Occupancy Is Not the Same as Response
A concentration–response curve can plateau below the maximum response that the same receptor system can produce, even when the agonist occupies nearly all available receptors. The distinction matters because receptor binding describes how much drug is bound, whereas a functional assay measures the response generated after binding and signal transduction. An agonist that produces a lower maximum under the same system conditions can therefore be a partial agonist, not simply a ligand that failed to reach sufficient concentration.
The operational model makes this distinction quantitative. It represents response as a function of concentration, affinity, and the efficiency with which receptor occupancy is converted into response. The focus here is the model’s implication for partial agonism: finite operational efficacy can produce a response ceiling below the system maximum, even as concentration—and thus occupancy—increases without bound.
Operational Model and Its Assumptions
For a single agonist at concentration \([A]\), write the operational model as
Here \(E([A])\) is the measured response, \(E_0\) is baseline response, \(E_m\) is the maximum response capacity above baseline in the specified system, \(K_A\) is the agonist affinity parameter, \(\tau\) is operational efficacy, and \(n\) is the transducer slope parameter. The response increment above baseline is \(E([A])-E_0\). In this tutorial, \(n\) is positive, and comparisons of agonists hold the system and assay conditions fixed unless stated otherwise.
The model is typically interpreted under equilibrium conditions with reversible binding to one receptor population, and with a stable relationship between receptor activation and measured response. It assumes that the concentrations used are relevant to the receptor site and that the assay reports a response that is not distorted by a changing baseline, toxicity, or an independent limiting step. These assumptions matter: a low observed plateau can arise from experimental or model problems as well as from genuinely lower operational efficacy.
For reversible one-site binding, the occupied fraction is
As \([A]\) becomes very large relative to \(K_A\), this fraction approaches 1. But occupancy is not the response equation: \(\tau\) describes how effectively occupied receptors drive the response, with \(n\) shaping the transduction relation. Thus near-complete occupancy need not imply a response equal to \(E_0+E_m\).
Why a Partial Agonist Has a Lower Plateau
The high-concentration limit follows directly from the operational model. Divide its numerator and denominator by \([A]^n\), then take the limit as \([A]\) tends to infinity. Since \(K_A/[A]\) tends to zero, the response increment approaches \(E_m\tau^n/(1+\tau^n)\).
For finite positive \(\tau\), the fraction \(\tau^n/(1+\tau^n)\) is less than 1. Therefore the limiting response is below \(E_0+E_m\). It increases as \(\tau\) increases and approaches \(E_m\) as \(\tau\) becomes very large. In this model, a full agonist is a useful functional comparator that approaches the system maximum under the stated conditions; “full” does not mean that another ligand cannot bind more completely.
The same result also clarifies the role of affinity. \(K_A\) appears in the concentration-dependent part of the equation, so it affects how the curve develops with concentration. It does not appear in the limiting response increment. A difference in \(K_A\) can shift the concentration range over which a response is observed, but a lower \(K_A\) alone does not explain a lower model-predicted plateau when \(\tau\), \(n\), and \(E_m\) are held fixed.
Worked Example: Near-Complete Occupancy, Lower Maximum
Suppose two hypothetical agonists are tested in the same system, with \(E_m=100\) response units above baseline, \(K_A=10\) nM, and \(n=1\). Let the first agonist have \(\tau=1\), and the comparator have \(\tau=9\). Calculate their limiting response increments and compare their predicted responses at 1,000 nM.
For \(n=1\), the model’s response increment is \(E_m\tau[A]/(K_A+(1+\tau)[A])\). At saturation, the first agonist reaches
The comparator reaches \(100(9)/(1+9)=90.0\) response units. At 1,000 nM, the occupancy for either agonist, given the shared \(K_A\), is
or about 99.0%. Yet their response increments differ. For the agonist with \(\tau=1\),
For the comparator with \(\tau=9\), the increment is \(100(9000)/(10+10(1000))=89.91\) response units. Both are nearly at their respective plateaus, although occupancy is the same. The lower maximum for the first agonist is explained by lower operational efficacy, not by incomplete occupancy.
Reading Efficacy and Potency Together
A lower maximum and a higher concentration requirement are different features of a concentration–response curve. In the operational model, affinity and operational efficacy both influence the curve’s position, while operational efficacy also determines its limiting response. For a fixed agonist’s own maximum response increment, the concentration giving half that increment is
To see why, express the response increment as a fraction of its limiting value. That fraction is \((1+\tau^n)[A]^n/((K_A+[A])^n+\tau^n[A]^n)\). Setting it to one half and rearranging gives \((K_A+[A])^n=(2+\tau^n)[A]^n\); solving for \([A]\) yields the expression above. For \(n=1\), it reduces to \(EC_{50}=K_A/(1+\tau)\).
This \(EC_{50}\) is defined relative to the agonist’s own model-predicted maximum, not to the system maximum \(E_m\). A partial agonist can therefore have a lower \(EC_{50}\) than another agonist and still have a lower maximum response. Potency, affinity, and operational efficacy should not be treated as interchangeable descriptions.
Worked Example: Calculating a Partial Agonist’s Plateau and EC50
Consider a hypothetical agonist with \(E_m=120\) response units, \(K_A=4\) µM, \(n=1\), and \(\tau=0.5\). Calculate its limiting response increment and its \(EC_{50}\). Then calculate occupancy and response increment at 20 µM.
The limiting increment is
The concentration producing half of this agonist’s own maximum is
At 20 µM, occupancy is \(20/(4+20)=0.8333\), or 83.33%. The response increment is
That response is below the 40.0-unit plateau, as expected at finite concentration. The calculation illustrates why occupancy and response cannot be equated: 83.33% occupancy is not 83.33% of the system’s maximum response. The response is instead determined by the operational model.
Comparisons, Diagnostics, and Limits
The operational model is useful because it relates occupancy, transduction, and maximal response in one framework. An occupancy-only account can describe binding but cannot, by itself, predict how occupied receptors translate into a measured effect. A simple empirical Hill curve can summarize a functional curve’s slope, midpoint, and plateau, but those fitted features do not by themselves distinguish affinity from operational efficacy. The operational model adds a mechanistic interpretation, subject to its assumptions; that interpretation should not be mistaken for direct proof of a particular molecular mechanism.
When a curve appears to have a low plateau, first ask whether a plateau has actually been observed. If concentrations do not extend sufficiently high, the apparent maximum may be an extrapolation. Check that the concentration range approaches saturation relative to the estimated \(K_A\), that the assay remains valid at high concentrations, and that response measurements are not limited by the instrument or an unrelated biological bottleneck. Replicate data and a fitted curve with uncertainty are more informative than a single highest-concentration observation.
Parameter interpretation also requires care. The fitted \(\tau\) is conditional on the system, including receptor abundance and downstream coupling, so it is not automatically a ligand property that remains unchanged across tissues. Comparisons that claim reduced efficacy should use the same system and assay conditions, or explicitly account for differences in system capacity. In addition, a fitted plateau can be difficult to distinguish from \(E_m\) when data do not identify the system maximum independently. A good numerical fit does not guarantee that every parameter is separately well identified.
This framework has limitations. Real systems may depart from equilibrium, contain heterogeneous receptors, exhibit time-dependent adaptation, or include multiple downstream processes. A low plateau can also reflect desensitization, toxicity, solubility limits, or assay ceiling effects rather than partial agonism. The operational model is a structured interpretation of functional data, not a substitute for checking the binding, experimental, and measurement assumptions that make that interpretation plausible.
Worked Example: Checking Whether a Low Plateau Supports Partial Agonism
In a hypothetical assay, a comparator agonist has an established system maximum \(E_m=80\) response units above baseline. A test agonist’s concentration–response data are consistent with \(n=1\) and a limiting increment of 32 units. The tested concentrations are also high relative to its estimated \(K_A\), and the assay remains within its validated measurement range. Under the operational model, estimate \(\tau\) and state what the result supports.
For \(n=1\), the limiting increment \(M\) satisfies \(M=E_m\tau/(1+\tau)\). Rearranging gives \(M(1+\tau)=E_m\tau\), hence \(\tau=M/(E_m-M)\). Substituting the hypothetical values,
The estimated operational efficacy is finite and below that of a comparator that approaches the system maximum. Because the data support a plateau, concentrations are high relative to \(K_A\), and the assay range is adequate, the lower limiting response is consistent with partial agonism in this system. It does not establish that the same \(\tau\) would apply in another tissue, nor does it prove the model assumptions; those remain part of the interpretation.
Common Mistake
A common error is to reason, “The agonist occupies nearly all receptors, so it must produce the maximum system response; a lower observed plateau means the concentration was not high enough.” The correction is that occupancy and transduction are distinct. In the operational model, occupancy can approach 1 while the response increment approaches \(E_m\tau^n/(1+\tau^n)\), which remains below \(E_m\) for finite \(\tau\). Insufficient concentration is one possible explanation for a low observed response, but it is not the only one and should not be assumed once a well-supported plateau has been reached.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For \(n=1\) and \(\tau=2\), what fraction of \(E_m\) is the limiting response increment?
- Can an agonist with high receptor occupancy still be a partial agonist? Explain using the operational model.
- For \(n=1\), \(K_A=6\) µM, and \(\tau=1\), calculate the model \(EC_{50}\) and the limiting increment as a fraction of \(E_m\).
- Why does a difference in \(K_A\), by itself, not change the limiting response increment in this model?
- Name two reasons, other than lower operational efficacy, that could produce an apparently low plateau in an assay.
Solutions. (1) The fraction is \(\tau/(1+\tau)=2/3=0.6667\), or 66.67% of \(E_m\). (2) Yes. Occupancy can approach 1 as concentration rises, while the response increment approaches \(E_m\tau^n/(1+\tau^n)\), which is below \(E_m\) for finite \(\tau\). (3) \(EC_{50}=K_A/(1+\tau)=6/2=3.000\) µM, and the limiting fraction is \(1/(1+1)=0.5000\), or 50.00% of \(E_m\). (4) \(K_A\) affects the concentration-dependent terms but cancels from the high-concentration limit, which is \(E_m\tau^n/(1+\tau^n)\). (5) Examples include an insufficient concentration range to establish a plateau, assay ceiling or measurement limitations, toxicity, or a solubility limit.