The protein clock,
with its assumptions showing.

This tool asks whether a declared RNA trajectory and protein-loss rate can account for a protein endpoint. It is a forward-model audit, not a fit, a delivery model, or a claim that time explains every disappointing result. The code is original; the biological foundations are established.

What is being modeled?

Let M be functional RNA per cell and P be the compatible mean protein amount per cell, or another observable for which the same constant effective-loss model is justified. With constant translation coefficient k_tr and effective loss rate λ, the unnormalized model is dP/dt = k_trM − λP. Pretreatment steady state requires k_trM₀ = λP₀. Defining m = M/M₀ and p = P/P₀ gives the implemented equation.

The integral states the mechanism: the protein pool contains surviving baseline protein plus surviving protein made subsequently. RNA is a driver, not an extracellular drug concentration. The link from RNA to synthesis is an assumption; bulk qPCR abundance need not equal translationally competent RNA, and incompatible normalizations can invalidate the comparison. The turnover literature distinguishes synthesis from loss and describes context-dependent and non-exponential behavior (Ross et al., 2021).

Which half-life?

In effective mode, any applicable dilution is already included. The application rejects a non-null doubling time in an imported effective-mode scenario. In intrinsic mode, a blank doubling time means no added dilution. A 48-hour intrinsic degradation half-life plus a 24-hour doubling time gives a 16-hour effective half-life, under the declared balanced-growth convention. These are synthetic parameters, not measurements of a named protein.

The growth term follows a population accounting argument: if total protein A obeys dA/dt = N k_trM − k_degA and cell number obeys dN/dt = μN, then P = A/N obeys dP/dt = k_trM − (k_deg + μ)P. This requires a compatible per-cell observable and a stable growth rate, not protein per well. If treatment changes division, cell size, cell composition, synthesis, or degradation, the same constant-λ normalization is no longer justified. Do not add dilution to a half-life already corrected for it. The distinction between true degradation and division-related dilution is discussed by Ross et al. (2021).

The first worked case

For an instantaneous, sustained fractional RNA level r, the model has a closed-form solution. The first preset uses r = 0.1 and an effective half-life of 48 hours; the initial jump is imposed, not a simulation of siRNA entry or RISC activity.

The predicted protein reduction is 26.36%, despite 90% RNA reduction. Reaching 50% protein remaining takes 56.16 hours under sustained suppression. Reaching the asymptotic RNA level itself takes infinite time in this step model; a threshold below it is never reached. These are synthetic calculations from the displayed assumptions, not published experimental results.

Recovery introduces a second timing trap

In the recovery preset, RNA remains at 0.1 through 24 hours and rises linearly to 1 by 36 hours. With the same 48-hour effective protein half-life, the central protein minimum is approximately 0.7003 at 32.00 hours. RNA recovery has already begun, yet protein is still decreasing until the two normalized quantities cross. This preset never reaches 50% protein remaining; simply waiting longer does not rescue that decision rule.

These relations follow from the implemented equation, not a claim that RNA and protein abundance are interchangeable. They show why a protein endpoint can retain a history that an RNA endpoint no longer displays. The tool evaluates internal extrema as well as endpoints; it explicitly labels a minimum at the observation boundary.

A limit on the timing explanation

Because the RNA driver is nonnegative and new synthesis is nonnegative, the integral representation gives an immediate lower bound. At a specified λ, even complete shutoff cannot remove the pre-existing protein pool faster than its declared loss rate.

The challenge preset enters a synthetic protein observation of 40% at 24 hours. With effective half-lives from 24 to 96 hours, even the fastest-rate complete-shutoff floor is 50%. The declared RNA-only mechanism and loss-rate range therefore cannot reproduce 40%. This is a conditional mathematical inconsistency, not proof of accelerated degradation: error, wrong rate bounds, baseline mismatch, mixed populations, or other violated assumptions remain alternatives.

For a fixed λ and constant nonnegative translation coefficient, decreasing translation all the way to zero still cannot cross this floor. A measured value below it requires measurement/normalization error or a violation affecting the initial pool or its loss, not merely stronger RNA suppression. The interface’s broad checklist is a prompt for investigation, not a unique mechanistic diagnosis.

Numerical method and inspection

The RNA driver is piecewise linear. No extrapolation beyond the last RNA point is permitted. Duplicate times are rejected; the initial value can represent an imposed jump at time zero, but later instantaneous jumps must be represented by a resolved transition or a different model.

The implementation uses expm1 for Q and a small-x series for the ramp term to avoid cancellation. Internal extrema solve m − p = 0 by bisection within each linear segment. There is at most one internal extremum per segment. Threshold roots are bisected on the resulting monotone pieces, then below-threshold intervals are merged. A zero-duration tangency is not reported as time below threshold.

The sensitivity band is the pointwise minimum and maximum of 31 logarithmically spaced half-lives plus the central half-life, with duplicates removed. It is a sampled parameter envelope, not an exact continuous-parameter bound or a confidence interval. After recovery, trajectories can cross; the band is not constructed by assuming the two endpoint half-lives always bound all intervening values. Changing the RNA driver or interpolation can produce uncertainty not represented by this band.

Plots and CSVs use 481 uniform time points plus input knots, central extrema, and the current inspection time. Inspection values and threshold roots use the analytical segment solution rather than interpolation on the plot grid. A scenario JSON stores version, assumptions, and inputs; imports validate the schema and recalculate outputs. CSVs contain numerical curves; retain their companion JSON for reproducibility. An entered protein observation is cleared when its inspection time changes, to prevent quietly assigning it to another time.

Surviving protein is not all the same history

The first term is the surviving initial pool, and the second is surviving protein synthesized after perturbation. Both are expressed relative to baseline, not relative to the remaining pool. The model can therefore show continued protein synthesis during net depletion without calling that observation a resistant subpopulation. This decomposition is mathematical, not a measured molecular-age distribution.

When waiting can and cannot help

At inspection time t_a, freeze RNA at r_a = m(t_a) only as an explicit counterfactual. The predicted initial state is p_a = p(t_a); the optional observed protein is not substituted. This does not change the main trajectory or extrapolate the supplied RNA data.

If the current protein exceeds the target and the held RNA level is lower than it, waiting can reach the target at the computed delay. If the held RNA is higher than the target, waiting cannot solve the problem under this model. Equality with the plateau is asymptotic, not finite-time success. A single RNA measurement does not prove that this level will persist; delay and residual synthesis can coexist.

Same endpoint, different suppression

The companion laboratory constructs two hypothetical instantaneous, sustained RNA steps. Each starts from p(0) = 1 and passes through the same protein endpoint y at t_a, but uses a different effective half-life. It does not fit or reproduce the measured RNA course in the main workbench.

For the synthetic 75%-protein anchor at 24 hours, a 48-hour half-life requires 14.64% residual RNA; a 12-hour half-life requires 66.67%. At 72 hours, those hypotheses predict 44.82% and 67.19% protein remaining. The former reaches 50% at about 61.03 hours; the latter never does. These are derived scenarios, not empirical target-specific parameters.

Their RNA inputs differ. If a reliable functional-RNA time course is already available, it may reject one or both without waiting for another protein endpoint. A matched protein readout does not establish indistinguishable biology. A longer half-life can even have a steeper decline after the common anchor because the synthesis input differs: the decline at that point is λ_i(y−r_i).

A separation rule, not statistical power

For these ordered, endpoint-matched sustained steps, the gap increases monotonically after the anchor. The model bisects the touching boundary, with disjoint intervals only after that time. At ±5 percentage points per prediction, the default boundary is approximately 45.14 hours. If the eventual plateau difference is no greater than twice the allowance, longer follow-up never satisfies this rule.

The allowance is an assumed comparison tolerance. It is not a confidence interval, a measured error distribution, a sample-size calculation, or a treatment-effect significance test. Parameter uncertainty and uncertain RNA histories are excluded from this comparison. Display shading is clipped to 0–100%. Read the complete mathematical contract for the monotonicity argument and numerical boundary handling.

RNA and turnover uncertainty without breaking the anchor

The uncertainty module retains the fixed comparison as a reference, but tests entire bounded parameter sets. RNA varies multiplicatively, effective half-life varies over a declared fold range, and the protein anchor has its own measurement tolerance. Only combinations consistent with the anchor are retained. This is set membership, not a likelihood fit or confidence interval; the defaults are illustrative, not a universal error model.

Relative qPCR quantities involve exponential conversion, normalization, amplification-efficiency estimation, and possible inter-run calibration. Their propagated error is not simply the standard deviation of raw technical Cq replicates (Hellemans et al., 2007). Enter a justified bound on the final normalized RNA ratio. The tool does not analyze raw qPCR data or reproduce qBase. Uncertain half-lives describe uncertain knowledge, not a simulated cell mixture; measurement limitations and context dependence are discussed by Ross et al. (2021).

Separate RNA centers are hypothetical inputs. Shared-RNA mode instead uses one independently justified RNA interval for both explanations, which can render one or both parameter sets empty. A single RNA point cannot establish a sustained functional RNA history. A zero RNA center stays zero under multiplicative uncertainty; below-quantification-limit RNA must not be treated as known zero.

Computing continuous parameter envelopes

At fixed time, each active-constraint segment is monotone in H. Continuous extrema therefore occur at feasible half-life endpoints or intersections of RNA and anchor bounds. The code evaluates those candidates directly, rather than estimating extremes from Monte Carlo draws. The full mathematical contract gives the derivation. Every exported extreme includes its generating RNA/half-life parameters. Different times can have different extremizers, so an envelope edge is not necessarily a physical trajectory.

Can a later readout separate every allowed prediction?

Shading shows parameter envelopes only. The decision adds prospective protein-readout allowance separately; it is not a significance test or statistical power calculation. The model keeps every bounded combination without assigning probabilities or assuming random errors are independent. Known correlations, including a shared baseline, can make this box treatment conservative. Omitted bias, recovery, changing translation, or changing turnover can make it inadequate.

With the default ±0.5 log₂ RNA bound, ×/÷1.25 half-life bounds, and ±3-point anchor tolerance, protein at 72 h spans 39.51–50.62% for A and 59.72–73.42% for B. Those ranges overlap after adding ±5-point readout allowances, even though the nominal curves separate. The first checked disjoint time is 75 h on the default 0.3-hour grid. This is a synthetic robustness result, not a sampling recommendation.

Follow-up search checks 481 grid times plus the inspection time. It does not report an exact earliest boundary or guarantee continuous separation between or after checked times. RNA-only and turnover-only comparisons retain the same anchor tolerance and are sensitivity checks, not additive variance components. Empty parameter sets indicate incompatibility of the inputs and model, not proof of a specific biological mechanism.

Reproducibility of the extension

The measured-driver audit, fixed matched-endpoint comparison, and uncertainty audit have separate JSON schemas and exports. Uncertainty JSON includes its fixed comparison and uncertainty inputs; importing it updates those two panels but leaves the main measured-RNA model unchanged. Original comparison JSON contains no uncertainty settings. Older main and comparison files remain readable. Exact-zero protein targets are never labeled as reached at finite time, even if numerical underflow makes a computed value round to zero. Download uncertainty computation.

One shared baseline, correlated readout errors

The separate measurement audit retains sustained-step biology and RNA/turnover bounds but replaces the earlier rectangular anchor and follow-up allowances. Let b be measured protein reference divided by true baseline, and express numerator errors relative to true baseline protein. One b applies across anchor and follow-up within a candidate trajectory. Shared quantities can induce correlation in derived results, so their dependencies belong in the measurement model (Kessel and Kacker, 2009).

An overestimated reference lowers normalized protein. It is not independently redrawn at each time or averaged away by adding technical replicates. Protein-reference bias does not rescale the RNA bounds. Residual errors must exclude the shared reference error already assigned to b. Each competing hypothesis can have its own feasible nuisance-factor explanation; the model does not claim both are simultaneously true.

The ellipse is a bounded, covariance-shaped error budget, not a fitted distribution. No probability coverage is assigned. A radius of 2 in two dimensions is not automatically 95%. Zero correlation retains an elliptical bound, not the previous rectangular allowances. At perfect positive or negative correlation the ellipse collapses to a line; zero follow-up scale gives zero follow-up numerator error. All combinations with the declared baseline interval are allowed.

Correlation links anchor and follow-up residuals, not A and B. This is a pairwise error model, not a full stochastic process for the time course. At the anchor time, the follow-up represents a new replicate, not reuse of the original datum.

Condition the error on the anchor

Only errors and RNA values consistent with the observed anchor survive. At each turnover/baseline node, the code solves the remaining extrema analytically. With E_t = 2^−t/H and Q_t = 1−E_t:

Endpoints and feasible stationary points give exact inner extrema to floating-point precision. The outer half-life/baseline domain is sampled: up to 65 × 33 base nodes, or 129 × 65 in fine mode, with additional feasible-boundary and center points. Every extreme has an exportable witness that reconstructs the anchor and prediction.

A numerical profile is not a separation certificate

The inspection reports the maximum coarse-to-fine change in its extrema. This is a diagnostic, not an error bound; zero change does not prove global convergence. Positive sampled gaps may disappear if an unsampled outer extremum is found. The plot uses 61 post-anchor times plus inspection and does not report an exact earliest discriminating follow-up. Observed predictions are not clipped to 0–100%; no old allowance is added again. Pointwise profile edges need not follow one physical trajectory.

In the synthetic default case at 72 h, fixing the baseline at one gives A 39.66–50.37% and B 59.82–73.85% observed protein. Allowing b = 1 ×/÷1.1 gives A 33.57–55.04% and B 55.01–75.37%: slight overlap, not separation. These figures use 1.5/2.5-percentage-point error scales, correlation 0.6, and joint radius 2. They are illustrative assumptions, not empirical assay performance.

What cancels in a ratio

The shared denominator cancels exactly; numerator errors remain. Cancellation does not identify turnover or suppression from an uncertain absolute anchor. Reference drift, distinct denominators, changed translation or turnover, and incorrect RNA histories require different models. Positive correlation is not promised to improve discrimination; the interface supplies a zero-correlation comparison instead of presuming its direction.

Measurement JSON includes comparison, uncertainty, and measurement assumptions. CSV includes generating parameters, and PNG states numerical limits. Imports recompute all comparison panels while leaving the main measured-RNA driver unchanged. Read the full derivation and numerical contract or inspect the computation.

Measurement reference: Kessel R, Kacker R. Correlation in Uncertainty of Measurement: A Discussion of State of the Art Techniques. Fundamental and Applied Metrology, Lisbon, September 6–11, 2009. NIST. Read the published conference paper. This supports covariance-aware propagation, not the synthetic bounds or a confidence interpretation of this tool.

What is the next measurement worth?

The design companion compares tighter baseline calibration, a smaller prospective numerator-error scale, and a later readout, first separately and then together. These are hypothetical changes, not equally priced alternatives or a formal experiment recommendation. All use the same original protein anchor and biological families.

The retained fractions a_b and a_e run from zero to one. Calibration narrows the log-width without moving the selected baseline center; it requires independent justification. Precision changes only the follow-up numerator scale, not historical anchor error, shared bias, biological bounds, correlation, or joint radius. No conversion to replicate count is assumed.

A later readout remains a separate prediction conditional on the original anchor. It is not a joint fit to two follow-up observations or a calculation of their combined information. Sustained RNA, constant turnover, and the same observation model must remain credible. Requested times outside the declared horizon are explicitly unavailable.

Smaller error bars do not imply nested conditional predictions

The anchor-conditioned coefficient A = σ_tρ − σ_aQ_t/Q_a shows why correlation matters. Reducing σ_t can weaken cancellation with anchor uncertainty; the conditional prediction set need not shrink monotonically. In a synthetic fixed-baseline case with ρ = 0.9, halving the default follow-up scale increases A’s prediction width from 6.58 to 7.12 percentage points at 72 h and reduces the sampled gap from 13.48 to 13.10 points. This is a sensitivity of the assumed covariance geometry, not a claim that improved assays are intrinsically harmful.

Map the assumption boundary

The 121-cell map varies only baseline fold width and residual correlation, holding the current inspection time, baseline center, error scales, and biological bounds fixed. It does not silently inherit proposed strategy changes. Each cell includes ranges, signed gap, refinement diagnostic, and generating witnesses. Signs distinguish positive sampled gaps from overlap; incompatible families are not reported as successes.

These are discrete sensitivity calculations, not fitted assay properties or confidence classifications. Outer turnover/baseline extrema are numerical profiles. Refinement change is not an error bound, positive cells are not certified separation, and nothing is guaranteed between cells. CSV and annotated PNG exports preserve the assumptions.

The ratio changes units, not the evidence

The same positive observed anchor divides both prediction intervals, so overlap status cannot change. Algebraic cancellation of the common denominator does not remove its role in selecting biological trajectories compatible with the absolute anchor. The ratio is not an independent measurement and must not be counted twice. Apparent decline refers to observed readouts, not true-protein depletion; negative values are not clipped.

With default inputs, retaining half the baseline log-width, halving follow-up error scale, and adding 24 h produces sampled gaps of −0.027 points for the current design, 3.984 for calibration, 2.645 for precision, 6.366 for the 96 h readout, and 12.102 for the combined changes. These are synthetic conditional outcomes, not a universal ranking of experiments.

Design JSON contains all comparison, uncertainty, measurement, and design inputs. Import recomputes these panels but leaves the main measured-RNA driver unchanged. Applying a selected map cell explicitly changes baseline width and correlation in the measurement audit and makes prior design results stale. See the complete derivation and interpretation limits and design computation.

Worked example: calibration against another day

A separate page, Calibrate, or wait another day?, recomputes two synthetic cases in the browser from example.js. In the plateau case (effective half-lives 16 h and 4 h, anchor 75% at 24 h, follow-up at 96 h, baseline ×/÷1.08, half-life bounds ×/÷1.08, RNA log₂ half-width 0.25, otherwise shipped defaults), the current design leaves the sampled ranges overlapping by about 0.75 pp; retaining half the baseline log-width yields a positive sampled gap of about 1.02 pp, while waiting 24 h reaches only −0.56 pp and waiting to the 288 h horizon reaches about −0.47 pp. In the shipped default, the ranking reverses: waiting 24 h yields about 6.37 pp against 3.98 pp for calibration. The mechanism is that a later readout can add only the separation the two central trajectories still have left to generate; once both sit near their plateaus r_A and r_B, time no longer acts on the shared baseline term that calibration shrinks. The page also bisects for the largest retained log-width fraction with a positive sampled gap and reports the gap on both sides of the crossing. These are numerical statements about declared bounds, not power, coverage, or a recommendation.

The page decomposes each sampled gap exactly into the central separation between the two hypotheses and the two reach terms that face each other: sampled gap = central separation − upper reach down − lower reach up, with a reported residual of zero. Calibration leaves the central separation untouched and shrinks only the reach; a later readout adds central separation while widening both reaches. Closed-form plateau diagnostics give the ceiling on central separation implied by the two sustained-RNA plateaus, the share realized at the current follow-up, the times at which 95% and 99% of it arrive, and the value of one more delay step. Exchange rates convert the two levers into each other — hours of waiting, retained baseline log-width, and replicates under an assumed 1/√n width scaling — and refuse with a reason when no in-horizon readout or exactly known baseline can match. A grid of slower half-lives against follow-up times shows that in every row where the winner flips, the crossover falls between 3.0 and 5.0 elapsed slower half-lives after shutoff, bracketed by adjacent columns and never interpolated. Every input is editable, and kinetics the declared anchor cannot produce are refused with the implied half-life ceiling rather than approximated.

What the tool cannot establish

• Mechanism identification: agreement with an endpoint does not uniquely identify translation, loss, delivery, or target engagement. No parameter fitting or statistical inference is implemented.
• Therapeutic efficacy: total protein is not necessarily active, accessible, surface-localized, or functionally limiting protein. No phenotype, target reserve, viability, or clinical outcome is modeled.
• Unmodeled biology: feedback, age-dependent degradation, trafficking, secretion, multiple protein pools, cell-state mixtures, and treatment-dependent turnover require extensions with their own evidence.
• Modality transfer: RNA-lowering perturbations with measured RNA are the intended context. The model cannot be applied unchanged to PROTACs, molecular glues, or translation inhibitors.
• Validation: tests verify the implementation against mathematical results and an independent integrator. They do not validate an individual target, assay, or biological system.

Published foundations

1. Bartlett DW, Davis ME. Insights into the kinetics of siRNA-mediated gene silencing from live-cell and live-animal bioluminescent imaging. Nucleic Acids Research. 2006;34(1):322–333. DOI: 10.1093/nar/gkj439. Read the published article. It grounds the distinction between RNA silencing and protein-response timing, but this tool does not reproduce its full model. Its division-related dilution of intracellular siRNA species is not treated here as direct validation of a protein-dilution term.
2. Ross AB, Langer JD, Jovanovic M. Proteome Turnover in the Spotlight: Approaches, Applications, and Perspectives. Molecular & Cellular Proteomics. 2021;20:100016. Published online December 2020. DOI: 10.1074/mcp.R120.002190. Read the review. It grounds the separation of synthesis, degradation, and dilution, and the biological limits of a single constant loss rate.

Additional uncertainty reference: Hellemans J, Mortier G, De Paepe A, Speleman F, Vandesompele J. qBase relative quantification framework and software for management and automated analysis of real-time quantitative PCR data. Genome Biology. 2007;8:R19. DOI 10.1186/gb-2007-8-2-r19; published article. This reference supports careful propagation of normalized RNA measurement error, not the numerical widths of the synthetic defaults.

All presets are synthetic. No experimental, employer, or confidential datasets are included. This is an independent educational implementation of openly described kinetics, not an endorsement by the cited authors or institutions.