Irreversible Receptor Binding

Why Irreversible Binding Depends on Time

For a rapidly reversible ligand, target engagement can often be described by association and dissociation, with occupancy changing as free-ligand concentration changes. Covalent or effectively irreversible ligands behave differently: once a target molecule is modified, it does not simply dissociate back into the active pool on the time scale of interest. Consequently, the fraction of target modified can continue to increase during sustained exposure, even when concentration is constant. It can also remain elevated after the drug has left the system.

Here the outcome is the fraction of the target that is covalently modified and functionally inactivated. This fraction is sometimes called irreversible occupancy, but it is not necessarily the fraction of all receptors occupied by any ligand. The model separates irreversible modification from pharmacological efficacy: a modified receptor is assumed to be inactivated, while the magnitude of any downstream response is outside this model.

The recovery term is a phenomenological representation of restoration of the active target pool. It assumes that recovered target replaces modified target so that the total target amount stays constant. It does not specify the biological process responsible for recovery. A later treatment of receptor turnover can represent that process more explicitly.

Model and Assumptions

The model assumes a well-mixed target pool with only two states: active and modified/inactivated. Drug modifies active target with a concentration-dependent rate, and modified target returns to the active pool with first-order rate constant \(k_{\mathrm{rec}}\). Modification is treated as irreversible on the modeled time scale: there is no dissociation term for the modified drug-target complex. The model also assumes that \(C(t)\) is the free concentration at the target site, not necessarily the measured plasma concentration.

A common effective inactivation-rate relationship is \(k_{\mathrm{obs}}(C)=k_{\mathrm{inact}}C/(K_I+C)\), where \(k_{\mathrm{inact}}\) is the maximum inactivation rate and \(K_I\) is a concentration scale. This relationship can arise when drug first forms a reversible encounter complex and then reacts, under conditions where formation of that complex is sufficiently rapid for an effective rate to be used. It is a modeling choice, not a universal law. If the mechanism or data do not support it, \(k_{\mathrm{obs}}\) should be specified or estimated using a more suitable model.

The rate of change in \(q(t)\) is the rate at which active target is modified, minus the rate at which modified target recovers:

For constant concentration \(C\), write \(k_{\mathrm{obs}}=k_{\mathrm{obs}}(C)\). Then \(\frac{dq}{dt}=k_{\mathrm{obs}}-(k_{\mathrm{obs}}+k_{\mathrm{rec}})q\). The constant-exposure solution from initial fraction \(q_0\) is

Here \(q_{\mathrm{ss}}\) is the long-term inactivated fraction under that constant concentration, not necessarily complete inactivation. It is below one when recovery is present. When exposure stops, \(C=0\), so \(k_{\mathrm{obs}}(0)=0\) and \(\frac{dq}{dt}=-k_{\mathrm{rec}}q\). Thus the washout-period solution, starting from \(q_w\), is \(q(t)=q_w e^{-k_{\mathrm{rec}}t}\), where \(t\) now measures time since washout.

Why the Constant-Exposure Solution Has This Form

For constant concentration, define \(\lambda=k_{\mathrm{obs}}+k_{\mathrm{rec}}\). The differential equation is \(dq/dt+\lambda q=k_{\mathrm{obs}}\). Multiplying by the integrating factor \(e^{\lambda t}\) gives \(\frac{d}{dt}(q e^{\lambda t})=k_{\mathrm{obs}}e^{\lambda t}\). Integrating from \(0\) to \(t\) yields \(q(t)e^{\lambda t}-q_0=(k_{\mathrm{obs}}/\lambda)(e^{\lambda t}-1)\). Rearranging gives the stated solution, because \(k_{\mathrm{obs}}/\lambda=q_{\mathrm{ss}}\). This also shows that the difference \(q(t)-q_{\mathrm{ss}}\) decays exponentially at rate \(\lambda\). If both rates are zero, there is no change and \(q(t)=q_0\); the formula involving \(q_{\mathrm{ss}}\) is not used because its denominator is zero.

Exposure Duration, Concentration, and Recovery

If recovery is negligible during exposure, set \(k_{\mathrm{rec}}=0\). When \(k_{\mathrm{obs}}>0\), at constant concentration the inactivated fraction approaches one as \(t\to\infty\), according to \(q(t)=1-(1-q_0)e^{-k_{\mathrm{obs}}t}\). If \(k_{\mathrm{obs}}=0\), then \(q(t)=q_0\). Longer exposure increases inactivation, but the increase becomes progressively smaller as fewer active target molecules remain. If recovery is appreciable, the competition between modification and recovery creates a plateau \(q_{\mathrm{ss}}\). Increasing exposure duration then moves \(q(t)\) toward that plateau rather than toward complete inactivation.

For a desired inactivated fraction \(q^*\), the constant-exposure model can also give the required duration. When \(q_0<q^*<q_{\mathrm{ss}}\),

The target level must be below \(q_{\mathrm{ss}}\) to be reached in finite time from below. The formula is not a way to obtain a finite duration for a target level above the plateau.

Diagnostics, Alternatives, and Limitations

Before interpreting a prediction, check that concentration is free concentration at the target site, that the exposure schedule is represented correctly, and that the rate constants use the same time units as the time variable. The model predicts \(0\leq q(t)\leq1\) when \(0\leq q_0\leq1\) and both rates are nonnegative. A fitted trajectory outside that range, or systematic deviations from the predicted approach to a plateau and subsequent recovery, suggests a coding error or a model that does not capture the data.

A key model check is whether the long-exposure plateau makes biological sense. If the data show continuing inactivation despite a substantial proposed recovery rate, the assumed balance may be wrong. Conversely, if apparent engagement declines during washout much faster or slower than \(e^{-k_{\mathrm{rec}}t}\), a single constant recovery rate may not describe the process. Changes in free concentration, drug loss, multiple target states, incomplete functional inactivation, and variation between target pools are possible explanations; the simple model does not distinguish among them.

Compared with reversible one-site binding, this model has no dissociation of the modified complex. A reversible model instead includes association and dissociation, and its occupancy can decline as ligand concentration falls even without target replacement. For irreversible binding, washout alone stops new modification but does not instantly remove existing modified target; decline requires recovery in the model. The irreversible model is appropriate only when dissociation is negligible over the time scale being studied. If modification is reversible on that time scale, a reversible or explicitly reversible-adduct model is preferable.

The concentration-response expression for \(k_{\mathrm{obs}}\) also has limits. It assumes a particular effective relationship between concentration and chemical inactivation. A changing or depleted free-drug concentration, slow encounter-complex formation, multiple binding modes, or time-varying reactivity can invalidate that relationship. In such settings, use a time-varying \(C(t)\) in the differential equation if the rate law remains appropriate, or choose a mechanistic model with additional states. The model predicts target state, not downstream response; equal inactivation fractions need not imply equal functional effects.