99 SMALL PROBLEMS · No. 02 COMPANION WORKED EXAMPLE · SYNTHETIC
Lower uptake. Higher productive delivery.
Entering the cell is not the same as completing the delivery.
Follow a prescribed ADC entry flux through routing, processing and payload escape. The example is executable; its parameters are invented. No efficacy or therapeutic ranking is inferred.
MODEL RESULT · NOT EXPERIMENTAL DATA
TOTAL EXPOSURE IS NOT ITS TIME HISTORY
Same area. Different biological questions.
Loading the example replaces all settings. Matching changes only B’s entry flux, not processing, loss, removal or the shared input schedule. It is not an equipotent-dose calculation. Changing the window or rates does not silently rematch.
A: solid teal. B: dashed amber. The right plot normalizes each curve to its own finite-window AUC, even when the two AUCs are unequal. It is not target occupancy or the fraction of molecules delivered.
| Within the selected 0–T window | A | B |
|---|
Peak and AUC quantile times are numerically refined, not selected from the 401 reported nodes. t10, t50 and t90 are times by which 10%, 50% and 90% of that case’s 0–T AUC has accumulated. They are not onset, EC50 or response percentiles. Centroid is exposure-weighted mean clock time; every descriptor is window-dependent.
This arbitrary cutoff is not a pharmacological threshold. Change it to challenge how the descriptive duration ranking depends on the chosen level; durations are restricted to the displayed window.
What equality of AUC leaves unresolved
For a linear, time-independent effect proportional only to integrated cytosolic amount, equal AUC would give equal values of that particular exposure summary. That is an assumption about the effect model, not a conclusion from the ADC transport model.
A peak-sensitive process, a requirement for sustained engagement, saturable target binding, damage repair or a time-varying susceptible state would require additional dynamics and independently supported parameters. These plots identify exposure differences that such hypotheses would need to confront; they do not select the correct hypothesis or predict which profile is more efficacious.
Equal 0–T AUC does not imply equal lifetime AUC, equal entered ADC, equal peak, equal onset, or identical late exposure. Use the cohort ledger below to see the exposure still to come. A longer chase can destroy equality that was deliberately constructed at 24 hours.
OPTIONAL PHARMACODYNAMIC HYPOTHESIS · NOT EFFICACY
The same area can meet a different biological clock.
AUC is a useful summary. It is not a mechanism. Hold the delivery profiles fixed, then ask which part of an apparent advantage comes from saturation, engagement kinetics or what remains when the assay is read. The comparison is useful precisely because those answers need not agree.
This is an original, synthetic PD overlay. D is an arbitrary excess signal, not DNA damage counts, viability, tumor response or a clinical endpoint. All PD parameters are shared between A and B.
K is an effective amount scale, not a molar Kd or EC50. At fixed K, changing b changes both association and disengagement speed. Presets replace only these four PD fields; delivery and AUC matching are untouched.
This is a one-way approximation, not a coupled binding mass balance. It can fail when target binding materially depletes free payload. No concentration conversion, target abundance or validation of that approximation is supplied.
Where does the endpoint ratio come from?
The same endpoint can reward a different history. This identity separates four dependent accounting terms; it does not identify four independent biological causes.
| Accounting term | A | B | B/A |
|---|
C = K∫Oeq dt / Q; M = ∫O dt / ∫Oeq dt; R = D(T) / (s∫O dt). C and R lie between 0 and 1; M can exceed 1 because kinetic engagement can persist after the equilibrium reference falls. M is not an efficiency.
Why more engagement is not free
Integrating the engagement equation gives Q = K∫O dt + (K/b)O(T) + ∫PO dt. Exposure is partitioned into an integrated-engagement term, residual engagement at the observation boundary, and exposure coinciding with already occupied targets. All three terms are nonnegative under this model, so ∫O dt ≤ Q/K even when ∫O dt exceeds ∫Oeq dt.
The overlap term is inferred by subtraction, not independently measured or integrated. This is an algebraic interpretation, not a second numerical validation or proof that occupied-target exposure is therapeutically wasted.
A solid teal; B dashed amber. Fractional engagement and arbitrary downstream signal are different quantities. No signal cutoff or killing rule is assumed.
The assay clock can change the ordering.
This scans the 401 reporting times. Brackets are not refined crossing times, and brief reversals may be missed. “Unresolved” uses a numerical tolerance, not a detection limit, equivalence margin or biological threshold. Equal AUC refers to the selected full window, not every earlier time.
| Conditional PD observable · within 0–T | A | B |
|---|
AU means arbitrary units. A large endpoint ratio can reflect near-complete recovery in the reference, not a large absolute effect. Compare the two absolute signals and their histories; no detection limit or uncertainty is modeled.
Three assumptions, three different questions
Linear control: dD/dt = s(P/K) − rD. With r = 0, D(T) = s·AUC/K, so equal AUC gives equal endpoints under these shared assumptions. With recovery, the endpoint depends on when exposure occurred, even without saturation.
Instantaneous saturation: Oeq = P/(K + P), with dDeq/dt = sOeq − rDeq. Integrating a saturable function is not equivalent to applying it to an integrated exposure. This is a counterfactual equilibrium reference, not a declaration that binding is fast.
Kinetic engagement: dO/dt = b[(P/K)(1 − O) − O]. Its instantaneous relaxation time is 1/[b(1 + P/K)]. At fixed K, slower b preserves the equilibrium curve while changing whether the profile can approach it. The unforced half-time ln(2)/b is not, by itself, proof of stronger action.
Recoverable signal: dD/dt = sO − rD, so D(T) = s∫₀ᵀ exp[−r(T−t)]O(t)dt. Recovery weights later engagement more strongly at the endpoint. Removing recovery gives s∫Odt, not s·payload AUC. The ledger distinguishes signal that formed from signal still present.
O and D start at zero. The target pool is fixed, payload is not depleted by binding, and feedback, cell-cycle dependence, target-specific catalysis, nonlinear repair and cell death are absent. Target renewal is not separately identified; treating b as including renewal requires replenishment of a constant pool in the unengaged state.
Different biological processes can produce similar delayed readouts. Time-resolved free payload, engagement and downstream measurements would be needed to distinguish them. This is a hypothesis generator, not a fitted model or an assay-validation claim.
Conceptual background: Vauquelin, 2016: binding kinetics versus affinity; Dayneka, Garg and Jusko, 1993: response production and dissipation. No numerical parameters, therapeutic constructs or code were imported from these publications. Full PD derivation and scope.
What would distinguish these explanations?
These are proposed discrimination tests, not validated protocols. A good experiment should threaten the preferred explanation, not merely provide another endpoint it can accommodate.
Saturation: does peak compression explain the result?
Compare engagement histories under measured target-accessible free-payload profiles that differ in shape. Test whether one equilibrium amount–engagement curve can explain both rising and falling phases. Equal nominal dose or equal total intracellular payload is not the matching condition used here. A large C ratio motivates this test; it does not establish saturation experimentally.
Engagement memory: does the same amount mean the same engagement?
Pair engagement and accessible-payload measurements on the rising and falling limbs at similar P. The finite-rate hypothesis permits different O at the same P. Such a loop would challenge a purely instantaneous mapping, but would not uniquely identify binding kinetics: delayed access, changing target pools and population mixtures remain alternative explanations outside this model.
Recovery: is the downstream tail still being driven?
Once engagement is demonstrably negligible, the declared model predicts D′ = −rD, hence exponential recovery. Continued signal formation from persistent O cannot be separated from slow recovery by a downstream trace alone. External washout is not proof that intracellular exposure or engagement has ended.
ADC interpretation: keep the missing steps visible.
A bispecific format is not assigned a routing benefit, and a cytotoxic payload is not assigned a killing coefficient. For irreversible target modification, catalytic damage, cell-cycle-dependent susceptibility or proliferative recovery, the present engagement/signal equations are insufficient. Add a mechanism only when its state, units, discriminating measurement and unsupported alternatives can be stated.
Practical output: a time-resolved free-payload → engagement → downstream-readout comparison, with an explicitly tested chase period. The tool does not optimize sampling times, fit parameters, model noise or establish assay validity.
Where the advantage ends
A one-parameter challenge, not an uncertainty interval. Hold all rates, yield and A fixed; change only B’s entry.
Processing is now a race, not a waiting room
The first button replaces the settings. The second preserves every other input, isolating the effect of this loss route; it requires processing rates of at least 0.001 h⁻¹ in both cases.
This sink means irreversible failure to generate the active-payload species modeled here. It does not identify a particular degradation pathway, transporter, catabolite or therapeutic mechanism.
THE CURRENT SIMULATION · AN EXACT MODEL DECOMPOSITION
Yield is not timing. Exposure is not efficacy.
Finite-window factor = fraction of eventual cohort AUC realized by the observation time. It is not the fraction of payload delivered. All factors compare B/A; the shared payload yield and input duration cancel.
| What the model distinguishes | A | B |
|---|
Why faster disappearance can mean worse delivery
In this constant-hazard model, an intact lysosomal molecule exits at total rate p + ℓ. Conditional on productive processing, its waiting time is still 1/(p + ℓ), while its chance of that productive exit is p/(p + ℓ). Increasing loss therefore shortens the wait among successful trajectories but reduces their number. A faster-looking successful subset does not establish a better route.
The remaining-exposure ledger weights each current compartment by its remaining productive probability and cytosolic residence. Irreversibly lost material receives no future credit. For continuous entry, the ledger deliberately asks what would happen if entry stopped now; it does not count future doses.
What would make this useful for a bispecific or alternative-format ADC?
Use a format-specific claim to choose the unresolved measurement, not to rename a fitted rate. If the claim is improved entry, distinguish bound from internalized material and constrain the entry time course. If it is productive trafficking, challenge the partition between intact conjugate, released lysosomal species and cytosolic payload. If it is prolonged activity, distinguish cytosolic residence from target engagement and downstream effect.
A paired-antigen audit can test the joint expression distribution. It cannot supply the joint occupancy, binding geometry, trafficking rates or co-internalization needed to convert a bispecific design into these inputs. The two tools remain separate until measurements justify their connection.
One AUC value cannot identify its six factors. Even an entire cytosolic trace can leave different internal mechanisms indistinguishable, as the built-in counterexample shows. Choose a species-resolved time course that separates the competing explanations; additional precision on an ambiguous endpoint does not remove the ambiguity.
Equal AUC also does not imply equal efficacy. This transport audit contains no effect model; the separate opt-in PD layer tests hypothetical engagement and an arbitrary recoverable signal, not measured damage, cell-cycle dependence or killing. AUC is an exposure summary whose pharmacodynamic relevance must be established separately, not a universal currency for comparing payloads.
A FIXED REFERENCE · TWO COMPETING RATES IN B
Where does the lower-entry advantage survive?
B above AB below ADashed line: tieCircle: current BDiamond: inspected point
Both axes include zero using log10(1 + rate / 0.001), not a standard logarithmic scale. Rates span 0–10 h⁻¹. Hatched cells have total exit below the supported 0.001 h⁻¹ minimum. Colors show an algebraic comparison, not a probability of success.
Click the map or enter exact rates to inspect without changing the simulation. Apply updates only B’s processing and intact-loss rates, then recomputes its time course. Every other model input stays fixed.
The full JSON report includes the 81 × 81 grid, current B point, fixed parameters and analytical boundary. The arbitrary displayed rate domain is not a biological plausibility range; region area has no probabilistic interpretation.
SAME RATES · THE ACTUAL OBSERVATION WINDOW
Does the advantage arrive in time?
B AUC above AB AUC below ASolid line: approximate AUC tieCircle: current BDiamond: inspected point
AUC is the integral of active cytosolic payload from zero to the selected observation time, in payload·h/cell. Both axes use log10(1 + rate / 0.001) over 0–10 h⁻¹. Hatched points have total exit below 0.001 h⁻¹. Color is a deterministic comparison, not a probability.
Click or enter exact rates to inspect without changing the run. Apply changes only B’s processing and intact-loss rates and recomputes both maps and the trajectories. Each map has its own temporary inspector; both current-point markers follow the applied B settings.
The 81 × 81 grid uses the same linear equations as the time-course solver, evaluated by a matrix exponential with the prescribed pulse boundary retained. The full JSON contains both maps and numerical-method metadata. No payload target occupancy, killing, efficacy, uncertainty distribution or fitted biological rates are inferred.
Separate the ledger from the signal
A stock is not a flux. Cumulative entry is not retained intracellular ADC, and none of these quantities is automatically a fluorescence signal.
| Quantity and units | A | B |
|---|
Retained intact ADC
E + L · ADC/cell
Active payload arrival
kesc × Plys · payload/cell/h
Active cytosolic payload
Pcyt · payload/cell, not concentration
Cytosolic exposure accumulated
Integral of Pcyt · payload·h/cell
A · solid B · dashed. All curves are computed from invented inputs; numerical trajectories are downloadable.
A beautiful trace is not a unique mechanism
This replaces both cases with a synthetic counterexample. Twice the routing fraction is offset by half the escape fraction, preserving the cytosolic time course while changing upstream inventories.
A separation here is a mathematical prediction, not proof that an assay can resolve it. No noise, measurement calibration or optimal experimental-design calculation is included. Numerical values also appear in the time-point table and full JSON/CSV exports.
Why even a time course can leave the mechanism unresolved
For zero initial stocks, the transfer function from internalization input J to cytosolic payload is:
s is the Laplace variable, with units h⁻¹. Matching these exit-rate sums and the numerator makes the cytosolic response identical for the same input history. The preset changes internal rates while preserving that function; a pulse alone cannot distinguish this pair. Failure to satisfy this sufficient condition does not establish identifiability.
Keep every assumption attached
The model is a route, not a response claim
E and L contain intact ADC; Plys and Pcyt contain payload. The competing sink from L is counted in ADC equivalents, distinct from released-payload loss out of Plys. Arrows are modeled first-order transfers, not measured rates; processing multiplies the count by yield ν.
The mechanism behind the reversal
For constant entry and the stated linear assumptions, steady cytosolic arrival equals entry × payload yield × routing fraction × processing-success fraction × escape fraction. B wins only when its downstream delivery factors more than compensate for its lower input.
The routing probability applies to one internalization episode. No receptor binding, repeated uptake, payload target binding, spatial bystander transfer or cell killing is modeled.
When faster processing changes eventual flux
With intact-loss set to zero, processing is the only exit from intact lysosomal ADC and its positive rate changes delay and inventory, not eventual throughput. With competing loss, the fraction processed successfully is kproc / (kproc + kloss,I); slower processing now allows more material to leave through the nonproductive sink.
A shorter mean residence time is not automatically better delivery: increasing loss shortens residence while reducing processing success. A pulse has zero eventual arrival flux, even when its accumulated delivery differs.
Which experiments would distinguish these explanations?
My proposed workflow separates surface-associated from internalized material, follows intact conjugate and released species over time, and uses a pulse–chase design to challenge routing and persistence. A total fluorescence plateau alone cannot identify every rate in this chain.
Cellular processing measurements and their proxy limitations are discussed by Maass et al., 2016; staged intracellular and multiscale calibration is described by Scheuher et al., 2024, online 2023. Those studies motivate the distinctions; they do not validate these invented parameters.
Full equations, assumptions, worked values and solver checks · Calculation code · Scientific reading guide