Identifiability of Affinity and Occupancy Parameters

Why Identifiability Matters

A binding model can fit observed data closely and still fail to determine the parameter values that generated them. For target occupancy, this matters because affinity, free drug concentration, and the amount of available target answer different scientific questions. If the data only determine a ratio or a curve-shaped combination of these quantities, reporting each as if it were separately estimated gives a false impression of what the experiment established.

The central issue is not whether an optimization algorithm returns a unique answer. It is whether the model and measurement design contain enough information to distinguish plausible parameter values. A numerical fit can return one point on a broad ridge of nearly equivalent fits; changing starting values or small details of the data may then change the reported parameters substantially.

Binding Model and Identifiability

Consider a reversible, one-to-one interaction between free drug and a homogeneous population of available binding sites. Assume binding is at equilibrium, free concentration is constant during measurement, and there is no cooperativity or irreversible binding. Let \(C_f\) be free drug concentration, \(K_D\) the equilibrium dissociation constant, and \(B_{\max}\) the maximal amount of specifically bound drug when all available sites are occupied. The model for bound amount \(B\) and fractional occupancy \(\theta\) is:

B = B_{\max}\frac{C_f}{K_D+C_f}, \qquad \theta = \frac{B}{B_{\max}}=\frac{C_f}{K_D+C_f}.

Here \(C_f\) and \(K_D\) must use the same concentration units; \(B\) and \(B_{\max}\) use the same binding-capacity units. The model treats the measured binding signal as a valid measure of \(B\), or the occupancy measurement as a valid estimate of \(\theta\). If free concentration is replaced by total concentration, that substitution requires a separate justification: free and total concentrations need not be equal.

Definition: A parameter is structurally identifiable if distinct values of that parameter cannot produce exactly the same model predictions under the stated design, assuming ideal noiseless data. It is practically identifiable if the available, finite and noisy observations constrain it sufficiently well to estimate with useful precision. Structural identifiability is a property of the model and design; practical identifiability also depends on the data and their precision.

Identifiability is always conditional on what is observed. A fractional occupancy measurement contains no direct information about \(B_{\max}\), because dividing bound amount by capacity removes that parameter. Likewise, if \(C_f\) is unknown, occupancy alone does not reveal an absolute concentration scale.

Worked Example: One Occupancy Measurement Leaves a Continuum of Answers

Suppose a hypothetical experiment reports fractional occupancy \(\theta=0.60\), but the free concentration during the measurement was not independently determined. Use the equilibrium model to describe the parameter combinations compatible with this result.

Starting from \(\theta=C_f/(K_D+C_f)\), rearrange:

\theta(K_D+C_f)=C_f, \qquad \theta K_D=C_f(1-\theta), \qquad \frac{C_f}{K_D}=\frac{\theta}{1-\theta}.

Substitution gives \(C_f/K_D=0.60/0.40=1.50\). Thus, any positive pair with \(C_f=1.50K_D\) predicts the observed occupancy. For example, \(C_f=15\) nM and \(K_D=10\) nM give \(15/(10+15)=0.60\); \(C_f=150\) nM and \(K_D=100\) nM give the same result. The data identify a ratio, not the separate concentration and affinity values. They provide no information about \(B_{\max}\) if only fractional occupancy was measured.

Exact Ambiguities in Sparse Data

A single measurement of absolute bound amount does not generally solve the problem. Even if \(C_f\) is known, one observation \(B\) gives one equation involving both \(B_{\max}\) and \(K_D\). For every admissible \(K_D>0\), the value

B_{\max}=B\frac{K_D+C_f}{C_f}

produces that same observation. A second concentration can add information, but only if the concentrations and measurements distinguish the curve's capacity from its curvature.

When free concentrations are unknown, there can be an exact scale ambiguity even with multiple observations. For any positive constant \(a\), replace every free concentration \(C_f\) with \(aC_f\) and replace \(K_D\) with \(aK_D\). Then:

\frac{aC_f}{aK_D+aC_f}=\frac{C_f}{K_D+C_f}.

Consequently, all predicted occupancies and bound amounts remain unchanged if \(B_{\max}\) is unchanged. Without independent information fixing the free-concentration scale, the data cannot distinguish these rescaled concentration-affinity pairs. A known administered dose does not by itself remove this ambiguity: a dose is not a direct measurement of the free concentration at the binding site.

Key result: With known \(C_f\), a fractional occupancy at one concentration identifies \(K_D=C_f(1-\theta)/\theta\), but not \(B_{\max}\). An absolute bound amount at one concentration generally identifies neither \(B_{\max}\) nor \(K_D\) separately. If the free-concentration scale is unknown, scaling \(C_f\) and \(K_D\) by the same positive factor leaves model predictions unchanged.

When Multiple Concentrations Identify the Curve

With known free concentrations and absolute bound amounts at multiple concentrations, the shape of the binding curve can separate capacity from affinity. For two distinct concentrations \(C_1\) and \(C_2\), with corresponding positive bound amounts \(B_1\) and \(B_2\), define \(R=B_2/B_1\). Under the stated model:

R=\frac{C_2(K_D+C_1)}{C_1(K_D+C_2)}.

Solving this equation for \(K_D\) gives the following result when the denominator is nonzero:

K_D=\frac{C_1C_2(R-1)}{C_2-RC_1}, \qquad B_{\max}=B_1\frac{K_D+C_1}{C_1}.

This calculation also exposes a design limitation: two points can yield a formal solution, but that does not guarantee a precise estimate. The concentration range must make the ratio informative, the measurements must be reliable, and the model assumptions must be appropriate. Replication and additional concentration levels allow the fit to be checked rather than determined by exactly two observations.

Worked Example: Two Concentrations Estimate Affinity and Capacity

In a hypothetical binding experiment, free concentrations are independently measured as \(C_1=2\) nM and \(C_2=8\) nM. Calibrated absolute bound amounts are \(B_1=33.33\) pmol/mg and \(B_2=66.67\) pmol/mg. Assume the equilibrium, one-to-one model and that both measurements are positive and expressed on the same binding-capacity scale. Estimate \(K_D\) and \(B_{\max}\).

First calculate \(R=66.67/33.33\approx2.000\). Substituting into the ratio formula:

K_D = \frac{(2\ \mathrm{nM})(8\ \mathrm{nM})(2.000-1)} {8\ \mathrm{nM}-(2.000)(2\ \mathrm{nM})} = \frac{16\ \mathrm{nM}^2}{4\ \mathrm{nM}} = 4.000\ \mathrm{nM}.

Then use the first observation to obtain capacity:

B_{\max} = (33.33\ \mathrm{pmol/mg}) \frac{4.000\ \mathrm{nM}+2\ \mathrm{nM}}{2\ \mathrm{nM}} = 99.99\ \mathrm{pmol/mg} \approx100.0\ \mathrm{pmol/mg}.

The values are internally consistent: at \(C_f=2\) nM the model predicts \(100(2)/(4+2)=33.33\) pmol/mg, and at \(C_f=8\) nM it predicts \(100(8)/(4+8)=66.67\) pmol/mg. These two distinct, known concentrations provide a formal solution under the assumed model. In a real design, more levels and replicates would be needed to assess precision and whether the curve describes the observations adequately.

Practical Identifiability and Concentration Range

Even when parameters are structurally identifiable, a poorly chosen concentration range can make them practically indistinguishable. The derivatives of the bound-amount model show where information is concentrated:

\frac{\partial B}{\partial B_{\max}}=\frac{C_f}{K_D+C_f}, \qquad \frac{\partial B}{\partial K_D} = -\frac{B_{\max}C_f}{(K_D+C_f)^2}.

At concentrations far below \(K_D\), the model is approximately \(B\approx(B_{\max}/K_D)C_f\). Measurements in this range mainly inform the ratio \(B_{\max}/K_D\); increasing capacity while increasing affinity's dissociation constant can leave predictions nearly unchanged. At concentrations far above \(K_D\), \(B\approx B_{\max}\), so the plateau informs capacity but changes little with \(K_D\). To estimate both parameters well, measurements should generally include concentrations around the transition, where the curve bends, as well as observations that help define the plateau.

Worked Example: Low-Concentration Data Can Leave a Broad Parameter Ridge

Suppose a hypothetical study measures absolute binding at known free concentrations of 0.2 and 0.4 nM, with an assay standard deviation of 0.2 pmol/mg. Compare two parameter pairs: \(B_{\max}=100\) pmol/mg and \(K_D=10\) nM, versus \(B_{\max}=200\) pmol/mg and \(K_D=20\) nM. Both satisfy \(B_{\max}/K_D=10\) pmol/mg per nM.

For the first pair, the predictions are:

B(0.2)=100\frac{0.2}{10+0.2}=1.961\ \mathrm{pmol/mg}, \qquad B(0.4)=100\frac{0.4}{10+0.4}=3.846\ \mathrm{pmol/mg}.

For the second pair:

B(0.2)=200\frac{0.2}{20+0.2}=1.980\ \mathrm{pmol/mg}, \qquad B(0.4)=200\frac{0.4}{20+0.4}=3.922\ \mathrm{pmol/mg}.

The predictions differ by only 0.0194 and 0.0754 pmol/mg, respectively, both smaller than the stated 0.2 pmol/mg standard deviation. These parameter pairs are not exactly equivalent at these nonzero concentrations, but the observations are unlikely to distinguish them precisely at this noise level. The concentration range is far below both candidate \(K_D\) values, so the data primarily constrain the shared ratio \(B_{\max}/K_D\), not each parameter separately.

Design and Diagnostic Checks

A useful design begins by deciding which quantities are measured directly and which are inferred. The following actions address different sources of ambiguity:

Measure free concentration independently. This anchors the concentration scale and helps distinguish \(K_D\) from an unknown concentration multiplier.
Use a concentration range that spans the curve. Include levels below, around, and above the anticipated \(K_D\), where feasible, so the data contain information about the low-concentration slope, transition, and plateau.
Measure absolute binding when capacity is a target. Fractional occupancy alone does not estimate \(B_{\max}\). Alternatively, obtain an independent capacity measurement under conditions that support its interpretation.
Use more than the minimum number of levels. Replicate measurements and additional concentrations help reveal whether a small number of points is driving the fitted parameters.
Check parameter stability. Compare estimates across reasonable starting values and inspect profile likelihoods or uncertainty summaries. A flat profile, very wide interval, or strong parameter correlation signals weak identification, even if the optimizer reports convergence.

A short identifiability check can be organized as a sequence:

State the observation model.
Specify whether the data are fractional occupancy or absolute bound amount, and whether the concentration is free, total, or latent.

Look for exact invariances.
Ask whether a transformation of \(C_f\), \(K_D\), or \(B_{\max}\) leaves all predictions unchanged. If it does, those quantities cannot all be separately identified from the current data.

Assess information in the observed range.
Determine whether the measurements cover the transition and plateau, or only a region where predictions depend mainly on a parameter combination.

Modify the design or narrow the claim.
Add an independent measurement or informative concentration levels, or report only the combination the data support.

Alternatives, Limitations, and Common Pitfalls

An experiment can address ambiguity through direct measurement, design, or external information. Measuring free concentration is useful when the concentration scale is uncertain; adding a saturating range is useful when capacity is uncertain; and an independent estimate of capacity can help isolate affinity from absolute binding data. Fitting the same sparse observations with a more flexible numerical method does not create information. Likewise, a Bayesian prior can yield a finite posterior estimate, but when the likelihood is structurally uninformative, the result may depend strongly on that prior and should not be described as data-only identification.

These conclusions apply to the specified equilibrium model. If binding is not at equilibrium, sites are heterogeneous, binding is cooperative, or the assay does not measure the assumed quantity, estimates may be misleading for reasons beyond identifiability. The model assumptions must therefore be considered before interpreting a seemingly precise parameter estimate. This tutorial's analysis does not establish that those assumptions hold; it shows what the model's data can identify if they do.

Common pitfall: “The optimizer converged and returned a \(K_D\) and \(B_{\max}\), so both are identified.” Convergence only means the algorithm found a numerical solution under its settings. If many parameter pairs give the same or nearly the same predictions, the separate estimates are not well determined. Check for exact invariances, examine parameter profiles and stability, and report the identifiable combination rather than overstating individual precision.

Identifiability is important because a parameter estimate is useful only when its meaning is supported by the experiment. When the design has not separated free concentration, affinity, and capacity, the scientifically defensible conclusion may be that occupancy or a parameter combination is estimated, while the individual mechanistic quantities remain uncertain.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

If \(\theta=0.75\) and free concentration is unknown, what ratio \(C_f/K_D\) is identified?
Why can fractional occupancy measurements alone not estimate \(B_{\max}\)?
With known free concentrations, what additional feature makes multiple binding observations useful for separating \(K_D\) and \(B_{\max}\)?
What information is most strongly informed when all measured concentrations are much smaller than \(K_D\)?
Why does optimizer convergence not establish practical identifiability?

Solutions. (1) \(C_f/K_D=\theta/(1-\theta)=0.75/0.25=3\). (2) Fractional occupancy is \(B/B_{\max}=C_f/(K_D+C_f)\), so \(B_{\max}\) cancels from the observation model. (3) Distinct known concentrations that reveal curve curvature, supported by measurements precise enough to distinguish predictions, allow capacity and affinity to be separated; data near the transition and plateau are particularly informative. (4) Since \(B\approx(B_{\max}/K_D)C_f\), the data mainly inform the ratio \(B_{\max}/K_D\). (5) Convergence gives a numerical solution, but a flat likelihood or near-equivalent parameter combinations can leave the estimates highly uncertain.