Why Availability and Occupancy Must Be Separated
A fractional occupancy is a proportion, not an amount of bound target. If two tissues have the same fraction of target sites occupied but different amounts of binding-competent target, they have different amounts of bound target. Conversely, a change in the measured amount of bound target does not by itself show that occupancy changed: the target pool may have grown or shrunk.
The distinction matters when interpreting binding measurements, comparing conditions with different target abundance, or constructing a mechanism-based model. The key is to state the denominator for every fraction. Here, target availability means the abundance of the target pool that is present and capable of binding the ligand under consideration. Fractional occupancy is defined within that pool. Target that is absent or not binding-competent is not included in that occupancy denominator.
The terminology can be ambiguous across assays and models. In particular, “available” is sometimes used informally to mean unoccupied. To avoid that ambiguity, distinguish the binding-competent pool \(A\) from its unoccupied amount \(A-B=A(1-\theta)\). In this tutorial, \(f_A\) describes the size of the binding-competent pool relative to a stated reference; it does not mean the fraction left unoccupied after ligand binding.
From Free Ligand to Bound-Target Amount
For a single reversible ligand binding to a homogeneous target pool at equilibrium, fractional occupancy is determined by the free ligand concentration and the dissociation constant. If \(C\) is free ligand concentration and \(K_D\) is the dissociation constant, the equilibrium relation is \(\theta=C/(K_D+C)\). This expression assumes one ligand-binding site per target unit, equivalent independent sites, reversible binding, and equilibrium. It uses free ligand concentration, not total ligand concentration.
The first equation gives a fraction, while the second converts that fraction to an amount. If the free ligand concentration and affinity remain constant, changing target availability changes \(B\) proportionally while leaving \(\theta\) unchanged. That conclusion depends on the free concentration actually remaining constant. If the system has substantial ligand depletion, changing target abundance can change free ligand concentration as well, and occupancy may change too. The binding mass balance is then needed to determine free concentration; the simple proportional comparison cannot be applied while holding \(C\) fixed without justification.
Worked Example: Converting Occupancy to Bound Target
Suppose a hypothetical sample has \(T_{\mathrm{ref}}=120\) pmol of target. Only 75% is binding-competent, so \(f_A=0.75\). At equilibrium, free ligand concentration is \(C=3.0\) nM and \(K_D=2.0\) nM. Assume the single-site equilibrium model applies and that the stated free concentration is maintained.
First calculate the available target amount:
Then calculate occupancy:
Finally, multiply occupancy by the available amount:
The unoccupied amount within the binding-competent pool is \(90-54=36.0\) pmol. The remaining \(120-90=30.0\) pmol is outside the defined available pool. It is therefore incorrect to call all \(66.0\) pmol “unoccupied available target”: only 36.0 pmol is unoccupied within that pool.
When Target Abundance Changes
Suppose target abundance changes between conditions. If the assay quantifies the binding-competent amount directly, use that amount as \(A\); if it reports abundance relative to a reference, first calculate \(A=f_A T_{\mathrm{ref}}\). In either case, the occupancy denominator must match the pool used to calculate the bound amount. A fraction measured relative to all target cannot be multiplied by a smaller accessible pool unless the fraction has been redefined for that pool.
The product \(B=A\theta\) also clarifies how bound target can change over time. If both availability and occupancy vary smoothly, differentiating the product gives:
The first term represents change in bound amount attributable to changing available target, evaluated at the current occupancy. The second represents change attributable to changing occupancy, evaluated at the current available amount. The equation is an accounting identity; it does not specify the biological processes that cause either change. Those processes require an appropriate model of target production, loss, binding, or other relevant mechanisms.
Worked Example: More Target, Same Fractional Occupancy
Consider two hypothetical conditions. In condition 1, total target abundance is 100 pmol and 80% is binding-competent. In condition 2, total target abundance is 150 pmol and the same fraction, 80%, is binding-competent. Assume the ligand conditions maintain \(C=4.0\) nM and \(K_D=4.0\) nM in both conditions, with equilibrium single-site binding.
In each condition, occupancy is:
Condition 1 has \(A_1=(0.80)(100\ \mathrm{pmol})=80\) pmol available, so \(B_1=(80\ \mathrm{pmol})(0.500)=40\) pmol. Condition 2 has \(A_2=(0.80)(150\ \mathrm{pmol})=120\) pmol available, so \(B_2=(120\ \mathrm{pmol})(0.500)=60\) pmol.
Bound target rises by 20 pmol, or 50% relative to condition 1, even though fractional occupancy remains 0.500 in both conditions. The interpretation relies on the assumed equal free ligand concentration. If the increase in target abundance depleted ligand, \(C\) might be lower in condition 2, changing \(\theta\) as well as \(A\).
Worked Example: Occupancy Falls While Bound Target Also Falls
At a hypothetical instant, the binding-competent target amount is \(A=100\) pmol and occupancy is \(\theta=0.25\). Suppose target availability is increasing at \(dA/dt=12\) pmol/h, while occupancy is decreasing at \(d\theta/dt=-0.040\ \mathrm{h}^{-1}\). Calculate the bound amount and its instantaneous rate of change.
The current bound amount is:
Apply the product rule:
Although the available target pool is growing, the decrease in occupancy is larger at this instant, so bound target is declining at 1.0 pmol/h. Considering only the increase in \(A\) would give the wrong direction for the net change.
Assumptions, Checks, and Alternative Summaries
Before using \(B=A\theta\), check that \(A\) and \(B\) refer to the same target pool, are expressed in compatible amount units, and refer to the same time and biological compartment. The product identity itself does not require equilibrium. However, calculating \(\theta\) from \(C/(K_D+C)\) does require the stated equilibrium and binding assumptions, including an appropriate free ligand concentration. A concentration of target and an amount of bound target cannot be multiplied directly unless the relevant volume conversion is made.
The main alternative summaries answer different questions. Fractional occupancy compares binding across conditions while normalizing for the size of the binding-competent pool. Bound amount reports how much target is bound, which can be more relevant when total bound target drives a downstream process. Reporting both is often informative: occupancy alone can conceal a large abundance change, while bound amount alone cannot distinguish an occupancy change from an abundance change.
Useful diagnostics include checking that \(0\leq\theta\leq1\), that \(0\leq B\leq A\), and that calculated unoccupied target \(A-B\) is nonnegative. A reported \(B>A\) signals inconsistent denominators, units, sampling times, or measurements, or a model that does not match the measured quantities. If observed bound amount changes while calculated occupancy is stable, investigate changes in target abundance before attributing the difference to altered affinity or ligand exposure. If occupancy is calculated from total rather than free ligand under depletion, the inferred occupancy may also be biased.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A sample has 200 fmol of reference target, of which 50% is binding-competent. If occupancy within that pool is 30%, what are the available, bound, and unoccupied amounts?
- If available target doubles while free ligand concentration and \(K_D\) remain unchanged under the equilibrium model, what happens to occupancy and bound amount?
- Explain why 40% occupancy does not necessarily mean that 40% of the reference target amount is bound.
- At an instant, \(A=80\) pmol, \(\theta=0.50\), \(dA/dt=-4\) pmol/h, and \(d\theta/dt=0.020\ \mathrm{h}^{-1}\). Calculate \(B\) and \(dB/dt\).
Solutions. (1) Available amount is \((0.50)(200)=100\) fmol. Bound amount is \((0.30)(100)=30\) fmol, and unoccupied available amount is \(100-30=70\) fmol. (2) Occupancy is unchanged because the assumed free concentration and \(K_D\) are unchanged; bound amount doubles because \(B=A\theta\). (3) Occupancy is the fraction of the binding-competent pool that is bound. If only a fraction of reference target is binding-competent, the bound fraction of reference target is \(f_A\theta\), not \(\theta\). (4) \(B=(80\ \mathrm{pmol})(0.50)=40\) pmol. The rate is \((0.50)(-4\ \mathrm{pmol/h})+(80\ \mathrm{pmol})(0.020\ \mathrm{h}^{-1})=-2+1.6=-0.400\) pmol/h.