How a restless bead becomes a force meter

Choose a direction before measuring a force

Take rightward as positive. The trap center is $X$ and the bead center is $x$ , both in nanometers. Near the focus, the force on the bead is $F_{\rm trap}=k_{\rm trap}(X-x)$ . Stiffness is measured in piconewtons per nanometer. A bead left of the focus feels a rightward force.

A tether attached on the left supplies the opposing force. At equilibrium the two forces balance. Without a tether, the mean bead position returns to the focus. Calibrating that simpler configuration helps us recognize the extra force when a molecule is attached.

1\;\mathrm{pN}=10^{-12}\;\mathrm{N},\qquad 1\;\mathrm{nm}=10^{-9}\;\mathrm{m}

Use compatible length, force and energy units throughout a calculation.

Reading: Wang et al. · Lang et al.

Water both damps and shakes the bead

Water molecules continually kick the bead. Viscous drag also removes its momentum quickly. On the millisecond time scale used here, we neglect inertia: the mean velocity under a force is that force divided by the drag coefficient. Thermal kicks keep the position fluctuating around its mean.

The trap potential near the focus is a bowl. Displacement costs energy quadratically. At thermal equilibrium, this positional degree of freedom stores half of $k_BT$ on average. Here $k_B$ is Boltzmann’s constant and $T$ is absolute temperature. The result converts position variance into stiffness.

U=\frac12 k_{\rm trap}(x-X)^2,\qquad \left\langle(x-X)^2\right\rangle=\frac{k_BT}{k_{\rm trap}}

The record must sample equilibrium and retain the relevant thermal fluctuations. A slow detector suppresses the measured variance.

Reading: M. Ribezzi-Crivellari & F. Ritort

Worked example with a ten-nanometer fluctuation

Use $k_BT=4.11\;\mathrm{pN\,nm}$ near room temperature. A free bead has a true positional standard deviation of $10.0\;\mathrm{nm}$ , giving a variance of $100\;\mathrm{nm^2}$ . Dividing thermal energy by that variance gives $k_{\rm trap}=0.0411\;\mathrm{pN\,nm^{-1}}$ .

Attach a tether. Its mean displacement from the trap center is now $40.0\;\mathrm{nm}$ to the left. In the linear trap range, the equilibrium tension is $1.64\;\mathrm{pN}$ . Before multiplying, check that trap stiffness and detector calibration have remained unchanged. Laser power, bead height and optical sensitivity can alter the calibration.

Reading: Wang et al.

Many samples can still contain little new information

Neighboring samples often belong to the same unfinished fluctuation. After a displacement, the mean offset relaxes exponentially. Its time constant is drag divided by stiffness. Doubling the sampling rate does not automatically double the independent information.

To assess random uncertainty, divide a long record into blocks much longer than the relaxation time and compare their estimates. An error shared by the entire calibration survives this averaging. Attaching DNA adds another restoring force; we will need to reconsider the fluctuations.

\tau=\frac{\gamma}{k_{\rm trap}},\qquad C(t)=\left\langle\delta x^2\right\rangle e^{-|t|/\tau}

Drag in piconewton milliseconds per nanometer gives a time in milliseconds. This is a linear overdamped model; nearby surfaces alter drag.

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

Try the prediction

Start with the untethered bead

Predict whether a stiffer trap increases or decreases the fluctuations. Change stiffness alone and compare the position record and power spectrum.
Keep the existing free-bead geometry. Record the calibration method before moving to the tether experiment in the next chapter.

Exercises and explanations

A detector reduces the true ten-nanometer standard deviation to eight nanometers. Does a direct equipartition calibration overestimate or underestimate stiffness?

Read the explanation

It overestimates stiffness. The variance falls from one hundred to sixty-four square nanometers, so the inferred stiffness is 1.56 times the true value. A longer record reduces sampling scatter but cannot restore frequencies removed by the detector.

What happens if you calibrate a tethered bead with the free-bead variance formula?

Read the explanation

The tether supplies an additional restoring force. The smaller variance makes the inferred trap stiffness include molecular stiffness. Chapter three separates the stiffnesses relevant to fluctuations and to pulling.

Sources and calculation scope

Wang et al., Stretching DNA with Optical Tweezers, Biophysical Journal 72, 1335–1346 (1997)

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.

Lang et al., An Automated Two-Dimensional Optical Force Clamp for Single Molecule Studies, Biophysical Journal 83, 491–501 (2002)

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.

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)

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.