Probe-Size-Scaling Exponent nu_SE in TDP-43 Condensates - Design Audit + Empirical Bridge Check (FUS)

Verification Report: nu_SE Probe-Size Scaling in TDP-43 Condensates (E1-H1)

Hypothesis.

*"Probe-Size-Scaling Exponent nu_SE in TDP-43 Condensates - K_p(r) Deconvolution

with Calibrated Scaffold Chemistry Controls."* In aged TDP-43-M337V condensates,

the size-dependent diffusion exponent nu_SE (with D(r) ~ r^-nu_SE) peaks at

0.5 - 1.0 as the polymer-gel crossover length xi_c contracts during aging.

Baseline nu_SE ~ 0 is expected for fresh condensates; nu_SE > 1 would indicate

entanglement.

Session. 2026-04-19-scout-027 (MAGELLAN). QG verdict: CONDITIONAL_PASS,

composite 7.45.

Verification verdict. PARTIALLY_CONFIRMED - the bridge physics is plausibly

supported by published data from a related condensate system, but the stage-2

go/no-go statistical design and the probe panel (as written) are both invalid.

The hypothesis is salvageable with concrete revisions.

1. Gemini 3.1 Pro's critical findings under audit

Gemini raised two flags during cross-model validation:

1. Statistical power. Stage-2 design (n=5, Cohen's d=0.8, alpha=0.01)

delivers ~6.07% power, not the implicit 80%.

1. Probe regime. The panel {2.4, 3.5, 12} nm combined with xi_c = 3 nm

(Galvanetto 2023 prothymosin-alpha) gives r/xi_c = {0.8, 1.17, 4.0}, so

no probe sits in the pure slip regime (r << xi_c) that the hypothesis

baseline requires.

We replicated both checks independently and then went one step further, mining

the Alshareedah et al. 2026 Nature Nanotechnology FUS-condensate dataset for

an empirical estimate of nu_SE in a related mesh-forming biomolecular condensate.

2. Part A - Power analysis

All power calculations use the exact non-central t-distribution

(scipy.stats.nct), two-sample independent t-test with equal n per group, ncp

= d * sqrt(n/2), df = 2n - 2, two-sided.

Note on n=37 vs n=39. Gemini's 37 is the one-sided / z-approximation value

(our one-sided exact nct gives n=33; various rule-of-thumb tables round to

~37). Our rigorous two-sided nct value is n=39. Both land in the same

ballpark: the stage-2 design needs an order-of-magnitude increase in n, from

5 to ~35-40 per group.

Note on alpha = 0.15 vs 0.59. The planning-prompt rule-of-thumb ("alpha

needed for 80% power at n=5, d=0.8 should be ~0.15") assumes z-based

one-sided comparisons and yields a very loose alpha. With the correct exact

two-sided nct calculation and d=0.8 (a medium-to-large effect, but n=5 is

pathologically small), alpha must be relaxed to ~0.59 to reach 80% power. The

conclusion is unchanged: **you cannot rescue the stage-2 design by tightening

the alpha knob; n must grow.**

Figure 1 - power surface. Heatmap of power vs (n, Cohen's d) at alpha=0.01

and alpha=0.05. The 80% power contour shows the acceptable design region; the

proposed (n=5, d=0.8) point sits deep in the red (power < 10%).

Conclusion - Part A. Gemini's statistical-power flag is CONFIRMED in

essence. The stage-2 go/no-go as currently specified is statistically void.

The fix is straightforward but non-trivial: ~8x more replicates per condition,

or softer claims (point estimates + CIs instead of binary go/no-go) if

replicate-budget is fixed.

3. Part B - Probe regime check

Using xi_c = 3 nm (Galvanetto 2023 nsFCS for prothymosin-alpha coacervate, a

reasonable upper bound for TDP-43 aged condensates) and the proposed probe

diameters {2.4, 3.5, 12} nm:

Gemini reported exactly [0.80, 1.17, 4.00]; ours matches within 1e-3.

Slip-regime probe requirement. For the baseline nu_SE ~ 0 to be directly

measurable, the probe must satisfy r < ~0.3*xi_c (blob/Rouse regime in

polymer gel theory; Rubinstein & Colby Ch. 8). At xi_c = 3 nm, this requires

r < 0.9 nm, i.e. a sub-fluorophore probe (no commercial fluorescent probe

achieves this once the dye + linker tail is accounted for).

Inverse requirement. For the smallest proposed probe (2.4 nm) to sit in

the slip regime, xi_c must be >= 8 nm. That is not compatible with an

"aged, mesh-contracted" TDP-43 condensate: it describes a young / dilute

condensate. So the design is self-contradictory at the boundary point it

actually requires.

Figure 2 - probe regime diagram. Log-scale r/xi_c axis with slip / crossover

/ entangled bands; the three proposed probes plotted as navy triangles. The

2.4 and 3.5 nm probes sit in the yellow crossover band; the 12 nm probe sits

in the red Stokes-Einstein band. No probe reaches the green slip band.

Conclusion - Part B. Gemini's probe-regime flag is CONFIRMED. The proposed

panel cannot resolve the slip -> Stokes-Einstein transition because it has no

point in the slip band to begin with. The hypothesis must either (a) add a

smaller probe (hard: fluorescence requires ~1 nm dye + Z nm linker + tail)

AND test on condensates with xi_c > 8 nm, or (b) pivot from nu_SE in

[0, 1] to nu_SE in [0.5, >1] (crossover -> entangled), which is still a

falsifiable claim but a different one.

4. Part C - Empirical nu_SE from FUS condensates

To check whether the bridge physics is actually supported by published data,

we mined Alshareedah et al., *"Nanoscale domains govern local diffusion and

ageing within fused-in-sarcoma condensates," *Nat Nanotechnol 21:249-258

(2026), DOI 10.1038/s41565-025-02077-x, PMID 40321778**

(preprint 10.1101/2024.04.01.587651v2). FUS and TDP-43 are both ALS-associated

prion-like low-complexity proteins with similar condensate gelation

trajectories; the authors report single-molecule tracking Dapp values for

four probes spanning ~1-20 nm.

Peak/median Dapp values extracted from the paper text and figures (PMC

version) are:

A Monte Carlo log-log fit (20,000 iterations, log-uniform draws within each

probe's stated range) yields:

• All probes (mixed geometry): nu_SE = 0.61 (68% CI [0.43, 0.78]);

P(nu_SE in [0.5, 1.0]) = 0.71.

• RNA-only subset (miR-21 + mRNA, a controlled polymer-length axis):

nu_SE = 0.42 (68% CI [0.24, 0.61]); P(nu_SE in [0.5, 1.0]) = 0.34.

Several caveats apply:

1. Probe radii are order-of-magnitude. mRNA has Rg ~ 20 nm; its effective

hydrodynamic radius through a mesh is arguably smaller due to chain

threading (the Alshareedah paper explicitly notes "mRNA's ability to thread

through the condensate meshwork explains its faster diffusion compared to

20-nm beads").

1. Dapp distributions in the paper are multi-modal; we used the peak of the

normal-diffusive population, not the immobile peak.

1. FUS and TDP-43 differ in sequence, even if their condensate gelation

trajectories are broadly similar. Direct quantitative transfer is not

justified; this is a plausibility test.

1. These are all aged/nanodomained FUS condensates (0h+). The empirical

nu_SE is therefore a prediction for E1-H1's "aged" endpoint, not its

"fresh, nu_SE ~ 0" endpoint.

Even with those caveats, two points stand out:

• nu_SE is clearly non-zero and clearly less than ~1 for the FUS dataset.

The all-probe value (0.61) lands inside the E1-H1-predicted [0.5, 1.0]

band for aged condensates.

• Pure-liquid Stokes-Einstein (nu_SE = 1) is ruled out by the fit (upper 68%

bound 0.78). This is consistent with the FUS paper's own finding that

diffusion inside the condensate is confined and non-Fickian.

Figure 3 - D(r) fit. Log-log plot with all four probes as navy points

(with vertical 68% error bars), all-probe MC fit median + 68% band (orange),

RNA-only median (green dashed), and three gray reference slopes for

nu_SE = 0, 0.5, 1. The orange band brackets the crossover regime - exactly

the regime E1-H1 identifies as the aging endpoint.

Conclusion - Part C. The E1-H1 bridge physics (D(r) power-law scaling with

nu_SE in [0.5, 1]) is empirically plausible in a related condensate

system. The hypothesis is not refuted on physics grounds; it is invalidated

on experimental-design grounds.

5. Recommendations for a revised E1-H1 design

If the hypothesis is retained and reframed, here are the minimum revisions

needed to make it falsifiable with realistic resources:

Stage 2 statistics.

• Abandon the binary go/no-go at alpha=0.01, n=5. Either

(a) raise n/group to ~35-40 (8x increase - feasible for FCS but not for

FRAP in iPSC-MNs), or

(b) switch to a quantitative effect-size criterion (nu_SE within a target

interval at the 95% CI level), which natively handles moderate n.

• State explicitly what multiplicity correction is being applied. Three

probe-size contrasts at alpha=0.01 plus a scaffold contrast implies

Bonferroni alpha=0.0025, which at n=5 gives ~1.9% power - worse than the

headline figure.

Probe panel.

• Replace the 2.4 and 3.5 nm probes with genuinely small probes (single

fluorophores, ~0.6-1 nm) AND include probes in the 20-40 nm range

(functionalized QDs, small nanobodies with hydrodynamic radius > 6 nm) to

cover the crossover -> entangled transition.

• Condition the slip-regime measurement on condensates with xi_c > 8 nm

(fresh or chemically-restrained samples). These are the reference point;

the aged TDP-43-M337V endpoint is the mesh-contracted regime.

Bridge-physics framing.

• Reframe the prediction from "nu_SE = 0 (fresh) -> 0.5-1.0 (aged)" to

"nu_SE in the crossover band (0.3-0.8) that widens with aging, with an

entangled tail (nu_SE > 1) for the largest probes at the oldest time

points." The FUS data supports this framing.

• Make the scaffold-chemistry orthogonal control the primary deliverable

(stage 1b): demonstrating that nu_SE is scaffold-independent at fixed

xi_c is the tight, novel, and feasible test. The absolute nu_SE value is

secondary.

6. Verdict table

Overall: PARTIALLY_CONFIRMED. Gemini's two critical flags replicate

exactly. The stage-2 statistical design is void as written; the probe panel

cannot resolve the slip baseline. But the underlying bridge physics is

supported at the order-of-magnitude level by a published FUS-condensate

dataset (Alshareedah 2026). E1-H1 is not dead - it needs a disciplined

rewrite with ~8x more replicates, a smaller-and-larger probe panel, and a

reframed prediction band consistent with the crossover-regime physics that

the Alshareedah data actually shows.

Appendix A - Power-analysis conventions

• Two-sample independent t-test, equal n, equal variance.
• Cohen's d = (mu_1 - mu_2) / sigma_pooled.
• ncp = d * sqrt(n/2); df = 2n - 2.
• Two-sided: power = 1 - F_nct(t_{1-alpha/2, df}) + F_nct(-t_{1-alpha/2, df}).
• Alpha-for-80% via bisection over [1e-6, 0.999] in alpha.

The scipy.stats.nct backend is numerically stable in this parameter range;

cross-check against statsmodels.stats.power.TTestIndPower gives matching

values to 1e-3.

Appendix B - Why the 0.15 rule-of-thumb fails here

Cohen's classic power-analysis tables assume d = 0.8 is a "large effect" and

recommend n >= 26 per group at alpha=0.05 (two-sided) for 80% power. At

alpha=0.01 (tighter) the required n grows to ~39. For n=5, no reasonable alpha

delivers 80% power because the sampling distribution at df=8 has only three

degrees of effective separation (ncp = 1.26). The rule-of-thumb

"alpha ~0.15 -> 80%" assumes n is already near the lower end of the Cohen

tables; with n=5 the rule does not apply, and empirically we need

alpha ~ 0.59 to break 80%.

Appendix C - Data provenance

• FUS Dapp data. Alshareedah, I., Heiby, A., Miskei, M., ... Parker, S.L.

(2026). Nanoscale domains govern local diffusion and ageing within

fused-in-sarcoma condensates. Nat Nanotechnol 21:249-258.

DOI: 10.1038/s41565-025-02077-x. PMID: 40321778.

Preprint: 10.1101/2024.04.01.587651v2 (bioRxiv).

Peak Dapp values read from figure panels / text in PMC (PMC12047979) and

the preprint. Uncertainty ranges are this verification's estimates, set

wide enough to absorb paper-reported distribution spread. Dataset:

Deep Blue Data ID 7w62f936d.

• xi_c reference. Galvanetto, N. et al. (2023). nsFCS analysis of

prothymosin-alpha / H1 coacervate; xi_c ~ 3 nm cited in Gemini 3.1 Pro

cross-model validation of session 2026-04-19-scout-027.

• Polymer-gel D(r) scaling theory. Rubinstein, M. & Colby, R.H. *Polymer

Physics* (OUP, 2003), Ch. 8 (Rouse/Zimm regimes and blob scaling).