FR19-B · Waiting for a molecule to open

Stochastic Unfolding Force Spectroscopy

Why does the same molecule unfold at different forces? Unfolding is a random escape over a molecular energy barrier. A stronger pull raises the escape rate, yet each molecule waits a different time. Pulling faster leaves less time to escape at low force and shifts the unfolding distribution upward.

Pull a folded domain with an ideal force clamp. Compare individual extension jumps with a population, then narrow the detector bandwidth to see how a jump arrives late.

Tension $30.8\,\mathrm{pN}$

Molecular extension $267\,\mathrm{nm}$

Folded population $0.2$

Rate inferred from detector $0.1\,\mathrm{s}^{-1}$

Pull, wait, compare

Force protocol

Loading rate

r\;(\mathrm{pN/s})

12

Clamp force

F_{\rm c}\;(\mathrm{pN})

18

Detector bandwidth

f_{\rm d}\;(\mathrm{Hz})

8

Replay time

t\;(\mathrm{s})

2.4

Drag the gold trap horizontally to change the pulling protocol. Each replay follows one saved molecule.

Kinetics and instrument assumptions

Positive force pulls to the right. Ideal feedback moves the trap to keep the prescribed tension, including when the domain opens. The bead is in quasistatic force balance; the detector filter acts only on the recorded position.

$k(F)=k_0e^{Fx^\ddagger/(k_{\rm B}T)}$

$k_0=0.1\,\mathrm{s}^{-1},\quad x^\ddagger=0.45\,\mathrm{nm},\quad k_{\rm B}T=4.114\,\mathrm{pN\,nm}$

$\kappa_{\rm trap}=0.3\,\mathrm{pN/nm},\quad \kappa_{\rm handle}=0.5\,\mathrm{pN/nm},\quad \Delta x=25\,\mathrm{nm}$

One irreversible first-passage event adds the fixed extension above. A linear handle, a fixed barrier distance and ideal feedback are teaching approximations. The bead and domain are enlarged; no protein-specific rates, refolding, viscous bead dynamics or finite feedback bandwidth are modeled.

The detector adds independent Gaussian position noise before a single-pole low-pass filter. All molecules are sampled every twenty milliseconds for six seconds. A trace that has not crossed half the extension jump by the end is right-censored.

Model assumptions

A single irreversible Bell transition releases a fixed extension in a linear handle. The force is imposed by ideal instantaneous feedback; bead drag, feedback delay, refolding and atomistic protein structure are excluded. Rates and geometry are teaching parameters. All records are synthetic.

128 synthetic molecules · fixed seed

One jump, a whole distribution

Extension of the selected molecule

Raw detectorFiltered detectorModel truth

How many remain folded?

Detected populationIdeal Bell survival

The default record contains an unfolding event. The gold curve follows the detector response.

Rate estimate, censoring and export

\widehat k_0=\frac{N_{\rm events}}{\sum_i\int_0^{t_i}e^{F(s)x^\ddagger/(k_{\rm B}T)}\,ds}

The barrier distance is supplied. For each saved filtered trace, subtract the filtered folded baseline and find the first half-jump crossing. The likelihood includes every trajectory, with the full six-second exposure for censored records. A slow detector shifts crossing times and biases this estimate. The export includes the seed, units, model version, imposed force protocol, exact event times, all raw and filtered positions, detected event times and both rate estimates.

The quoted standard error uses the event-count approximation under the assumed Bell model. It excludes barrier-distance uncertainty, force calibration error, baseline misspecification and detector bias.

Continue the single-molecule course

Start with the essentials

Focus question: Why does the same molecule unfold at different forces?
One-sentence intuition: An unfolding force records both a random waiting time and the force applied during that wait.

Core mathematical model

Force changes the opening rate

k(F)=k_0\exp\!\left(\frac{Fx^\ddagger}{k_{\rm B}T}\right)

The zero-force rate has units of inverse seconds. The barrier distance is measured along the pulling direction in nanometers. Positive tension lowers the barrier in this single-path Bell approximation.

Survival depends on the whole protocol

S(t)=\exp\!\left[-\int_0^t k(F(s))\,ds\right],\qquad F(t)=F_0+rt

Survival means the domain remains folded. During a force ramp, a molecule can open at any time; the cumulative rate determines the distribution of its opening force.

Ideal force balance locates the bead

x_{\rm mol}=x_{\rm folded}+\frac{F}{\kappa_{\rm handle}}+u\Delta x,\qquad x_{\rm trap}-x_{\rm bead}=\frac{F}{\kappa_{\rm trap}}

The molecular extension ends at the bead attachment. The indicator is zero before opening and one afterwards. Ideal feedback moves the trap with the extending molecule and keeps the prescribed tension.

Common difficulties

A peak is not a breaking threshold

Typical misconceptionThe peak of the unfolding-force distribution is the force required to open every molecule.

Better mental modelThe peak is the most frequent result under one loading protocol. Changing the loading rate changes the time available to cross the barrier and shifts that peak.

Keep the unfinished trajectories

Typical misconceptionA molecule that stays folded contributes no useful information.

Better mental modelIt survived the entire observation interval. Leaving it out removes long waits and usually makes opening appear faster than it was.

Run the experiment

01
Predict what a slower pull changes
Compare Give it time and Pull quickly. Watch the population curve against tension, then choose Next molecule several times.
What to observe: A slower ramp leaves more time to open at lower forces. Individual molecules still open at different points because their waiting times are random.
02
Separate waiting from increasing force
Choose Hold the force. Predict the shape of the folded fraction before watching the full replay.
What to observe: The force is constant, yet different molecules open at different times. The ideal population follows an exponential survival curve.
03
Make the detector slow
Choose A slow detector, then increase Detector bandwidth without changing the force protocol. Compare the two curves and the inferred rate.
What to observe: The domain opens at the same model time, but the filtered position reaches its half-jump threshold later at low bandwidth. Treating that delay as molecular waiting biases the estimated rate downward.

Stochastic Unfolding Force Spectroscopy

One jump, a whole distribution

Extension of the selected molecule

How many remain folded?

Why unfolding force has a distribution

Core mathematical model

Force changes the opening rate

Survival depends on the whole protocol

Ideal force balance locates the bead

Common difficulties

A peak is not a breaking threshold

Keep the unfinished trajectories

Run the experiment

Predict what a slower pull changes

Separate waiting from increasing force

Make the detector slow

Related experiments and articles

Thermal motion, molecular elasticity, unfolding and motors