force. Its displacement makes an invisible event visible. The trap, tether and detector all contribute to that signal. Understanding them first lets us ask a sharper question about the molecule itself.

$A_n$ Chemical transition $B_n$ Mechanical step $A_{n+1}$

Original schematic of the teaching cycle. A chemical transition changes state at one position; a mechanical transition advances to the next position. Both edges allow reverse transitions.

Begin with spring forces and thermal fluctuations. Build chain elasticity and series compliance before studying random escape. Then turn escape rates into a reversible chemical cycle. The final chapters ask which hidden features survive finite bandwidth, and whether a model predicts a load it has not seen before.

[Shared textbook foundations](https://github.com/yangzichao/sandboxphysics/issues/27) · [FR25 statistical-physics prerequisites as needed](https://github.com/yangzichao/sandboxphysics/issues/55) · [FR19 models and acceptance](https://github.com/yangzichao/sandboxphysics/issues/49)

1. L1 · 1

   ## [How a restless bead becomes a force meter](https://sandboxphysics.com/stories/single-molecule/bead-noise/)

   Use thermal motion to calibrate an optical trap, derive its relaxation time, and follow calibration errors into a molecular force measurement.

2. L2 · 2

   ## [Why DNA gets harder to stretch](https://sandboxphysics.com/stories/single-molecule/dna-elasticity/)

   Build an entropic force–extension law, define persistence length, and check the weak-stretching and near-contour-length limits.

3. L2 · 3

   ## [How much of a trap movement stretches DNA?](https://sandboxphysics.com/stories/single-molecule/series-compliance/)

   Solve bead–tether force balance, distinguish pulling stiffness from fluctuation stiffness, and propagate calibration and bandwidth errors.

4. L2 · 4

   ## [Why a molecule unfolds at different forces](https://sandboxphysics.com/stories/single-molecule/unfolding-hazard/)

   Derive survival and first-passage distributions under a force clamp and a ramp, and retain the information in censored records.

5. L3 · 5

   ## [Inferring a barrier from many unfolding events](https://sandboxphysics.com/stories/single-molecule/loading-rate-inference/)

   Derive the loading-rate shift of rupture forces, fit censored records, and diagnose bias from compliance and finite detector response.

6. L3 · 6

   ## [How ATP gives random transitions a direction](https://sandboxphysics.com/stories/single-molecule/chemical-motor/)

   Construct a reversible two-state motor, impose local detailed balance, and derive its stationary speed and stall load.

7. L3 · 7

   ## [What remains measurable when steps blur together?](https://sandboxphysics.com/stories/single-molecule/hidden-steps/)

   Separate motor jumps, bead dynamics and detector filtering, then distinguish velocity estimation from step-size and hidden-state inference.

8. L4 · 8

   ## [Can a motor model predict a new load protocol?](https://sandboxphysics.com/stories/single-molecule/predicting-protocols/)

   Recompute pinned single-cycle thermodynamic relations, compare kinetically distinct models, and test a held-out load protocol.

## Experiments along the route

- [Calibrate the untethered bead](https://sandboxphysics.com/labs/optical-tweezers-calibration/)
- [Stretch DNA through a compliant optical trap](https://sandboxphysics.com/labs/dna-tether-elasticity/)
- [Collect random unfolding events under different loads](https://sandboxphysics.com/labs/stochastic-unfolding-force/)
- [Drive a reversible motor and challenge it with a load step](https://sandboxphysics.com/labs/atp-molecular-motor/)

The calibration experiment keeps its existing untethered-bead model. The three new experiments generate seeded synthetic records from stated low-dimensional models. The sources below connect these calculations to original apparatus and theory; no historical measurement is presented as simulated acquisition.

## Recalculate the saved examples

The default synthetic records include units, parameters, model versions, seeds and separate raw and detector channels. Their manifest records checksums. The unfolding file retains one full trace and event/censoring metadata for the ensemble.

- [Data manifest and checksums](https://sandboxphysics.com/data/fr19/manifest.json)
- [Default DNA tether record](https://sandboxphysics.com/data/fr19/dna-tether-default.json)
- [Default unfolding trace and ensemble events](https://sandboxphysics.com/data/fr19/unfolding-default.json)
- [Default motor events and detector samples](https://sandboxphysics.com/data/fr19/motor-default.json)

## 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. [Wang et al., Stretching DNA with Optical Tweezers, Biophysical Journal 72, 1335–1346 (1997)](https://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/ufk_papers/DNA/wang.pdf)

   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.

3. [Marko & Siggia, Stretching DNA, Macromolecules 28, 8759–8770 (1995)](https://doi.org/10.1021/ma00130a008)

   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.

4. [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.

5. [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.

6. [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.

7. [K. Svoboda & S. M. Block, Force and Velocity Measured for Single Kinesin Molecules, Cell 77(5), 773–784 (1994)](https://doi.org/10.1016/0092-8674(94%2990060-4)

   Figures 1 and 4–5 describe the bead assay and linker compliance. The published experiment used a stationary trap. Our ideal-clamp two-state motor illustrates chemical affinity and load response with independently chosen rates.

8. [U. Seifert, Stochastic Thermodynamics, Fluctuation Theorems, and Molecular Machines, arXiv:1205.4176v1 (2012)](https://arxiv.org/html/1205.4176v1)

   Section 9.4.5, equations (240)–(244), gives local detailed balance. Section 10.5, equation (273), gives unicyclic input/output power. Our limited recomputation applies these relations to a chosen network. Figure 3 illustrates a general multicycle network; our two-state diagram is an original illustration.

9. [M. Ribezzi-Crivellari & F. Ritort, Force Spectroscopy with Dual-Trap Optical Tweezers: Molecular Stiffness Measurements and Coupled Fluctuations Analysis, Biophysical Journal 103(9), 1919–1928 (2012)](https://doi.org/10.1016/j.bpj.2012.09.022)

   Variance analysis and misalignment-induced coupling in a dual-trap assay. Our calculation retains one bead, collinear motion and a local harmonic tether, with an explicitly defined detector response.