A better Z′ can hide a worse measurement

The bright controls have become unusually consistent. Their coefficient of variation has dropped, the Z-prime factor has improved, and the plate passes QC. Before crediting the assay optimization, check whether the reader is still measuring those wells proportionally.

In a luminescence assay, detector compression can make different bright signals look more alike. Under some conditions, that improves Z′ while making the measurements less accurate. The controls look tidier because part of their variation has been lost in detection.

What Z-prime measures

The Z-prime factor, usually written Z′, compares the separation of two control populations with their variability. Using H and L for high- and low-signal controls:

Here, μ is the mean and σ is the standard deviation. Larger separation raises Z′; larger standard deviations lower it. The original formulation by Zhang, Chung and Oldenburg remains a useful way to assess screening controls. [1]

The formula contains no check that light input and reported signal are proportional. It uses the numbers the reader supplies. It also says nothing about the response between the two controls, where many compounds will fall.

Simply reducing every value by 20% cannot improve Z′. Both the standard deviations and the difference between the means fall by 20%, leaving their ratio unchanged. The problem considered here is nonlinear compression: the brighter signals lose a greater fraction of their counts.

How photon counting loses proportionality

Photon counting works by resolving and counting individual detector pulses. The detector and its electronics need a finite interval to distinguish successive events. At sufficiently high arrival rates, pulses overlap and some events are missed. Hamamatsu describes this count-rate dependence and uses a paralyzable detector model in its photon-counting simulator. [2]

Compression can begin while the output is still increasing. A numerical result below an obvious ceiling therefore does not establish linearity. Warning behavior depends on the instrument: Hamamatsu describes both photon-counting heads with over-light detection and heads without it. [3] Check what a reader's warning actually monitors and how that relates to its specified linear range.

This also explains why shortening the integration time may fail to help. For a stable glow, counting for one-tenth as long reduces the accumulated counts, but it does not reduce the rate at which pulses arrive. A remedy for a total-count limit is not necessarily a remedy for pulse overlap. Whether a particular setting changes either limit depends on the acquisition system.

A worked example of misleadingly good controls

Take five constructed values in each control group. The high control has a mean of 100 and a sample standard deviation of 10. The low control has a mean of 10 and a standard deviation of 1. Their Z′ is:

Now pass those same values through an illustrative compression curve, f(x) = x × exp(−0.00223144x). This is the shape of the paralyzable mean-response model, expressed in arbitrary units and chosen so that an input of 100 produces an output of 80. It represents no particular reader. [2]

The inputs have not become more reproducible. Compression has reduced the spread of the high controls enough to outweigh the loss of control separation. Z′ rises from about 0.63 to 0.69, and the high-control CV falls from 10% to about 7.8%.

The same curve distorts intermediate values. An input of 55 lies halfway between the original control means and represents 50% inhibition under high-to-low normalization. After compression, it reads about 48.65. Normalize that value against the compressed control means in the table and it reports about 44.5% inhibition.

The curve is still increasing across this example, so it preserves the order of noiseless inputs. Distorted effect sizes do not require shuffled rankings. A cutoff applied to normalized inhibition can move a compound between hit and non-hit categories even if every well retains its place in the ranking.

These are deterministic calculations on invented well values. They illustrate distortion of between-well differences, without adding photon-counting noise or modeling an instrument's correction electronics. Nor does compression always raise Z′: it changes both the spread and the separation. If low-control variability dominates, the loss of separation can make Z′ worse.

The general statistical concern is already recognized in the Assay Guidance Manual, which notes that saturation artifacts can reduce control variability in imaging assays. [4] Imaging saturation and photon-counting losses have different physical causes, but both can make control variability a misleading guide to measurement quality.

Test detector linearity without changing the chemistry

A falling slope in a concentration series is a reason to investigate, but it does not identify the detector as the cause. The assay itself may be nonlinear. In CellTiter-Glo, for example, Promega notes that ATP content per cell can change with cell density, altering the relationship between cell number and luminescence. [5]

Optical attenuation helps separate those possibilities. It changes the light reaching the detector while leaving the sample chemistry in place. Hamamatsu describes a check using a neutral-density filter with known transmission: a 10% transmitting filter should reduce the measured count rate to one-tenth within the linear range. [3]

For a microplate reader, use a manufacturer-supported attenuation setting or a suitable calibrated verification device. A piece of dark plastic over the plate is not a calibrated filter. Transmission must be known over the relevant emission wavelengths, and the arrangement must preserve the measurement geometry.

Compare several brightness levels, including the brightest expected samples and intermediate controls. Account for dark/background contributions. Confirm whether the software reports raw attenuated output or automatically compensates for transmission; the expected numerical ratio differs. Hold temperature and assay age steady, and balance measurement order so that glow decay does not masquerade as nonlinearity.

If you use dilution instead, keep the final volume and matrix composition matched as far as the assay permits. Verify that the dilution changes light production proportionally. Diluting an active luciferase mixture can change substrate conditions, inhibitors, and reaction kinetics; diluting cells can change biology. A dilution series tests the combined assay-and-reader response unless those contributions are controlled.

Changing the settings requires another check

Lowering PMT gain is not interchangeable with reducing the light arriving at a photon counter. It changes pulse amplitudes and can affect which pulses cross the counting threshold. [3] Use the instrument's validated settings and assess linearity again; a smaller displayed number alone is insufficient.

Readers can extend useful range through faster counting electronics, appropriate count-loss correction, optical attenuation, or an alternative acquisition mode. Hamamatsu documents both count-rate correction and the hardware dependence of counting linearity. [3] The relevant specification is the linear range of the measurement mode you will use, with an acceptable error over the signals your assay produces.

An advertised range spanning many orders of magnitude is incomplete information without the test conditions. Ask whether it applies within one acquisition setting, how measurements are combined when settings change, and what happens at the transition. If two modes overlap, compare them there using stable samples. An alternative mode needs its own linearity and background checks.

Choose settings that preserve weak signals while accommodating the brightest plausible wells. Verify intermediate responses before comparing Z′. Otherwise, optimization can favor a setting that suppresses bright-well variation at the expense of accurate effect sizes.

What to include in assay validation

Inspect the raw control distributions. Save means, standard deviations, CVs, acquisition settings, and the individual well values alongside Z′. Unexpected tightening of the bright controls deserves investigation, but does not by itself prove compression.

Include intermediate controls or a reference concentration series. The two endpoints can look acceptable while the response between them is distorted. The Assay Guidance Manual recommends examining raw trends and reference compounds at multiple concentrations during pilot screening. [6]

Check more than one brightness range. Cover ordinary controls, the brightest expected samples, and enough higher signal to test the margin you intend to rely on.

Repeat the relevant checks after changes. A brighter reagent formulation, a different plate, or new acquisition settings can move the assay outside the range you previously established.

Investigate suspect runs before correcting them. Preserve the original data. Establish the response experimentally and assess whether stored samples can be remeasured under suitable conditions; dividing everything by a guessed recovery factor will not undo nonlinear compression.

References