Incoprea research · leads batch L · 3 November 2026

Molecular and cellular measurement: 200 leads and three computations

0 · The structure being generalised

A measured rate is a quotient. In the notation used throughout this page,

reported rate  =  (signal per unit time) / (denominator)

The numerator is what the instrument sees. The denominator is a claim about the system: how much catalyst is present, how many molecules are in the illuminated volume, what fraction of the protein is labelled, how much of the compartment the probe can reach. Four properties of that quotient organise the four sections:

• The identity of the denominator — active sites or total protein, activity or concentration, molecules or fluorescence units. Section Q.
• The distribution of the denominator — a single molecule count or a cell-level average, and what the mean of that distribution does not constrain. Section R.
• The perturbation of the denominator — a tag, a dye or a fixation step that changes the quantity it is used to measure. Section S.
• The accessibility of the denominator — whether the concentration at the site equals the concentration in the bulk. Section T.

The three computations at the end of the page take one member of each of the first three families and push it until it fails: the active fraction (Q), the cell-to-cell distribution (R), and the diffusion coefficient as a stand-in for the accessibility of a compartment (T). All three are simulations or closed-form arithmetic with stipulated parameters. No measured data appear anywhere on this page.

Q · Concentration, activity and the denominator problem in rate measurements

Every rate is a ratio whose denominator is itself a measurement. The leads below ask what the denominator is, how it is obtained, and what happens when it is replaced by a proxy: total protein for active sites, molarity for activity, nominal for effective volume. The first computation on this page is the simplest member of the family.

R · Single-molecule and single-cell statistics: what a mean hides

A single-cell measurement replaces an ensemble average with a distribution, and a distribution carries parameters that a mean does not. The leads below ask which of those parameters are identifiable, which are degenerate with the detection efficiency, and which are artefacts of the threshold, the gate, or the bin width.

S · Labelling, tags and perturbations that change what they measure

A label is a perturbation with a readout attached. The leads below separate three effects that are usually reported as one: the label changing the molecule, the label changing the population, and the detection changing the label.

T · Transport in crowded and confined media

A concentration at a reaction site is only equal to a bulk concentration when transport is fast compared with the reaction and the medium is not structured. The leads below ask when the diffusion coefficient is a constant, when it is a function of the lag window, and when the measured quantity is a geometry rather than a mobility.

Tests run

Three computations, all in incoprea_nov3/code/biomeasure_tests.py, writing incoprea_nov3/results/biomeasure.json and a plain-text run log biomeasure_log.txt. Test 1 is closed-form arithmetic plus a seeded Monte Carlo; tests 2 and 3 are seeded simulations with stipulated parameters. The Student-t quantiles are computed from a continued-fraction incomplete beta, so no statistics package was used. No measured data are used anywhere.

1 · What an unmeasured active fraction does to a turnover number

A reported catalytic constant is k_app = v / [E]_total, while the quantity of interest is k_true = v / [E]_active. Writing the active fraction as f = [E]_active / [E]_total gives k_app = f · k_true exactly, so the bias is multiplicative and the correction is 1/f. The true value is stipulated at 25.0 s⁻¹; nothing here is measured.

What the table says. Over the range shown the reported value spans 20.0× (1.250 to 25.000 s⁻¹) for one enzyme with one true rate. The elasticity d ln k_app / d ln f is 1.000000 at every point — computed here by a centred difference at each f — so a 10% error in the active-site titration is a 10% error in the corrected value, with no attenuation. This is the opposite of the reference-ratio case, where the elasticity is bounded below one; a denominator is not a correction factor and its uncertainty is not damped.

The ranking reversal. Take two enzymes with true kcat of 120 and 80 s⁻¹, so A is 1.50× faster. Give B an active fraction of 0.85 and vary A's. The reported ratio crosses 1 at fA* = fB·kB/kA = 0.5667, and below that value the slower enzyme is reported as the faster one.

How often, in a population of preparations. Draw each preparation's active fraction from Beta(4,4), which has mean 0.50 and standard deviation 0.1667, and compare two enzymes whose true ratio is r. 20000 trials.

A 5-fold true difference still reverses 0.49% of the time, and a 1.5-fold difference reverses 21.35% of the time. At a true ratio of 1.1 the reported ranking is little better than a coin toss (42.20%), so a comparison between two preparations drawn from that distribution carries almost no information about which enzyme is faster. The same Monte Carlo gives a mean within-experiment CV of the reported k_cat of 0.3307 against the population CV of the active fraction, 0.3333. The per-trial identity CV(k·f) = CV(f) holds to 3.3e-16, as it must for a pure scale factor, so all of the spread in reported k_cat across preparations is active-fraction spread and none of it is a rate difference.

And the titration itself. If the inhibitor's stoichiometry is wrong by a factor τ, the measured active fraction is τ·f and the reported constant becomes k_true/τ: the error is exactly the titration error, inverted and undamped.

The expectation this contradicted. The intended finding was that the active fraction is a nuisance scale factor: annoying, correctable, and irrelevant to conclusions drawn between enzymes. It is correctable but not irrelevant, because a correction applied after the comparison cannot restore an ordering that has already flipped. A 30% error in the titration produces a -23.1% error in the reported constant, because the reported value is k_true/τ and nothing else — no leverage either way. The denominator is the one place in this family of measurements where an uncertainty passes through untouched, and it is the one place where the usual advice (measure a ratio, and the systematic cancels) does not apply.

2 · Cell-to-cell variability and the power to detect a fold change

Two groups of n cells, a true fold change of 2.0×, single-cell values drawn log-normal with biological CV as tabulated and a technical CV of 0.10 on the log scale, analysed with a pooled-variance two-sample t-test on log values at α = 0.05. 4000 simulations per cell; the critical value in each cell is the exact Student-t quantile for the degrees of freedom, not a normal approximation. Power is the rejection rate.

What the table says. At a CV of 0.5 the power at three cells per group is 0.268 — the experiment fails three times in four. At a CV of 0.8 it is 0.158 — three times the α-level rejection rate, and the regime in which many single-cell claims of a 2-fold effect are made. Forty cells per group at CV 1.6 reach only 0.770. The null check (δ = 0, CV 0.5, n = 10) rejects at 0.0525, so the test is calibrated.

The same data on the linear scale. Analysing untransformed values with the same test loses power everywhere the distribution is skewed, and the loss grows with the CV: at CV 1.6 and n = 40 the linear-scale power is 0.538 against 0.770 on the log scale. The log transform is not cosmetic here — it is the difference between a test of ratios and a test of means whose standard error is dominated by the tail.

The inverse question, which is the one an experimenter asks. Smallest fold change detectable at 80% power, at the same technical CV:

A cell marked >16× exceeded the bisection bound of 16 and is a lower bound. Three cells per group at CV 0.5 can only detect a 4.37× change at 80% power. Forty cells at CV 1.6 can detect 2.05×. Between those two cells lies the whole practical range of single-cell work, and the cost of the CV is paid in powers of the cell number, not in units of it.

Cells required for 80% power at a 2-fold effect.

What this table says that the first one does not. Technical noise is almost irrelevant once the biological CV exceeds it: at CV 0.5 the technical variance is 4.3% of the total, and raising the technical CV from 0 to 0.10 leaves the required n at 9 in both cases — no change at all in the rounded integers. At CV 0.05 the technical variance is 80% of the total, and that is the only regime in which improving the assay is worth more than counting more cells. A laboratory that responds to a failed experiment by improving its measurement precision is therefore doing the right thing only when its biological variability is already small — and it usually is not.

The floor no test can cross. A rank test with n cells per group has 2/C(2n,n) as its smallest attainable two-sided p. At n = 3 that is 0.10, so no effect, however large, can be called significant at α = 0.05 with three cells per group and a rank test. At n = 5 it is 0.0079. The parametric test escapes this bound only by assuming a distribution, which is the assumption most single-cell data violate.

The expectation this contradicted. The working assumption was that the power problem is a problem of small n, and that the remedy is always more cells. The grid says something narrower and less comfortable: the power depends on the product n·CV⁻², so going from CV 0.5 to CV 1.6 is equivalent to throwing away 90% of the cells collected. And at CV 0.05, where the cells are cheap, the binding constraint is the technical noise and not the biology — the opposite diagnosis from the one the same laboratory would make on the same day about a different marker.

3 · Diffusion in a crowded medium against a constant-diffusion model

A two-dimensional square lattice of 160 × 160 sites with a fraction φ blocked at random, unit-step walkers with four neighbours, periodic boundaries, and a blocked target meaning the walker stays. 1200 walkers, 4096 steps, 3 independent obstacle realisations averaged per point. The unit-step walk has D0 = 1/4 per step, so the reduced coefficient is D/D0 = MSD(τ)/τ. The dynamics are stipulated: this is an obstruction lattice, not a model of any particular cytoplasm, and the only physics in it is excluded volume.

Internal check. At φ = 0 the fitted exponent is 1.000, the reduced coefficient is 1.0000, and the constant-D fit residual is 0.06% — a free walker is recovered to within the finite-lattice artefact. One artefact survives into the table and is worth naming: the raw ratio D(8)/D(1024) at φ = 0 is 1.0112, because minimum-image folding on a 160-site torus costs about 1% of the long-lag slope. Each column is normalised by its own φ = 0 value, so that offset is removed by construction and the null for the short/long ratio is exactly 1.000.

Two models of the same curve. The obstruction model D/D0 = exp(−αφ) and the percolation form D/D0 = (1 − φ/φc)μ were fitted to the same reduced coefficients by least squares in log space, both over all points and over the sub-threshold points alone.

The single-exponent form is not merely a worse fit, it is the wrong shape: the apparent exponent computed point by point rises from 2.42 at φ = 0.05 to 6.22 at φ = 0.45, so no single α describes the curve it is fitted to. A literature that reports "α ≈ 2 for cytoplasm" is reporting the average of a function that doubles over the accessible range.

The lag-window problem. The fifth and sixth columns of the main table are the same trajectory fitted over two different lag windows. The table below instead takes the instantaneous slope coefficient at τ = 8 and at τ = 1024, and their ratio:

At φ = 0.30 the discrepancy between the short-lag and long-lag coefficients is 75.1%; at φ = 0.40 it is 259% and at 0.45 it is 501%. Fitting MSD = 4Dt over τ = 4–64 and again over τ = 128–1024 returns coefficients differing by up to 3.1× on the same trajectory, and the instantaneous slope ratio between τ = 8 and τ = 1024 reaches 6.0×. Nothing changed except the window. The last column of the main table shows why: the walker at φ = 0.40 has MSD(1024)/MSD(512) = 1.63 against the value 2 that a diffusive process requires, so even the longest lag available here has not reached the diffusive limit. At φ = 0.45 the free fraction is 0.55, below the site-percolation threshold of the square lattice (pc = 0.5927, so φ_c = 0.407), and the walker is confined to a finite cluster: the "diffusion coefficient" is then a measurement of cluster size.

The expectation this contradicted. The prediction going in was that the obstruction model would be adequate — it is the model everyone quotes — and that the interesting failure would be the one everybody already knows about, namely subdiffusion in the middle of the lag range. Both parts were wrong in a specific way. The obstruction model fails by a factor of 8.8 in log-residual against the percolation form, and it fails because its exponent is not constant (2.42 to 6.22). The subdiffusive regime is real — the exponent drops to 0.770 at φ = 0.40 — but it is a consequence of the same structure that makes the percolation form fit, not a separate phenomenon. And the largest practical error, 259% at φ = 0.40, comes not from subdiffusion but from using a long-lag coefficient at a short lag: a constant-D model that is fitted correctly and extrapolated wrongly.

Counts

Tags are editorial: test marks a lead that can be attacked with a computation, a dataset, or a re-analysis already to hand; note marks one that needs a literature sample, a new measurement, or an institutional change. A "test" tag is not evidence that the lead is true, and none of the 200 has been tested here.

Limits

• Test 1 is an identity, not a result. k_app = f·k_true is true by the definition of f. What is non-trivial is the consequence: the elasticity is 1, so the correction cannot damp its own input, and the ranking-reversal probability of 0.213 at a true ratio of 1.5 follows from the stipulated Beta(4,4) active-fraction distribution. That distribution is a stipulation, not a measurement; a narrower distribution lowers the reversal probability roughly in proportion to its width.
• Test 2 assumes log-normal single-cell values and equal variances. Real single-cell distributions are often zero-inflated or bimodal, and the pooled-variance t-test is not the test most laboratories would use. The qualitative scaling — power set by n·CV⁻² — survives those changes, but the tabulated powers do not.
• Test 2's technical CV is stipulated at 0.10. Nothing here derives it. The row at CV_tech = 0.25 in the requirement table is the only place a different value appears, and it shows the same conclusion: technical noise matters only when the biological CV is small.
• Test 3 is a lattice with a single obstacle type. There are no hydrodynamic interactions between the walker and the crowders, no attractive interactions, and no polydispersity of obstacle size. All three are known to matter for real cytoplasm, and the last of them is the one most likely to change the fitted percolation threshold.
• Test 3's fitted φ_c is not the geometric threshold. The fit returns 0.468 against the exact square-lattice value of 0.407. The discrepancy is not noise: it is the diffusive limit not being reached below the threshold within 4096 steps, which is the same effect the lag-window table demonstrates. Treat the fitted φ_c as a fit parameter, not as a critical point.
• The window ratio is a worst case within the sampled grid. φ = 0.45 is above the percolation threshold and its coefficients are not diffusion coefficients at all. The number 3.1× is quoted as the largest value the grid produced, not as a typical one; below φ = 0.20 the same ratio is under 1.3.
• No literature search. Several leads are certainly known under other names — activity coefficients, inner-filter corrections, the Saffman–Delbrück relation, the Thiele modulus, Fano-factor decomposition — and are stated here as falsifiable claims rather than as discoveries.
• All 200 leads are untested. The tags describe intent, not status. Nothing on this page is a measurement; the only numbers are closed-form evaluations and seeded simulations, and every one of them was produced by the script named above.

If one sentence survives: a molecular or cellular rate is only as good as its denominator, and the three denominators that fail most quietly are the active fraction (which can reorder a comparison rather than merely scale it), the single-cell distribution (which sets the power through n·CV⁻² and not through n), and the diffusion coefficient (which is a function of the lag window before it is a property of the medium).

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