Selectivity in two-antigen T-cell engagers is a statement about time, not about affinity

# Selectivity in two-antigen T-cell engagers is a statement about time, not about affinity

Ben Hizer

Luminary Health, Luminary AI Technologies, Indiana, USA · ben@luminary-tech.ai

Running title: Relaxation times set engager selectivity

Keywords: T-cell engager · bispecific antibody · dual-antigen targeting · avidity · binding kinetics · kinetic proofreading · tumor selectivity

Two-antigen T-cell engagers are built to kill only cells carrying both of their tumor targets, and the selectivity they achieve is universally attributed to affinity and avidity. We show that the standard equilibrium account of that mechanism cannot reproduce the one kind of experiment that tests it: carried through to a fixed-effect endpoint it predicts selectivity flat across an affinity series where the measurement moves a hundredfold, and it does so for every value of its free parameter. The missing ingredient is in the source data already. When the two cell populations in one well fall on opposite sides of the assay clock, the ratio of their potencies becomes linear in the gating arm's equilibrium constant — a parameter-free prediction about the shape of a selectivity series, which two published series satisfy and a flat equilibrium trend does not. The design variable is a separation of relaxation times, and the handle on it is the dissociation rate rather than the affinity.

The word selectivity denotes at least three distinct quantities in the T-cell engager literature — an apparent-affinity ratio R_K, a fixed-effect concentration ratio R_p, and an asymmetric window W — which are routinely compared across papers as though they were one. We separate them and show that R_p carries a capacity factor which makes it depend on the effect level chosen, persists even at matched receptor density because a dual-antigen binding curve is not a Langmuir, and diverges as a cell approaches the bound density an effect requires. Carrying a standard confinement-style equilibrium binding model through to a fixed-effect endpoint then fails structurally rather than by misfitting: across its entire admissible parameter range it predicts selectivity flat to within 5% where the measurement changes 98-fold, because at equilibrium the gating arm's affinity multiplies both cell populations' bound density alike. The correction is present in the source data. The same report gives association and dissociation rates separately and a 6 h assay, and those numbers place the two populations in different kinetic regimes in one well: singly bound complexes turn over 6 to 780 times within the assay, while bridged complexes relax over hours and are still filling when the plate is read. The single-positive cell's endpoint is therefore an equilibrium quantity proportional to K_B, the dual-positive cell's a delivery quantity proportional to 1/(k_on·t), and their ratio must be linear in K_B — a log–log slope of 1, against the equilibrium model's 0. A three-species kinetic model that conserves free receptors, respects detailed balance and relaxes exactly to the equilibrium model predicts 1.009; the measurement gives 1.068 (95% interval 0.54 to 1.60), and no assumption swept moves the prediction by more than 0.19 of a unit. A second, independently published series — different antigen pair, trispecific format, primary T-cell effectors, isogenic knockout comparator, twelvefold longer exposure — gives 1.210 on three points, which alone excludes nothing; fitted with the first on one shared slope, nine points give +1.119 (0.67 to 1.57), excluding zero and containing one. The design variable that follows is a separation of relaxation times across the exposure window, on which the handle is k_off rather than K_D. Applied to a third dataset the criterion changes the variable rather than the answer: the footprint that sets the effective local concentration is 3 × 10⁻⁴ µm², so the governing density is local rather than per-cell, and any account in which potency is set by target-cell binding against a fixed bound-density threshold predicts 143- to 22,642-fold across that panel's antigen range where 4.4-fold is measured. Building the two-cell conjugate those models omit yields two further bounds: adhesion saturates at four engager links, placing the inferred threshold 201-fold above anything conjugation requires, and an effective-concentration account of epitope height is capped at the ratio of intermembrane reach distances in required density. We state three falsifying predictions; one needed no new experiment and has been tested, and passed. Nothing here has been synthesized or measured by us.

T-cell engagers have changed the treatment of hematologic cancers and have struggled in solid tumors, where few antigens are tumor-restricted and on-target off-tumor toxicity has halted trials. One response is to require two tumor antigens, so that only a cell presenting both partners retains the construct long enough to assemble a synapse. Selectivity then comes from co-expression rather than from abundance.

The architecture is in hand, in two forms. A Ly6E × B7-H4 × CD3 trispecific, monovalent for each tumor antigen, controlled tumor growth in 6 of 8 mice at 5 mg/kg without weight loss, while the same format carrying only one tumor arm — or the two single-antigen constructs given together — failed entirely.¹ A second design adds synapse gating: a high-affinity anchoring arm deliberately chosen for an epitope and geometry that cannot assemble a productive synapse, paired with an affinity-attenuated gating arm placed next to the CD3 domain, reaching 502-fold selectivity in vitro on the optimized pair and selective tumor control in vivo.²⁰ A separate route, single-antigen avidity from a bivalent tumor arm, is in the clinic.²

Both are engineering results without a quantitative account. The selectivity in them is attributed to cross-arm avidity and to synapse formation, which name the mechanism without fixing any number: neither tells you what selectivity a given pair of arms will produce, which arm to weaken, by how much, or what the answer depends on. That is the gap addressed here, and the two published affinity series are what make it checkable.

The binding physics is also in hand. Rhoden and colleagues derived the effective antigen concentration available to a tethered second arm by treating the bound antibody as sampling a hemisphere of radius equal to its own arm length over a surface of evenly distributed receptors.¹⁶ Sengers and colleagues went further, building ODE and Monte Carlo models in which lateral diffusion of the second target is rate-limiting, and validating them against flow-cytometric binding of an anti-CD4/CD70 bispecific series.¹⁷

What is missing sits between those two bodies of work. Rhoden measures receptor occupancy; Sengers measures median fluorescence intensity. Both stop at binding. Neither converts bound density into the quantity that selectivity claims are actually made in — a ratio of concentrations producing a fixed biological effect. That conversion is not a formality. It is where the capacity of a cell, the effect level chosen and the steepness of the response all enter, where three different numbers the field calls selectivity come apart, and — as Section 6 shows — where the timescale of the assay turns out to decide the answer.

This paper does five things. It separates those quantities and shows how large the differences are. It states the binding model they must be derived from, carefully enough to carry an endpoint. It shows that the equilibrium form of that model fails on published data for an identifiable reason. It supplies the missing regime from rate constants the same source already reports, yielding a parameter-free prediction about the shape of a selectivity series and a design criterion written in lifetimes rather than affinities. And it tests that prediction on a second series it had no part in building, in a system where the ladder holds k_on approximately fixed — then states a design variable the framework does not contain at all, which the same source shows to be comparable in size to the one it does.

A note on how the published work used here is treated. Every number quoted from a source is taken from that source's tables, not from its running text, and is re-derived from those tables in the verification script described in Section 11. Three of the sources state something in their text that their own tables do not support, and in each case the table is used and the discrepancy is reported where it arises. This is not a criticism of those papers, whose tables are complete enough to check; it is a statement about which numbers a reader of this one is entitled to rely on.

Scope. No construct has been built, no binding has been measured, no cell assay or animal study performed, and no claim is made about efficacy, cytokine release, or any clinical endpoint.

2.1 States, conservation, and what N counts

Consider an engager with one arm against antigen A, one against antigen B, and one against CD3, each monovalent. Write surface species as densities per µm²: x held by arm A only, y held by arm B only, z bridging both. With total antigen densities a_T and b_T and free fractions f_A and f_B,

free A = a_T·f_A = a_T − x − z free B = b_T·f_B = b_T − y − z
x = a_T·f_A·C/K_A y = b_T·f_B·C/K_B
z = x·C_eff,B·f_B/K_B = y·C_eff,A·f_A/K_A

The first binding event is from solution at free concentration C; the second is intramolecular, so a tethered arm sees its partner at an effective local concentration. This is Rhoden's construction:¹⁶ a hemisphere of radius L over a surface of evenly distributed receptors contains σπL² receptors in a volume (2/3)πL³, so

C_eff = 3σ / (2·L·N_A)

which is 22.2 µM at 89 accessible receptors/µm² with L = 10 nm. Substituting conservation gives two coupled equations, solved by iteration:

f_A = 1/[1 + (C/K_A)(1 + C_eff,B·f_B/K_B)] f_B = 1/[1 + (C/K_B)(1 + C_eff,A·f_A/K_A)]

Throughout, N counts bound engager molecules per µm²: N = x + y + z. Every bound engager presents exactly one CD3 arm whatever route held it, so the quantity counted is molecules — not arms, and not occupied receptors.

N is target-bound engager presenting a CD3 arm; it is not productive T-cell engagement. Lytic and stimulatory synapses have different requirements,⁷ epitope distance from the membrane changes potency independently of affinity,⁹ and contact geometry and antibody flexibility jointly set killing.⁸ Using N as the input to a killing function is therefore a modeling assumption, not a derivation. Section 6.2 turns that assumption into a bounded one by showing which conclusions survive any monotonic choice of the map.

Writing the model in rate constants. Everything above is an equilibrium statement, and Section 6.2 needs the underlying rates. Each arm has k_on and k_off with K = k_off/k_on; a tethered arm rebinds at k_r, the series combination of waiting for a partner to diffuse into reach and reacting with it once there. The quasi-equilibrium forms are recovered when every complex turns over many times within the observation window, which is the condition Section 6.1's failure violates.

2.2 Detailed balance, and why conservation is worth writing out

The two routes to the bridged state must agree, which requires a_T·C_eff,B = b_T·C_eff,A. With C_eff defined per partner density this is 3a_T·b_T = 3b_T·a_T, so it holds identically and the model is symmetric under relabelling the arms by construction.

That symmetry is worth having explicitly, because a natural shortcut loses it. Writing bound density as a sum of Langmuir terms with the avidity correction applied to one arm,

N = σ_A·C/(C + K_A/f) + σ_B·C/(C + K_B), with f = 1 + C_eff/K_B

gives the correct dilute limit — as C → 0 both reduce to C·[a_T/K_A + b_T/K_B + a_T·C_eff,B/(K_A·K_B)], agreeing to within 2% below 10 pM and to one part in 10⁴ below 0.1 pM — but handles finite occupancy inconsistently. At equal densities (89/µm²) and C = 10 nM, with affinities of 10 nM and 1000 nM, it returns 86.2 one way round and 129.7 the other, a 1.50-fold change from renaming the arms where no physical asymmetry exists; the state model returns 77.7 both ways. It also overstates bound density at finite concentration by up to 1.6-fold. An expression that changes by half again when the arms are renamed cannot be used to compare arm assignments, which is what such expressions are for.

2.3 From binding to effect

Killing is taken as a Hill function of bound density, E = N^h/(N^h + N₅₀^h). The density required for effect p is

N_p = N₅₀·(p/(1−p))^(1/h)

N₅₀ is a half-effect density, not a floor. A ceiling on attainable bound density caps the response plateau; it does not abolish response. For a cell whose maximum attainable density is N_max, that plateau is E_max = N_max^h/(N_max^h + N₅₀^h):

Table 1. Attainable plateau under a density ceiling, at N₅₀ = 10/µm² and h = 2.

So a density ceiling makes an effect level unattainable — none of these reaches 50% — while leaving a finite and possibly undetectable response. Section 8, arm D is what would measure it.

Table 2. Required bound density N_p (/µm²) at N₅₀ = 10/µm², by effect level and Hill coefficient.

The dependence on h runs opposite to intuition: a steeper response makes high effect levels reachable at lower density.

For a cell engaged by one arm, N(C) = σC/(K + C), so reaching N_p requires

EC_p = K·N_p/(σ − N_p), defined only for σ > N_p

and between two such cells at the same N_p,

R_p = (K_N/K_T) · (σ_T − N_p)/(σ_N − N_p)

Table 3. The three quantities.

3.1 The capacity factor

R_p equals R_K only at matched receptor density. Away from it the capacity factor survives and the ratio moves with the effect level.

Table 4. The capacity factor at R_K = 150×.

An EC₂₀ ratio and an EC₅₀ ratio are therefore different numbers, which matters because published series in this area report EC₂₀.

The capacity factor is also unbounded. As σ_N approaches N_p from above, R_p diverges: at N_p = 1, σ_T = 2 and σ_N = 1.00001, the density ratio is 2× and R_p is 100,000×. A modest density ratio can produce an enormous fixed-effect ratio, and beyond that point the effect is unattainable rather than merely expensive.

And the single-arm result does not transfer to the design case. The dual-positive binding curve is not a Langmuir, so the dual-positive and single-positive concentration–response curves are not horizontal translations of one another even at matched antigen density: their capacities and shapes both differ.

Table 5. EC_p computed directly from the state model at a_T = b_T = 89/µm², K_A = K_B = 149 nM, N₅₀ = 10/µm², h = 2.

Effect-dependence survives at matched density, varying 1.40-fold here. Every dual-positive EC value below is computed from the state model rather than from the single-arm formula, which applies only where one arm can bind.

3.2 The window, and when the Hill slope cancels

For two normalized concentration–response Hill curves sharing a slope, W = R·81^(−1/h) with 81 = (0.9/0.1)², and the slope cancels exactly from a fixed-effect ratio. Both conditions are load-bearing and neither is automatic.

Applying a Hill function to a nonlinear binding curve does not produce a Hill function in concentration with the same coefficient: the concentration–response slope equals the synapse-number slope only when the threshold sits well below saturation, and it differs between cells of different density. The cancellation is not general even in the single-arm case, because N_p = N₅₀·(p/(1−p))^(1/h) itself depends on h.

Table 6. Fixed-effect ratio at p = 0.2 with equal K, σ_T = 50 and σ_N = 10 per µm². The variation is entirely through N_p.

At the design point of Table 5, computing EC₁₀ on the single-positive cell and EC₉₀ on the dual-positive cell directly from the state model gives W = 7.8× at h = 2, against fixed-effect ratios of 102× to 145× — an order of magnitude apart, from the same two curves. And W depends on h far more sharply than R_p does, because it is taken at two different effect levels on two different curves.

Table 7. The window, and the fixed-effect ratio, against the Hill coefficient.

At h = 1 there is no window at all — the normal tissue reaches 10% killing below the dose the tumor needs for 90% — and at h = 4 it is 35-fold, from the same two curves. At h = 0.75 the 90% level is unattainable on either. The last row is flat because at p = 0.5 the required density is N₅₀ whatever h is, so the exponent cancels exactly; that is the shared-slope condition of this section, met at one effect level and nowhere else. Any window or fixed-effect figure must state the shape assumptions it rests on, and a window must state h. The ratio of killed fractions at a shared dose is a fourth quantity again: it goes as R^h and is a ratio of two small numbers at a dose that kills almost nothing.

One consequence is used repeatedly below, and it is narrower than it first appears. Within a single pair of curves compared at one effect level, whatever monotonic map carries bound density to killing enters the ratio only through the single number N_p: R_p = B_N⁻¹(N_p)/B_T⁻¹(N_p), where B_T and B_N are the two binding curves. That is a reduction, not a cancellation — sharing an argument does not make a ratio independent of it. What makes the central result of Section 6.2 robust is the separate, numerical finding that the predicted slope moves by less than 0.1 of a unit as N_p is swept across a 58-fold range. The Hill coefficient is not needed; it is also not shown to be irrelevant in general.

The model says how to build a given ratio. Pharmacokinetics says how large a ratio the regimen demands. Writing θ_n and θ_t for the occupancy permitted on normal tissue and required on tumor,

K_N/K_T ≥ (C_normal,peak / C_tumor,trough) · [θ_t/(1−θ_t)] · [(1−θ_n)/θ_n]

Table 8. Required apparent-affinity ratio at θ_t = 0.5.

Two cautions. The first factor is a ratio of free concentrations at the two tissues; using a single compartment's temporal peak-to-trough span in its place presumes a relationship between them that this model does not establish, and relative arm affinities are themselves known to change biodistribution.⁶ And the inequality is stated on R_K, an apparent-affinity ratio; converting it to a requirement on an observed EC ratio reintroduces the capacity factor of Section 3.1. It is an occupancy requirement, and becomes a statement about efficacy and injury only once occupancy targets are tied to those endpoints.

Dividing the dual-positive dilute-limit slope by each single-positive slope gives two apparent-affinity ratios that act in opposition:

vs an A-only cell: R_A = 1 + (K_A + C_eff)/K_B — falls as K_B rises
vs a B-only cell: R_B = 1 + (K_B + C_eff)/K_A — rises as K_B rises

The governing value is the smaller, and setting R_A = R_B gives (K_A − K_B)(K_A + K_B + C_eff) = 0, so the two cross exactly at K_B = K_A.

Table 9. The two dilute-limit ratios at K_A = 149 nM.

Below the crossing the governing ratio is flat: the gating arm stops contributing and the anchoring arm sets the ratio alone, rising without bound as it tightens.

Three things keep this from being a design rule. These are dilute-limit apparent-affinity ratios, and Section 3.1 shows fixed-effect ratios computed from the full curve differ from them and depend on the effect level. Nothing in these equations stops K from falling, so the unbounded rise is a statement about the model rather than about the molecule. And Section 6.1 shows the same model does not reproduce the one published series in which an arm is varied. Section 6.2 supplies the rule these equations cannot, and it is written in a different variable.

6.1 The equilibrium account fails, structurally

Mazor et al. built an anti-CD4/CD70 monovalent bispecific IgG and weakened the anti-CD4 arm 78-fold by CDR mutagenesis.³ It is the dataset this framework needs: three cell populations, both receptor densities by calibrated ABC, both arm affinities measured in the bispecific format, association and dissociation rate constants reported separately, and selectivity reported as a fixed-effect concentration ratio.

Table 10. Measured selectivity, from that source's Tables 2 and 3.

Fitting the equilibrium model with a single free parameter — N₂₀, the bound-engager density giving 20% kill — by least squares on log EC₂₀ across all six constructs and both cell types gives N₂₀ = 76.77/µm², and predicts selectivity of 1,397× to 1,409×: flat where the measurement rises 98-fold.

This is not a fitting artifact. Sweeping N₂₀ across the entire admissible range — bounded above by the single-positive cell's capacity of 120.96/µm², beyond which EC is undefined — the prediction is flat everywhere.

Table 11. The structural-failure sweep.

No value of the free parameter produces the measured trend. The failure is structural.

And the cause is one term. The bridged species is z = x·C_eff,B·f_B/K_B, so the gating arm's affinity multiplies the dual-positive cell's bound density exactly as it multiplies the single-positive cell's. At the fitted parameter, across the 78-fold affinity range:

Table 12. The two populations move together under the equilibrium model.

The two move together, so their ratio cannot change. The measurements say otherwise in both domains. From parental to the weakest variant, that source's on-target EC₂₀ moves 1.3-fold (1.5-fold across the full set of six) while non-target EC₂₀ moves 124-fold; and Sengers's flow-cytometric binding curves show dual-positive binding virtually unaffected by attenuating the CD4 arm while single-positive binding is progressively eliminated.¹⁷

This is a property of the equilibrium assumption, not of the confinement term, the conservation bookkeeping or the response function, all of which survive the next section unchanged.

6.2 Two kinetic regimes in one well

The same paper contains what is needed to fix it. Its Table 2 reports k_on and k_off separately for every variant, and the Methods state a 6 h ADCC assay. Those two facts decide whether a quasi-equilibrium treatment is admissible — and for one of the two cell populations it is not.

How long does each complex last? A singly bound complex escapes as soon as its one arm releases: lifetime 1/k_off,B. A bridged complex escapes only if one arm releases and the second releases before the first can rebind, which for rebinding rate k_r gives an escape rate of order k_off,A·k_off,B/k_r. Here a bound arm has σπL² = 0.046 partners within reach on average — fewer than one — so rebinding is the series combination of waiting for a receptor to diffuse in (k_D = 29.9 s⁻¹) and reacting once it has (k_on·C_eff = 10.9 s⁻¹ at a representative k_on = 3 × 10⁵ M⁻¹s⁻¹), giving k_r = 8.0 s⁻¹. C_eff is 36.5 µM at this cell's CD4 density of 146.4/µm², not the 22.2 µM of the Section 2 design point.

Table 13. Complex lifetimes from that source's Table 2, against its 6 h assay.

The two populations are in different kinetic regimes in the same well. The single-positive cell turns its complexes over 6 times at the parental arm and 780 times at the weakest, so for five of the six variants it is comfortably equilibrated; the parental is the marginal case, and it is the one point where the equilibrium Langmuir below is doing the most work. The dual-positive cell is nowhere near equilibrium: 41% of bridged complexes survive the whole assay even at the weakest arm, and 99% at the parental.

What that implies, before any arithmetic. The single-positive cell's EC₂₀ is an equilibrium quantity, EC₂₀ = N₂₀·K_B/(σ − N₂₀), proportional to K_B. The dual-positive cell's is a delivery quantity — how much engager arrives and is trapped in time t — set by the on-rates and the clock and containing no K_B at all. Dividing them, selectivity is linear in K_B: on log axes the slope is exactly 1, where the equilibrium account of Section 6.1 predicts 0. Two models, a full unit of slope apart, on a published series.

Table 14. Measured selectivity across all six variants, as the ratio of the two columns of that source's Table 3.

The measured log–log slope is 1.068 with r² = 0.887, against a kinetic prediction of 1.008 and an equilibrium prediction of 0.

The quantitative version, with nothing fitted to the trend. The threshold density comes from the equilibrated cell alone, by inverting the Langmuir at each of its six measured points: N₂₀ = 43.2, 20.2, 23.0, 40.3, 57.5, 56.8/µm², mean 40.16/µm². That 2.8-fold spread across a 78-fold affinity range is the first limitation of the shared-threshold assumption, and it is carried forward rather than averaged away.

Carrying that mean — and nothing else from that cell — to the dual-positive population requires a kinetic model of it. We use one that conserves free receptors explicitly, in the three species of Section 2:

free A = a_T − x − z free B = b_T − y − z
dx/dt = k_on,A·C·(a_T−x−z) − k_off,A·x − [k_on,B·C_eff,B·φ_B]·f_B·x + [k_off,B·φ_B]·z
dy/dt = k_on,B·C·(b_T−y−z) − k_off,B·y − [k_on,A·C_eff,A·φ_A]·f_A·y + [k_off,A·φ_A]·z
dz/dt = [k_on,B·C_eff,B·φ_B]·f_B·x + [k_on,A·C_eff,A·φ_A]·f_A·y − [k_off,B·φ_B + k_off,A·φ_A]·z

Two features are load-bearing. Rebinding is scaled by the free-receptor fraction, not by total density, so a filling surface slows its own bridging. And the diffusion factor φ multiplies the forward and reverse rates of the intramolecular step equally, in the Collins–Kimball form, so it cancels from every equilibrium ratio: diffusion changes how fast the system arrives, not where. Applying it to the forward rate alone would shift the equilibrium, which is not what a transport limitation does.

The model is consistent with Section 2 by construction, and verifiably so. Integrated to long times at fixed concentration it reproduces the equilibrium state model species by species, to integration tolerance, at every variant and across four orders of magnitude in concentration. A reduced one-variable equation in N alone, which is simpler and which we also ran, has a high-concentration limit of (k_A·a_T + k_B·b_T)/(k_A + k_B) = 154.4/µm² — above the smaller receptor capacity of 146.4/µm². It agrees with the conserved model at the densities used here and should not be carried to saturation.

Table 15. The conserved kinetic model against the measurement.

Table 16. Slopes, series I.

Six points give the measured slope a wide interval. It excludes zero and contains one. That is the comparison this dataset supports: the data are compatible with the kinetic prediction and inconsistent with a flat trend, and they do not identify the mechanism uniquely. Repeating both regressions against K = k_off/k_on rather than the source's rounded K_D column gives 1.016 and 1.078 and changes nothing.

Absolute levels are over-predicted by up to 4.6-fold, and the residual is structured rather than constant. Predicted over measured runs 1.92, 4.62, 4.67, 3.56, 1.34, 1.45 across the series in order of K_B: a 3.5-fold spread that is smallest at both ends and largest in the middle. A single level constant would therefore not absorb it. The trend is right, the individual values are not, and the shape of the discrepancy says the model's curvature is also slightly wrong — most plausibly in the dual-positive EC₂₀, which the model holds nearly flat at 0.025–0.036 nM while the criterion's limiting form holds it exactly flat.

One feature of this ladder is not what its source's text says it is. That text reports no significant change in association rates across the variants. Its own Table 2, in the DuetMab column used here, gives k_on of 2.8, 2.0, 2.8, 5.4, 5.6 and 5.2 × 10⁵ M⁻¹s⁻¹ in order of weakening affinity — a 2.8-fold span, not monotonic, and worth 0.24 of a log–log slope unit against the full unit separating the two accounts. The conserved kinetic model uses the per-variant rates, so its prediction of 1.009 already carries this. What it means is that the measured slope is a dependence on the arm's chemistry as a whole and not on K_B alone, which Section 10 records.

An independent series, in a system built on purpose. The analysis above was constructed on one dataset. A second, published separately and designed around a different architecture, supplies the same test. In a synapse-gated ROR1 × EGFR × CD3 trispecific, the anchoring arm is chosen because it cannot form a productive synapse on its own, and the gating arm is an anti-EGFR affinity ladder.²⁰

Table 17. Series II, from that source's Tables S1 and S6.

Both columns are from that source's supplementary Table S6, on the ROR1/EGFR rows; the main text reports only the ratios, and rounds the first of them to 3. The K_B values are from its Table S1, from the ROR1/EGFR rows rather than from the R347/EGFR control rows, whose v2 entry reads 8.0 rather than 8.1. The columns confirm the potency statement as a measurement rather than a rounding: the dual-positive cell moves 10-fold and the single-positive cell 1,436-fold.

Nothing about this series was available to the model. The antigen pair is different, the format is trispecific rather than bispecific, the effector is primary human T cells rather than NK cells, the comparator is an isogenic knockout, and the exposure is 72 h rather than 6 h — twelve times longer. The two series are independent as datasets and as systems; they are not independent as laboratories, since they share a senior author and institution, and Section 10 records that.

The measured log–log slope is 1.210 with r² = 0.829, where equilibrium requires 0. Three bounds on that figure matter more than the figure. The printed precision of Table S6 alone — one or two significant figures on most entries — puts the slope anywhere from 1.08 to 1.35 before any statistics. The same table's second block, which compares the gated molecule against its single-antigen control on the parental line rather than comparing the gated molecule across two lines, gives 1.274 on the same abscissa. And three points put a 95% interval of −5.8 to +8.2 on any one of these, so this series alone excludes nothing; what it supplies is a second system, and the pooled fit below is where the discrimination lives.

It is also a ladder of the right kind, and the kind that exposes the criterion's weakest point. That source's Table S1 reports k_on and k_off for all three arms: along the EGFR ladder k_off rises 42-fold while k_on moves 1.79-fold and not monotonically, so the affinity change is carried mainly by the dissociation rate and the criterion's delivery term 1/(k_on·t) is nearly, though not exactly, held fixed by the experiment. The 1.79-fold in k_on is 0.15 of a slope unit and is not negligible against the differences being tested. The anchoring arm is constant across the ladder (K_D 8.7–9.9 nM) and so is the CD3 arm (47.3–48.0 nM), so the ladder does vary one arm and only one arm.

It does not follow that the dual-positive branch is held fixed, and this is where the limiting form shows its edge. The bridged complex escapes at a rate of order k_off,A·k_off,B/k_r, so a ladder that raises k_off,B by 42-fold raises the escape rate by about as much and un-traps the bridged state as it goes. Slope 1 assumes that state trapped; an off-rate ladder is precisely the operation that erodes the assumption along its own abscissa. The data show that erosion directly.

Table 18. The two branches of series II, as endpoint-to-endpoint slopes across the ladder, which decompose exactly.

Dual-positive potency falls about 10-fold across the 46-fold affinity span, where equilibrium requires 46-fold and the limiting criterion requires none. Both branches deviate, the equilibrated one by more, and only their difference lands near the prediction. That difference, +1.30, is the endpoint slope; the three-point regression slope of 1.210 quoted above is a different statistic and is not the difference of the two branch slopes.

Fitting the two series together with one shared slope and a separate intercept each — they cannot share an intercept, having nothing else in common:

Table 19. The pooled fit.

Nine points from two systems exclude a flat trend and contain slope 1. The interval is still wide; what has changed is that it is no longer one dataset. Propagating the printed precision of Table S6 through the pooled fit moves the common slope between 1.07 and 1.17.

The fitted intercepts are not independent evidence and are not offered as any. Under a forced common slope m they satisfy b_I − b_II = 0.597 − 0.489·m identically, so their difference is a deterministic function of the slope and vanishes at m = 1.220. The fitted 1.119 sits near that crossing, which is why the two prefactors come out close; fitted with separate slopes they differ 1.52-fold, and evaluated at the predicted m = 1 they differ 1.28-fold. The levels of the two systems are consistent with one another and nothing more is claimed from them.

A third dataset fails the shared-threshold assumption, and this one fails it in one place. The single-positive cell of the second series carries the gating antigen alone, so the engager binds it monovalently and the threshold inversion of this section applies to it exactly as it does to the first series — and that cell is genuinely equilibrated, its EGFR-arm complexes turning over 36 to 1,530 times inside the 72 h assay. Inverting the Langmuir at each point gives the fractional occupancy the endpoint requires.

Table 20. The third threshold failure.

A fixed threshold calibrated on the first two predicts the third at 0.58 nM. The measurement is 20.1 nM — a 35-fold failure, and it holds between 33- and 37-fold across every digit of rounding in both the EC₅₀ and K_D columns.

Four things do not account for it, each checked rather than dismissed. Ligand depletion in the well is under 2% and moves the ratio the wrong way. The competitive occupancy of the effector arm by free engager — the prozone term — costs a factor of 1.4 where 35 is needed. A falling on-rate cannot enter an equilibrium endpoint at all. And no single threshold fits all three points under Section 3.1's capacity factor: calibrated on v3 instead, the first two are over-predicted thirtyfold each. The value itself is not scatter; the same molecule and cell appear six times in that source's Table S8 at 16.4 to 21.1 nM.

What it is and is not. It is a third independent failure of a fixed bound-density account, alongside the structural failure of Section 6.1 and the antigen-count panel of Section 6.4, and it is the cleanest of the three because it needs no regression and no comparison between cells. It is not evidence for a lifetime threshold. Along this ladder k_off and K_B are collinear to r = 0.997, so a persistence clock and a plain affinity dependence are the same regression and cannot be separated; and the agreement of the first two points carries no information beyond the slope of that leg, being algebraically the same statement.

One alternative was proposed and is now constrained. The inversion assumes all three EC₅₀ values mark the same absolute effect. If the reported value is instead half of each curve's own plateau, and the weakest variant's plateau on this cell is depressed, the third point sits at a different effect level by construction and the gap needs no further explanation. Nothing in the source states which convention was used, and the third variant's EC₅₀ sits at 0.42 times the effector arm's own equilibrium constant, which is where a hook would be looked for.

The model contains exactly one mechanism that can cap a curve's top without capping binding: competitive occupancy of the effector arm by free engager, which makes bridges rare once both surfaces are independently saturated. Computing it on the published constants shows it is nowhere near engaging here. A tethered effector arm within reach of its partner sees an effective local concentration of about 10 mM against a 47.5 nM constant, a ratio of 2 × 10⁵, so capture is certain unless free engager has taken all but one part in 10⁵ of the available sites. At the measured EC₅₀ it has taken 30%. The three curves' tops agree to four decimal places, and they continue to agree across effector densities from 20 to 600/µm², reaches from 8 to 40 nm and effector constants from 20 to 200 nM. Read as half of its own top rather than half of an absolute, the model places the weakest variant at 0.64 nM against 20.1 measured — the same failure, undiminished.

So the convention cannot absorb the gap, on this model. That is a narrowing rather than a settlement: the model is ours, and a published curve that visibly tops out lower than its siblings would overturn it. A mechanism that caps a top without capping binding and is not this one — effector exhaustion within a 72 h assay is the obvious candidate — is outside the model entirely.

What the answer depends on. Three inputs are not taken from the source: lateral diffusion, well volume, and a 10 µm cell diameter, since that source reports counts per cell rather than densities. Depletion is 2.4–3.3% of free concentration. Each was swept with every quantity that depends on it moved at the same time — accessible fraction scales the densities and therefore also C_eff, the diffusive capture rate, φ and the escape rate; cell diameter scales those and the inferred threshold as well.

Table 21. Sensitivity of the predicted slope, with couplings moved together.

The widest excursion is 0.19 of a unit, against the full unit that separates the two accounts. Lateral diffusion now barely matters at all, which is a consequence of the detailed-balance-respecting form: a transport limitation that cancels from the equilibrium can only move the timescale. None of these inputs is an explicit function of K_B, though that alone would not guarantee a slope is unaffected; the sweep is what shows it.

On the binding-to-killing step. Section 3.2 shows that a fixed-effect ratio depends on the map only through the single number N_p. It does not follow that the map's shape cancels: R_p = B_N⁻¹(N_p)/B_T⁻¹(N_p), and sharing an argument does not make a ratio independent of it. What the last row above shows is insensitivity — across a 58-fold range of N_p the slope stays between 0.93 and 1.02 — which is sensitivity evidence and not a cancellation theorem. And N₂₀ itself is inferred from cytotoxicity under binding assumptions; it is not a measured bound density.

The criterion, stated on the right timescale. A complex's survival time and a population's relaxation time are different quantities, and it is the second that decides whether a timed solution differs from an equilibrium one. For this model the relaxation time at the predicted endpoint is

τ_relax = 1 / [C·(k_on,A + k_on,B) + k_esc]

Table 22. Relaxation times against the 6 h exposure.

The dual-positive population is not in a clean irreversible-trapping limit. It is trapped to intermediate — relaxation takes 5 to 19 hours against a 6 hour assay, so the system is partway to equilibrium when the plate is read. That is enough to separate it from the single-positive population, which relaxes in seconds to minutes and is fully equilibrated. It is not enough to call the bridged state irreversible, and the slope-1 result is a limiting prediction that the finite escape rate perturbs rather than a general consequence of the published rates.

So the criterion is better written on relaxation times and without a double inequality: the two populations must sit on opposite sides of the exposure time. Below that condition both are trapped and the ratio collapses toward a ratio of on-rates; above it both equilibrate and it collapses to the flat equilibrium value of Section 6.1. The engineering target is a separation of relaxation times across the exposure window, and k_off is the handle on it rather than K_D. The mechanism is the bookkeeping of kinetic proofreading¹⁵ with the assay clock in the role of the delay.

This is also why attenuating the CD3 arm and attenuating a tumor arm are not the same operation, a distinction that review of CD3 affinity attenuation has found difficult to resolve on affinity grounds alone.⁵

Three predictions, and what each would and would not show.

1. Shorten the assay. Selectivity should fall as t approaches the dual-positive population's relaxation time. The equilibrium account contains no t. But time-dependent killing does not by itself establish time-dependent binding — killing can accumulate over time under equilibrium binding — so binding must be measured alongside killing at each timepoint, or the test does not discriminate.
2. Apply the same ladder to the anchoring arm. Measured against the gating-antigen-only cell, that cell never sees the anchoring arm, so its EC₂₀ cannot move; only the dual-positive cell can. The kinetic account predicts a 72-fold rise from the gating ladder and no trend from the anchoring one. An equilibrium account also predicts different consequences for the two ladders against that fixed comparator — the arms are not interchangeable there either — so what this test compares is the size and sign of two specific predictions, not presence against absence.
3. The third population, below, which needs no new experiment.

Each of the first two is one plate of an assay already run, with its timer changed or its arms swapped. Neither has been run: the series above varied affinity at fixed exposure, and an earlier series from the same group varied format and valence,¹⁰ so the clock has never been the variable in either. The two published exposures that do differ — 6 h and 72 h — give slopes of 1.07 and 1.21, and the criterion predicts the slope to fall toward zero as exposure grows past the dual-positive population's relaxation time. That is a tension and it is reported as one. It is not a test: the two systems differ in antigen pair, format, effector type, arm ladder and comparator, and relaxation time is a property of the arms rather than of the clock, so nothing here holds fixed what prediction 1 requires to be fixed. One plate with four timers does.

A third population, and a prediction already collected. That source's mixture contains three cell types and the paper tabulates EC₂₀ for two. The third is CD4⁻CD70⁺ B cells at 98.7/µm² of the anchoring antigen and under 100 copies of the gating antigen. That cell is held by the anchoring arm alone, and the anchoring arm is the same molecule in every variant, so the absence of any effect of the mutations on that population is forced, with no freedom in it. That is the strong part of the prediction. The level — EC₂₀ = N₂₀·K_A/(σ_CD70,B − N₂₀) = 17.2 nM — is weaker, because it transfers a threshold inferred on T lymphocytes to a B-cell line. The complex lifetime there is 204 s against a 21,600 s assay, so the equilibrium form applies without qualification.

One cell sees the mutated arm and one does not, so one curve must move and the other must not. Both accounts in this paper require it: under the equilibrium model the third population's EC₂₀ is proportional to the anchoring arm's constant, which no variant changes. So this is not a test between the two accounts. It is a test of the structural assumption they share — that each cell is held by the arms its own antigens present — and a failure would take both down.

That test has been run. The population is plotted in all six panels of the source's Figure 4A and tabulated in none of them. Digitizing the three series by color and expressing each trace in frame-fractional coordinates — the six panels are printed on identical axes, so position within the frame is comparable across them with no absolute calibration — gives, across the right half of each frame where the curves rise:

Table 23. Arm zero: the two series measured the same way in the same frames.

A factor of 9.9 between them. The curve that cannot move does not move, and the curve that must move does — by ten times as much, in the same measurement.

The single-positive series is the internal control for the method, and it passes twice over: it separates, and it separates in the published order of that source's Table 3 — 0.236, 0.102, 0.090, 0.044, 0.030, 0.035 at the same point in each frame, which is the published ranking recovered without reference to it.

What this does not deliver is the level. Section 6.2 predicts EC₂₀ = 17.2 nM there, and that number cannot be read reliably off this figure: two calibration routes disagree by 1.8-fold, and the one anchored on the printed concentrations fails its own consistency check, inferring a top dose of 174 to 310 nM where the source plots about 70. The shape prediction is what this test returns, and the level stays open. A lineage-dependent threshold would appear as a constant offset on exactly that number, so it remains the informative measurement when the underlying data can be obtained.

Scope. This series is ADCC with NK effectors, not a CD3 engager; the targeting-arm physics is shared, the effector step is not. The dataset separately establishes that reformatting the same weakened arms as bivalent IgGs lost the selectivity, because a bivalent arm acquires its own avidity term — and its own long-lived bridged state — on a single-positive cell.

6.3 Single-antigen avidity, and its realized exponent

A single-antigen engager's engagement ratio is the antigen density ratio for a monovalent tumor-antigen arm. A bivalent arm has apparent constant K/(1 + C_eff(σ)/K) with C_eff ∝ σ, so capture is superlinear in density — but not the square.

Table 24. The realized avidity exponent.

The exponent is not a constant of the architecture: saturation pulls it from 1.74 at a twofold ratio to 1.45 at thirtyfold, so quoting σ² overstates such a construct by 1.2-fold, 2.7-fold and 6.4-fold across those three rows. And because avidity reads absolute density through C_eff, a crossover between architectures quoted as a density ratio is ill-formed — the same ratio gives different answers depending where it sits, so no single crossover density ratio exists.

An exponent only helps when the base exceeds one. If the dangerous normal tissue carries the antigen near tumor density, the ratio is near 1 and any exponent on 1 is 1.

A two-antigen construct is not thereby density-independent: its ratio is flat in the density ratio, but its magnitude runs on the tumor's absolute accessible density through C_eff — 150× at 89 receptors/µm², 51× at 30 and 18× at 10, an eightfold span from one molecule.

6.4 Similar antigen counts, 43,000-fold different killing

De Tavernier et al. report a biparatopic bivalent HER2 × TCRαβ engager with HER2 quantified by calibrated flow on four breast lines and primary human cardiomyocytes.⁴ Across 143-fold in total HER2 antibody binding sites per cell, IC₅₀ varies 4.4-fold and does not order by that count.

Table 25. Three rows of that source's Table 1.

A 1.17-fold ratio of antigen counts per cell, and more than 43,000-fold in killing — a lower bound, since the cardiomyocyte IC₅₀ was never reached. It is an in-vitro cytotoxicity separation, not a therapeutic window; the same paper's cytokine-related in-vitro indices run 7 to 174 for IL6 and IL10.

The quantity this framework needs is local density, and neither counts per cell nor whole-cell average density is that quantity. C_eff is built on a hemisphere of radius L = 10 nm, whose membrane footprint is πL² = 3.1 × 10⁻⁴ µm² — one part in 2.2 million of a 700 µm² cell. At a cell-average 10 receptors/µm² that footprint contains 0.003 of a partner, so the second arm is almost never in contact and C_eff describes waiting rather than concentration. Averaging antigen over a whole cell to compute it is valid only if the antigen is uniform at the 10 nm scale.

It is not, and the source imaged that directly: Airyscan super-resolution confocal of BT-549 against the cardiomyocytes found comparable HER2 at the cell surface but distinct HER2-enriched areas on BT-549, quantified as a frequency distribution of surface objects sorted by area.⁴ That imaging cannot reach the scale this account requires, and the gap is large. Its plotted objects run to 17.3 µm² on BT-549 against 5.2 µm² on the cardiomyocyte, a 3.3-fold difference in the largest object at nearly matched object counts. Airyscan resolves about 120–140 nm laterally, so its smallest admissible object is 0.011–0.015 µm² — 36- to 49-fold larger than the 3.1 × 10⁻⁴ µm² footprint C_eff integrates over. The largest object it reports is 55,000-fold larger than that footprint and 0.86 times the whole 20 µm² contact assumed in Section 6.5. So these are domains the size of a synapse, not clusters within one arm's reach, and the measurement is consistent with any arrangement at all below its resolution floor. The numbers are in that figure; the length scale is not.

What local organization can buy, holding everything else fixed. Section 6.2's criterion applies here too, with the monovalent complex in the role of the single-arm state and the bivalent complex in the role of the bridged one, and the assay running 4 to 5 days. Solving the same timed model for a cell of fixed total antigen, fixed capacity and fixed valence, varying only how that antigen is distributed across local densities from 10 to 3,000 per µm²:

Table 26. Local organization against a different intervention.

Locality alone buys up to 537-fold, against about 140-fold for a density-driven avidity ratio at the most extreme pairing this dataset admits. Fourfold more, and for a reason: at one end of the range EC is an equilibrium quantity in an apparent affinity and at the other it is a delivery quantity in 1/(k_on·t), so the two ends are different functions of concentration rather than two values of one. The second column is a different intervention — removing the second arm's engagement is not something clustering does — and brackets the regime rather than measuring it.

It is still far short. Against >43,478-fold observed, locality spans 537-fold and the bracket 4,868-fold. The enrichment needed to reach the regime boundary runs from none at all to about 500-fold depending on the arm's k_off, which at the top of that range is 44% of hexagonal close packing for receptors 10 nm across — a diameter, not the 10 nm reach radius used for the footprint above. That cap moves as 1/d² and HER2's ectodomain footprint is not measured here, so geometry does not exclude the mechanism, which is all a packing ceiling can establish. The arms here are modeled as identical, and the source reports that the two paratopes differ in k_off.⁴

And a whole class of explanation can be ruled out. Any account in which potency is set by binding to the target cell against a bound-density threshold that does not itself scale with antigen makes potency a steep function of antigen density. The measurement is flat and unordered.

Table 27. What a target-binding account predicts across that panel.

Both figures treat the four lines as sharing a 700 µm² surface, so they convert a count range into a density range; per-line areas would compress them, though not by two to four orders of magnitude.

Capture limitation is the one binding account that could flatten a panel, since a cell studded with absorbing patches saturates once n·s exceeds πR.¹⁸ It does not apply here. A perfectly absorbing 1 nm receptor would react at 9.6 × 10⁷ M⁻¹s⁻¹; a real antibody arm reacts at about 5 × 10⁵, 193-fold slower, so it behaves as a perfect absorber of 5.2 pm radius and saturation would need 3 × 10⁶ receptors — above every line in the panel. Below saturation, capture is linear in receptor number and predicts a 143-fold spread. And a capture rate cannot set a plateau the system reached long ago: the monovalent complex turns over every 1,000 s against a 108 h assay.

The conclusion is negative, and it is the useful kind. Under the assumptions set out below, the target-binding models tested do not reproduce the weak relationship between receptor count and potency across this panel. Several alternative explanations remain; two binding-side ones are worth naming because they are the easiest to test. The first is a threshold that scales with antigen — if the trigger is a fixed fraction of surface HER2 bridged rather than a fixed density, EC is flat in σ identically; Section 2.1 states that the map from bound density to killing is assumed rather than derived, and this is precisely where that assumption bites, because the comparison is across cells rather than between two curves on one pair. The second is binding on the effector side: the TCRαβ arm is the same molecule against the same T cells for every line, so a potency set there is flat and unordered by construction.

A third possibility is not binding at all: effector throughput limiting target clearance at sub-saturating engager concentration. A simple cap on total kills would flatten the plateau, not the IC₅₀, and at E:T 5:1 over 108 h a serially killing T-cell pool is in large excess — so the throughput has to bind at concentrations below saturation for this to move an IC₅₀, which is the specific thing an E:T series would test.

It also changes what can be claimed about the cardiomyocyte. If target-cell binding against a fixed threshold is not rate-determining for four tumor lines spanning 143-fold in antigen, a binding-side explanation for a fifth cell needs an argument this panel does not supply.

What the dataset establishes. Total antigen counts do not identify the mechanism, and neither does whole-cell binding of any kind. The governing variable on the binding side is local density within one arm's reach, and no published imaging of this system resolves that scale; and the panel's flatness says the binding side may not be governing at all. Four measurements would constrain it considerably: the arms' k_off, the density inside the enriched areas measured at 10–50 nm rather than at the diffraction limit, the assay's detection threshold, and one E:T series.

An independent system separates geometry from affinity directly. The negative result above is inferred from a panel in which affinity was constant and antigen count varied. The complementary experiment — antigen fixed, affinity comparable, geometry varied — has been run. The same study that supplies Section 6.2's second series built a HER2-anchored version of its platform and compared two anti-HER2 anchoring clones of comparable cellular binding against a panel of dual-positive lines.²⁰ The informative comparison is not between the two gated molecules but between the two anchoring arms on their own, formatted as single-antigen engagers against an inert partner — same format, same cells, same assay.

Table 28. The anchoring arms alone. The first row is from that source's Table S7; the second is plotted in its Fig. S10b and not tabulated there, and its text reports that molecule below 25% killing on all four lines.

An anchoring arm is an engager or it is not, and epitope height is what decides. A membrane-proximal anchoring arm kills single-antigen cells on its own at 30 to 140 pM. The same target, comparable cellular binding, a membrane-distal epitope: no measurable killing at all.

The consequence for a gate is a ceiling. With the trastuzumab anchor, the gated molecule is only 10- to 15-fold more potent than its own anchoring arm alone, so the gate buys at most that. With the clone-4380 anchor the ratio is censored — the anchoring arm never reaches an endpoint, so no ceiling is measured.

A caution on how the source reports selectivity. Its fold-change parameter is computed against two different comparator molecules depending on which arm is under discussion — the trastuzumab construct against the anchoring-only control, the clone-4380 construct against the active-only control — and its text says so. Those two figures cannot be divided by one another. Reconstructed from its Table S7 on a matched comparator the two gated molecules are indistinguishable, at 0.67 to 1.00, which is why the comparison above is drawn between the anchoring arms alone.

No binding-affinity model here contains the variable the authors moved. Neither the equilibrium model of Section 2 nor the kinetic model of Section 6.2 carries a coordinate normal to the membrane, so neither can distinguish an anchoring arm that triggers from one that does not at the same affinity.

What a controlled distance series shows. Leithner et al. built four IgG1 engagers carrying the same anti-HER2 and anti-CD3ε paratopes against the same membrane-proximal epitope, and measured the span of the assembled CD3–engager–HER2 complex by small-angle X-ray scattering: 11.4, 12.6, 18.0 and 18.9 nm.⁸ They measured cell–cell adhesion and engager–cell binding directly and found no format-specific difference in either. Against the 12.8 nm CD2–CD58 ruler, the two short formats recruit co-stimulation and exclude a 14–16.5 nm probe; the two long ones do not.

Table 29. What moves across that distance series. No numeric EC₅₀ is published in that source, so these are ordinal statements.

Distance at fixed affinity and fixed adhesion moves the plateau and the signaling, and barely moves the EC₅₀. That is a different place for a geometric variable to act than any model in this paper allows, and it is consistent with Section 2.3: a ceiling on attainable bound density caps the response plateau without abolishing response.

Three cautions, and they matter. The source reports HER2 affinities spanning 80 to 250 pM without per-format values, so affinity is comparable rather than fixed. It identifies flexibility of the assembly as a third determinant of comparable magnitude to distance, and the ordering within each distance pair is set by flexibility rather than by distance. And no numeric EC₅₀ is published, so "barely moves" is an ordinal statement. The series varies two things, not one.

The synapse-gated study supplies the count-side result from the other direction. Its optimized construct, measured against its own matched single-antigen control, holds selectivity at 733× to 950× across four lines chosen for different densities of both antigens, and the authors state that the density ratio has limited impact. Those four figures are computed here from its Table S7; its text rounds them to 800–1,000. Section 6.3 derives the problem for a binding account directly: a gate's magnitude runs on absolute accessible density through C_eff, an eightfold span from one molecule between 10 and 89 receptors per µm². This panel, the panel above and the affinity series are three independent arrivals at the same place.

Inputs for this section, none of which are in the source: a 700 µm² surface for every tumor line and 2,000 µm² for the cardiomyocyte; k_on = 5 × 10⁵ M⁻¹s⁻¹ with k_off swept over two orders of magnitude, since the source reports affinities without tabulating rates; a threshold of 1 bound engager molecule per µm², well below the densities involved; and two identical arms, where the source reports that the two paratopes differ in k_off. Results are therefore reported as boundaries in local density and as spans, not as predicted potencies. One caution on the capture-limitation rejection: the saturation threshold scales as 1/k_on, so a nanobody arm faster than about 1.5 × 10⁶ M⁻¹s⁻¹ would bring the largest line to saturation. The timescale argument is independent of k_on and carries that conclusion on its own.

Section 6.2's criterion does not transfer wholesale. That analysis rests on the two cell populations converting bound engager into killing identically, which held for two lymphocyte populations from one donor mixture and plainly does not hold between a breast carcinoma line and a primary cardiomyocyte. The arms' rate constants are not tabulated in the main text. And the assay runs 4 to 5 days against 6 hours for series I, 16 to 20 times longer, which on the criterion moves both cell types toward the equilibrated regime. What survives the transfer is the variable: local density within an arm's reach, not counts per cell.

6.5 The object every model here is missing, and the one thing it settles

Sections 2 through 6.4 stop at the target cell's surface. They compute a bound-engager density and then assume a monotonic map from that density to killing — Section 2.1 says so, Section 10 lists it, and Section 8 arm C exists to measure the exponent it needs. But the engager has a third arm, and that arm binds a second cell. The thing that gets killed is not a loaded target cell. It is a conjugate: two cells held together by n engager molecules, which has to survive long enough for a lytic hit.

That object has a lifetime, exactly as the surface complexes of Section 6.2 do. An engager inside the contact bridges when at least one target arm and the effector arm are both engaged; the cells separate when all n are simultaneously unbridged, which is a first passage to zero on a birth–death chain and is solved exactly here. On the published rate constants, with an assumed 14 nm intermembrane reach and a 20 µm² contact:

Table 30. Conjugate lifetime against link number.

Adhesion saturates at four links. Lifetime is exponential in link number, so there is no adhesive benefit from the fifth engager onward: the two cells already stay together longer than the experiment runs.

That settles what the threshold of Section 6.2 is not. Four links in a 20 µm² contact is 0.20 bound engagers per µm². Section 6.2 infers 40.16 per µm², 201-fold higher. The inferred threshold is therefore not an adhesion requirement. Holding two cells together costs a handful of molecules. Whatever 40 per µm² measures, it is downstream of conjugation — a receptor-triggering requirement wearing a bound-density label.

This is a constraint on Section 6.4's three surviving accounts, and it points the same way: away from target-cell binding and toward the effector side. The escape from it is stated rather than buried. The saturating link number is fixed by the chain; the density it corresponds to is that number divided by the contact area, and the contact area is not measured. A contact of 0.10 µm² — a disc 356 nm across — reconciles the two exactly. That is the scale of a microvillus tip or an initial receptor microcluster, not of a mature synapse, and this work cannot choose between them. A second bound makes the high-density reading awkward in any case: at 40 per µm² a 20 µm² contact would hold 803 engagers against 1,200 effector receptors, 67% occupancy, at which the links are not independent and this model overstates conjugate stability. It may not be used there.

And it closes a class of explanation for Section 6.4's geometric result. Epitope height sets the intermembrane distance L, which sets the effective concentration across the contact, C_eff = 3σ/(2·L·N_A). So the conjugate is the first object here that contains the degree of freedom Section 6.4 reports. How much can it buy? The lifetime goes as p_u^(−n)/(n·k_b), so the required link number goes as ln(t·k_b·n)/ln(1/p_u). In the weak-binding limit ln(1/p_u) → p_b → k_on·C_eff/k_off ∝ 1/L, so the required number — and therefore the required density — is linear in L. In the strong-binding limit it is pinned at its floor and does not move at all. Hence

N(L₂)/N(L₁) ≤ L₂/L₁, with equality only in the weak-binding limit.

Table 31. The bound, swept over eight orders of magnitude in effector affinity. The 10 nm to 40 nm gap is chosen to show the bound saturating; the span a real engager series occupies is narrower, 11.4 to 18.9 nm,⁸ where the bound is about 1.5-fold.

The ratio saturates at exactly L₂/L₁ and never exceeds it, at any effector affinity. Across the 11.4 to 18.9 nm spans a real engager series actually occupies,⁸ that bound is about 1.5-fold.

What the bound bounds, and what it does not. It bounds the required bound density at a fixed effect. It does not bound a selectivity ratio, and the difference is Section 3.1's: the capacity factor diverges as the attainable density on a cell approaches the density an effect requires, so a bounded rise in required density can produce an unbounded fixed-effect ratio by pushing one cell through its own ceiling. An avidity account is therefore constrained here, not excluded: a bound on required density is not a bound on selectivity.

The bound is still the useful part: it is parameter-free — independent of contact area, assay clock and every arm's chemistry — and it says that whatever an effective-concentration mechanism contributes to the geometric gate, it contributes it through the capacity term rather than through apparent affinity. Alongside it sits the account this arithmetic does not touch: kinetic segregation, in which a close contact excludes the large phosphatase ectodomains and the exclusion is itself the trigger. Section 6.4's controlled distance series is consistent with that — distance at fixed adhesion moves the plateau and the signaling, not the EC₅₀ — and so is the threshold result above, since both put the geometric variable downstream of how much engager is bound.

What this object was built for and did not deliver. It was built to derive the binding-to-killing map and with it the Hill coefficient. It does not supply one, and the way the first attempt went wrong is worth stating because it is a general trap. Treating n as a real number and interpolating survival between integers gives an apparent exponent near 8 — but the midpoint sits at two to three links, so the entire 20-to-80% range lies inside one integer interval, and the 8 is the slope of the interpolation rule. Making n the Poisson variable it physically is, and allowing the assay the repeated conjugation attempts it permits, both correct it.

Table 32. The retracted Hill coefficient and the two corrections that remove it.

About 2, which is what bulk killing curves already show. The object predicts nothing new here and cannot supply h. Arm C of Section 8 still has to measure it. A model evaluated where it holds two or three quanta of its own state variable has no shape to report, and a smooth interpolation will hand you one anyway.

Inputs this section needs and the sources lack: the contact area, the persistence time for a lytic hit, and the effector arm's rate constants, none of which the source characterizes. The saturation result depends on all three and is reported with the contact-area escape above; the gap bound depends on none of them, which is why it is stated as a bound and the other as a conditional.

A two-antigen gate earns its complexity only where it is the only thing protecting the dose-limiting tissue:

(i) the tumor co-expresses both antigens · (ii) each antigen alone has a dose-limiting normal tissue · (iii) that tissue expresses only one of the two · (iv) the accessible densities are comparable · (v) from Section 6.2, the two populations' relaxation times must straddle the exposure window · (vi) from Section 6.4, the anchoring arm must be unable to nucleate a synapse on its own

Requirement (ii) does the work and is the one usually skipped. BCMA × GPRC5D fails it. GPRC5D's single-positive tissue is real — nail, skin and tongue, where talquetamab's dose-limiting toxicities land¹⁴ — and the densities match. But BCMA is confined to plasma cells, so a plain BCMA × CD3 engager already spares keratinized tissue, and clinically it does. The pair remains well chosen for antigen-escape coverage, a different objective this arithmetic does not address.

Requirement (v) is new here and is a kinetic screen that can be applied before any cell work: from k_on and k_off of both candidate arms and an estimate of the exposure window, the two populations' relaxation times either straddle it or they do not. It also sets what to optimize — a slow anchoring arm and a fast-releasing gating arm at whatever K_D the chemistry gives — rather than an affinity target. The physiological limits of bispecific targeting specificity have been analyzed on affinity grounds;¹¹ this adds a dimension those analyses do not contain.

Requirement (vi) is not kinetic and is not visible in any model here, and the measurement behind it is the two-clone comparison of Section 6.4: comparable anchoring binding, a membrane-distal epitope instead of a membrane-proximal one, and the difference between an arm that kills on its own at 30 to 140 pM and one that never reaches an endpoint.²⁰ Section 6.5 narrows what (vi) can be about: an avidity account of epitope height is capped at the ratio of reach distances in required density — about 1.5-fold over the spans a real engager occupies — so whatever converts an anchoring arm from inert to active is not acting through apparent affinity, and the screen cannot be a binding calculation. Where (v) is screened from rate constants, (vi) is screened from epitope mapping and ectodomain geometry, and it is settled by a negative assay: the candidate anchoring arm formatted as an ordinary single-antigen engager should kill the anchor-only cell badly. A high-affinity anchoring arm that is also a competent synapse-former converts the gate into two engagers in one molecule. The two screens are independent and both are cheap relative to a cell campaign, which is the practical content of this section: the gate can be screened on the arms' rate constants and their epitope heights before anything is built. Screened, not specified — (v) returns straddle or no-straddle, and (vi) rests on an experimental result rather than on any model here.

On normal tissue. Dicara et al. surveyed 28 normal tissues for their pair and found one antigen or the other at weak-to-moderate intensity in 11, with co-expression in exactly one — normal breast.¹ That sets a reachable bar. But scoring two antigens on duplicate cores of a tissue microarray is not staining both on the same cells, and cannot exclude a co-expressing minority population inside a tissue scored positive for one and negative for the other. The relevant statement is a single-cell one, and it is the one the model is most sensitive to: against a normal cell expressing both antigens the gate's advantage collapses as densities converge. Co-expression removes the assumption that the second antigen is absent; what protection remains must be evaluated. Delayed-onset toxicity in a B7-H4 CAR-T study is a reminder that the relevant normal expression may not be visible in the tissues first surveyed.¹³

And tumor co-expression is a selection statistic, not a gate. In that series 54% of breast tumors co-expressed the pair, but only 23% at 2+ for both and 7% at 3+ for both, with the two antigens uncorrelated across tumors.

On format. In the geometries modeled here the two routes to avidity are mutually exclusive: a bivalent anchor acquires avidity — and the long-lived bridged state of Section 6.2 — on a cell carrying only the anchor antigen, which is the tissue a two-antigen gate protects. So single-antigen avidity requires bivalency while two-antigen gating requires monovalency on both arms. Whether another geometry evades this is not ruled out, and Dicara's finding that domain order alone changes both potency and tolerability¹ is a reminder that the configuration space is larger than the one modeled.

Nothing here is experimental, and the fastest way to settle it is small. The three predictions of Section 6.2 and the open questions of Section 6.4 can be run as one study on cells already in use.

Design. The six published anti-CD4/CD70 variants, or any equivalent series spanning ~80-fold in one arm's K_D, against the same three-population cell mixture.

Table 33. The proposed study.

What each outcome means. If arm A shows no t-dependence, the kinetic account is wrong and Section 6.2 falls. If arm B shows the two arms interchangeable, the same. If arm C returns h, every window figure in this literature becomes computable rather than assumed, and the shared-slope condition of Section 3.2 becomes checkable instead of stipulated. Arm D costs one plate and converts the leading candidate mechanism of Section 6.4 from a suggestion into a measurement.

Arm E · the height. A ladder of anchoring clones against epitopes at increasing distance from the membrane, at matched affinity, with the gating arm held fixed. Two points of this already exist — a membrane-proximal and a membrane-distal anti-HER2 clone, one an engager on its own and one not²⁰ — and three or four would turn requirement (vi) of Section 7 from a binary screen into a design curve. The readout that matters is the anchoring arm alone, formatted against an inert partner, since that is the measurement the two existing points come from and the only one with a matched comparator. It is the one arm here that no model in this paper can predict, which is why it is worth running.

Where to run arms A and B. They are cheaper in the trispecific system of the second series than in the original one, for two reasons: its comparator is an isogenic knockout rather than a mixed population, so the two curves come off the same parent; and its gating ladder holds k_on approximately fixed, so arm A varies the clock against a panel in which the delivery term is already pinned. Arm B is not free anywhere: swapping two anchoring clones of different epitope is not an affinity ladder on the anchoring arm, and in that platform such a swap confounds requirement (vi) with the kinetic effect. It still needs its own ladder of variants.

Arm zero is done and is not part of this study. The third prediction of Section 6.2 — that the anchoring-antigen-only population cannot move across the affinity series — needed no new experiment, has been run against the published curves, and passed: that series is ten times flatter than the single-positive series measured the same way in the same frames. Its level is still open and would need the underlying data rather than a figure.

What is deliberately not here. No new construct is required for arms A, C or D. Arm B requires one new ladder of variants on the other arm, which is the same CDR-mutagenesis campaign the original study already performed once. Arm E requires clones that exist but have not been placed in a series.

Three numbers in this field are all called selectivity, and the differences between them are comparable to the effects being reported. The capacity factor makes the effect level part of a fixed-effect ratio's definition, survives even at matched density because a dual-antigen curve is not a Langmuir, and becomes unbounded as a cell approaches the density an effect requires. The shape conditions behind a window are rarely stated and do not generally hold.

The main result is that a selectivity endpoint is a statement about time. Carrying an equilibrium confinement model to a fixed-effect ratio fails structurally, and the reason is not an error in the confinement term or the conservation bookkeeping, both of which survive. It is that the model places both cell populations at equilibrium when the published rate constants place only one of them there. A singly bound complex and a bridged complex differ in lifetime by a factor of 640 to 880 across this series — between two and three orders of magnitude — and whether each falls on the short or the long side of the exposure window is what makes the two cells distinguishable at all. Once that is admitted, a prediction with no free parameter in its shape follows — selectivity linear in K_B — and a conserved kinetic model gives 1.009 against a measured 1.068. The prediction then met a second series it had no part in building, in a system differing in antigen pair, format, effector, comparator and exposure, and one whose ladder holds k_on approximately fixed: measured slope 1.210 on three points, and 1.119 (0.67 to 1.57) with the two fitted on a common slope. The interval excludes the equilibrium account's zero and contains the kinetic account's one. That is the claim the data support: a finite-time binding model reproduces the shape of two independent series substantially better than the particular equilibrium model tested. It is not a demonstration that this is the only mechanism that could, and the structure in the residual — largest in the middle of the first series, and leaving both branches of the second deviating from their limiting forms, the equilibrated one by more — says the two-regime picture is the right decomposition rather than a finished description.

The design consequence is a change of variable. Affinity is the wrong knob, or rather it is the right knob read in the wrong units: what the gate needs is k_off on the gating arm high enough that the single-arm complex turns over inside the exposure window, and k_off on the anchoring arm low enough that the bridged complex does not. Holding K_D fixed on both arms and walking k_off across the range an antibody arm can occupy moves predicted selectivity sevenfold, monotonically across the range tested. An equilibrium treatment sees one molecule across that whole range. The criterion's upper bound is a statement about sweeping K_B at fixed on-rate, not about this sweep: here the single-positive cell's EC₂₀ is pinned by a fixed K_D, so there is nothing for it to collapse to. It also means the answer depends on the exposure window, so an in-vitro selectivity figure and an in-vivo one are not the same quantity even for the same molecule — which is a prediction, and an uncomfortable one.

On the four datasets. The two affinity series are accounted for in shape. The first carries one fitted level constant and 2.0-fold point scatter about it; no model is run on the second, where only the slope is compared. The avidity arithmetic bounds what a bivalent arm can buy and shows the realized exponent running 1.45 to 1.74 rather than 2. The antigen-count dataset is not explained, and the arithmetic says which explanations remain: any account in which potency is set by target-cell binding against a fixed bound-density threshold predicts 143-fold to 22,642-fold across that panel's 143-fold antigen range, where 4.4-fold is measured. Three candidates survive — a threshold that scales with antigen, binding on the effector side, and effector throughput at sub-saturating concentration — and an E:T series separates them. Local organization alone spans 537-fold against a measurement exceeding 43,000-fold, and that gap is the honest state of it.

The same change of variable runs through both. Section 6.2 says the quantity that decides selectivity is a lifetime measured against an exposure time; Section 6.4 says the quantity that sets that lifetime is the density within one arm's reach, a footprint of 3 × 10⁻⁴ µm². Neither is a number anyone reports. Papers in this area report affinities and counts per cell, and both of those are averages over the wrong variable — one over time and one over space.

And there is a third variable neither model contains. Two anchoring clones against the same antigen with comparable cellular binding, differing in how far their epitope sits from the membrane, differ in whether they are engagers at all: one kills on its own at 30 to 140 pM and one never reaches an endpoint. A gate whose anchoring arm is itself an engager is capped at 10- to 15-fold by that arm; a gate whose anchoring arm is inert has no measured cap. A two-antigen gate therefore has two gates in series, a kinetic one on lifetimes and a geometric one on which arm can nucleate a synapse. The framework here screens the first and can only report the second. A complete model would place the synapse, not only the complex. Section 6.5 takes one step in that direction and reports what it finds: the conjugate the engager holds together saturates its adhesion at four links, which puts the inferred threshold 201-fold above anything conjugation requires and therefore downstream of it; and an effective-concentration account of epitope height is bounded by the ratio of reach distances, which constrains avidity as the route and leaves kinetic segregation — a signaling mechanism — standing. Both of those are exclusions. The same section tried to derive the Hill coefficient and could not, which is reported with the artifact that nearly made it look otherwise.

The research question we would put first is which measurements explain selective killing when total antigen counts are similar. Answering it requires binding, productive T-cell engagement and killing measured separately on the same pair of cells, and — on the strength of Section 6.2 — it requires them measured as a function of time, which almost no published study reports.

Stated at the level the arithmetic supports rather than as a list of worries.

• Nothing experimental. No construct, no binding measurement, no cell assay, no animal. Section 8 states the study and it has not been run. Arm zero is the one exception and it is a re-reading of published figures, not an experiment: it returns the shape of the third population's response and not its level, and the level is the half of that prediction that would have tested the shared threshold across lineages.
• Section 6.2 matches a trend and not a level, and the trend's own interval is wide. The common-slope interval is 0.67 to 1.57 on nine points from two systems: it excludes a flat trend and contains the prediction, and it does not single out this mechanism. A 1.12 slope and a 1.00 slope are not distinguished by these data. Absolute selectivity is over-predicted by up to 4.6-fold, and a 2.4-fold level constant in the reduced treatment is equally well read as an accessible fraction of 0.41 or an effective exposure of 2.5 h — arm A of Section 8 separates those, since the first is independent of assay duration and the second proportional to it.
• The level residual is structured, and that is not explained. Predicted over measured runs 1.92, 4.62, 4.67, 3.56, 1.34, 1.45 in order of K_B — smallest at both ends, largest in the middle, a 3.5-fold spread that no single multiplicative constant absorbs.
• Neither series is a pure off-rate ladder, and one source's text says otherwise than its own table. On series I the DuetMab column of that source's Table 2 gives k_on spanning 2.8-fold and not monotonically, worth 0.24 of a slope unit, while its Results state that association rates do not change significantly. On series II k_on spans 1.79-fold, worth 0.15 of a unit, which that source's Table S1 shows directly. The conserved kinetic model uses the per-variant rates, so the predicted 1.009 carries this; the consequence is that the measured slopes are dependences on the arms' chemistry as a whole and not on K_B alone.
• Neither branch is correctly described on its own. On the second series the dual-positive branch runs at a slope of 0.60 where the limiting criterion requires 0, and the single-positive branch at 1.90 where the Langmuir used for it requires 1.0. The off-rate ladder is the likely reason for the first — raising k_off,B raises the bridged escape rate and un-traps the state the limiting form assumes trapped — and nothing here explains the second. The criterion survives as a statement about the difference of the two branches and not as a description of either.
• The two published exposures run the wrong way. 6 h gives 1.07 and 72 h gives 1.21, where the criterion requires the slope to fall as exposure grows. Everything else differs between those systems, so this is not evidence against the criterion — but it is also not the confirmation a timed series would be, and prediction 1 remains untested. There is a further reason to doubt that the two are comparable at all: in the second series every complex lifetime is minutes to hours against a 72 h assay, so no population there is anywhere near the exposure window and the clock the criterion names cannot be what separates them.
• The two series are independent as systems but not as laboratories. They share no antigen pair, format, effector, comparator or exposure, and the second was not used to build the model. They do share a senior author and institution, so a common design philosophy or a common assay practice is not excluded as a source of the agreement in shape.
• The third failure of a fixed threshold is narrowed but not closed. Section 6.2's 35-fold miss assumes all three EC₅₀ values mark the same absolute effect. The one mechanism in this model that could depress a curve's top without depressing binding is competitive occupancy of the effector arm, and it is 2 × 10⁵-fold from engaging at these concentrations; the three tops agree to four decimal places across every input swept. That removes the convention as an explanation within the model, and the model is ours. A published curve topping out lower than its siblings would restore it, as would any top-capping mechanism this model omits — effector exhaustion inside a 72 h assay being the obvious one.
• No lifetime claim in this work is separable from affinity. Along both affinity ladders k_off and K_B are collinear — r = 0.997 on the second series, 0.985 on the first — so a persistence threshold and a plain dependence on the equilibrium constant are the same regression. The criterion of Section 6.2 is stated in lifetimes because that is what the two-regime argument is about, and the data cannot discriminate the two readings. Arm A of Section 8 is the only thing that can, since an assay clock is not collinear with anything.
• The second series' numbers are one or two significant figures. Its EC₅₀ values are reported as 0.004, 0.01 and 0.04 nM on the dual-positive cell; the printed precision alone puts the three-point slope anywhere from 1.08 to 1.35 and the common slope from 1.07 to 1.17, and the same table's two blocks disagree on the dual-positive span (6-fold against 10-fold). No figure from that series should be quoted to three digits, and none is.
• The dual-positive population is not cleanly trapped. It relaxes in 5 to 19 hours against a 6 hour assay, so it is partway to equilibrium when the plate is read. Slope 1 is the limiting prediction of complete trapping; the conserved model's 1.009 carries the finite escape rate, but the regime is intermediate and is not described otherwise.
• The binding-to-killing map is reduced, not cancelled. A fixed-effect ratio depends on it only through the scalar N_p, and the sweep shows the slope insensitive to N_p across a 58-fold range — that is sensitivity evidence, not a cancellation theorem. Two things remain assumed: that both populations share the same map, which Section 6.4 shows cannot be carried between cell lineages; and that N₂₀ is a single number, when the six measured points imply values spanning 2.8-fold.
• The effector side is not modeled, and Section 6.4 suggests it may dominate. It sets absolute potency, could hold the whole 2.4-fold level offset, and is the leading candidate for the flat antigen-count panel. For Section 6.2's result it would need to act on the two populations differently and as a function of K_B — the unmutated effector arm responding to a mutation in the targeting arm — which has no mechanism behind it. Everything else it could do is a level effect already inside the sweeps.
• No Hill coefficient has been measured for any system of this kind. The central result does not need one — the exponent cancels from a ratio at a single effect level — but a window does, and Table 7 shows W running from 0.01× to 35× across the range this literature assumes without measuring. That is the single most valuable missing number, and two effect levels on one curve supply it.
• Section 6.4 identifies no mechanism. What it does establish is narrower than it sounds: potency across that panel is not set by target-cell binding against a fixed bound-density threshold, which predicts 143-fold to 22,642-fold where 4.4-fold is measured. Three accounts survive that — an antigen-scaling threshold, effector-side binding, and effector throughput — and the section does not choose between them. Local organization alone spans 537-fold against a measurement exceeding 43,000-fold. Four measurements would constrain it considerably: the arms' k_off, the density inside the enriched areas, the assay's detection threshold, and one E:T series. The first is in the source's own data and unreported; the second is not, because the imaging that would supply it resolves 120–140 nm against a footprint of 3.1 × 10⁻⁴ µm² and so sits 36- to 49-fold above the scale in question. An earlier reading of this paper treated its object-area distribution as the measurement that would decide the local-density account. It is not, and nothing published on this system is. The exclusions above are conditional on the assumed areas, rates, accessibility, threshold, identical-arm behavior and the applicability of this kinetic model; listing those assumptions does not make the spans exhaustive.
• Section 6.5's results are two exclusions, not a model. It does not predict a selectivity, a threshold or a Hill coefficient, and it says so. Its saturation result depends on a contact area the source does not measure — a 0.10 µm² contact, a microcluster rather than a mature synapse, reverses it, and the escape is printed rather than buried. Its gap bound depends on no such input and is the stronger of the two.
• The conjugate object supplies no Hill coefficient, and a naive reading of it supplies a false one. Interpolating survival between integer link numbers, at a midpoint of two to three links, yields a confident 8 that is the slope of the interpolation rule rather than a property of the object. Poisson link number and the repeated conjugation attempts a 6 h assay permits both take it to about 2, which is what bulk curves already show. No h follows from this object, and arm C still has to measure it.
• No model here has a coordinate normal to the membrane, and Section 6.4 reports that coordinate deciding whether an anchoring arm is an engager at all. That comparison is two clones rather than a ladder; their rate constants are unreported and Section 9's sevenfold k_off effect at fixed K_D is not excluded as a contributor; the clone-4380 result is a censored bound, since no endpoint was reached and that molecule is not tabulated in the source's Table S7 at all; and the controlled distance series that supports the mechanism varies flexibility as well as distance, by its own account in comparable measure. Requirement (vi) of Section 7 is therefore stated on an experimental result rather than derived, and the gate it describes sits outside the framework the rest of this paper builds. A model that contained it would have to place the synapse rather than only the complex.
• Three of the sources say something in their text that their own tables do not support, and each has been resolved in favor of the table. One reports a fold-change parameter computed against two different comparator molecules depending on which arm is under discussion, so those two figures cannot be divided by one another; on a matched comparator the two molecules are indistinguishable at 0.67 to 1.00. The same source rounds one selectivity ratio in its text to 3 where its Table S6 gives 3.5, and quotes a four-line panel at 800–1,000× where its Table S7 gives 733–950×. A third states that association rates do not change across its affinity ladder where its own Table 2 shows a 2.8-fold span. None of these is a large error in its own paper; all three would have propagated into this one had its numbers been read from prose.

Three inputs are not measured for these cells — receptor lateral diffusion, well volume, cell diameter — and each is bounded rather than guessed, with every quantity that depends on it moved together in the sweep. Lateral diffusion has no single correct value: surface CD4 on living T lymphocytes resolves into two distinct populations, one freely diffusing and one confined,¹⁹ which is why it is swept over two orders of magnitude; in the detailed-balance-respecting form it moves the slope by 0.001. Depletion runs 1.5% to 2.9% across the plausible 100–250 µL range, and 5.9% even at 50 µL. Cell diameter scales every density, C_eff, the diffusive rate, the escape rate and the inferred threshold; moved together across 7 to 12 µm it changes the slope from 1.06 to 0.97. None of the three is an explicit function of K_B — which does not by itself guarantee a slope is unaffected, and is why each was swept rather than argued away.

Computation. Results are computed in Python scripts provided with their stored outputs. states.py carries the equilibrium model: surface states and conservation, the detailed-balance check, the fixed-point solve for f_A and f_B by damped iteration to a tolerance of 10⁻¹³ on both free fractions, the relabelling test, the concentration sweep, the response function and N_p, EC_p and the capacity factor, and the governing-ratio analysis. priorart.py reproduces Rhoden's effective-concentration construction from its stated geometry and runs the structural-failure sweep of Section 6.1. conserved.py carries Section 6.2: the three-species kinetic model with free-receptor conservation and a Collins–Kimball diffusion factor applied equally to the forward and reverse intramolecular rates, its verification against the equilibrium model of Section 2, the slope test with confidence intervals, the relaxation-time analysis and the sensitivity sweeps with couplings moved together. It is integrated with LSODA at rtol 10⁻⁹ and atol 10⁻¹² and inverted for concentration by bisection in log C. kinetic.py carries the reduced one-variable treatment, the complex lifetimes and the threshold inversion. design.py carries the quantities no other script computes — the window W, the realized avidity exponent point by point, the gate's dependence on absolute density, the fixed-K_D rate-constant sweep, the arm-swap comparison, and the two physical readings of the Section 6.2 offset as slopes. locality.py carries Section 6.4: the footprint of C_eff, the capture-limitation calculation, and the timed solution across local densities. closure.py holds the exact first-passage check against the closed form and against a stochastic simulation, the third-population prediction, the window as a function of h, the enumeration of effector-step mechanisms and which of them could move a slope, and the input sweeps. armzero.py carries the third-population test: it renders the source's figure at 400 dpi, separates the three series by color, and compares them in frame-fractional coordinates, which needs no axis calibration because the six panels share one printed scale. It also records the two absolute calibration routes that disagree, and why neither is used. series2kinetics.py carries the threshold inversion on the second series' single-positive cell, its rounding propagation, and the four alternatives checked against it. segregation.py carries Section 6.4's controlled distance series and the anchoring-arm-only comparison, and fcp_axes.py reconstructs the source's fold-change parameters from its Table S7 on both comparator axes, which is what establishes that the two figures its text reports are not divisible by one another. conjugate.py carries Section 6.5: the per-engager state chain in a cell–cell contact and its exact two-state reduction preserving both the stationary bridged fraction and the bridged dwell time; the exact mean first passage to zero bridged links, checked against a direct solution of the first-passage linear system; the saturation link number and the contact-area enumeration around it; the gap bound swept over eight orders of magnitude in effector affinity; and the retracted Hill coefficient with the two corrections that remove it. The object is explicitly not valid at the inferred threshold, where a 20 µm² contact would hold 803 engagers against 1,200 effector receptors and the links are not independent. secondset.py carries the independent series of Section 6.2: the slope on the published trispecific affinity ladder, the common-slope fit with one intercept per series and its interval on 6 degrees of freedom, and the matched-comparator epitope-height comparison of Section 6.4. It also runs a third, weaker test on the earlier format-and-valence series,¹⁰ whose two reported endpoints give a slope of +0.510 — between the two accounts. That endpoint is direct growth inhibition with no effector cells, so the criterion of Section 6.2 does not apply to it, and it is reported here rather than counted. sources2026.py, cd3review.py and chain.py hold the remaining arithmetic of Sections 4, 6.3 and 6.4. review6.py and review7.py are verification passes. Figure scripts recompute the quantities they display rather than reading them from the text.

Provenance of every quoted number. verify_primary.py transcribes the relevant source tables verbatim, with the table cited beside each block, and re-derives every figure quoted in this paper from them. It reads neither the manuscript nor any source's running text. Where a source's prose and its own table disagree, it prints both and the table is used: three such disagreements are recorded in Section 10, together with the three figures earlier drafts took from prose and the corrections that replace them. The tables used are: Tables 1, 2 and 3 of ref. 3 (receptor densities, rate constants in both formats, concentrations for 20% cytotoxicity); Table 1 of ref. 4 (HER2 antibody binding sites per cell and IC₅₀); and Tables S1, S6, S7 and S8 of ref. 20 (binding kinetics of all three arms for every construct, cytotoxicity EC₅₀ on the parent and the isogenic knockout, cytotoxicity EC₅₀ across four lines for six molecules, and the same molecule and cell measured six times). Numbers whose provenance is not a table — the ordinal statements from ref. 8, which publishes no numeric EC₅₀, and one censored control in ref. 20 that is plotted rather than tabulated — are labeled as such wherever they appear.

Fitting. Section 6.1 fits one parameter, N₂₀, by minimizing squared log error in EC₂₀ across six constructs and two cell types, over a range bounded above by the single-positive cell's capacity (120.96/µm²); the sweep reports predictions across that whole range rather than at the optimum alone. Section 6.2 fits nothing to the trend under test: N₂₀ is obtained by inverting the Langmuir on the single-positive population and averaging, and one global multiplicative constant is fitted to the level of the dual-positive predictions and reported as such.

The second series. Affinities (0.7, 8.1, 32 nM) are from Table S1 of ref. 20, read from its ROR1/EGFR rows; selectivity (3.5, 13.0, 502.5) is the ratio of the two EC₅₀ columns of its Table S6, where selectivity is EC₅₀ on the single-positive knockout over EC₅₀ on the isogenic dual-positive parent in a 72 h cytotoxicity assay with primary human effectors at E:T 10:1. Rate constants are tabulated in that Table S1 for all three arms of every construct, and receptor densities per cell line in its Table 2. Along the EGFR ladder k_off runs 1.40 × 10⁻⁴, 1.20 × 10⁻³ and 5.90 × 10⁻³ s⁻¹ while k_on runs 2.10, 2.50 and 1.40 × 10⁵ M⁻¹s⁻¹, so the affinity change is carried mainly but not entirely by the dissociation rate: k_on moves 1.79-fold and not monotonically, which is 0.15 of a slope unit on its own. The anchoring arm is constant (K_D 8.7–9.9 nM) and so is the CD3 arm (47.3–48.0 nM), so the ladder does vary one arm and only one arm. No kinetic model is run on this series — its receptor densities, arm geometry and effector step differ from the first series in ways the model would have to assume — so the slope is the only quantity comparable without running it. The slope's predicted value is not itself free of those inputs: it is conditional on the exposure window relative to the dual-positive relaxation time, which is the tension Section 6.2 reports. The common-slope fit is ordinary least squares on log₁₀-transformed values with one slope and one indicator intercept per series, interval from the t distribution on n − 3 = 6 degrees of freedom. The printed precision of Table S6 is propagated by recomputing the slope at every corner of the resulting box.

Kinetics. Rate constants for series I are taken from Table 2 of ref. 3, DuetMab columns. Its printed K_D column is rounded and disagrees with k_off/k_on by −5.0% to +3.2%; a kinetic model uses the rates, an equilibrium model the ratio, and the slope test was repeated against both abscissae (1.016 and 1.078 against 1.009 and 1.068). Bridged-complex escape is a first-passage calculation over the two routes out of the bridged state, validated against the exact three-equation mean-first-passage solution of the four-state chain and against a stochastic simulation of it, with tethered-arm rebinding treated as two resistances in series: a two-dimensional diffusive capture rate 2πDσ/ln(b/a), with b the mean half-spacing and a = L, and a reaction rate k_on·C_eff. The timed solution integrates dN/dt = C·[k_on,A(a_T − N) + k_on,B(b_T − N)] − k_esc·N, which is linear in N, in closed form and inverts it for C by bisection in log C.

Constants. N_A = 6.02214076 × 10²³ mol⁻¹; arm reach L = 10 nm unless stated (Rhoden uses r_IgG = 12.5 nm, giving C_eff = 17.7 µM rather than 22.2 µM at the same density); accessible density σ = 89 receptors/µm² where a design point is needed, from 6.3 × 10⁴ accessible receptors on a 15 µm sphere.¹² Not measured for these cells, and swept: receptor lateral diffusion (central value 0.05 µm²/s, swept over two orders of magnitude, with the heterogeneity of surface CD4 the reason¹⁹), well volume (100 µL central, 50–250 µL swept), lymphocyte diameter (10 µm central, 7–12 µm swept, and it cancels). For Section 6.4 the HER2 arm's rate constants are not tabulated in the source, so that section reports boundaries in local density across a plausible k_off range rather than predicted values.

Use of AI tools. The derivations, code and figures in this paper were produced in collaboration with Claude (Anthropic), used as a computational and analytical instrument under the author's direction. Every result is accompanied by the script that computes it and by the stored output of that script, so that no claim here rests on an assertion that cannot be re-executed.

Data and code availability. All scripts, their stored outputs, the figure-generating scripts and the manuscript source accompany this preprint. No new experimental data were generated.

Competing interests. None declared.

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