---
title: "How much of a trap movement stretches DNA?"
description: "Solve bead–tether force balance, distinguish pulling stiffness from fluctuation stiffness, and propagate calibration and bandwidth errors."
canonical_url: "https://sandboxphysics.com/stories/single-molecule/series-compliance/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/single-molecule/series-compliance/"
last_modified: "2026-10-28"
---

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

L2 · Chapter 3

# How much of a trap movement stretches DNA?

Sandbox Physics · 2026-10-28

Moving the focus outward divides displacement between extending DNA and pulling the bead away from the focus. Both elements carry the same equilibrium force. Solving that balance separates the softness of the molecule from the softness of the instrument.

## Before you begin

Read chapters one and two. Local stiffness is the tangent slope at an operating point. Coordinates here run from the fixed attachment to the bead attachment, with bead radius subtracted.

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

- Recover molecular extension from force balance.
- Derive the stiffnesses governing pulling and fluctuations.
- Explain calibration and detector-bandwidth bias.

## One force balance determines both readouts

Put the fixed attachment at zero. Molecular extension is $z$ and the effective trap-center separation is $D$. Positive tension pulls the bead left. The trap pulls right with displacement $D-z$. At equilibrium the trap force equals DNA tension.

For a specified separation, trap stiffness and chain, this equation has one unknown. DNA force rises with extension as trap force falls. Their intersection determines the bead position and tension shown in the scene.

$$
k_{\rm trap}(D-z)=F_{\rm DNA}(z),\qquad z=D-\frac{F}{k_{\rm trap}}
$$

Bead radius is subtracted in this collinear geometry. A real surface assay also requires attachment height and pulling angle.

Reading: Wang et al.

## Pulling the endpoint puts two springs in series

Increase force slightly around one operating point. DNA extends by the force increment divided by its local stiffness $k_{\rm DNA}$. The bead’s offset from the focus increases by the increment divided by trap stiffness. Add the two displacements to obtain the endpoint movement.

The compliances therefore add. The softer element takes more displacement. At endpoint speed $v$, the local quasistatic loading rate is the effective stiffness times speed. It changes as DNA stiffens.

$$
\frac1{k_{\rm series}}=\frac1{k_{\rm trap}}+\frac1{k_{\rm DNA}},\qquad r=\frac{dF}{dt}\simeq k_{\rm series}v
$$

This is a local linearization. A large pull requires solving the nonlinear equilibrium again.

Reading: Wang et al. · M. Ribezzi-Crivellari & F. Ritort

## For a fluctuating bead, restoring forces add

Now hold both outer endpoints fixed and let the bead fluctuate right. DNA stretches, increasing its leftward pull. The rightward trap force decreases, adding another leftward restoring increment. Fluctuations therefore experience the sum of the stiffnesses.

The pulling experiment moved an outer endpoint; the fluctuation experiment held both endpoints fixed. Their different boundary conditions explain the different stiffnesses. Calling the fluctuation stiffness a trap calibration overestimates later molecular forces.

$$
\left\langle\delta z^2\right\rangle=\frac{k_BT}{k_{\rm trap}+k_{\rm DNA}},\qquad \tau=\frac{\gamma}{k_{\rm trap}+k_{\rm DNA}}
$$

This assumes small equilibrium fluctuations and constant drag. Strong nonlinearities, attachment rotation and surface hydrodynamics need a larger model.

Reading: M. Ribezzi-Crivellari & F. Ritort

## Worked example and error estimates

Choose $k_{\rm trap}=0.050\;\mathrm{pN\,nm^{-1}}$ and $k_{\rm DNA}=0.010\;\mathrm{pN\,nm^{-1}}$. The series stiffness is $0.00833\;\mathrm{pN\,nm^{-1}}$, whereas fluctuation stiffness is $0.0600\;\mathrm{pN\,nm^{-1}}$. An endpoint speed of $100\;\mathrm{nm\,s^{-1}}$ gives a local loading rate of $0.833\;\mathrm{pN\,s^{-1}}$.

A force of $1.00\;\mathrm{pN}$ displaces the bead by $20.0\;\mathrm{nm}$ from the focus. A ten-percent stiffness error gives a ten-percent force error. A $2.0\;\mathrm{nm}$ position-zero error adds $0.100\;\mathrm{pN}$ bias. Uncertain chain parameters change the reference model and cannot be averaged away as independent sampling noise.

Reading: Wang et al.

## Filtering changes the variance you can observe

A first-order detector suppresses fast variations. Multiply the fluctuation spectrum by the squared frequency response and integrate to predict the recorded variance. The linear bead model gives the reduction below, where $f_c$ is the bead corner frequency and $f_d$ is detector cutoff.

The Lab saves the bead position and the filtered record separately. Recomputing variance from those saved samples separates model truth, detector response and a bandwidth correction.

$$
|H(f)|^2=\frac1{1+(f/f_d)^2},\qquad \frac{\operatorname{Var}(z_{\rm det})}{\operatorname{Var}(z)}=\frac{f_d}{f_c+f_d}
$$

The ratio is the continuous-time stationary result for an Ornstein–Uhlenbeck process passed through a first-order low-pass filter. Finite sampling introduces additional aliasing and record-length effects.

Normalized power Frequency / bead corner frequency

- Bead
- Filtered detector

Analytic linear-model reference. Teal is bead fluctuation power; amber includes a detector with cutoff twice the bead corner frequency. The detector removes high-frequency power.

Reading: Lang et al. · M. Ribezzi-Crivellari & F. Ritort

## Reading one frozen default record

This synthetic default uses trap separation $840\;\mathrm{nm}$ and stiffness $0.025\;\mathrm{pN\,nm^{-1}}$, with DNA contour length $1000\;\mathrm{nm}$ and persistence length $50\;\mathrm{nm}$. Calibration gain is one, bandwidth is $1000\;\mathrm{Hz}$ and seed is $1937$. It saves $8192$ samples at $0.025\;\mathrm{ms}$ intervals.

The model column comes from equilibrium and its linearized fluctuation process. The record column is a mean or variance computed directly from saved detector samples. Their small difference reflects finite correlated sampling. A new seed changes the record column; longer records do not remove a shared calibration error.

Synthetic DNA default: model references and saved detector statistics are distinct.

| Quantity | Model reference | Saved record |
| --- | --- | --- |
| Tension $(\mathrm{pN})$ | $0.64284$ | $0.64885$ |
| Extension $(\mathrm{nm})$ | $814.286$ | $814.046$ |
| Filtered extension variance $(\mathrm{nm^2})$ | $85.2347$ | $83.6603$ |

Reading: Sandbox Physics

## Try the prediction

### [Separate pulling from fluctuations](https://sandboxphysics.com/labs/dna-tether-elasticity/)

1. At fixed focus position, predict what a stiffer trap does to tension and fluctuation amplitude. Check the scene and the separate data record.
2. Compare A stretched chain with The trap gives way. Then keep the seed fixed and change trap stiffness alone. Check that extension and force share the same equilibrium.
3. Change detector bandwidth alone. Identify the random error that a longer record reduces and the calibration bias that remains.

## Exercises and explanations

1. If DNA is much stiffer than the trap, where does most additional focus movement go?

   Read the explanation

   Into the bead’s offset from the focus. The stiff molecule extends little under a force increment, so the series stiffness approaches the softer trap stiffness.

2. What is the recorded variance when detector cutoff equals bead corner frequency?

   Read the explanation

   Half the true variance in this first-order model. Standard deviation falls by the square root of two. An uncorrected equipartition calculation overestimates the total stiffness by a factor of two.

[Previous: Why DNA gets harder to stretch](https://sandboxphysics.com/stories/single-molecule/dna-elasticity/) [Next: Why a molecule unfolds at different forces](https://sandboxphysics.com/stories/single-molecule/unfolding-hazard/)

## 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. [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.

4. [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.