Occupancy Versus Functional Potency in Cell Engagers

Three Outputs, Three Different Questions

For a cell engager, target binding is not the same as productive engagement, and productive engagement is not the same as target-cell killing. An occupancy assay asks how much target antigen is bound by the drug. A bridge assay asks how many drug molecules connect a target cell to an effector cell. A killing assay asks what fraction of target cells die over a specified period. These quantities may be related, but none can be substituted for another without a model and supporting assumptions.

The distinction matters when comparing constructs, doses, or assay conditions. Two conditions can have the same fraction of target antigen occupied while differing in the fraction of bound molecules whose second arm binds an effector cell. Even if bridge abundance is matched, killing can differ when downstream lytic capacity or target susceptibility differs. Thus, occupancy is useful evidence of target binding, but it is not by itself a measure of functional potency.

Definition: Target occupancy, denoted \(\theta_T\), is the fraction of accessible target-antigen sites occupied by the target-binding arm at a specified time. Bridge abundance, denoted \(B\), is the number of productive engager-mediated connections between target and effector cells, defined here per target-cell–effector-cell contact. Killing, denoted \(K(t)\), is the fraction of target cells killed by time \(t\) under specified assay conditions. Functional potency is the concentration required to achieve a stated level of a specified functional response in a specified assay; it is not a binding affinity or an occupancy value.

An occupancy measurement may count target-bound engager whether or not the engager’s other arm is bound to an effector cell. It may also average over cells and contacts that differ in target abundance or accessibility. The precise operational meaning therefore depends on how the assay detects bound drug and defines its denominator. A reported occupancy percentage should be interpreted with that measurement definition in view.

A Minimal Model Linking Occupancy, Bridges, and Killing

A simple equilibrium approximation helps separate the outputs. Let \(C\) be the free engager concentration, and let \(K_T\) be the dissociation constant for its target-binding arm. If target binding is at equilibrium, free drug is not appreciably depleted by binding, and accessible sites behave as equivalent independent sites, then target-arm occupancy is

\theta_T = \frac{C}{K_T+C}.

This expression describes target-arm binding, not cell bridging. To represent the second binding event, let \(q_E\) be the probability that a target-bound engager binds an effector-cell antigen in the contact, and let \(g\) be the fraction of target-bound molecules that are geometrically able to form a productive bridge. In a simplified conditional model, \(q_E=\rho_E/(K_E+\rho_E)\), where \(\rho_E\) is an effective local concentration of accessible effector antigen and \(K_E\) is the corresponding dissociation constant. Let \(N_T\) be the number of accessible target-antigen sites per target cell. Then expected bridge abundance is approximated by

B = N_T\theta_T gq_E.

Here \(B\) is an expected number of bridges per cell-cell contact. This approximation assumes that each productive bridge uses one target-bound engager, that the factors \(g\) and \(q_E\) can be treated as fixed for the condition being modeled, and that depletion, steric interference, and variation among contacts do not dominate. It makes the key distinction explicit: occupancy is one factor in bridge abundance, but geometry and effector-arm binding also matter.

For a compact functional model, suppose bridges generate an instantaneous target-cell killing hazard

h(B)=h_{\max}\frac{B}{B_{50}+B}, \qquad K(t)=1-\exp[-h(B)t].

The hazard \(h\) has units of inverse time; \(h_{\max}\) is its limiting value, and \(B_{50}\) is the bridge abundance at which the hazard is half of \(h_{\max}\). The survival expression assumes a constant hazard during the interval, no target-cell replacement, and a homogeneous population with the same hazard. It is a deliberately simplified link from bridge abundance to killing, not a universal biological law.

Worked Example: Equal Occupancy, Different Bridge Abundance and Killing

Consider two hypothetical conditions with the same target occupancy, \(\theta_T=0.80\), and \(N_T=10{,}000\) accessible target sites per target cell. Suppose both have \(g=0.75\), but their conditional effector-arm binding probabilities differ: \(q_E=0.20\) in condition A and \(q_E=0.80\) in condition B. Let \(B_{50}=2{,}000\) bridges per contact and \(h_{\max}=0.60\) per hour.

For condition A,

B_A=10{,}000(0.80)(0.75)(0.20)=1{,}200 \quad\text{bridges per contact}.

For condition B,

B_B=10{,}000(0.80)(0.75)(0.80)=4{,}800 \quad\text{bridges per contact}.

The occupancies are identical, but bridge abundance is four times larger in condition B. The predicted hazards are

h_A=0.60\frac{1{,}200}{2{,}000+1{,}200} =0.225\ \text{h}^{-1}, \qquad h_B=0.60\frac{4{,}800}{2{,}000+4{,}800} \approx0.4235\ \text{h}^{-1}.

Under the constant-hazard assumption, after four hours the predicted killed fractions are

K_A(4)=1-e^{-(0.225)(4)}\approx0.593, \qquad K_B(4)=1-e^{-(0.4235)(4)}\approx0.816.

Thus the model predicts approximately 59.3% killing in A and 81.6% in B, despite equal target occupancy. The difference arises from the assumed probability of forming the second, effector-binding interaction. These predictions are conditional on the model assumptions and are not empirical observations.

Why Bridge Abundance Still Does Not Fix Killing

The bridge-to-killing relationship also depends on biology downstream of bridge formation. Effector-cell activation state, lytic capacity, target-cell susceptibility, and the duration of productive contacts can alter killing at a given bridge abundance. The earlier material on contact-time dependence in immune synapse formation treats contact duration as a dynamic influence; here it is enough to recognize that a bridge count measured at one time need not summarize the duration or quality of all contacts.

Worked Example: Equal Bridges, Different Killing Capacity

Suppose two hypothetical conditions each have \(B=3{,}000\) bridges per contact and \(B_{50}=2{,}000\). Condition A has \(h_{\max}=0.60\ \text{h}^{-1}\), while condition B has \(h_{\max}=0.20\ \text{h}^{-1}\), representing different effective lytic capacities. The bridge-dependent fraction is

\frac{B}{B_{50}+B}=\frac{3{,}000}{2{,}000+3{,}000}=0.60.

Therefore,

h_A=(0.60)(0.60)=0.36\ \text{h}^{-1}, \qquad h_B=(0.20)(0.60)=0.12\ \text{h}^{-1}.

After three hours, the corresponding killed fractions are

K_A(3)=1-e^{-(0.36)(3)}\approx0.660, \qquad K_B(3)=1-e^{-(0.12)(3)}\approx0.302.

The model predicts approximately 66.0% killing in A and 30.2% in B. Equal bridge abundance does not imply equal killing when the parameters governing the bridge-to-killing step differ. In an experiment, this difference could reflect effector function or target susceptibility; bridge counts alone cannot identify which explanation applies.

Deriving Functional Potency in the Simplified Model

The model also shows why functional potency need not match occupancy potency. Write \(M=N_Tgq_E\), the maximum bridge abundance as target occupancy approaches one. Then \(B=M\theta_T\). For this derivation, define functional EC50 as the concentration at which the modeled instantaneous killing hazard is half its own asymptotic maximum for that condition. This is an assay- and model-specific definition.

As \(C\) increases without bound, \(\theta_T\) approaches one, \(B\) approaches \(M\), and the limiting hazard is \(h_{\max}M/(B_{50}+M)\). At half of that limit, canceling the common factor \(h_{\max}M\) gives

\frac{\theta_T}{B_{50}+M\theta_T} = \frac{1}{2(B_{50}+M)}.

Multiplying both sides by the positive denominators and collecting terms yields

2\theta_T(B_{50}+M)=B_{50}+M\theta_T, \qquad \theta_T=\frac{B_{50}}{M+2B_{50}}.

Since \(\theta_T=C/(K_T+C)\), solving for \(C\) gives

C_{50,\mathrm{functional}} =K_T\frac{\theta_T}{1-\theta_T} =K_T\frac{B_{50}}{M+B_{50}}.

This result depends on the stated model and definition. In particular, changing \(M\) can change functional EC50 even when \(K_T\), the target-binding dissociation constant, is unchanged. It also changes the asymptotic hazard, so reporting EC50 without the maximal response can conceal an important difference in efficacy.

Worked Example: Same Target Binding, Different Functional EC50

Use hypothetical values \(K_T=10\ \text{nM}\), \(N_T=10{,}000\), \(g=0.50\), and \(B_{50}=2{,}000\). Compare conditions with \(q_E=0.50\) and \(q_E=0.80\). Their maximum bridge abundances are \(M_A=10{,}000(0.50)(0.50)=2{,}500\) and \(M_B=10{,}000(0.50)(0.80)=4{,}000\) bridges per contact. The derived functional EC50 values are

C_{50,A}=10\frac{2{,}000}{2{,}500+2{,}000} \approx4.444\ \text{nM}, \qquad C_{50,B}=10\frac{2{,}000}{4{,}000+2{,}000} \approx3.333\ \text{nM}.

The target-binding parameter \(K_T\) is the same in both conditions, but the modeled functional EC50 differs because bridge-forming capacity differs. At \(C=10\ \text{nM}\), each condition has the same target occupancy:

\theta_T=\frac{10}{10+10}=0.50.

Yet bridge abundance is \(B_A=2{,}500(0.50)=1{,}250\) in A and \(B_B=4{,}000(0.50)=2{,}000\) in B. With \(h_{\max}=0.50\ \text{h}^{-1}\), the predicted hazards are \(0.50(1{,}250/3{,}250)\approx0.1923\ \text{h}^{-1}\) and \(0.50(2{,}000/4{,}000)=0.2500\ \text{h}^{-1}\). Over two hours, predicted killing is \(1-e^{-0.1923(2)}\approx0.319\) in A and \(1-e^{-0.2500(2)}\approx0.393\) in B. Equal occupancy therefore coexists with different bridge abundance, killing rate, and functional potency in this example.

Assumptions, Diagnostics, and Alternatives

The equations are useful for organizing hypotheses, not for making every cell-engager assay follow a fixed curve. Their assumptions should be checked against the experimental design and data. The occupancy equation assumes equilibrium and effectively constant free concentration. The bridge equation compresses spatial geometry and second-arm binding into fixed factors. The killing equation assumes a constant hazard and homogeneous target cells. If these conditions are implausible, the calculated quantities remain model-based summaries rather than direct biological truths.

Useful diagnostics include comparing occupancy and bridge measurements over the same concentration range and at matched times; checking whether measured bridge abundance rises with occupancy as the proposed model predicts; and plotting killing against both occupancy and bridge abundance. If conditions with similar \(B\) have systematically different killing, the bridge-to-killing parameters likely vary, or the model omits relevant biology. If occupancy and bridge abundance do not relate as expected, investigate assay definitions, accessibility, contact formation, and whether equilibrium or fixed geometry is a defensible approximation. Replicate variability and cell-to-cell heterogeneity can also make an average occupancy a poor description of the cells contributing most to killing.

An occupancy-only analysis is direct and valuable for assessing target binding, but it cannot establish productive bridging or cytotoxic function. A bridge assay adds mechanistic resolution, but a bridge is still not a dead target cell. A killing assay measures the integrated functional outcome, yet by itself may not identify whether changes arose from binding, bridging, effector function, or target susceptibility. Joint measurements provide stronger discrimination than any single readout. Alternatively, an empirical concentration-response curve can summarize functional potency and maximal response without claiming a mechanistic bridge model; it is less explanatory about why conditions differ.

A common mistake is to reason, “Both conditions have 80% target occupancy, so they should kill the same fraction of target cells.” That conclusion does not follow: occupancy does not specify the fraction of occupied engager with a productively bound second arm, nor the killing capacity associated with each bridge. The correction is to treat occupancy, bridge abundance, and killing as separate outputs, then test or model the links between them. Functional EC50 should likewise be reported with the assay endpoint, exposure duration, and maximal response, rather than interpreted as a direct measure of target-binding affinity.

Key takeaway: Equal target occupancy constrains only one part of a cell-engager mechanism. Bridge formation and the bridge-to-killing relationship can each differ, so occupancy is not a sufficient surrogate for functional potency or killing.

Check Your Understanding

Use the definitions and simplified model above. The questions are followed by concise solutions.

If \(N_T=8{,}000\), \(\theta_T=0.75\), \(g=0.50\), and \(q_E=0.40\), what is the predicted bridge abundance?
Two conditions have equal occupancy but different \(q_E\). Which model output changes directly, and why does that not by itself determine killing?
Under a constant hazard of \(0.20\ \text{h}^{-1}\), what fraction of targets is predicted to be killed after four hours?
In the functional EC50 derivation, what does changing \(M\) represent, and why can it change functional EC50 without changing \(K_T\)?
Name one limitation of using occupancy alone and one limitation of using a killing assay alone to explain a difference between conditions.

Solutions. (1) \(B=8{,}000(0.75)(0.50)(0.40)=1{,}200\) bridges per contact. (2) Bridge abundance changes directly because \(B=N_T\theta_Tgq_E\); killing also depends on the bridge-to-killing relationship, so the changed bridge count does not uniquely specify killing. (3) \(K(4)=1-e^{-(0.20)(4)}=1-e^{-0.8}\approx0.551\), or 55.1%. (4) \(M=N_Tgq_E\) is the model’s maximum bridge abundance; changing it changes the bridge response at a given target occupancy and therefore the concentration needed for half of that condition’s asymptotic hazard. (5) Occupancy alone does not show whether the second arm forms productive bridges. Killing alone measures a functional outcome but does not identify whether its change came from binding, bridging, effector function, or target susceptibility.