Identifiability in a Target-Mediated Binding Model

Why the Observation and the Mass Balance Both Matter

A free-drug concentration is not a direct measurement of every process that determines drug-target binding. Some information depends on whether measurements capture the binding transient or only equilibrium, and on whether the target pool is finite or effectively constant. Writing the mass balances makes these distinctions explicit: the same free-drug output can identify separate kinetic rates in one model and only a rate combination in another.

We first analyze an idealized, closed system containing free drug \(D\), free target \(R\), and complex \(C\). Binding is reversible, and there is no drug clearance, target turnover, or complex internalization during observation. Concentrations are absolute and calibrated. The total drug concentration \(D_T\) is known, while the total target concentration \(R_T\) may be known or unknown but is constant. These assumptions are important: conclusions from this model do not automatically carry over to an open system with elimination or target turnover.

The state equations and conservation relations are

Suppose the measured output is free drug \(D(t)\). If \(D_T\) is known, then \(C(t)=D_T-D(t)\) follows from drug mass balance, even if the assay does not measure complex directly. Likewise, \(R(t)=R_T-C(t)\) if \(R_T\) is known. The key question is therefore not simply which state is observed, but what the mass balances allow us to reconstruct and whether the observations contain a transient.

Finite-Pool Binding: Information in a Transient

Substitute both conservation relations into the equation for complex:

This is a quadratic polynomial in the observable-derived quantity \(C\). With an ideal, continuously observed, nonconstant transient, its three coefficients can in principle be recovered from the relationship between \(dC/dt\) and \(C\). Denote the coefficients of \(C^2\), \(C\), and the constant term by \(a_2\), \(a_1\), and \(a_0\), respectively. Then \(a_2=k_{\mathrm{on}}\), \(a_1=-[k_{\mathrm{on}}(D_T+R_T)+k_{\mathrm{off}}]\), and \(a_0=k_{\mathrm{on}}D_TR_T\). If \(D_T\) is known and positive, these give

Thus, in this ideal finite-pool model, free-drug observations alone can generically identify both kinetic rate constants, even when \(R_T\) was not known in advance. “Generically” matters: there must be a measurable binding transient, the concentration scale and \(D_T\) must be known, and the parameters must be in a nondegenerate range. At a single constant equilibrium state, there is no transient from which to recover those polynomial coefficients.

Equilibrium Measurements Identify Affinity, Not the Clock

At equilibrium, \(dC/dt=0\). For positive free drug and free target, the mass-action equation becomes \(k_{\mathrm{on}}DR=k_{\mathrm{off}}C\), or

This relation contains the ratio of the two kinetic rates, not their absolute time scale. Multiplying both \(k_{\mathrm{on}}\) and \(k_{\mathrm{off}}\) by the same positive factor leaves the equilibrium unchanged. If total drug and target are known, equilibrium free-drug data can determine \(K_D\) through mass balance, but they cannot separate the two rates. Multiple equilibrium measurements at different total concentrations can add information about pool sizes or affinity; they still do not reveal the shared rate scale without transient information or another time-sensitive measurement.

Target-Excess Approximation: A Rate Product

A different conclusion applies when free target is treated as approximately constant. This target-excess or negligible-depletion approximation can be useful when binding removes only a small fraction of the available target. Let that nearly constant free-target concentration be \(R_0\). Because total drug is conserved, \(C=D_T-D\). The free-drug equation reduces to

For a complete transient with known \(D_T\), this first-order equation identifies the dissociation rate from the plateau and the sum of rates from the exponential rate constant. If \(D_\infty\) is the limiting free-drug concentration and \(\lambda\) is the positive decay rate, then \(D_\infty=D_Tk_{\mathrm{off}}/\lambda\) and \(\lambda=k_{\mathrm{on}}R_0+k_{\mathrm{off}}\). Hence the identifiable quantities are \(k_{\mathrm{off}}\) and the product \(k_{\mathrm{on}}R_0\), not \(k_{\mathrm{on}}\) and \(R_0\) separately. For any \(q>0\), replacing \(k_{\mathrm{on}}\) by \(qk_{\mathrm{on}}\) and \(R_0\) by \(R_0/q\) leaves the free-drug trajectory unchanged.

Choosing an Output and Diagnosing the Analysis

The value of a target-related output depends on the model and on what is already recoverable from mass balance. In the ideal finite-pool model, with known \(D_T\) and an informative transient, free drug reconstructs \(C\), and the transient can identify \(k_{\mathrm{on}}\), \(k_{\mathrm{off}}\), and \(R_T\). A simultaneous complex measurement is then redundant in the structural sense. It may still be useful in practice to check the assay scale or model fit. In the target-excess approximation, measuring \(R_0\) can separate \(k_{\mathrm{on}}\) from the product \(k_{\mathrm{on}}R_0\). Measuring only a drug-derived complex concentration does not resolve that confounding.

Several checks help establish whether the stated conclusions apply:

• Check that the experiment observes a transient. If only equilibrium values are available, the absolute rate scale is missing; equilibrium identifies affinity, not separate kinetic rates.
• Check the mass-balance quantities. Unknown or incorrectly scaled \(D_T\), unrecorded dilution, or loss of drug breaks the reconstruction \(C=D_T-D\).
• Check the target-excess approximation. If target depletion is appreciable, treating \(R\) as \(R_0\) can distort the inferred rate product. The full finite-pool equations may be needed.
• Check the output scale and time scale. A multiplicative concentration-calibration error or an incorrect time unit changes rate estimates and can invalidate the assumed absolute-scale analysis.
• Check for omitted processes. Drug elimination, target synthesis or degradation, and complex internalization are absent from the closed reversible model. If present, they must be represented before interpreting fitted binding rates.

The finite-pool transient approach and the target-excess approximation are not competing answers to the same equations: they are different models with different assumptions. Equilibrium analysis is more limited about kinetic rates but can be appropriate when only steady-state measurements are collected. A target-related assay is most useful when it measures a quantity not already determined from observed drug and known totals, such as free target in the target-excess setting. No output automatically fixes a misspecified model.

Limits of the Conclusions

Structural identifiability asks whether ideal, noise-free observations under a specified model determine parameter values uniquely. It does not guarantee that sparse, noisy measurements estimate them precisely. In particular, derivative-based recovery of quadratic coefficients can be highly sensitive to measurement error, and a transient that is very small or very fast may be poorly resolved. Those issues concern practical information in the data, not the algebraic distinction between a unique parameter and an exact confounded combination.

The results here also rely on reversible binding and known concentration calibration. If the experiment includes elimination or turnover, the mass balances change; a free-drug trajectory may then combine binding and loss processes in ways not captured by these equations. The appropriate next step is to write the expanded model’s balances and observation equations, rather than to transfer an identifiability conclusion from the closed system unchanged.