# Antigen-tail Check ## Scientific model and first-release specification Data-Rich, Insight-Poor · 99 Small Problems · No. 02 Useful models for assumptions with expensive ambitions. Version: 0.1.0-alpha · Research-use preview ## The question Are two expression-matched populations actually comparable? Antigen-tail Check tests the narrower proposition that matching an expression summary is sufficient to match an expression distribution, then shows how a specified response mapping changes the functional interpretation. The tool has two layers that must not be confused. The first is descriptive: compare the observed or deliberately constructed distributions, their quantiles, their low-expression fractions and their cumulative-distribution separation. The second is conditional: apply a shared, explicitly hypothetical response function to each distribution. The second layer cannot discover a biological response law from expression alone. Use it when selecting or comparing assay-cell populations, investigating a clone or lot change, or deciding whether a population average has concealed a consequential subpopulation. The intended decision is whether the distribution deserves a functional comparison or a better measurement, not whether a therapeutic will succeed. ## The scientific basis ### Antigen recognition is not one endpoint Watanabe and colleagues studied antigen-density dependence in a CD20 CAR-T system and found different expression requirements for lysis versus activation, cytokine production and proliferation; their lowest tested CD20 expression group, approximately 240 specific antibody-binding sites per cell, underwent partial lysis but did not behave like the higher-expression groups across activation endpoints ([Watanabe et al., 2015](https://academic.oup.com/jimmunol/article/194/3/911/7972840?guestAccessKey=)). The paper also reports an approximately 1,000-site half-maximal cytotoxicity region, illustrating why the lowest detectable response, a response midpoint and complete eradication are not interchangeable thresholds ([Watanabe et al., 2015](https://academic.oup.com/jimmunol/article/194/3/911/7972840?guestAccessKey=)). Those findings motivate endpoint-specific comparison, not a universal numerical cutoff. No threshold, construct, sequence or fitted parameter from that study is implemented here. ### Cell-autonomous expression is not all of ADC pharmacology Wood and colleagues model antigen-positive and antigen-negative populations, payload transport and bystander effects, showing that the consequences of heterogeneous target expression depend on interactions among target-positive cells, target-negative cells and payload disposition ([Wood et al., 2025](https://pmc.ncbi.nlm.nih.gov/articles/PMC12370192/)). Antigen-tail Check does not reproduce those ODEs or borrow their therapeutic parameters; the study explains why a marginal expression distribution is an informative starting point but an incomplete ADC-response model. For an ADC, interpreting the Hill scenario as cell killing would require evidence that a shared, cell-autonomous expression-to-killing relationship is adequate under the chosen dose, exposure time, growth conditions and payload context. Bystander transfer, unequal payload sensitivity or spatial organization can invalidate that interpretation; payload-transfer and growth dependencies are explicit in the published ADC model ([Wood et al., 2025](https://pmc.ncbi.nlm.nih.gov/articles/PMC12370192/)). ### Fc-dependent binding does not establish functional susceptibility Tan and colleagues show how immune-complex valency and Fc composition affect receptor binding, and distinguish binding from effector response in their experimental and modeling systems ([Tan et al., 2023](https://pmc.ncbi.nlm.nih.gov/articles/PMC10404157/)). That paper is used as a boundary on interpretation, not as direct evidence for an antigen-tail threshold or a population-heterogeneity result. The tool therefore does not advertise an “ADCC threshold,” an “ADC threshold,” or a “CAR threshold.” It asks how two distributions differ under a user-specified cutoff or common response mapping. A therapeutic mechanism supplies the experimental question; it does not license the same response law across modalities. ### The measurement scale is part of the model ABC measures antibodies bound per cell under a specified staining definition, while MESF and ERF are fluorescence-equivalence measures; calibration, standardization and low-signal resolution are distinct concerns ([Wang and Hoffman, 2017](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=921423)). None of those labels, by itself, establishes the number of therapeutically accessible receptors per cell. Both populations need a compatible mapping from the biological quantity of interest to the reported signal. A common unit label does not resolve different staining reagents, binding stoichiometry, epitope accessibility, gating, background distributions or instrument behavior; the calibration reference discusses these distinctions and the special difficulty of resolving dim populations ([Wang and Hoffman, 2017](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=921423)). In particular, the app is an audit of per-cell abundance proxies, not a measurement of molecules per square micrometer. Equal abundance does not mathematically imply equal surface density when cell surface areas differ. That distinction must be settled experimentally when the proposed mechanism depends on local surface organization. ## Mathematical definition The following derivations define this implementation. They are general probability calculations, not a claim of a new biological law. ### Distribution, weights and arithmetic mean For population \(j \in \{A,B\}\), let \(a_{ji}\geq0\) be an expression value and \(c_{ji}\) its positive integer multiplicity. An event-level file has \(c_{ji}=1\). \[ n_j=\sum_i c_{ji},\qquad w_{ji}=\frac{c_{ji}}{n_j},\qquad \mu_j=\sum_iw_{ji}a_{ji}. \] Repeated identical values are merged exactly. Counts represent observed events or synthetic multiplicities, never biological replicate numbers. Histogram-bin centers with bin counts are not exact observations and are outside this release's input contract. The descriptive standard deviation is \[ s_j=\sqrt{\sum_iw_{ji}(a_{ji}-\mu_j)^2},\qquad CV_j=s_j/\mu_j. \] This is the weighted population standard deviation of the supplied distribution, not an unbiased estimator of a biological population parameter. CV is undefined at zero mean. The quantile \(Q_j(p)=\inf\{t:F_j(t)\geq p\}\) is reported at 5%, 50% and 95%. The symmetric mean mismatch is \[ \delta_\mu=100\frac{|\mu_A-\mu_B|}{(\mu_A+\mu_B)/2}. \] When both means are zero, the mismatch is defined as zero. “Mean matched” means only that this descriptive mismatch is within the user-selected tolerance. It is not a statistical equivalence test. ### The low-expression tail \[ F_j(t)=\sum_iw_{ji}\mathbf1(a_{ji}\leq t),\qquad \Delta_F(t)=F_B(t)-F_A(t). \] The cutoff is inclusive: a cell exactly at \(T\) belongs to the at-or-below-\(T\) fraction. A hard response interpretation would therefore use \(g_T(a)=\mathbf1(a>T)\), whose unresponsive fraction is exactly \(F_j(T)\). This convention is explicit because ties can be material in coarse or synthetic data. For a prespecified interval \([L,U]\), the largest descriptive tail difference is \[ D_{[L,U]}=\sup_{L\leq t\leq U}|F_A(t)-F_B(t)|. \] The software evaluates both interval endpoints and every empirical jump within the interval. It does not approximate this statistic from the 101 plotted sweep points. If \(D_{[L,U]}\) exceeds the user's tail-gap margin, that margin fails for the supplied distributions. Passing it establishes neither biological equivalence nor a population-level confidence statement. The global maximum \[ D=\sup_{t\geq0}|F_A(t)-F_B(t)| \] is the two-sample Kolmogorov–Smirnov distance used descriptively. No KS hypothesis test or p-value is performed. ### A response bound that does not require fitting a Hill curve Suppose \(g\) is the same nondecreasing function for both populations and its values lie in \([0,1]\). Then \[ |\mathbb E_A[g(a)]-\mathbb E_B[g(a)]|\leq D. \] For a differentiable \(g\), the result follows from \[ \mathbb E_j[g(a)] =g(0)+\int_0^\infty g'(t)\,[1-F_j(t)]\,dt, \] so that \[ |\mathbb E_A[g]-\mathbb E_B[g]| \leq D\int_0^\infty g'(t)\,dt \leq D. \] Monotone step functions give the corresponding general result; for a finite empirical distribution a step at a maximally separated cutoff attains the bound. Thus the global CDF distance is a worst-case response-gap bound over this function class. The qualifications are essential. A small global D limits response differences only under a shared, bounded, monotone mapping. It does not constrain differences caused by different biology, nonmonotone responses, different drug exposures or population interactions. A small interval-specific D does not constrain unexamined thresholds; the global bound cannot be replaced by the interval-specific value without additional restrictions on where g changes. ### Order, crossing and why “more heterogeneous” is not a mechanism If \(F_A(t)\leq F_B(t)\) for all t, then A is first-order stochastically higher in expression than B, and every shared nondecreasing response has an average at least as high in A. If the CDFs cross, no universal ranking exists over that entire class of response mappings. Two distinct nonnegative distributions with exactly the same finite mean cannot have strict first-order stochastic dominance. Their equal integrated survival functions force any nontrivial CDF difference to change sign. The equal-mean case therefore makes threshold sensitivity a mathematical consequence, not a display artifact. This does not imply that every Hill family or every experimentally plausible parameter range exhibits a reversal. It means that a ranking based on the mean alone has not resolved the question. ### The optional Hill scenario \[ g(a;T,h)=\frac{a^h}{a^h+T^h},\qquad T>0,\ h>0, \] \[ R_j(T,h)=1-\sum_iw_{ji}g(a_{ji};T,h). \] R is labeled the “unresponsive share under a hypothetical mapping.” It is not automatically the fraction of live cells after treatment. T is the half-response expression level because \(g(T)=1/2\), not the lowest expression that can produce a detectable response. The slope h is a phenomenological shape parameter, not an estimate of receptor number, molecular stoichiometry or cooperativity. The mean-only comparator is \[ R^{\mathrm{mean}}_j=1-g(\mu_j;T,h). \] The difference between \(R_j\) and \(R^{\mathrm{mean}}_j\) is the aggregation error under this assumed function. No response parameters are fitted and the defaults are invented. At h = 1, g is concave and averaging expression first overestimates average response by Jensen's inequality. For h > 1, curvature changes sign: \[ g''(a)= \frac{hT^h a^{h-2}\left[(h-1)T^h-(h+1)a^h\right]} {(a^h+T^h)^3},\qquad a>0. \] The inflection is at \(a=T[(h-1)/(h+1)]^{1/h}\). Heterogeneity therefore does not always reduce response; its effect depends on where expression lies relative to the response curve. A slogan about variance cannot substitute for integrating the actual distribution. As h grows without bound, g approaches a hard step away from a = T, but g(T) remains 1/2. If there is an atom at T, the limiting unresponsive share is \(\Pr(a