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Sandbox Physics
Single-molecule physics from bead motion to motors

L3 · Chapter 7

What remains measurable when steps blur together?

Sandbox Physics ·

A blurred staircase can retain an accurate long-time speed while losing information about step size and intermediate states. The motor jumps first, the bead follows, and the detector turns bead motion into a record. Treating these layers separately reveals which information survives.

The bead takes time to follow

Write motor position as xmx_m and bead position as xbx_b. An effective linker of stiffness kbk_b pulls the bead toward the motor, with a static offset caused by load. After a step, the mean bead position relaxes on τb=γ/kb\tau_b=\gamma/k_b.

The Lab applies an ideal prescribed motor load and treats the bead as a noisy linear follower. Its instantaneous fluctuations do not feed back into jump rates. A full assay model would couple stochastic tension, motor chemistry and the feedback controller.

γx˙b=kb(xm−xb)−F+ξ(t),⟨ξ(t)ξ(t′)⟩=2γkBTδ(t−t′)\gamma\dot x_b=k_b(x_m-x_b)-F+\xi(t),\qquad \langle\xi(t)\xi(t')\rangle=2\gamma k_BT\delta(t-t')
The noise strength gives equilibrium variance equal to thermal energy divided by linker stiffness for a fixed motor.

Reading: Lang et al. · U. Seifert

Two responses round a step into a ramp

Start with a continuous-time reference: the bead and detector each have a response time, τb\tau_b and τd\tau_d. For a sinusoid, gain is output amplitude divided by input amplitude. Each gain falls at high frequency; their product gives the total. Angular frequency is ω=2πf\omega=2\pi f, with ff in hertz.

If a typical dwell is 10  ms10\;\mathrm{ms} and detector response is 20  ms20\;\mathrm{ms}, another step can arrive before the previous one settles. Faster playback cannot restore these frequencies. Greater bandwidth may also admit more thermal noise.

The motor Lab uses a specified sampled observation model: continuous thermal bead motion is sampled every 2  ms2\;\mathrm{ms} and then digitally filtered. The second expression gives that digital detector response. Ω=2πfΔt\Omega=2\pi f\Delta t is phase advance per sample, and fdf_d is its half-power frequency. No analog anti-alias filter precedes sampling.

∣Hcontinuous(ω)∣2=1(1+ω2τb2)(1+ω2τd2)|H_{\rm continuous}(\omega)|^2=\frac1{(1+\omega^2\tau_b^2)(1+\omega^2\tau_d^2)}
This is squared gain for two continuous first-order stages, a reference explaining amplitude suppression.
∣Hdigital(Ω)∣2=(1−a)21+a2−2acos⁡Ω,a=1C+C2−1,C=2−cos⁡(2πfdΔt)|H_{\rm digital}(\Omega)|^2=\frac{(1-a)^2}{1+a^2-2a\cos\Omega},\qquad a=\frac1{C+\sqrt{C^2-1}},\quad C=2-\cos(2\pi f_d\Delta t)
The digital response has half power at the specified cutoff. Filtering cannot recover bead fluctuations already aliased by sampling. Saved events are model truth; filtered samples are synthetic detector data.

Reading: Lang et al.

Mean speed does not determine step size

A line fitted to a sufficiently long position record estimates average speed. Filtering changes rapid variations and causes transient lag near initialization or a protocol switch. Report the fitting interval and either avoid or model these transients.

A trace advances 160  nm160\;\mathrm{nm} during 2.00  s2.00\;\mathrm{s}, giving an endpoint speed of 80.0  nm s−180.0\;\mathrm{nm\,s^{-1}}. This could arise from ten net eight-nanometer steps per second or twenty net four-nanometer steps. The mean alone cannot distinguish them.

Estimating step size needs resolved events, an adequate noise model and suitable sampling. Thresholds can merge neighboring steps or split thermal excursions into false events. The Lab’s true step size checks its generator; it is not an independent measurement from filtered samples.

Reading: K. Svoboda & S. M. Block · Lang et al.

Dwell times can expose an intermediate state

Consider a limit with negligible reverse rates and two sequential exponential waits, of rates aa and bb. A complete dwell is their sum. Its variance is smaller than that of a single exponential with the same mean.

Variance divided by squared mean is the squared coefficient of variation. Equal slow stages give one half. When one stage is much slower, the value approaches one. A value near one leaves another possibility: the other wait may be too fast to resolve.

⟨t⟩=1a+1b,Var⁡(t)=1a2+1b2,CV2=a−2+b−2(a−1+b−1)2\langle t\rangle=\frac1a+\frac1b,\qquad \operatorname{Var}(t)=\frac1{a^2}+\frac1{b^2},\qquad \mathrm{CV}^2=\frac{a^{-2}+b^{-2}}{(a^{-1}+b^{-1})^2}
This limit assumes accurate detection of complete cycles and negligible reversals. Backsteps or missed events require recalculating the distribution of the observed dwells.

Reading: U. Seifert

Retain uncertainty with the record

Thermal noise, finite step counts and random dwells affect a velocity estimate. Independent seeded records reveal finite-record scatter. A shared displacement-calibration error, omitted response or missed-event process requires separate treatment.

Save load protocol, seed, sample interval and response parameters with the raw events and detector samples. Another reader can then change the fit window or detection method. The next chapter asks a stronger question: can the same parameters predict a new load protocol?

Reading: Lang et al. · U. Seifert

Try the prediction

Pass a step record through a slower detector

  1. Compare Clear forward steps with Blurred steps. Only detector bandwidth changes; predict which features disappear.
  2. Compare motor events, bead motion and detector samples. Export and fit velocity using only detector positions, then compare with the disclosed model truth.
  3. Choose A sudden heavier load. Inspect the exported fitting windows and fit separate intervals before and after the switch. Explain what one line across both intervals would mix.

Exercises and explanations

  1. Does an accurate average velocity establish correct step detection?

    Read the explanation

    No. Merging pairs of steps can preserve total displacement and average speed while doubling apparent step size. Check the step distribution, missed-event rate and noise separately.

  2. What is the squared coefficient of variation for two equal sequential exponential waits?

    Read the explanation

    One half. Both mean and variance double relative to one stage, so variance divided by squared mean halves. Reversals, missed events and heterogeneity change this result.

Sources and calculation scope

  1. 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.

  2. K. Svoboda & S. M. Block, Force and Velocity Measured for Single Kinesin Molecules, Cell 77(5), 773–784 (1994)

    Figures 1 and 4–5 describe the bead assay and linker compliance. The published experiment used a stationary trap. Our ideal-clamp two-state motor illustrates chemical affinity and load response with independently chosen rates.

  3. U. Seifert, Stochastic Thermodynamics, Fluctuation Theorems, and Molecular Machines, arXiv:1205.4176v1 (2012)

    Section 9.4.5, equations (240)–(244), gives local detailed balance. Section 10.5, equation (273), gives unicyclic input/output power. Our limited recomputation applies these relations to a chosen network. Figure 3 illustrates a general multicycle network; our two-state diagram is an original illustration.

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