Search-limited
Effectors spend most of their time searching. Per-target killing follows effector density, so at a fixed ratio a denser well kills faster.
99 SMALL PROBLEMS / NO. 05
Encounter Check
Same E:T ratio.
Same experiment?
Useful models for assumptions with expensive ambitions.
Fixed ratio, fixed well, different density
EFFECTOR-ATTRIBUTED LYSISIf the ratio were enough, color would be constant along each diagonal
LYSIS AT END OF WINDOW| Quantity | Value | Meaning |
|---|
WHEN IS THE RATIO ENOUGH?
Search time decides which variable matters.
Each effector alternates between searching and handling a target. When searching dominates, the killing rate per target scales with effector density and the E:T ratio is not a sufficient description. When handling dominates, effectors are saturated with targets and the fraction killed follows the ratio and the clock. The search fraction below is computed at time zero from the declared rates.
0 · handling-limited, ratio governs1 · search-limited, density governs
Mixed
Search and handling take comparable time. Density and ratio both move the endpoint, and neither alone describes the condition.
Handling-limited
Effectors are saturated with targets. Each kills at close to its maximum rate, so the fraction killed tracks the ratio and the clock.
Single-use ceiling
Effectors do not return after killing. Failed contacts can be retried, so lysis creeps toward the ratio itself and can never exceed it, however long the assay runs.
DISCRIMINATING EXPERIMENT 1
Change the medium, keep the cells.
Suspension cells settle within minutes, so many co-cultures meet on the well bottom rather than in the liquid. The two hypotheses below are matched to give identical lysis in the reference well. They disagree about what a volume change or a plate-format change should do. If lysis follows medium volume at fixed counts, the bulk description survives; if it follows bottom area, the settled one does.
| Condition at fixed cell counts | Settled hypothesis | Bulk hypothesis | Separation | Bottom coverage |
|---|
A PRACTICAL DECISION
Fewer targets. Which well reproduces the reference?
When fewer target cells are available, keeping the ratio in the same well also lowers both densities. In this model only preserving both densities preserves the experiment, which means changing the well, not the ratio. Crowding and settled-area assumptions still have to hold in the smaller well.
| Design | E per well | T per well | E:T | Medium µL | Area mm² | Lysis | vs reference |
|---|
DISCRIMINATING EXPERIMENT 2
Same lysis, more contacts.
A lysis endpoint integrates contact frequency and per-contact kill probability. The rival below meets targets more often but kills less per conjugate, with its kill probability re-solved to reproduce the reference lysis exactly. A fixed-ratio density series separates the two only weakly. Measuring conjugates at the contact read time separates them directly.
| Density at fixed ratio | Lysis, declared | Lysis, rival | Conjugated targets, declared | Conjugated targets, rival |
|---|
DISCRIMINATING EXPERIMENT 3
Same dose, different free drug.
A bispecific links two cells only when one arm holds a target antigen and the other finds a free effector receptor. Density now enters twice. More cells per area mean more contacts, which raises the plateau. More cells per volume bind more drug, which leaves less free drug at the same nominal concentration and moves the curve right. The two effects can offset each other at a working dose, so a flat response across density does not show that the ratio suffices.
The bispecific bridge is off. Switch it on in 04, or load the example “Same ratio, same dose.” Everything above then uses the drug-dependent kill probability.
Dose response at a fixed ratio and three densities
EFFECTOR-ATTRIBUTED LYSIS| Density at fixed ratio | Plateau | EC50, free | EC50, total | Drug on cells at EC50 | Free fraction at dose | Lysis at dose |
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THE USEFUL OUTPUT
A plate that separates density from ratio.
Cross E:T ratio with target count and medium volume. Ratio-only thinking predicts flat rows across target counts. Density-dependent killing predicts a slope. The volume factor tests which density applies. Predictions use the declared mechanism under both matched geometries; wells are laid out in row order for a 96-well plate.
| # | E:T | E | T | µL | pM | Free | Settled | Bulk | Conjugated at read | Search fraction | Coverage |
|---|
WHAT THIS MODEL IS NOT
Boundaries, stated plainly.
Scope. A well-mixed, short-window, mass-action model of one effector population meeting one target population, with one contact state, one kill transition, optional effector recovery, and first-order background death. It excludes proliferation, activation delay, exhaustion, cytokine effects, multicellular conjugates, cooperative killing, target heterogeneity, motility-limited or spatially aggregated contact, and adherent geometry. Each could change the ranking.
The bridge. Drug binding is monovalent to each cell pool, at equilibrium, and fixed at time zero. Bridges are assumed rare relative to singly bound drug in the mass balance. Surface cross-linking constants are taken as proportional to the solution affinities, with no cooperativity, avidity, receptor internalization, shedding, or soluble antigen. The drug changes what a contact does, not how often contacts happen. Relative EC50 is measured against each well’s own plateau.
Inference. Rate constants are declared, not fitted. There is no confidence coverage, no power calculation, and no cost weighting. The indices describe time zero. A lysis endpoint alone cannot identify the encounter coefficient and the per-contact kill probability separately; that is the point of the contact read.
Verification. The code is tested against analytic limits, conservation laws, and scaling identities. Software verification is not biological validation. Presets are synthetic and do not describe any specific molecule, cell product, or assay.