Worked Target-Engagement Time-Course Analysis

From Concentration Through Time to Target Engagement

A concentration measured at one instant does not, by itself, determine how much target is occupied at that instant. Occupancy also reflects what happened before: receptors need time to bind ligand, and bound ligand needs time to dissociate. A time-course analysis makes this distinction explicit by comparing the occupancy predicted from binding kinetics with the occupancy that would be predicted if binding were always at equilibrium.

The worked analysis below uses a hypothetical free-drug concentration profile, a reversible one-site binding model, and six hypothetical occupancy-assay observations. The calculations are conditional on the stated model and inputs; agreement with these observations would not establish that the model is uniquely correct.

Definition: Let $\theta(t)$ be the fraction of available receptors occupied at time $t$, and let $C(t)$ be the free ligand concentration. Under one-site reversible binding with association rate constant $k_{\mathrm{on}}$ and dissociation rate constant $k_{\mathrm{off}}$, occupancy follows $$ \frac{d\theta}{dt}=k_{\mathrm{on}}C(t)\bigl(1-\theta(t)\bigr)-k_{\mathrm{off}}\theta(t). $$ Here $k_{\mathrm{on}}$ has units of concentration$^{-1}$ time$^{-1}$, $k_{\mathrm{off}}$ has units of time$^{-1}$, and $0\leq\theta(t)\leq1$. The instantaneous equilibrium occupancy at a fixed concentration $C$ is $\theta_{\mathrm{eq}}=C/(C+K_D)$, where $K_D=k_{\mathrm{off}}/k_{\mathrm{on}}$.

This model assumes a well-mixed target compartment with a known free concentration profile; a single reversible binding site; constant rate constants; and no ligand depletion, competing ligand, or change in receptor availability over the analyzed interval. The measured occupancy is assumed to represent the same receptor population described by $\theta(t)$. These assumptions matter: if, for example, the assay reflects a changing receptor pool, an occupancy fraction may not follow this equation even when the binding chemistry is otherwise reasonable.

Solving Within a Constant-Concentration Interval

When $C(t)=C$ is constant over an interval, define $\lambda=k_{\mathrm{on}}C+k_{\mathrm{off}}$. The differential equation becomes $d\theta/dt=\lambda(\theta_{\mathrm{eq}}-\theta)$. If occupancy at the start of the interval is $\theta_0$, its solution after elapsed time $\Delta t$ is:

\theta(\Delta t)=\theta_{\mathrm{eq}}+ \bigl(\theta_0-\theta_{\mathrm{eq}}\bigr)e^{-\lambda\Delta t}, \qquad \theta_{\mathrm{eq}}=\frac{k_{\mathrm{on}}C}{k_{\mathrm{on}}C+k_{\mathrm{off}}}.

If $\theta_0=\theta_{\mathrm{eq}}$, the differential equation gives the constant solution $\theta(t)=\theta_{\mathrm{eq}}$. Otherwise, this expression is obtained by separating variables: $d(\theta-\theta_{\mathrm{eq}})/(\theta-\theta_{\mathrm{eq}})=-\lambda\,dt$. Integration from $0$ to $\Delta t$ gives $\ln\!\left|\{\theta(\Delta t)-\theta_{\mathrm{eq}}\}/\{\theta_0-\theta_{\mathrm{eq}}\}\right|=-\lambda\Delta t$; exponentiating and rearranging gives the stated solution. At a concentration change, occupancy itself remains continuous. The new interval uses the occupancy at the transition as its starting value, but has a new $\theta_{\mathrm{eq}}$ and $\lambda$.

The equilibrium expression is not a second kinetic trajectory. It is the limiting occupancy if concentration were held fixed long enough for binding to equilibrate. Its use as an instantaneous prediction therefore requires binding to be fast relative to concentration changes and sampling, or otherwise requires evidence that the system is near equilibrium.

Worked Example: Calculate Occupancy During the First Exposure

Consider the hypothetical parameter set $k_{\mathrm{on}}=0.100\ \mathrm{nM}^{-1}\mathrm{h}^{-1}$ and $k_{\mathrm{off}}=0.067684\ \mathrm{h}^{-1}$, with initial occupancy $\theta(0)=0.200$. Free concentration is $3.23\ \mathrm{nM}$ from 0 to 2 hours. Calculate the occupancy at 1 hour and check the 2-hour value.

For this interval, $K_D=k_{\mathrm{off}}/k_{\mathrm{on}}=0.67684\ \mathrm{nM}$, $\lambda=(0.100)(3.23)+0.067684=0.390684\ \mathrm{h}^{-1}$, and $\theta_{\mathrm{eq}}=0.323/(0.390684)=0.826755$. At 1 hour:

\theta(1)=0.826755+(0.200-0.826755)e^{-0.390684} \approx 0.40270.

At 2 hours, the same substitution gives $\theta(2)\approx0.53984$. The occupancy is below the equilibrium value of about 0.827 because the initial occupancy was much lower and the exposure has lasted only 2 hours. This is a kinetic lag, not an arithmetic discrepancy.

Worked Time-Course Analysis

The hypothetical concentration profile is $3.23\ \mathrm{nM}$ from 0 to 2 hours, zero from 2 to 4 hours, and $1.38\ \mathrm{nM}$ from 4 to 6 hours. The parameter values and initial occupancy are those in the preceding example. Treat the concentrations as free ligand concentrations and use the occupancy at each transition as the initial condition for the next interval. The predictions below are rounded for display; residual calculations use the unrounded predictions. The 2-hour sample is taken just before concentration falls to zero.

Worked Example: Propagate Occupancy Across Concentration Changes

At 2 hours, occupancy is approximately 0.53984. During the next interval $C=0$, so $k_{\mathrm{on}}C=0$, $\lambda=k_{\mathrm{off}}=0.067684\ \mathrm{h}^{-1}$, and equilibrium occupancy is zero. Thus:

\theta(3)=0+(0.53984-0)e^{-0.067684(1)} \approx 0.50451, \qquad \theta(4)=0.53984e^{-0.067684(2)} \approx 0.47149.

Occupancy does not become zero immediately when concentration falls to zero. Bound ligand dissociates over time; in the absence of rebinding, the occupied fraction declines exponentially at rate $k_{\mathrm{off}}$.

From 4 to 6 hours, $C=1.38\ \mathrm{nM}$. The interval has $\lambda=(0.100)(1.38)+0.067684=0.205684\ \mathrm{h}^{-1}$, and $\theta_{\mathrm{eq}}=0.138/0.205684\approx0.670932$. Starting from $\theta(4)\approx0.47149$:

\theta(6)=0.670932+(0.47149-0.670932)e^{-0.205684(2)} \approx 0.53876.

The complete kinetic predictions are therefore approximately 0.3112 at 0.5 hours, 0.40270 at 1 hour, 0.53984 at 2 hours, 0.50451 at 3 hours, 0.47149 at 4 hours, and 0.53876 at 6 hours. Values are rounded for display; propagation uses unrounded endpoints.

Suppose the hypothetical direct occupancy assay gives the following values. A residual is defined as observed occupancy minus model-predicted occupancy, so a positive residual means the observation is above the model prediction. The observations are illustrative and are not real study data.

Time (h)	Free concentration (nM)	Kinetic prediction	Observed occupancy	Residual	Instantaneous equilibrium prediction
0.5	3.23	0.3112	0.3312	0.0200	0.8268
1	3.23	0.40270	0.3829	-0.01980	0.8268
2	3.23	0.53984	0.5600	0.02016	0.8268
3	0	0.50451	0.55000	0.04549	0
4	1.38*	0.47149	0.45163	-0.01986	0.6709*
6	1.38	0.53876	0.50000	-0.03876	0.6709

The asterisks mark the concentration immediately after the 4-hour transition: the kinetic prediction at 4 hours is continuous with the preceding zero-concentration interval, while the equilibrium column reports the equilibrium for the new $1.38\ \mathrm{nM}$ concentration. This distinction makes explicit why a sudden concentration change need not cause an equally sudden occupancy change.

Worked Example: Compare Kinetic and Equilibrium Predictions

At 3 hours, concentration is zero. The kinetic prediction is $0.53984e^{-0.067684(1)}\approx0.50451$, whereas equilibrium occupancy at $C=0$ is $0/(0+0.67684)=0$. The two predictions answer different questions: the kinetic value accounts for receptors that remain occupied after the drug concentration has fallen; the equilibrium value describes the long-run result if concentration stays at zero.

At 6 hours, the concentration has been $1.38\ \mathrm{nM}$ for 2 hours. The equilibrium value is $1.38/(1.38+0.67684)\approx0.6709$, while the kinetic prediction is about 0.5388. The model has not yet reached equilibrium. It would be incorrect to replace the kinetic prediction with the equilibrium value, about 0.6709, merely because concentration is known at the sampling time.

Quantifying Agreement and Interpreting Discrepancies

The residuals in the table are signed, dimensionless occupancy fractions. For example, at 3 hours the observation is 0.55000 and the prediction is 0.50451, so the residual is $0.55000-0.50451=0.04549$, or 4.549 percentage points of occupancy. At 6 hours it is $0.50000-0.53876=-0.03876$, meaning the observation is 3.876 percentage points below the prediction.

Worked Example: Calculate the Residual RMSE

A descriptive summary of prediction error is the root mean squared error (RMSE), the square root of the mean squared residual across the observations. It is not a confidence interval and does not account for assay uncertainty. For these six residuals, the squared sum is:

$$ (0.0200)^2+(0.01980)^2+(0.02016)^2+(0.04549)^2+(0.01986)^2+(0.03876)^2 =0.0004000000+0.0003920400+0.0004064256+0.0020693401+0.0003944196+0.0015023376 =0.0051645629. $$

Therefore:

\mathrm{RMSE}=\sqrt{\frac{0.0051645629}{6}} =\sqrt{0.0008607605} \approx0.02934.

The typical squared-error scale summarized by this statistic corresponds to about 0.02934 occupancy fraction, or 2.934 percentage points. RMSE does not reveal whether errors are systematically positive or negative, so the residuals should also be examined in time order.

A time-ordered residual pattern can be more informative than a single summary. Here, the 3-hour residual is positive, while the 4- and 6-hour residuals are negative. With only six hypothetical measurements, this pattern is not enough to identify its cause. Possible explanations include measurement variability, imperfect concentration inputs, uncertainty in binding parameters, or a model assumption that does not adequately represent the measured target. A residual should not be assigned a biological explanation without additional evidence.

Check the inputs.
Confirm concentration units, sampling times, transition times, initial occupancy, and whether reported concentrations are free or total drug.

Check the propagation.
Use the concentration and rate constants for each interval, carry occupancy continuously across transitions, and verify that predictions remain between zero and one.

Inspect residuals over time.
Look for sustained bias, abrupt discrepancies near concentration changes, or isolated deviations; compare them with plausible assay variability before changing the model.

Revisit assumptions only with a reason.
If discrepancies persist, examine whether the assumed concentration profile, binding rates, or fixed available-target population is appropriate for the measurement.

This calculation differs from fitting a receptor-binding time course: the parameters here are specified, and the task is to propagate their predictions and assess discrepancies, not to estimate parameters from data. It also differs from using an equilibrium occupancy calculation alone. Equilibrium is simpler when exposure is effectively constant long enough for binding to equilibrate; the kinetic calculation is needed when concentration changes on a timescale relevant to association or dissociation. Neither approach makes the assay error disappear, and neither establishes that the assumed concentration at the target site is known without uncertainty.

Common Mistakes and Practical Limits

Using the wrong observation for a residual. At 3 hours the stated observation is 0.550, not the 2-hour observation of 0.560. Thus $r(3)=0.550-0.50451=0.04549$. Substituting 0.560 would change the residual and every summary based on it.
Resetting occupancy at a concentration transition. A change in concentration changes the derivative and equilibrium value, not the current occupied fraction. Carry the end-of-interval occupancy forward.
Treating equilibrium as the instantaneous response. The equilibrium calculation is a limiting value at a fixed concentration. It can substantially differ from occupancy during transient exposure or washout.
Interpreting a small RMSE as proof of a correct model. RMSE summarizes prediction error for the selected observations only; it does not check whether assumptions hold or whether another model could also explain the data.

In practice, concentration and assay measurements have uncertainty, sampling may be sparse, and the kinetic parameters may not be known precisely. The displayed calculation treats all inputs as fixed and uses unweighted residuals. It does not quantify parameter uncertainty, infer a cause for any discrepancy, or account for different measurement precision across time points. Those limitations should shape how strongly the result is interpreted.

Key takeaway: For a changing concentration profile, propagate occupancy interval by interval using the kinetic model and preserve continuity at transitions. Compare the result with both observed occupancy and the clearly distinguished equilibrium limit; interpret residual summaries as diagnostics, not as proof of mechanism.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If concentration falls to zero at a time when occupancy is 0.40, what determines the subsequent rate of occupancy loss under the stated model?
At 3 hours, calculate the residual using observed occupancy 0.550 and predicted occupancy 0.50451. State its direction and size in percentage points.
Why is the equilibrium prediction of zero at zero concentration not the kinetic prediction at 3 hours?
What does the RMSE of 0.02934 summarize, and what does it not establish?
Name two checks to make if observed and predicted occupancy repeatedly disagree after concentration transitions.

Solutions. (1) With $C=0$, association is zero and $d\theta/dt=-k_{\mathrm{off}}\theta$; occupied receptors dissociate exponentially, so the loss rate depends on the current occupancy and $k_{\mathrm{off}}$. (2) $r=0.550-0.50451=0.04549$, a positive residual: the observation is 4.549 percentage points above prediction. (3) Zero is the long-run equilibrium when concentration remains zero. At 3 hours, receptors are still occupied because dissociation takes time. (4) RMSE summarizes the root mean squared prediction error for these six observations, about 0.02934 occupancy fraction. It does not establish that the model is correct, quantify uncertainty, or identify the cause of discrepancies. (5) Check the timing and units of the concentration profile, verify interval-by-interval propagation and continuity, and assess whether the assay and fixed-target assumptions are appropriate.