Antigen Density Thresholds for Selective Cell Killing

From Antigen Density to a Killing Criterion

A cell engager can bind both a target-cell antigen and a T-cell receptor, but binding alone does not guarantee that a target cell receives enough functional engagement to be killed. The relevant question here is narrower and quantitative: under a specified model, how do antigen density and engager concentration jointly determine an effective bridge density, and when can that density meet a killing criterion on target cells without exceeding a sparing criterion on low-antigen cells?

The calculation below is a deliberately simplified, static model. It treats concentration as constant during the contact interval, assumes the two binding contributions can be represented by independent occupancy terms, and treats the local number of available T cells and contact geometry as fixed. These assumptions make the threshold transparent; they do not make it universal. The earlier material on Occupancy Versus Functional Potency in Cell Engagers distinguishes molecular occupancy from a functional response. Here, a bridge-density threshold supplies a specified functional criterion rather than assuming that occupancy itself equals killing.

Definition: Let $\rho$ denote target-antigen density, in antigen molecules per square micrometre of target-cell surface, and let $C$ denote free engager concentration, in nM. Let $K_T$ and $K_E$, also in nM, be effective dissociation constants for the target-binding and effector-cell-binding arms. In the independent-occupancy approximation, define $$ \theta_T(C)=\frac{C}{K_T+C}, \qquad \theta_E(C)=\frac{C}{K_E+C}. $$ The effective bridge density is $$ B(\rho,C)=g\,\rho\,\theta_T(C)\theta_E(C), $$ where $g$ is a dimensionless factor representing the fraction of doubly occupied configurations that are geometrically and functionally available as bridges under the specified contact conditions. $B$ is expressed as effective bridges per square micrometre.

The product of occupancies is a modeling approximation, not a general identity for a multivalent binding network. It stands in for the probability that both binding contributions are present in a productive configuration. The factor $g$ absorbs a fixed geometric and contact-efficiency contribution; it must be estimated or calibrated for the setting and should not be interpreted as a universal molecular constant. The model also assumes that the free concentration at the cell surface is adequately represented by $C$, and that target antigen is not depleted by binding.

Threshold definition: Specify a bridge-density criterion $B_{\mathrm{kill}}$ for meeting the target-cell killing requirement and a maximum acceptable density $B_{\mathrm{spare}}$ for low-antigen cells. A target cell meets the modeled killing criterion when $B(\rho,C)\geq B_{\mathrm{kill}}$. A low-antigen cell meets the modeled sparing criterion when $B(\rho,C)\leq B_{\mathrm{spare}}$. These are operational model criteria, not claims that every cell responds deterministically at one exact molecular count.

Deriving the Joint Threshold

For a fixed concentration, the model is linear in antigen density. If $g\,\theta_T(C)\theta_E(C)>0$, rearranging the killing condition gives the minimum antigen density that meets the criterion:

g\,\rho\,\theta_T(C)\theta_E(C)\geq B_{\mathrm{kill}} \quad\Longleftrightarrow\quad \rho\geq \rho_{\mathrm{kill}}(C) =\frac{B_{\mathrm{kill}}}{g\,\theta_T(C)\theta_E(C)}.

Likewise, the sparing condition is equivalent to an upper bound on antigen density at that concentration:

g\,\rho\,\theta_T(C)\theta_E(C)\leq B_{\mathrm{spare}} \quad\Longleftrightarrow\quad \rho\leq \rho_{\mathrm{spare}}(C) =\frac{B_{\mathrm{spare}}}{g\,\theta_T(C)\theta_E(C)}.

Because both occupancies increase with $C$, the bridge-density coefficient $g\,\theta_T(C)\theta_E(C)$ increases with concentration and approaches $g$ as concentration becomes very large. Thus, within this model, increasing concentration lowers the antigen density needed to cross the killing criterion, but it also increases bridge density on low-antigen cells. Selectivity is therefore a joint constraint, not a property of concentration alone.

For a target population with antigen density $\rho_H$ and a low-antigen population with density $\rho_L$, where $\rho_H>\rho_L$, the concentration must satisfy both:

g\,\rho_H\,\theta_T(C)\theta_E(C)\geq B_{\mathrm{kill}}, \qquad g\,\rho_L\,\theta_T(C)\theta_E(C)\leq B_{\mathrm{spare}}.

Equivalently, the common concentration-dependent coefficient must lie between two bounds:

\frac{B_{\mathrm{kill}}}{\rho_H} \leq g\,\theta_T(C)\theta_E(C) \leq \frac{B_{\mathrm{spare}}}{\rho_L}.

A necessary condition for any selective concentration window is therefore:

\frac{\rho_H}{\rho_L}\geq \frac{B_{\mathrm{kill}}}{B_{\mathrm{spare}}}.

This follows by requiring the lower bound not to exceed the upper bound and multiplying by the positive quantity $\rho_H\rho_L/(B_{\mathrm{kill}}B_{\mathrm{spare}})$. It is necessary but not always sufficient: the permitted coefficient must also be reachable, and its lower bound cannot exceed the model's maximum coefficient $g$.

Worked Concentration Window

Worked Example: Finding Concentrations That Kill High-Antigen Cells and Spare Low-Antigen Cells

Suppose a hypothetical assay is summarized by $K_T=2.0$ nM, $K_E=5.0$ nM, and $g=0.60$. The high-antigen cells have $\rho_H=800$ molecules per square micrometre, and the low-antigen cells have $\rho_L=100$ molecules per square micrometre. Use the operational criteria $B_{\mathrm{kill}}=120$ effective bridges per square micrometre and $B_{\mathrm{spare}}=30$ effective bridges per square micrometre. Find the concentration range satisfying both criteria.

The common coefficient is $q(C)=g\,\theta_T(C)\theta_E(C)$. For killing the high-antigen cells, it must be at least $120/800=0.1500$. For sparing the low-antigen cells, it must be at most $30/100=0.3000$. Hence $0.1500\leq q(C)\leq0.3000$.

At the lower boundary, $q(C)=0.1500$, so $\theta_T(C)\theta_E(C)=0.1500/0.60=0.2500$. Substituting the occupancy equations:

\frac{C^2}{(C+2)(C+5)}=0.2500 \quad\Longrightarrow\quad C^2=0.2500(C^2+7C+10) \quad\Longrightarrow\quad 3C^2-7C-10=0.

The positive root is $C=(7+\sqrt{169})/6=3.333$ nM. At the upper boundary, $q(C)=0.3000$, giving an occupancy product of $0.5000$:

\frac{C^2}{(C+2)(C+5)}=0.5000 \quad\Longrightarrow\quad C^2-7C-10=0.

The positive root is $C=(7+\sqrt{89})/2=8.217$ nM. Since $q(C)$ increases with concentration in this model, the qualifying interval is approximately $3.333$ to $8.217$ nM, including the boundaries under the stated criteria.

As a check, take $C=5.00$ nM. Then $\theta_T=5/(2+5)=0.7143$, $\theta_E=5/(5+5)=0.5000$, and $q=0.60(0.7143)(0.5000)=0.2143$. The predicted densities are $B_H=800(0.2143)=171.4$ bridges per square micrometre and $B_L=100(0.2143)=21.43$ bridges per square micrometre. The first exceeds 120 and the second is below 30, so both criteria are met at 5.00 nM.

When Antigen Contrast Is Not Enough

Worked Example: Detecting an Impossible Selectivity Window

Keep the same hypothetical affinity and geometry parameters and the same criteria, but suppose the low-antigen cells have $\rho_L=250$ molecules per square micrometre while the target cells remain at $\rho_H=800$. Determine whether any concentration can satisfy both requirements.

The target-killing requirement implies $q(C)\geq120/800=0.1500$. The sparing requirement implies $q(C)\leq30/250=0.1200$. No value of $q(C)$ can satisfy both inequalities, so no concentration in this model achieves the requested separation.

The same conclusion follows from the necessary density-ratio condition: $\rho_H/\rho_L=800/250=3.200$, whereas $B_{\mathrm{kill}}/B_{\mathrm{spare}}=120/30=4.000$. The available antigen contrast is too small relative to the separation demanded by the two criteria. Raising concentration cannot solve this conflict because it raises the bridge-density coefficient for both populations by the same factor.

This impossibility result depends on the shared coefficient $q(C)$: the two cell populations differ only in antigen density, and all other modeled conditions are held fixed. If the populations differ in contact geometry, effector availability, or other functional determinants, a single common coefficient may no longer describe both. That is a reason to revise the model, not a reason to ignore a failed inequality.

Calculating the Density Requirement at a Fixed Concentration

Worked Example: Minimum Target Density at a Chosen Exposure

Use the hypothetical parameters $K_T=2.0$ nM, $K_E=5.0$ nM, and $g=0.60$, with $B_{\mathrm{kill}}=120$ bridges per square micrometre. At $C=4.00$ nM, calculate the minimum antigen density that meets the killing criterion.

The occupancies are $\theta_T=4/(2+4)=0.6667$ and $\theta_E=4/(5+4)=0.4444$. Therefore $q=0.60(0.6667)(0.4444)=0.1778$. The density threshold is:

\rho_{\mathrm{kill}}(4.00) =\frac{120\ \text{bridges}/\mu\text{m}^2}{0.1778} =675.0\ \text{antigen molecules}/\mu\text{m}^2.

A cell with density $700$ molecules per square micrometre has predicted $B=700(0.1778)=124.4$ bridges per square micrometre and meets the criterion. A cell with density $600$ has predicted $B=600(0.1778)=106.7$, which is below the criterion. These are model classifications for the specified conditions, not guarantees of individual-cell outcomes.

Assumptions, Diagnostics, and Alternatives

The threshold calculation is most useful when its assumptions can be checked against the experimental setting. At minimum, establish that concentration is approximately constant over the modeled contact period; the two effective occupancy terms are reasonable approximations; antigen density and affinity parameters refer to the relevant cell and assay conditions; and the factor $g$ does not change substantially between the high- and low-antigen populations. The thresholds $B_{\mathrm{kill}}$ and $B_{\mathrm{spare}}$ must also be defined from an explicit response criterion, rather than selected after seeing which concentration gives a desired conclusion.

Several diagnostics can reveal a poor fit or a failed assumption:

Predictions do not scale approximately with antigen density. At fixed concentration, the model predicts $B\propto\rho$. Systematic departures suggest that a constant bridge fraction, an unsaturated target pool, or comparable cell-contact conditions may be implausible.
Observed killing is not ordered by predicted bridge density. If cells with higher predicted $B$ do not generally show at least as much killing under comparable conditions, bridge density alone is missing important determinants or the response criterion is not transferable.
The inferred coefficient differs by cell population. A different effective $g$ for target and low-antigen cells undermines the common-coefficient selectivity test. Check whether contact geometry or effector-cell availability differs before interpreting antigen density as the sole cause.
Apparent thresholds move with assay duration or concentration history. A static $C$ model does not represent time-varying exposure or accumulated signaling. Use a time-dependent formulation if those changes are material.
High concentrations fail to follow the predicted monotone trend. The present equations predict increasing, saturating bridge density. A reproducible nonmonotone response signals that the model's assumptions are insufficient; do not force it into this threshold calculation.

The main alternatives answer related but different questions. An occupancy-only analysis can estimate how much of a binding site is occupied, but does not by itself define a bridge density or a killing criterion. An empirical concentration-response curve can estimate a concentration associated with a chosen response in a particular assay, but may not explain how changing antigen density shifts that concentration. The present mechanistic approximation makes that dependence explicit and allows a density-based selectivity check. A more detailed kinetic or state-based binding model can represent association, dissociation, changing concentrations, or additional binding states, but requires more parameters and data. The appropriate choice depends on whether the purpose is a transparent screening calculation or a richer account of the measured system.

The model has important limits. The product of two occupancy terms does not resolve the full spatial arrangement of receptors and antigens, and the factor $g$ can hide substantial biology. A single threshold also compresses a graded and potentially heterogeneous cell response into a criterion. Antigen density varies across cells, while affinity estimates and response thresholds carry uncertainty; a predicted concentration window should therefore be tested over plausible parameter ranges before being treated as robust. Finally, satisfying the mathematical inequalities establishes only that the model's two operational criteria are met. It does not establish clinical selectivity, safety, or causation in a patient.

Practical Calculation Sequence

Specify the two cell populations and criteria.
Record $\rho_H$, $\rho_L$, $B_{\mathrm{kill}}$, and $B_{\mathrm{spare}}$, with units and a clear basis for each threshold.

Define the concentration-dependent coefficient.
Calculate $q(C)=g[C/(K_T+C)][C/(K_E+C)]$ using parameters appropriate to the modeled conditions.

Check feasibility before selecting a concentration.
Verify $\rho_H/\rho_L\geq B_{\mathrm{kill}}/B_{\mathrm{spare}}$, and check that the required coefficient is no greater than its maximum $g$.

Find and verify the concentration interval.
Solve $q(C)\geq B_{\mathrm{kill}}/\rho_H$ and $q(C)\leq B_{\mathrm{spare}}/\rho_L$, then substitute a concentration in the proposed interval back into both bridge-density calculations.

Key takeaway: Under the independent-occupancy, fixed-contact approximation, bridge density is antigen density multiplied by a concentration-dependent coefficient. A selective concentration window exists only when the high-to-low antigen-density ratio is large enough for the chosen killing and sparing criteria, and when the required coefficient is physically reachable in the model.

Check Your Understanding

Use the model and definitions above. Answers with calculations are provided after the questions.

At fixed $C$, if antigen density doubles and all other model quantities remain fixed, what happens to predicted bridge density?
A target population has $\rho_H=600$ and a low-antigen population has $\rho_L=100$, both in molecules per square micrometre. If $B_{\mathrm{kill}}=90$ and $B_{\mathrm{spare}}=20$ bridges per square micrometre, does the necessary density-ratio condition hold?
With $K_T=2$ nM, $K_E=5$ nM, $g=0.60$, and $C=3$ nM, calculate $q(C)$.
Why does increasing concentration not resolve an impossible selectivity inequality when the two cell populations share the same $q(C)$?

Solutions. (1) $B=g\rho\theta_T\theta_E$, so doubling $\rho$ doubles $B$. (2) The density ratio is $600/100=6.000$, while the required ratio is $90/20=4.500$; the necessary condition holds, though reachability and the actual concentration interval still need checking. (3) $\theta_T=3/5=0.6000$, $\theta_E=3/8=0.3750$, so $q=0.60(0.6000)(0.3750)=0.1350$. (4) The same coefficient multiplies both densities. If killing requires $q$ to exceed a lower bound that is already greater than the sparing upper bound, no single value of $q$, and therefore no concentration producing it, can meet both conditions.