FR19-A · A molecular spring

Stretch a Single DNA Molecule

How much of a pull stretches DNA, and how much moves the bead? The imposed distance is shared between DNA extension and the bead’s displacement from the trap center. A softer trap takes up more of that distance, so the same apparatus separation produces less molecular tension.

Drag an optical trap away from a fixed anchor. Watch the DNA straighten and the bead fall behind the focus, then compare the force with the filtered detector record.

Mean equilibrium tension $0.643\,\mathrm{pN}$

Force from saved samples $0.649\,\mathrm{pN}$

DNA extension $814\,\mathrm{nm}$

Series stiffness $0.00516\,\mathrm{pN}/\mathrm{nm}$

Pull against the chain

Trap separation

D/\mathrm{nm}

840

Trap stiffness

k_{\rm t}/(\mathrm{pN}/\mathrm{nm})

0.025

Persistence length

L_{\rm p}/\mathrm{nm}

50

Detector bandwidth

f_{\rm d}/\mathrm{Hz}

1000

Drag the bright trap ring to pull the bead. The bead’s displacement from that ring measures the optical force.

Calibration and random record

Distance calibration gain

g

1

Record seed

s

1937

The gain rescales displacement from the trap center in the stored detector channel. It changes inferred force and extension, while the physical bead and chain stay the same.

Geometry, units and assumptions

The DNA joins the fixed anchor to the bead’s left surface. Separation is measured to the trap center after subtracting the bead radius. Distances and bead radius use different drawing scales; the displayed excursion scale adapts to keep the complete stored record in frame.

$L=1000\,\mathrm{nm},\quad T=298\,\mathrm K,\quad a=500\,\mathrm{nm}$

Water viscosity is one millipascal second. Stokes drag assumes a spherical bead far from surfaces. A fixed contour length and persistence length describe an inextensible, untwisted chain. No internal polymer modes, hydrodynamic wall corrections, salt dependence or overstretching are included.

The bead fluctuates in the local harmonic approximation around the nonlinear mean equilibrium. The chain drawing indicates extension; it is not a sampled molecular conformation. Playback slows the stored record to twelve seconds.

Use the left and right arrow keys on the scene to move the trap.

Model assumptions

The Marko–Siggia interpolation for an inextensible worm-like chain sets the mean tension. A locally harmonic, one-dimensional bead model produces thermal motion and a first-order detector response. The strand is schematic; bead size and thermal motion are magnified. DNA internal relaxation, walls, salt changes, torsion, intrinsic stretch and overstretching are omitted. All records are synthetic.

Synthetic detector record

The instrument shares the pull

Fluctuations, bandwidth and uncertainty

\sigma_{\rm record}=10\,\mathrm{nm}

The recorded variance is computed from the saved calibrated detector samples. The reference variance comes from integrating the bead spectrum through a first-order low-pass detector. A finite record has sampling scatter; the reference is an ensemble expectation.

$\sigma_{\rm det}^2=\frac{k_{\rm B}T}{k_{\rm t}+k_{\rm m}}\frac{2\pi f_{\rm d}}{2\pi f_{\rm d}+1/\tau},\qquad \tau=\frac{\gamma}{k_{\rm t}+k_{\rm m}}$

The formula uses seconds and hertz. Exported times use milliseconds. The seed fixes stochastic scatter; calibration gain introduces systematic scale error. Temperature, drag and the distance zero are held known. The model does not include instrumental drift or independent electronic noise.

The saved file includes physical parameters, seed, model version, raw model positions, filtered positions, recorded forces and the estimates shown here.

Continue with single-molecule biophysics

Start with the essentials

Focus question: Why does a softer trap make the same DNA look softer?
One-sentence intuition: DNA extension and bead displacement add to the imposed separation, so the apparatus and molecule act as springs in series.

Core mathematical model

Force balance along the tether

F_{\rm WLC}(x)=k_{\rm t}(D-x)

The anchor is at zero and pulling to the right is positive. The extension is the distance to the bead attachment; the focus coordinate uses the same bead-radius correction. Force is in piconewtons and distance in nanometers.

The worm-like chain interpolation

F_{\rm WLC}(x)=\frac{k_{\rm B}T}{L_{\rm p}}\left[\frac{1}{4(1-x/L)^2}-\frac14+\frac{x}{L}\right]

Contour length is the length along the chain; persistence length measures how far its direction remains correlated. Pulling first removes bends. Near the contour length, removing the remaining bends costs much more force.

Two different stiffnesses

k_{\rm series}=\frac{k_{\rm t}k_{\rm m}}{k_{\rm t}+k_{\rm m}},\qquad \langle\delta x^2\rangle=\frac{k_{\rm B}T}{k_{\rm t}+k_{\rm m}},\quad k_{\rm m}=\frac{\mathrm dF_{\rm WLC}}{\mathrm dx}

Slowly moving the focus probes series stiffness. At fixed focus, the bead feels both restoring slopes, so its local thermal variance depends on their sum. The fluctuation formula is a local harmonic approximation.

Common difficulties

A quieter bead can mean a slower detector

Typical misconceptionA smooth detector trace proves that the DNA has become stiffer.

Better mental modelCompare the bandwidth with the bead relaxation time. The detector can remove high-frequency motion before the samples are stored.

Run the experiment

01
Watch the distance split
Drag the bright trap ring to the right. Compare A stretched chain with The trap gives way.
What to observe: The bead follows the focus incompletely. A softer trap absorbs more of the imposed displacement, so the same apparatus separation produces less DNA tension.
02
Find the steep part
Choose Hard to straighten. Reduce separation, then bring it back up while watching the reference curve below.
What to observe: Extension approaches the fixed contour length. Equal increments of extension require rapidly increasing force.
03
Slow the detector
Compare A stretched chain and Motion disappears. Keep the physical parameters fixed and vary only Detector bandwidth.
What to observe: The saved detector trace loses fast fluctuations, but the underlying bead record and equilibrium force stay the same. A variance estimate that ignores filtering overestimates the total restoring stiffness.

Stretch a Single DNA Molecule

The instrument shares the pull

A bead and a molecule share the displacement

Core mathematical model

Force balance along the tether

The worm-like chain interpolation

Two different stiffnesses

Common difficulties

A quieter bead can mean a slower detector

Run the experiment

Watch the distance split

Find the steep part

Slow the detector

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