Why Test the Equilibrium Approximation?
An equilibrium occupancy formula gives a simple relationship between free drug concentration and the fraction of targets bound. But target binding takes time. If concentration changes appreciably before occupancy can adjust, the formula describes where occupancy would settle under a fixed concentration—not necessarily the occupancy at that moment. The key question is therefore whether binding relaxation is fast relative to changes in free concentration.
This is a test of model adequacy, not a statistical hypothesis test with \(H_0\), \(H_a\), and a p-value. It combines the binding rates, the exposure time course, initial conditions, and the accuracy needed for the application. A useful approximation for one analysis may be inadequate for another that requires more precise occupancy estimates.
Model and Time Scales
Consider one reversible ligand binding one homogeneous target site. Let \(C(t)\) be the free ligand concentration, and let \(\theta(t)\) be the fraction of target occupied, between zero and one. Assume that \(C(t)\) is known and is not materially changed by binding, the target is available, and binding follows mass-action association and dissociation with constant rate parameters. For this model,
Here \(k_{\mathrm{on}}\) has units of concentration\(^{-1}\) time\(^{-1}\), and \(k_{\mathrm{off}}\) has units of time\(^{-1}\). At a fixed concentration, setting the derivative to zero gives the dissociation constant \(K_D=k_{\mathrm{off}}/k_{\mathrm{on}}\) and the equilibrium occupancy.
At fixed \(C\), the kinetic equation can also be written as \(d\theta/dt=\lambda(C)(\theta_{\mathrm{eq}}-\theta)\), where \(\lambda(C)=k_{\mathrm{on}}C+k_{\mathrm{off}}\). The corresponding binding relaxation time is \(\tau_{\mathrm{bind}}(C)=1/\lambda(C)\). After a change in concentration, the difference from the new equilibrium decays exponentially with this time constant, provided concentration then remains fixed.
A concentration-change time scale, when \(C(t)>0\) and its derivative is nonzero, is \(T_C=|C/(dC/dt)|\). Comparing \(\tau_{\mathrm{bind}}\) with \(T_C\) is a useful first screen: a much smaller binding time favors quasi-equilibrium. This comparison is not by itself a guarantee. The occupancy response depends on the rate of change of \(\theta_{\mathrm{eq}}\), and the initial occupancy may be far from equilibrium even when concentration subsequently changes slowly.
Why the Time-Scale Comparison Works
The approximation can be evaluated more directly by defining the tracking error \(e(t)=\theta(t)-\theta_{\mathrm{eq}}(C(t))\). Substituting the kinetic equation and differentiating this definition gives
The first term pulls occupancy toward equilibrium; the second reflects the movement of the equilibrium value itself. Solving this linear equation from time \(t_0\) yields
This expression separates two sources of error: the decaying effect of an initial mismatch and the ongoing lag behind a moving equilibrium. If \(\lambda(C(t))\geq\lambda_{\min}>0\) and \(|d\theta_{\mathrm{eq}}/dt|\leq M\) over an interval, then
The bound follows by taking absolute values in the solution, using \(\lambda\geq\lambda_{\min}\), and integrating the resulting exponential: \(\int_{t_0}^{t}e^{-\lambda_{\min}(t-s)}M\,ds=(M/\lambda_{\min})(1-e^{-\lambda_{\min}(t-t_0)})\). Thus, a small initial error that has decayed and a small ratio \(M/\lambda_{\min}\) support the approximation to a specified absolute occupancy tolerance. The result also explains why no single time-scale ratio is sufficient in every situation: initial conditions and the rate of change of equilibrium matter too.
Worked Example: Screening a Rising Concentration
Suppose \(k_{\mathrm{on}}=0.15\ \mathrm{nM}^{-1}\mathrm{h}^{-1}\), \(k_{\mathrm{off}}=0.30\ \mathrm{h}^{-1}\), and at the time of interest \(C=6\ \mathrm{nM}\), rising at \(3\ \mathrm{nM}/\mathrm{h}\). Assess whether quasi-equilibrium is a strong assumption for occupancy estimates requiring about \(0.02\) absolute precision.
First, \(K_D=k_{\mathrm{off}}/k_{\mathrm{on}}=0.30/0.15=2\ \mathrm{nM}\). The instantaneous equilibrium occupancy is \(6/(2+6)=0.750\). The binding relaxation rate is
The local concentration-change time is \(T_C=6/(3)=2.00\ \mathrm{h}\), so \(\tau_{\mathrm{bind}}/T_C=0.417\). Binding is not much faster than the concentration change by this screen. More directly, differentiating the equilibrium formula gives \(d\theta_{\mathrm{eq}}/dt=[K_D/(K_D+C)^2](dC/dt)\), hence
A local, slowly varying estimate of the lag is \(-\,(d\theta_{\mathrm{eq}}/dt)/\lambda=-0.09375/1.20=-0.0781\) occupancy units. This is an approximation, not an exact occupancy calculation, but its magnitude is several times the \(0.02\) tolerance. The time-scale comparison and the local lag estimate both warn against treating \(0.750\) as a sufficiently accurate dynamic occupancy without checking a kinetic model.
Initial Transients and Step Changes
An abrupt concentration step is a particularly clear failure case. Concentration changes immediately, but occupancy cannot jump under the reversible kinetic model: its derivative is finite. The concentration-change time scale at the jump is effectively zero, so the instantaneous equilibrium formula cannot describe the transient just after the step. If concentration is then held constant, occupancy approaches the new equilibrium exponentially. Waiting several binding time constants can make the residual mismatch small, but whether it is small enough depends on the required precision.
Worked Example: Occupancy After a Concentration Step
A hypothetical system has \(k_{\mathrm{on}}=0.10\ \mathrm{nM}^{-1}\mathrm{h}^{-1}\) and \(k_{\mathrm{off}}=0.20\ \mathrm{h}^{-1}\). Initially \(C=0\) and \(\theta=0\). At time zero, free concentration steps to \(8\ \mathrm{nM}\) and remains there. Calculate occupancy at \(0.5\) and \(3\) hours, and compare it with the equilibrium approximation.
Here \(K_D=0.20/0.10=2\ \mathrm{nM}\), so the post-step equilibrium occupancy is \(8/(2+8)=0.800\). The post-step relaxation rate is \((0.10)(8)+0.20=1.00\ \mathrm{h}^{-1}\), giving \(\tau_{\mathrm{bind}}=1.00\ \mathrm{h}\). Solving the constant-concentration equation with \(\theta(0)=0\) gives
At \(0.5\) hours, \(\theta=0.800(1-e^{-0.5})=0.800(0.3935)=0.3148\). The instantaneous equilibrium approximation would give \(0.800\), an overestimate of \(0.4852\) occupancy units. At \(3\) hours, \(\theta=0.800(1-e^{-3})=0.800(0.9502)=0.7602\), still \(0.0398\) below equilibrium. After five time constants, the occupancy is \(0.800(1-e^{-5})=0.7946\), within \(0.0054\) of equilibrium. The approximation is therefore poor early after the step and more accurate later, but the acceptable cutoff depends on the chosen error tolerance.
Direction of Systematic Error
When the system starts at equilibrium and concentration then rises, the equilibrium occupancy rises too. The tracking-error equation gives \(de/dt=-d\theta_{\mathrm{eq}}/dt<0\) at the instant the rise begins, so actual occupancy falls below the moving equilibrium value. Under a falling concentration, \(d\theta_{\mathrm{eq}}/dt<0\), so \(de/dt>0\): actual occupancy tends to exceed the instantaneous equilibrium value. These are systematic lag errors, not random deviations that necessarily cancel.
Worked Example: A Small but Directional Error During Declining Exposure
Suppose \(k_{\mathrm{on}}=0.20\ \mathrm{nM}^{-1}\mathrm{h}^{-1}\), \(k_{\mathrm{off}}=0.20\ \mathrm{h}^{-1}\), and the concentration is \(4\ \mathrm{nM}\) at a moment when it is declining at \(0.20\ \mathrm{nM}/\mathrm{h}\). Assume occupancy equals equilibrium at that moment. Find the local direction and scale of the tracking error.
The dissociation constant is \(K_D=1\ \mathrm{nM}\), and the instantaneous equilibrium occupancy is \(4/(1+4)=0.800\). The binding rate is \(\lambda=(0.20)(4)+0.20=1.00\ \mathrm{h}^{-1}\), so \(\tau_{\mathrm{bind}}=1.00\ \mathrm{h}\). The equilibrium occupancy is falling at
Because the initial error is zero, the error equation gives \(de/dt=-d\theta_{\mathrm{eq}}/dt=+0.00800\ \mathrm{h}^{-1}\) at that moment. Actual occupancy therefore begins to sit above instantaneous equilibrium occupancy. If the decline is slow enough for local tracking to be established, the first-order lag estimate is \(-\,(d\theta_{\mathrm{eq}}/dt)/\lambda=+0.00800\) occupancy units. This estimate is small and is not the exact error immediately at the stated moment; the exact initial error is zero. The sign, however, follows directly from the differential equation. Even a small positive lag can matter if an analysis requires tighter precision or uses many observations during decline.
Diagnostics, Alternatives, and Limitations
A practical assessment starts with the kinetic parameters and the predicted free concentration time course, not with occupancy data alone. Calculate \(\tau_{\mathrm{bind}}\) across the concentration range of interest, because \(\lambda\) changes with \(C\), and compare it with concentration and equilibrium-occupancy change rates. Check whether the system has had time to lose any initial mismatch. If a numerical occupancy tolerance is important, compare an approximate lag or a simulated kinetic-model difference against that tolerance.
Confirm that reversible, one-site binding and the assumed free concentration are suitable for the question.
At each relevant concentration, use \(\tau_{\mathrm{bind}}=1/(k_{\mathrm{on}}C+k_{\mathrm{off}})\).
Compare relaxation with concentration change, evaluate \(d\theta_{\mathrm{eq}}/dt\), and account for any initial difference from equilibrium.
If the expected lag is not negligible relative to the tolerated occupancy error, use the dynamic model or report the approximation’s limitation.
With observations, a systematic loop in occupancy plotted against concentration over rising and falling exposure can reveal lag: occupancy at the same concentration may differ depending on whether concentration is increasing or decreasing. Residuals from an equilibrium-based model may also show time-ordered patterns—underprediction during decline or overprediction during a rise. Such patterns are diagnostic clues, not proof by themselves: measurement error, incorrect concentration estimates, or other model misspecification can produce structured residuals too.
| Approach | What it represents | When it is useful | Main limitation |
|---|---|---|---|
| Quasi-equilibrium algebraic formula | Occupancy is set to \(C/(K_D+C)\) at each time. | Binding is fast relative to exposure change and initial mismatch has decayed. | Cannot represent binding lag or step-change transients. |
| Dynamic kinetic binding model | Occupancy evolves according to association and dissociation rates. | Exposure changes on a time scale comparable to binding, or transient occupancy matters. | Requires rate information and a specified concentration time course. |
The time-scale test has limits. A small \(\tau_{\mathrm{bind}}/T_C\) is a useful screen, not a universal cutoff; the required precision, time-varying rates, and starting occupancy still matter. The single-site equation also relies on its stated binding assumptions. If free concentration is not known because binding materially depletes ligand, or if the target is not adequately represented by one homogeneous reversible site, the model itself requires revision before its quasi-equilibrium approximation can be judged.
Common Mistake
A common mistake is to conclude that equilibrium occupancy is valid whenever concentration changes slowly at the current time. That reasoning ignores both how quickly binding relaxes and whether occupancy began near equilibrium. The correction is to compare the concentration-dependent binding time with the exposure time scale, examine the movement of equilibrium occupancy, and check that initial transients have decayed. A slow change can still be too fast for very slow binding; conversely, a rapid initial transient may become negligible after sufficient time at constant concentration.
Check Your Understanding
Use the reversible one-site model and distinguish an exact result from a time-scale approximation.
- If \(k_{\mathrm{on}}=0.25\ \mathrm{nM}^{-1}\mathrm{h}^{-1}\), \(k_{\mathrm{off}}=0.50\ \mathrm{h}^{-1}\), and \(C=2\ \mathrm{nM}\), calculate \(K_D\), \(\theta_{\mathrm{eq}}\), and \(\tau_{\mathrm{bind}}\).
- At fixed concentration after a step, what fraction of the initial occupancy difference from the new equilibrium remains after two binding time constants?
- Starting at equilibrium, concentration begins to fall. Does actual occupancy initially lie above or below instantaneous equilibrium occupancy? Explain using the error equation.
- Why does a good comparison of binding and concentration time scales not, by itself, justify the approximation if initial occupancy is far from equilibrium?
Solutions. (1) \(K_D=0.50/0.25=2\ \mathrm{nM}\), \(\theta_{\mathrm{eq}}=2/(2+2)=0.500\), and \(\lambda=(0.25)(2)+0.50=1.00\ \mathrm{h}^{-1}\), so \(\tau_{\mathrm{bind}}=1.00\ \mathrm{h}\). (2) The remaining fraction is \(e^{-2}=0.1353\), or \(13.53\%\), because the fixed-concentration difference decays exponentially. (3) It initially lies below equilibrium: falling concentration makes \(d\theta_{\mathrm{eq}}/dt<0\), so from \(de/dt=-d\theta_{\mathrm{eq}}/dt\) at \(e=0\), the error becomes positive. (4) A large initial mismatch contributes the decaying term \(e(t_0)\exp(-\int_{t_0}^{t}\lambda\,du)\); it can remain important until enough binding time has elapsed.