FR20 · Bending and pulling

Pulling a Membrane Tether

Why does a long membrane tube resist pulling with nearly constant force? A long tube settles at the radius where bending and membrane tension balance. Adding another length then costs the same energy, so the static pulling force stays constant. Pulling at finite speed adds a drag force that relaxes when the trap stops.

Drag an optical trap to extend a preformed membrane tube. Watch the bead lag behind the trap, then let the pulling force settle as the tube grows.

Tube radius $58.6\,\mathrm{nm}$

Trap force $8.83\,\mathrm{pN}$

Tube length $1.60\,\mathrm{\mu m}$

Bead displacement $147\,\mathrm{nm}$

Pull, then hold

Bending rigidity

20

\kappa/(k_{\mathrm B}T)

Membrane tension

0.012

\sigma\;(\mathrm{pN/nm})

Trap stiffness

0.06

k_{\mathrm{trap}}\;(\mathrm{pN/nm})

Pulling distance

1400

\Delta X\;(\mathrm{nm})

Trap speed

500

v\;(\mathrm{nm/s})

Drag the amber trap to pull the bead. Pause to examine its displacement.

Model, units and sources

The tube is already formed. The reservoir holds tension fixed; the optical trap is linear. A specified effective drag describes dissipative motion.

$k_{\mathrm B}T=4.114\,\mathrm{pN\,nm},\quad\zeta=0.012\,\mathrm{pN\,s/nm}$

Derényi, Jülicher & Prost (2002), Eqs. (1)–(2)

Derive the pulling force

Model assumptions

A preformed long cylindrical tether exchanges lipid with a flat reservoir at fixed tension. Bending rigidity and tension set its radius and static force. A linear trap and specified effective drag set the pulling dynamics. Pressure difference, spontaneous curvature, tube nucleation, neck mechanics, finite reservoir depletion and bead Brownian motion are omitted. Tube and bead dimensions share one length scale; the neck and reservoir edge are schematic. All traces are model simulations.

Force in piconewtons over time in seconds

How the pulling record is calculated

An exact linear propagator advances the bead under a stationary trap, a constant-speed pull and a final hold. Export includes all times, positions, forces, units and model version. The bead drag is a chosen teaching parameter, not a calibration of a specific instrument.

Start with the essentials

Focus question: Why does a long membrane tube resist pulling with nearly constant force?
One-sentence intuition: Once a tube has a fixed radius, each extra length costs the same membrane energy.

Core mathematical model

Energy per tube length

E/L=\pi\kappa/r+2\pi\sigma r

The first term penalizes a narrow, strongly curved tube. The second charges tension for the membrane area added to it.

Radius and static pulling force

r_0=\sqrt{\kappa/(2\sigma)},\qquad f_0=2\pi\sqrt{2\kappa\sigma}

Minimize the energy with respect to radius, then differentiate it with respect to length. These are the long-cylinder, zero-pressure results.

What the trap measures

F=k_{\mathrm{trap}}(X-L),\qquad\zeta\dot L=F-f_0

Trap position and tube end position use the same axial origin; a constant bead-radius offset is suppressed in both. The specified drag turns excess trap force into tube motion.

Common difficulties

A plateau and a moving bead

Typical misconceptionThe optical force must always equal the static tube force.

Better mental modelDuring a finite-speed pull, the bead needs extra force to overcome drag. When the trap stops, the bead catches up and the force relaxes.

Run the experiment

01
Watch the bead lag
Watch A growing tube, then pause during the pull.
What to observe: The amber trap centre sits ahead of the cyan bead. Their separation sets the force in the trace below.
02
Change tension
Choose Tension makes a thin tube.
What to observe: The tube becomes thinner and its static pulling force rises.
03
Move the trap yourself
Drag the amber trap handle, then release it.
What to observe: The bead follows with a delay, then settles at the displacement required by the static force.

Pulling a Membrane Tether

Pulling faster costs extra force

A tube trades bending for area

Core mathematical model

Energy per tube length

Radius and static pulling force

What the trap measures

Common difficulties

A plateau and a moving bead

Run the experiment

Watch the bead lag

Change tension

Move the trap yourself

Related experiments and articles

From curvature to active membrane measurements