Target Turnover and Free Versus Total Target

Why Free and Total Target Can Tell Different Stories

Antibody binding changes how much target is free and how much is bound in an antibody–target complex. A measurement that detects only unbound target can therefore change even when the amount of target in the system has not changed. Conversely, a measurement that detects both free and complexed target can rise as binding redistributes target into a form that the assay also counts. To interpret either measurement, the model must distinguish target turnover from binding.

We will use a minimal model in which target is synthesized continuously, free target is degraded, and antibody binding is reversible. The free antibody concentration is treated as prescribed rather than calculated from a pharmacokinetic model. This isolates the target balance and the distinction between free and total target.

Concentrations of \(R\), \(C\), and \(D\) have the same units, such as nM. The units of \(k_{\mathrm{syn}}\) are concentration per time; \(k_{\mathrm{deg}}\), \(k_{\mathrm{off}}\), and \(k_{\mathrm{on}}D\) have units of inverse time. \(K_D\) has units of concentration. We assume one-to-one binding, constant synthesis, first-order degradation of free target, and no separate loss of complex in this minimal model.

Mass Balance and Baseline Target

With \(D\) specified as a function of time, the target equations are

The binding terms remove free target when a complex forms and return it when a complex dissociates. Adding the equations cancels these internal transfers:

This balance is the central accounting result: reversible binding alone does not create or destroy total target. In this model, total target changes through synthesis and degradation of free target. The equation is not \(dR_{\mathrm{tot}}/dt=k_{\mathrm{syn}}-k_{\mathrm{deg}}R_{\mathrm{tot}}\), because that would also apply the specified degradation rate to the bound target, contrary to the model assumptions.

Before treatment, \(D=0\) and \(C=0\). At baseline equilibrium, synthesis balances free-target degradation. Write the baseline concentration as \(R_0\):

This baseline relation is useful for checking units and model inputs: a synthesis rate divided by a first-order rate constant gives a concentration. It also provides a reference for assessing treatment-associated changes.

Binding Equilibrium and the Free–Total Distinction

At binding equilibrium, formation and dissociation of complex balance:

For a constant, externally prescribed free-antibody concentration \(D\), a steady state of the full target model also requires \(dR_{\mathrm{tot}}/dt=0\). The mass balance then gives \(R=R_0\). Thus,

These steady-state results have a specific interpretation. Under the stated assumptions, antibody binding increases the steady-state total target by adding a bound pool, but does not change the steady-state free-target concentration. This does not mean that free target cannot change after treatment: binding can rapidly redistribute target, producing a transient fall in free target while total target is initially conserved. Synthesis can then replenish free target over time.

The steady-state bound fraction of total target follows from the same equations:

This fraction concerns the target pool, not the fraction of antibody bound. It is 0.5 when \(D=K_D\), and approaches 1 as \(D\) becomes much larger than \(K_D\), provided the assumptions of the model remain appropriate.

Time Course After a Change in Antibody

The full differential equations describe binding and turnover on their respective timescales. A useful approximation applies when binding and dissociation are fast relative to target turnover, so that binding remains near equilibrium as free antibody changes. This is the rapid-binding-equilibrium approximation. It is an assumption about timescales, not a consequence of target turnover alone.

Under this approximation, \(C=DR/K_D\), so \(R_{\mathrm{tot}}=R(1+D/K_D)\). Combining this relation with the target mass balance gives

For constant \(D\), the factor multiplying \(R\) is constant. The equation reduces to

Its steady state is \(R_0=k_{\mathrm{syn}}/k_{\mathrm{deg}}\), as before. If \(D\) steps from zero to a constant value at time zero, rapid binding redistributes the pre-existing target immediately: the total target does not jump, while \(R\) and \(C\) do. Thereafter, target synthesis and free-target degradation move the system toward its new steady state.

Reading Assay Results and Checking the Model

A free-target assay is intended to quantify unbound target, \(R\). A total-target assay is intended to quantify both \(R\) and \(C\), typically after a measurement procedure that detects the complexed form as well. Assay selectivity and sample handling matter: the model's \(R\) and \(R_{\mathrm{tot}}\) are defined quantities, while an actual assay may not measure them perfectly.

A useful diagnostic is to compare free and total measurements with the expected baseline and with each other. For example, a total-target value below the measured free-target value is inconsistent with the definition \(R_{\mathrm{tot}}=R+C\), unless measurement error or differing assay definitions explain it. A measured steady-state free target that differs substantially from the baseline prediction may indicate that assumptions such as unchanged synthesis, free-target degradation, or externally fixed free antibody are not adequate.

The full kinetic equations are the main alternative to the rapid-binding-equilibrium approximation. They retain finite association and dissociation rates and are appropriate when binding is not fast relative to turnover or when early-time behavior matters. The approximation is simpler for interpreting slow target turnover, but may misrepresent the initial transient if binding is not sufficiently fast. Both approaches here assume free antibody is known; a coupled drug-disposition model would be needed if free antibody changes in response to binding or clearance.

Common Pitfall and Model Limitations

Wrong reasoning: “Antibody binding lowers free target, so it must lower total target by the same amount.” This treats binding as target destruction. In the model, association transfers target from \(R\) to \(C\), and dissociation transfers it back; adding the balances shows that these terms cancel. The correction is to calculate \(R_{\mathrm{tot}}=R+C\) and account separately for synthesis and degradation.

The conclusion is conditional. This minimal model assumes that complex is not independently removed and that degradation acts only on free target. If complex is lost, target may be removed while bound; if target synthesis or degradation changes with treatment, the baseline turnover balance may change as well. Those mechanisms require different equations and can produce different free- and total-target responses. Model evaluation should therefore ask whether observed time courses and assay definitions are compatible with the assumed turnover and binding processes, rather than treating the equations as universal.