99 SMALL PROBLEMS / NO. 04
The RNA got the message.
Has the protein had time?
Useful models for assumptions with expensive ambitions.
THE READOUT IS NOT THE PERTURBATION
| RNA | Protein | Sweep range | Shutoff floor* |
|---|
*Central loss rate, complete synthesis shutoff at time zero. A mathematical lower bound under this model, not an assay detection limit. For the full rate range, see the audit below.
WHAT THE CLOCK DOES, AND DOES NOT, EXPLAIN
DELAY VERSUS A RESIDUAL-SYNTHESIS LIMIT
These are model-resolved contributions relative to baseline, not measured molecular ages. New protein can be made even while total protein is decreasing.
This hypothetical continuation uses the central loss rate and the predicted protein state, not the optional observed value. It is explicitly separate from the supplied RNA course. A single RNA point does not establish a plateau.
WITHIN THIS WINDOW
At an interior extremum, RNA and protein fractions are equal: m(t) = p(t). A window-edge minimum is not evidence that the biological nadir was captured.
A DECLARED DECISION RULE
Central half-life only. A crossing is a prediction to test, not a recommended sampling time or a statistical guarantee. Isolated tangencies are not counted as time below threshold.
THE MATCHED-ENDPOINT LABORATORY
A slowly turning-over protein under deep sustained suppression can match a faster-turnover protein under weak sustained suppression at one time. Construct both explanations, then ask what RNA or later protein measurement would separate them.
| Scenario A | Scenario B | Difference | Allowances |
|---|
WHAT WOULD DISTINGUISH THEM?
The separation rule holds only for these fixed hypotheses and the declared allowance. It does not account for uncertain half-lives, RNA histories, baseline error, or correlated assay errors. Neither scenario is a fitted explanation of your experiment.
WHEN THE PARAMETERS ARE NOT KNOWN EXACTLY
A clean separation between two curves can disappear when the inputs have room to move. Propagate RNA and turnover bounds, retain only combinations consistent with the protein anchor, then ask what remains distinguishable.
| A protein | B protein | B − A range |
|---|
ROBUSTNESS / NOT SIGNIFICANCE
Vary RNA alone, turnover alone, or both. All three retain the same anchor tolerance; these are sensitivity comparisons, not additive variance components or priorities for experimental spending.
| Bounds varied | A | B |
|---|
Pointwise extrema may use different parameter combinations at different times; an envelope edge is not necessarily one physical trajectory. Unmodeled RNA recovery, changing translation, changing loss, and assay bias are not absorbed by these bounds.
This JSON includes the fixed comparison and uncertainty inputs. The original comparison JSON does not contain uncertainty settings. Inspect the uncertainty code ↓ · Read the uncertainty math
WHEN THE MEASUREMENTS SHARE AN ASSUMPTION
Could delayed depletion and incomplete suppression look distinct only because the protein baseline was treated as certain? Carry one unknown reference through both readouts, then inspect the effect of correlated numerator errors.
A separate measurement model, not another layer of error bars. It inherits RNA and turnover bounds, but replaces the rectangular anchor and follow-up allowances above. Default widths and correlation are synthetic, not calibrated assay performance.
A / solid boundaryB / dashed boundary
Shading includes the joint measurement-error budget. Boundaries are pointwise profiles, not individual trajectories. They are not clipped to 0–100%. Changing follow-up synchronizes all comparison panels.
| Comparison | A | B | Gap |
|---|
The zero-correlation reference retains an elliptical joint bound, not an independent-error rectangle. Positive correlation need not always improve separation. Profiling allows each competing explanation its own feasible baseline value; it does not assert that both explanations are simultaneously true.
| Extreme | H (h) | RNA | b | Anchor error | Follow-up error |
|---|
With exactly the same denominator, Y(t)/Y(anchor) = [P(t)+e(t)]/[P(anchor)+e(anchor)]. The baseline cancels from this ratio; numerator error does not. Nor does cancellation identify RNA suppression or turnover from the anchor. Different denominators, reference drift, and biological changes require a different model.
At the anchor time, the predicted follow-up is a new replicate readout, not a reuse of the identical measured datum. This audit describes pairwise anchor/follow-up error geometry, not a jointly specified stochastic process for the entire time course.
Inspect measurement code ↓ · Read the measurement math. This numerical sweep cannot certify continuous-parameter separation. Correlation and shared-input propagation are grounded in Kessel and Kacker (2009); the chosen bounds are not derived from that publication.
THREE WAYS TO ASK WHAT THE NEXT MEASUREMENT IS WORTH
A later endpoint is not automatically a more informative endpoint. Compare explicit improvements, inspect where baseline uncertainty and correlation change the answer, and test whether a clever ratio actually adds information.
Prospective design sensitivity, not experimental optimization. These changes have unequal and unspecified costs; the tool assigns no statistical power, probability of success, or preferred experiment. The later-readout option predicts one candidate follow-up from the original anchor, not the joint information from two observed follow-ups. A worked example shows one synthetic case where calibration separates the hypotheses and waiting cannot, and one where the ranking reverses.
DESIGN COMPARISON
Observed normalized protein ranges. Gap and change in gap are percentage points; positive sampled gap means disjoint sampled ranges. Refinement change is not an error bound. Scroll the table sideways on smaller screens to inspect every column.
| Design | Time (h) | A | B | Gap | Δ gap | Refinement (pp) |
|---|
A smaller follow-up error scale need not improve discrimination monotonically when anchor conditioning and residual correlation interact. An incompatible family is not a successful experiment or proof of the other mechanism. Compare changes only under assumptions you can justify.
BASELINE / CORRELATION MAP
+ Positive sampled gap− Overlap or touch× Incompatible family
Click a cell or use the keyboard-accessible selectors below. Color strength encodes absolute gap, scaled within this map; signs distinguish separation from overlap. Empty sets are not colored as successes.
| Extreme | H (h) | RNA | b | Anchor error | Follow-up error |
|---|
The map is computed at the current inspection time and current error scales, not the proposed later time or improved precision. Every cell uses the selected standard/fine outer sweep. Small positive gaps are not certified separation; inspect refinement and repeat with finer resolution.
ANCHOR-RELATIVE DIAGNOSTIC
| Family | Normalized protein | Follow-up / anchor | Apparent decline from anchor |
|---|
A common denominator cancels algebraically in each candidate ratio. It does not remove its role in deciding which RNA/turnover trajectories can explain the absolute anchor. Different references or drift break this cancellation; unresolved numerator errors remain. No baseline-free biological fit is performed.
Design JSON preserves the comparison, RNA/turnover bounds, measurement assumptions, and design controls. Existing measurement JSON does not include these new controls. Inspect design code ↓ · Read the design math
THE BIOLOGY IN THE EQUATION
The measured RNA course is an input, not a simulated consequence of dose. Existing protein remains until it is lost; new synthesis continues in proportion to the RNA that remains.
dp/dt = λ [m(t) − p(t)] · p(0) = 1
MECHANISM / EVIDENCE / INFERENCE
Bartlett & Davis (2006) examined siRNA-silencing kinetics and modeled effects of protein half-life. Their complete model includes delivery and intracellular siRNA processes that this tool does not reproduce.
Ross, Langer & Jovanovic (2021) distinguish degradation from dilution and discuss why turnover measurements depend on biological and experimental context.
Stable pretreatment baseline; constant translation per functional RNA; constant first-order protein loss; compatible RNA/protein normalization; a single effective pool. Linear interpolation is a numerical assumption, not evidence for the unmeasured RNA trajectory.
The half-life range is supplied by you. There is no target-specific parameter database, delivery inference, automatic fit, or inference of cell killing.
Withhold a later protein measurement and predict it from RNA plus independently justified turnover. Add RNA measurements where the driver is undersampled. A failed prediction motivates checking translation, protein loss, normalization, or population composition; it does not identify which one changed.
Do not transfer this equation unchanged to targeted protein degraders, translation inhibitors, secreted multicompartment proteins, or treatment-altered growth.