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Sandbox Physics
Single-molecule physics from bead motion to motors

L4 · Chapter 8

Can a motor model predict a new load protocol?

Sandbox Physics ·

Two models can give the same mean speed at one load and disagree at another. A fitted curve is therefore a starting point. Freeze the parameters and predict a protocol excluded from fitting to expose kinetic differences. A small reversible network makes this test reproducible.

Specify the scope of the reproduction

We pin Seifert’s arXiv:1205.4176v1. Section 9.4.5, equation (244), constrains local detailed balance for motor transitions. Section 10.5, equation (273), expresses single-cycle input and output power through forward cycle current and affinity. We recompute these relations for chosen two-state rates.

Figure 3 illustrates a more general multicycle network. Our periodic two-state graph is a separate minimal realization. We draw our own diagram and do not label synthetic traces as the original Wang or Svoboda experiments. Published relations, numerical checks of teaching parameters and open biological questions have different roles.

Reading: U. Seifert · Wang et al. · K. Svoboda & S. M. Block

Solve state probabilities independently

Set aside the trajectory sampler. Let the total rate leaving AA be a=u0+w1a=u_0+w_1 and the total leaving BB be b=w0+u1b=w_0+u_1. The probability balance is a linear differential equation. It gives both stationary occupancy and relaxation after a load switch.

The offline benchmark solves probability balance independently and compares its speed with the browser model. It checks equilibrium, stall and reversal. Agreement tests numerical implementation; it does not establish that the two-state assumptions describe a real protein.

p˙A=b−(a+b)pA,pAss=ba+b,pA(t)=pAss+[pA(0)−pAss]e−(a+b)t\dot p_A=b-(a+b)p_A,\quad p_A^{\rm ss}=\frac b{a+b},\quad p_A(t)=p_A^{\rm ss}+[p_A(0)-p_A^{\rm ss}]e^{-(a+b)t}
Apply this solution on each constant-load segment. Occupancy is continuous at the switch, while rates change. Mechanical forward minus reverse flow gives instantaneous mean velocity.

Reading: U. Seifert

Turn the power relation into a numerical check

Separate the current numerator into forward and reverse cycle contributions, j+j^+ and j−j^-, sharing denominator SS. Their ratio is fixed by affinity; their difference is net current. Multiplying current by mechanical work or chemical free energy gives equation (273) for this network.

The frozen benchmark records parameters, units, methods and outputs. Rerunning it compares current direction, its zero, and input minus output power, all from the same rates.

j+=u0u1S,j−=w0w1S,j−j+=e−Aj^+=\frac{u_0u_1}{S},\qquad j^-=\frac{w_0w_1}{S},\qquad \frac{j^-}{j^+}=e^{-\mathcal A}
These cycle contributions belong to the two-state graph; the denominator is the sum of all four rates.
J=j+(1−e−A),Pout=j+(1−e−A)FdJ=j^+(1-e^{-\mathcal A}),\qquad P_{\rm out}=j^+(1-e^{-\mathcal A})Fd
Chemical and mechanical counts on a finite trajectory can differ by an endpoint state. Their net currents agree in stationary averages.

Reading: U. Seifert

The same speed can hide different waits

One stationary speed cannot determine four rates. Consider two explicit candidates. Model A uses chapter six rates. Model B multiplies both chemical rates by 0.150.15 and both mechanical rates by 1.44769241.4476924, preserving every edge ratio. At 100  μM100\;\mathrm{\mu M} ATP and 1.50  pN1.50\;\mathrm{pN} load, both give 104.628  nm s−1104.628\;\mathrm{nm\,s^{-1}}.

Freeze the parameters and raise load to 3.00  pN3.00\;\mathrm{pN}. Each starts in its own calibrated stationary distribution, retained at the switch. Model A initially continues forward; B initially moves backward on average. Both eventually advance, with different relaxation times. The mean displacements in the table come from integrating the master equation.

This protocol distinguishes the selected candidates in their predictions. The first 10  ms10\;\mathrm{ms} produces only a fraction of a nanometer of mean displacement, much smaller than single-bead thermal fluctuations. A measurement must pass predictions through the same bead and detector response, average independent records, and check whether noise and bandwidth permit that difference to be resolved.

Match one clamp velocity, freeze the parameters, then predict a new load. All values are teaching-model calculations.
ObservableModel AModel B
Calibration speed (nm s−1)(\mathrm{nm\,s^{-1}})104.628104.628104.628104.628
Speed just after switch (nm s−1)(\mathrm{nm\,s^{-1}})19.883419.8834−11.8260-11.8260
Post-switch relaxation (ms)(\mathrm{ms})3.1513.15115.10515.105
New-load long-time speed (nm s−1)(\mathrm{nm\,s^{-1}})23.711923.711924.681224.6812
Mean displacement in ten ms (nm)(\mathrm{nm})0.225560.22556−0.02019-0.02019

Reading: U. Seifert

A repeatable prediction test

First collect repeated constant-load records to estimate the scatter of speed estimates. Freeze parameters and the observation model before inspecting a second protocol. Use a sufficiently long load-step experiment and retain events, bead motion and detector samples.

Pass predictions through the same response model before comparing detector records. An early transient mismatch can reflect response time or initial occupancy. A persistent speed mismatch motivates checks of force-dependent rates, chemical conditions and extra cycles. General hidden-state inference needs additional conditions, adequate noise models and independent experimental data.

Reading: Lang et al. · U. Seifert

Try the prediction

Reserve the load step for prediction

  1. Record constant-load speed and between-record scatter. Predict the direction of change under a larger opposing load.
  2. Choose A sudden heavier load and open the protocol controls to inspect its post-step load. This browser preset shows model A; the table’s two-model comparison is computed by the offline benchmark.
  3. Export seed and version. Fit separate windows before and after the switch, then change detector bandwidth to test the conclusion.

Transfer the prediction test to unfolding

  1. Determine parameters under a ramp and predict survival at constant force.
  2. Choose Hold the force without changing molecular parameters. Compare events and censoring before adding another barrier.

Exercises and explanations

  1. Why can two models satisfy detailed balance constraints and match one velocity but predict different load responses?

    Read the explanation

    Rate ratios and one velocity constrain only combinations of rates. State occupancies, dwell times and force dependence can still differ. A load switch tests both stationary change and relaxation.

  2. The latent synthetic trajectory agrees with prediction but the detector trace does not. What should you inspect first?

    Read the explanation

    Inspect bead and detector response, sampling, fit windows and noise. Latent truth is available in a synthetic calculation. Real experiments need independent calibration and multiple protocols to separate instrumental from molecular causes.

Sources and calculation scope

  1. 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.

  2. 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.

  3. 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.

  4. 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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