# D23 — E4-H209: Two stall forces differ by a coupling free energy, d(F_r − F_v) = k_BT ln r(F_v)

**Batch:** 9 · **Selected rank:** 23/30 · **Tractability:** 8/10 · **Significance:** 7/10

## 1. Original hypothesis and refined claim

The original distinguishes a mechanical stall force (zero mean velocity) from a thermodynamic one (equal forward and backward step rates). It claims they coincide for a tightly coupled motor and differ for one with a slip cycle, that gap × step size is the coupling free energy, that the sign is fixed, and that one dwell-time dataset gives both.

The sign is right; three supports are not. (i) The nulls differ only if the mechanical coordinate has a channel not resolved into steps. If slip is a discrete backstep of the same size d, then v = d(k₊ − k₋) identically, v = 0 ⟺ k₊ = k₋, and the nulls coincide by definition — the hypothesis is empty, not false. Content requires continuous slip, a parallel pathway of different step size, or a loss of grip uncounted as a step. (ii) The ratio null needs no slip: it measures Δμ/d, which a tightly coupled motor supplies already — one of the two numbers is chemical, not mechanical. (iii) "One dwell-time dataset" understates the requirement — the loads must span both nulls, and F_r = Δμ/d is fixed by the assay's nucleotide potentials, not by the experimenter.

Repaired claim, two equations. Local detailed balance at the stepping edge gives

  ln r(F) = (Δμ − Fd)/k_BT,  r ≡ k₊/k₋   (1)

so r = 1 at F_r = Δμ/d, and the slope of ln r against F is −d/k_BT with no free parameter. The velocity null solves v = d(k₊ − k₋) − v_s = 0 with slip velocity v_s ≥ 0, so

  k_BT ln r(F_v) = Δμ − F_v d = d(F_r − F_v) ≡ ΔG_c ≥ 0,   η ≡ F_v/F_r = 1 − ΔG_c/Δμ   (2)

The force gap times d is the free energy per cycle that slip dissipates, read from the step ratio at the velocity null without knowing Δμ.

## 2. Refined scientific description

A single motor under controlled load, with position resolution to identify forward and backward steps. Two zeros are measured on one axis. The ratio null is thermodynamic: the stepping edge's free-energy change vanishes there, a property of states, immune to everything downstream. The velocity null is kinetic: the mean displacement rate vanishes, which depends on every channel that moves the motor, including ones producing no countable step. Their gap is therefore a free energy per cycle, not a calibration error, and measures how much of Δμ never reaches the mechanical coordinate.

## 3. Verified supporting literature and prior-art assessment

[EST] Local detailed balance: the ratio of forward to backward rates across an edge is the exponential of the free-energy change across it, so that ratio is fixed by state functions alone. [EST] The mechanical stall force as the zero-velocity load. [EST] Dwell-time analysis resolving forward and backward steps at high load with sub-nanometre resolution. [REC] Slip, futile hydrolysis and load-dependent detachment reported in several motor systems, the microscopic slip channel differing between them. [REC] Force-clamp traps with sub-piconewton force and millisecond time resolution. [UNV] Whether any motor has a slip channel with nonzero velocity exactly at its mechanical stall — the empirical crux — is not established; reported "slip" is usually detachment, a different object.

The literatures are separate: single-molecule mechanics reports one stall force and treats the other as an abstraction, while stochastic thermodynamics writes the ratio null and never measures it. Because the numbers are used interchangeably, a difference would be read as calibration error rather than a thermodynamic quantity — a gap in interpretation, not capability.

## 4. Proposed physical or mathematical model

Two channels on one coordinate. Discrete stepping: k₊(F) = k₀exp(−θFd/k_BT), k₋(F) = k₀exp((1−θ)Fd/k_BT)exp(−Δμ/k_BT), which reproduces (1) for any θ. Continuous slip: v_s(F) ≥ 0, monotone increasing, zero at zero load. Then

  v(F) = d[k₊(F) − k₋(F)] − v_s(F).   (3)

At F = F_r the bracket vanishes and v = −v_s(F_r) < 0, while at F = 0 the bracket is positive and, if it exceeds v_s(0), v > 0. Continuity gives F_v < F_r, hence ΔG_c > 0, which is the sign claim derived rather than asserted. If v_s ≡ 0 the two zeros coincide. Measured are v(F), whose zero is F_v, and r(F), a straight line on a log axis whose zero is F_r; (2) ties them. The model is a *relation between two independently measurable zeros*, not a fit.

## 5. Variables, assumptions and boundary conditions

Symbols: F load (pN); d step size; k₊, k₋ step rates; r their ratio; Δμ chemical free energy per cycle; F_r, F_v the two null forces; v mean velocity; v_s slip velocity; ΔG_c coupling free energy; η = F_v/F_r coupling efficiency; k_BT = 4.11 pN nm at 300 K.

Assumptions: (a) Markovian stepping, one load-dependent edge spanning all of Δμ; (b) slip monotone in F, independent of stepping direction; (c) detection efficiency above 0.9 at every load; (d) no detachment within a dwell. Small parameter: v_s(F_v)/(d k₊(F_v)) ≪ 1. Domain: 0.4 F_r < F < 1.4 F_r, d known to 2%.

## 6. Quantitative predictions

Slope of ln r against F: −d/k_BT = −1.95 pN⁻¹ for d = 8 nm, −1.2 pN⁻¹ for d = 5 nm — fixed by step size and temperature alone, so a 10% measurement of it tests local detailed balance at one edge.

With Δμ ≈ 20 k_BT ≈ 83 pN nm (estimate; the assay value depends on ATP, ADP and phosphate and must be measured, not assumed), F_r = Δμ/d = 10.3 pN for d = 8 nm. A motor whose velocity null sits at 0.7 F_r has ΔG_c = 6.0 k_BT ≈ 25 pN nm, an efficiency of 0.70. Force resolution of 0.5 pN gives ΔG_c to 4 pN nm ≈ 1 k_BT, 0.1 pN to 0.2 k_BT, and η to ±0.05 — one k_BT of coupling loss per cycle is resolvable with a standard trap.

Competing background: trap compliance. With a series spring, the bead transmits a fraction κ_m/(κ_m + κ_t) of the motor's motion — 86% for κ_m = 0.3 and κ_t = 0.05 pN nm⁻¹ — so slip inside the compliance is attenuated and forces are misassigned by 14%: 1.4 pN at 10 pN, 11 pN nm ≈ 2.8 k_BT, above the one-k_BT scale the assay otherwise resolves. This decides whether an existing dataset can support the claim.

## 7. Comparison against conventional explanations

(a) *Trap compliance.* The measured force is the trap force, not the motor force; a series spring makes both nulls stiffness-dependent, and their difference is what a sceptic attributes the gap to. Separator: repeat at two stiffnesses, or in force clamp. A compliance gap scales with κ_t and collapses as κ_t → ∞; a slip gap does not move, being a free energy.

(b) *Rare backsteps at high load.* Backward steps are exponentially suppressed near F_r, so r comes from few events and is biased toward 1, biasing the gap toward zero and *mimicking* tight coupling. Separator: report r(F_v) with confidence interval and event counts; a gap surviving with N₋ > 100 backsteps and a likelihood-ratio test is not an artefact.

(c) *Detachment rather than slip.* If the motor leaves the track under load and rejoins elsewhere, the mean velocity includes a diffusive segment that is no thermodynamic slip channel. Separator: the free-energy accounting holds if the motor returns to the same site, but the *interpretation* as coupling loss fails; detachment gives an abrupt threshold, slip a smooth v_s(F).

## 8. Specific experimental or computational method

One dataset: force-clamp optical-trap records of a single processive motor at six to ten loads spanning both nulls, with sub-nanometre position and 1 ms sampling. Method: hidden-Markov step detection at fixed bandwidth, extraction of per-step dwell times and displacements, and a maximum-likelihood fit of ln r(F) to a straight line and of v(F) to equation (3). Control arm: a tightly coupled motor, or the same motor under conditions that suppress slip, for which F_v and F_r must coincide within error. Nuisance parameter held fixed: trap stiffness within a series, and step size taken from the position histogram rather than assumed.

## 9. Necessary controls

(i) Stiffness series — the decisive control. (ii) ATP/ADP ratio series, which moves Δμ and hence F_r while F_v moves differently; a gap that tracks chemistry is a free energy, one that does not is an artefact. (iii) Detection-bandwidth halving, which must not change ΔG_c beyond its error.

## 10. Sources of measurement or numerical error

Dominant: compliance, 2.8 k_BT above — three times the one-k_BT resolution of section 6. Second: force calibration, 5% of 7 pN = 0.35 pN, giving 2.8 pN nm = 0.7 k_BT in ΔG_c. Third: statistical error on ln r at F_v, ≈ 1/√N₋, which is 10% (0.4 k_BT) at a hundred backsteps. Fourth: step-size calibration, 2% of ΔG_c, negligible. Fifth: Δμ constant across the scan, which fails if nucleotide pools drift; a 10% drift is 2 k_BT and moves F_r by 1 pN. Only compliance is both large and systematic, which is why the stiffness series is not optional.

## 11. Falsification criteria

(1) ln r versus F is not linear, or its slope departs from −d/k_BT by more than the force calibration allows: the single-edge picture fails and the ratio null is not at Δμ/d. (2) r(F_v) = 1 within error for a motor known to slip: the nulls coincide, the hypothesis is dead. (3) The gap is independent of the slip rate as varied through ATP/ADP: it is not a coupling free energy. (4) The gap scales with trap stiffness: compliance. (5) ΔG_c < 0 or ΔG_c > Δμ (η outside [0, 1]): the model or the data are wrong, the second law requiring η ∈ [0, 1].

## 12. Expected scientific contribution

It partitions a motor's free-energy budget into mechanical and slip channels with one dataset, converting a definitional ambiguity — which stall force a paper means — into a measurable ratio. The coupling efficiency becomes F_v/F_r, a number any force scan supplies, and the ratio null an in-situ Δμ requiring no nucleotide assay.

## 13. Resource and feasibility assessment

Apparatus: none new; force-clamp optical-trap data of the required quality exist for several motors (estimate, unverified: two to four published datasets per motor). Reanalysis: three to six months for one person, mostly step-detection validation. Software: an HMM step finder, a maximum-likelihood line fit, a bootstrap; all open-source. The binding constraint is coverage — most published records sit near zero load, where neither null is in range — so the realistic first target is a dataset already spanning high load; if none exists this is a proposal for a new measurement, not a reanalysis.

## 14. Reproducibility plan

(1) Ingest per-step dwell times and displacements, reporting detection bandwidth and threshold. (2) Fit ln r against F by maximum likelihood; report slope, intercept, covariance; check slope = −d/k_BT. (3) Fit v(F) to (3) and locate F_v with a bootstrap interval. (4) Evaluate ΔG_c = k_BT ln r(F_v) and η = F_v/F_r, propagating force calibration and step size. (5) Report the stiffness and ATP/ADP series as separate tables. Seeds: HMM initialisation and bootstrap resampling recorded. Replication: an independent group recovers the same ΔG_c within its stated error from the same public record, and a gap consistent with zero for the control motor.

## 15. Principal remaining uncertainties

Most likely killer: r(F_v) ≈ 1 for every real motor, slip being either absent or discrete. If slip is a same-size backstep the nulls coincide exactly and there is no observable — a possibility the original's own model permits and never confronts. Second: F_r may lie above the accessible range for a tightly coupled, high-Δμ motor. Third: compliance may exceed the effect in every existing dataset, so the claim needs a stiffer trap and shorter linker before it can be tested at all.