A Generalization of the Ternary Binding Model to Membrane-Confined Systems With Finite Copy Number

The standard Douglass ternary binding model (TBM) for three-body equilibria assumes a well-mixed, three-dimensional solution. When applied to bispecific T-cell engagers (BiTEs), however, the productive trimeric complex forms not in bulk solution but within a nanoscale membrane synapse with finite receptor copy numbers. We present a generalization of the TBM to membrane-confined systems that replaces the macroscopic bulk volume with a coarse-grained reactive contact volume defined by synapse geometry and microvillus topology, and extends the deterministic equilibrium to a stochastic description via the chemical master equation. The framework preserves the original algebra while restoring its representational capacity for the regime in which therapeutic activity occurs. A key finding is that conventional bulk mapping places the system in the affinity-limited regime, where antigen density is mathematically inert and the TBM predicts identical dose-response regardless of target expression. Membrane confinement shifts effective antigen concentration by six orders of magnitude-from $\sim {10}^{-3}$ nM to $\sim {10}^3$ nM-restoring antigen density as a governing variable for trimer formation. Using blinatumomab (anti-CD19 BiTE) as a case study, we introduce the absolute formation dose $d\left(N^{*}\right)$ : the drug concentration required to produce a fixed number of ternary complexes sufficient for T-cell activation. This metric replaces the conventional $T{F}_{50}$ , which normalizes each cell line to its own maximum, erasing the density dependence that confinement rescues. For NALM-6 and HAL-01 cell lines (CD19 density ratio $\approx 2.5$ ), the framework predicts a corresponding $\approx 2.5$ -fold difference in required dose-a prediction structurally invisible to the bulk formulation.