Population Variability in Target Affinity

Why Population Affinity Changes Occupancy

At a fixed free-drug concentration, two people with different target affinities need not have the same fractional occupancy. Even if their concentrations are identical, a person with a larger dissociation constant \(K_d\) will generally have lower equilibrium occupancy. A population model therefore predicts a distribution of occupancies, not just one value. That distinction matters when a population mean is used to summarize target engagement or when the spread of individual responses is scientifically important.

The derivations here concern a reversible, one-site binding model at equilibrium. The free concentration is assumed to be constant and shared across subjects, and binding is assumed to be noncooperative. Concentration and \(K_d\) must use the same units. In this setting, each subject’s affinity determines their equilibrium occupancy. If concentration changes over time and binding has not equilibrated, occupancy also depends on that subject’s \(k_{on}\) and \(k_{off}\) and on the concentration history.

A Lognormal Model for Between-Subject Affinity

A common population model represents positive \(K_d\) values using a normal distribution on the log scale:

Here \(K_{d,\mathrm{typ}}\) is the typical value: under this model it is the median and geometric mean of the population’s \(K_d\) distribution. The parameter \(\omega_{Kd}\) is the standard deviation of log affinity, a dimensionless measure of between-subject variability. This model implies that \(K_{d,i}\) is lognormally distributed and positive. It does not say that the arithmetic population mean \(K_d\) equals \(K_{d,\mathrm{typ}}\); in fact, \(E[K_{d,i}]=K_{d,\mathrm{typ}}\exp(\omega_{Kd}^{2}/2)\).

Substituting the lognormal model into the occupancy equation gives a useful result. Taking the logit, defined by \(\operatorname{logit}(\theta)=\log\{\theta/(1-\theta)\}\), transforms occupancy into a linear function of log affinity:

Thus the logit of individual occupancy is normal, with mean \(\log(C/K_{d,\mathrm{typ}})\) and variance \(\omega_{Kd}^{2}\). Equivalently, occupancy has a logit-normal distribution. Its quantiles can be calculated directly. If \(z_q\) is the \(q\)th standard-normal quantile, then:

The median occupancy is the occupancy at \(K_{d,\mathrm{typ}}\), because the median random effect is zero. The population mean occupancy, however, is generally not occupancy at the typical \(K_d\). It is the expectation of a nonlinear function:

where \(\phi\) is the standard normal density. This integral can be evaluated by numerical quadrature or by averaging simulated occupancies. In general, it does not simplify to \(C/(C+K_{d,\mathrm{typ}})\). A special case is \(C=K_{d,\mathrm{typ}}\): the occupancy distribution is symmetric around 0.5 on the logit scale, so its mean and median are both 0.5.

Variability in Association and Dissociation Rates

Affinity can also be modeled through the binding-rate constants. For reversible binding at equilibrium, \(K_{d,i}=k_{off,i}/k_{on,i}\), where \(k_{on,i}\) has units of concentration\(^{-1}\) time\(^{-1}\) and \(k_{off,i}\) has units of time\(^{-1}\). A population model for both rates can be written:

If the random effects are jointly normal, their difference is normal too:

The variance of log affinity is therefore:

This identity shows why the correlation between rate effects matters. When the two random effects tend to rise together, their difference can vary less than either rate alone. When they tend to move in opposite directions, their difference can vary more. The typical ratio \(k_{off,\mathrm{typ}}/k_{on,\mathrm{typ}}\) is the median \(K_d\) under this model; it is not necessarily the arithmetic mean of individual \(K_d\) values.

Population Summaries Are Not Interchangeable

A population report should specify which summary it gives. The median occupancy describes the middle of the occupancy distribution; the mean occupancy describes the expected fraction bound for a randomly selected subject under the model. Quantiles describe its spread. The occupancy calculated at the typical \(K_d\) equals the median occupancy under the lognormal model, but generally not the mean. Likewise, calculating occupancy at the arithmetic mean \(K_d\) is not a substitute for averaging individual occupancies.

Assumptions, Diagnostics, and Alternatives

The lognormal-affinity model is useful when positive affinities vary continuously and a normal random-effects approximation on the log scale is plausible. Its occupancy-distribution result relies on the equilibrium one-site equation and a common free concentration. If subjects have different concentrations, their occupancy distribution combines concentration and affinity variability; it cannot be obtained from affinity variability alone. If the free concentration is not known or is replaced by total plasma concentration without justification, the calculated occupancy may not represent target-site occupancy.

Diagnostics should assess both the affinity model and the binding assumptions. Useful checks include plotting estimated individual log \(K_d\) values or random effects, comparing their spread and tails with the assumed normal distribution, and checking whether observed occupancy summaries across concentrations agree with model predictions. At concentrations far below or above the typical \(K_d\), predicted occupancies may cluster near 0 or 1; that does not prove that affinity variability is absent, because the occupancy function compresses differences near its bounds. If estimates show distinct subgroups, a single normal distribution on log \(K_d\) may be inadequate. If concentrations vary across subjects, inspect whether the model accounts for that variation rather than attributing it to affinity.

An affinity-only equilibrium model is parsimonious when the scientific question concerns steady-state occupancy. A model with separate \(k_{on}\) and \(k_{off}\) random effects is more informative when time courses or changing concentrations matter, since subjects with the same \(K_d\) can have different binding time courses. A discrete mixture or stratified model can be more appropriate when there is evidence for distinct affinity subpopulations rather than a continuous distribution. Each alternative needs adequate data and a clear scientific rationale; added random effects do not automatically improve inference.

A common mistake is to calculate \(C/(C+\bar K_d)\) and label it the mean population occupancy. That calculation gives occupancy at the arithmetic mean affinity, not the average of individual occupancies. The correction is to calculate each subject’s occupancy under the model and average those values, or integrate over the modeled affinity distribution. Another mistake is to treat a shared administered dose as a shared free concentration. Dose does not ensure equal unbound concentration across subjects; the common-concentration result applies only when the free concentration at the target is in fact held fixed or appropriately modeled.