The force needed to pull a membrane tube

A cylinder fed by a membrane reservoir

Take a cylinder of radius $r$ and length $L$ , leaving out its end connections. Choose the pulling direction and force $f$ positive along the axis. Principal curvatures are $1/r$ and zero. Set spontaneous curvature and pressure difference to zero. A sufficiently large reservoir supplies membrane at specified tension $\sigma$ .

The tube area is $2\pi rL$ . Multiply the bending energy density by this area and add the tension cost of taking area from the reservoir. This fixed-tension condition differs from the fixed-total-area condition in the shape experiment.

G(r,L)=\left(\frac{\pi\kappa}{r}+2\pi\sigma r\right)L

Bending favors a broad tube and tension a narrow one. End regions, finite reservoir area, pressure, spontaneous curvature and tube nucleation need additional modeling.

Source: I. Derényi · W. Helfrich

Minimize radius, then differentiate length

At fixed length, differentiate with respect to radius. The bending slope is negative and the tension slope positive. Setting their sum to zero gives the equilibrium radius. A positive second derivative confirms a minimum against radial changes.

Insert this radius into the energy per length. Total energy grows linearly with length, so $f_0=\partial G/\partial L$ is constant. The plateau describes an already formed, sufficiently long tube. Initial protrusion from a membrane has a different shape and end-region force.

-\frac{\pi\kappa}{r_0^2}+2\pi\sigma=0,\qquad r_0=\sqrt{\frac{\kappa}{2\sigma}}

Rigidity has energy units and tension has energy per area; the square root of their ratio is a length.

f_0=2\pi\sqrt{2\kappa\sigma}=\frac{2\pi\kappa}{r_0}

Energy per length is force. The two expressions also give an independently checkable product relation.

Source: I. Derényi · W. Helfrich

Moving the instrument also stretches it

Connect the pulling probe to a linear spring of stiffness $k$ . Let $X$ be the axial reservoir-to-support distance and $L$ the effective tube length. The spring extension is $x=X-L$ . Static balance equates its force with the tube force.

On a long-tube plateau, spring extension stays fixed, so further support motion becomes tube length. A softer instrument needs a larger displacement to supply the same force. Stop using the cylindrical limit when the remaining tube approaches the scale of the end regions. The experiment begins with an existing long tube; pull stroke adds to the initial support coordinate. During motion, drag allows instantaneous spring force to differ from the plateau until the tube settles.

G_{\rm total}=f_0L+\frac{k}{2}(X-L)^2,\qquad x=\frac{f_0}{k},\qquad L=X-\frac{f_0}{k}

This one-dimensional quasistatic series model assumes a spring-like probe and a preexisting tube that remains sufficiently long.

Worked example of force and displacement

Choose teaching parameters $\kappa=80\,\mathrm{pN\,nm}$ and $\sigma=0.010\,\mathrm{pN\,nm^{-1}}$ . Then $r_0=\sqrt{80/0.020}\,\mathrm{nm}\simeq63.2\,\mathrm{nm}$ and $f_0\simeq7.95\,\mathrm{pN}$ . This radius is much smaller than a micrometer-scale vesicle, so apparatus drawings often magnify the tube diameter.

For $k=0.050\,\mathrm{pN\,nm^{-1}}$ and $X=2.00\,\mu\mathrm m$ , spring displacement is approximately $159\,\mathrm{nm}$ and tube length $1.84\,\mu\mathrm m$ . Treating the support distance as tube length would overcount by the spring displacement.

Doubling tension narrows the tube and raises force. Doubling rigidity broadens it and also raises force. Observing both quantities helps distinguish the causes.

Recover two parameters from two measurements

Plateau force alone determines the product $\kappa\sigma$ . Add an independent radius measurement: multiplying force by radius gives rigidity and dividing force by radius gives tension. Use a radius measured independently of the assumed rigidity for this second observable.

\kappa=\frac{f_0r_0}{2\pi},\qquad \sigma=\frac{f_0}{4\pi r_0}

These expressions retain the long-cylinder, zero-pressure and zero-spontaneous-curvature assumptions. Probe stiffness and radius calibration errors propagate into the estimates.

Try the reasoning

Turn probe displacement into pulling force

Predict how increased tension changes diameter, force and probe displacement.
Hold rigidity fixed and vary tension. Then change instrument stiffness and compare displacement at the same force.
Drag the pulling end and compare support position, spring displacement and tube length. Explain why the plateau force does not grow with long-tube length.

Study an independent probe calibration

Inspect probe thermal motion and response in the specified calibration model.
Explain why membrane tube equations cannot determine instrument stiffness and list the quantities needed before converting displacement to force.

Exercises and explanations

Quadruple tension at fixed rigidity. How do radius and plateau force change?

Read the explanation

Radius halves and force doubles. Taking area becomes more expensive, favoring a narrower tube that requires a larger pulling force.

Measured force increases. Does this alone establish increased rigidity?

Read the explanation

No. Increased tension also raises plateau force. Measure radius or tension independently to distinguish the two.

Why cannot extrapolating the long-tube formula to zero length explain a nucleation peak?

Read the explanation

The derivation removes end regions, which dominate the short-tube problem. Their shape needs a fuller calculation. Constant plateau force describes the long-tube limit.

Sources and calculated scope

W. Helfrich, Elastic Properties of Lipid Bilayers: Theory and Possible Experiments, Z. Naturforsch. C 28, 693–703 (1973)

Original curvature-elasticity paper, pages 693–703. We use its local quadratic curvature energy with an explicitly stated sphere-positive curvature convention. The sphere, tube and small-slope examples are derived here; no original figure or measured material parameter is copied.

I. Derényi, F. Jülicher and J. Prost, Formation and Interaction of Membrane Tubes, arXiv:cond-mat/0205630v1; Physical Review Letters 88, 238101 (2002)

Version 1, 29 May 2002, equations (1)–(2): the Helfrich energy and the long cylindrical tube radius and force. We recalculate this analytic limit and add a specified linear instrument spring. The nucleation peak, short-tube profiles, tube interactions and published figure points are outside the reproduced scope.