Choosing Sampling Times for Binding Kinetics

Why Sampling Times Matter

A binding time course can contain a rapid initial change, a slower approach to equilibrium, or both. If measurements begin only after the rapid change is complete, the data may show an apparently gradual response while missing the early phase entirely. If measurements end too soon, a slow phase may look like a plateau even though binding is still changing. Sampling times should therefore be chosen in relation to the rates the experiment is intended to observe.

For a simple one-site model, association at constant free ligand concentration follows one exponential time scale, and dissociation after ligand removal follows another. These time scales provide a practical design guide: sample before and around each expected time constant, then continue long enough to observe the slower change or a convincing plateau. This guide does not guarantee that rates can be estimated precisely, but it helps expose schedules that cannot capture the expected dynamics.

Time Constants for Association and Dissociation

Let \(\theta(t)\) denote the fraction of receptors occupied at time \(t\), \(C\) the constant free ligand concentration during association, \(k_{\mathrm{on}}\) the association rate constant, and \(k_{\mathrm{off}}\) the dissociation rate constant. The simple binding model assumes a homogeneous receptor population, a single binding site, constant free ligand concentration during association, and no changes in receptor abundance or binding properties over the observation period. It also assumes that the measured quantity tracks occupancy without a substantial delay.

Under these assumptions, association is described by

\frac{d\theta}{dt} = k_{\mathrm{on}}C(1-\theta)-k_{\mathrm{off}}\theta = k_{\mathrm{on}}C-(k_{\mathrm{on}}C+k_{\mathrm{off}})\theta.

When \(k_{\mathrm{on}}C+k_{\mathrm{off}}>0\), define the observed association rate and equilibrium occupancy as

k_{\mathrm{obs}}=k_{\mathrm{on}}C+k_{\mathrm{off}}, \qquad \theta_{\mathrm{eq}}=\frac{k_{\mathrm{on}}C}{k_{\mathrm{obs}}}.

If \(k_{\mathrm{obs}}>0\), the association time constant is \(\tau_{\mathrm{assoc}}=1/k_{\mathrm{obs}}\). A time constant is the time required to complete about 63.2% of the total change toward the limiting value in a single-exponential process. The dissociation time constant is \(\tau_{\mathrm{diss}}=1/k_{\mathrm{off}}\) when \(k_{\mathrm{off}}>0\). If the relevant rate is zero, the corresponding time constant is not finite: that process has no change on a finite time scale under the stated model.

Definition: A binding time constant is the reciprocal of the positive rate governing the exponential change of interest. For association at constant \(C\), \(\tau_{\mathrm{assoc}}=1/(k_{\mathrm{on}}C+k_{\mathrm{off}})\) when \(k_{\mathrm{obs}}>0\). For dissociation after free ligand is removed, \(\tau_{\mathrm{diss}}=1/k_{\mathrm{off}}\) when \(k_{\mathrm{off}}>0\). These are different quantities and need not be equal.

For association, rearranging the differential equation gives \(d\theta/dt=k_{\mathrm{obs}}(\theta_{\mathrm{eq}}-\theta)\). If \(\theta(0)\ne\theta_{\mathrm{eq}}\), separation and integration yield the exponential solution below. If instead \(\theta(0)=\theta_{\mathrm{eq}}\), the differential equation gives \(d\theta/dt=0\), so occupancy remains constant; the same expression also gives that constant solution.

\theta(t)=\theta_{\mathrm{eq}}+ \bigl(\theta(0)-\theta_{\mathrm{eq}}\bigr)e^{-k_{\mathrm{obs}}t}.

After free ligand is removed, assume that rebinding is negligible and that the receptor population does not change. Then \(d\theta/dt=-k_{\mathrm{off}}\theta\), so for \(k_{\mathrm{off}}>0\),

\theta(t)=\theta(0)e^{-k_{\mathrm{off}}t}.

These equations guide sampling times, not by themselves the interpretation of every real assay. A delayed or indirect biomarker can lag occupancy, and changes in free ligand concentration can alter the association dynamics. Such effects must be considered before treating an observed signal time course as if it directly followed these equations.

Placing Samples Around the Expected Change

For a single exponential, the fraction of the total change completed by time \(t\) is \(1-e^{-t/\tau}\). The fraction remaining in a dissociation process is \(e^{-t/\tau_{\mathrm{diss}}}\). Thus, by one time constant, about 63.2% of the change has occurred; by three time constants, about 95.0%; and by five, about 99.3%. The same exponential fractions apply to the distance remaining from the association equilibrium.

These landmarks suggest a schedule with a baseline, multiple early observations, points around the expected time constant, and later observations extending toward three to five time constants when feasible. If both rapid and slow phases are expected, the schedule needs early points for the rapid phase and sufficiently late points for the slow one. Logarithmically spaced times can cover a broad range efficiently, while additional samples can be concentrated around a phase of particular interest. No spacing rule replaces a reasoned choice based on the expected rate range, assay precision, and experimental constraints.

Worked Example: Scheduling an Association Time Course

Suppose a hypothetical ligand is held at \(C=5.00\ \mathrm{nM}\), with \(k_{\mathrm{on}}=0.0800\ \mathrm{nM}^{-1}\mathrm{h}^{-1}\) and \(k_{\mathrm{off}}=0.100\ \mathrm{h}^{-1}\). Assume occupancy starts at zero and the model assumptions above hold. Calculate the association time constant and expected occupancy at selected times.

First calculate the observed rate and equilibrium occupancy:

k_{\mathrm{obs}}=(0.0800)(5.00)+0.100 =0.500\ \mathrm{h}^{-1}, \qquad \theta_{\mathrm{eq}}=\frac{(0.0800)(5.00)}{0.500}=0.800.

Therefore, \(\tau_{\mathrm{assoc}}=1/0.500=2.00\ \mathrm{h}\). With \(\theta(0)=0\), the predicted time course is

\theta(t)=0.800(1-e^{-0.500t}).

At \(t=0.25\ \mathrm{h}\), \(\theta=0.800(1-e^{-0.125})=0.0940\). At \(t=1.00\ \mathrm{h}\), \(\theta=0.800(1-e^{-0.500})=0.315\). At \(t=2.00\ \mathrm{h}\), \(\theta=0.800(1-e^{-1})=0.506\). At \(t=6.00\ \mathrm{h}\), \(\theta=0.800(1-e^{-3})=0.760\). These are fractions of receptors occupied, not percentages; for example, 0.506 corresponds to 50.6% occupancy.

A schedule with only a baseline and a six-hour sample would show the net change but poorly locate its timing. The quarter-hour and one-hour samples help observe the approach to the two-hour time constant, while the six-hour sample is near three time constants and checks the approach to equilibrium. Additional samples around two hours would better describe the central part of the transition.

Worked Example: Covering Dissociation Over Several Time Constants

In a hypothetical washout experiment, suppose occupancy at the start of dissociation is \(0.750\) and \(k_{\mathrm{off}}=0.0400\ \mathrm{min}^{-1}\). Assume free ligand is removed rapidly enough that rebinding is negligible. Then \(\tau_{\mathrm{diss}}=1/0.0400=25.0\ \mathrm{min}\), and

\theta(t)=0.750e^{-0.0400t}.

At 5.00 minutes, \(\theta=0.750e^{-0.200}=0.614\). At 25.0 minutes, \(\theta=0.750e^{-1}=0.276\). At 75.0 minutes, \(\theta=0.750e^{-3}=0.0373\). At 125 minutes, \(\theta=0.750e^{-5}=0.00505\). These values show why a sample near one time constant is informative about the main decline, while later samples help establish whether the signal is nearing its lower limit.

A schedule beginning at 75 minutes would miss most of the dissociation transition: only 4.98% of the initial occupancy would remain by then. Samples before and around 25 minutes are needed to capture the decline. Whether the very late 125-minute point is useful depends on assay sensitivity and the scientific objective; a predicted occupancy close to zero may be difficult to distinguish from the assay's measurement floor.

When Fast and Slow Phases Are Both Expected

A single homogeneous one-site model predicts one exponential association time scale at constant \(C\). A time course with two apparent phases may instead arise from more than one kinetic mode, a heterogeneous system, changing experimental conditions, or an indirect readout. The following calculation is an illustrative two-component signal, not a claim that every two-phase curve has a unique biological explanation. Its purpose is to show how a sampling schedule can miss one phase.

Worked Example: Detecting a Rapid Component Before It Disappears

Consider a hypothetical normalized association signal with a fast component of amplitude \(0.600\) and rate \(2.00\ \mathrm{min}^{-1}\), and a slow component of amplitude \(0.400\) and rate \(0.100\ \mathrm{min}^{-1}\):

y(t)=0.600(1-e^{-2.00t})+0.400(1-e^{-0.100t}).

The corresponding time constants are \(0.500\ \mathrm{min}\) and \(10.0\ \mathrm{min}\). At \(t=0.100\ \mathrm{min}\), the predicted signal is \(0.600(1-e^{-0.200})+0.400(1-e^{-0.0100})=0.113\). At \(t=1.00\ \mathrm{min}\), it is \(0.600(1-e^{-2.00})+0.400(1-e^{-0.100})=0.557\). At \(t=10.0\ \mathrm{min}\), it is \(0.600(1-e^{-20})+0.400(1-e^{-1})=0.853\), and at \(t=30.0\ \mathrm{min}\), it is approximately \(0.980\).

If samples are collected only at 0, 10, and 30 minutes, the fast component is already essentially complete at the first nonzero sample: \(1-e^{-20}\) is effectively 1. Those observations can reveal later change, but cannot show when the fast component occurred. A more informative design includes several early samples, such as 0.10, 0.25, 0.50, and 1.00 minutes, plus later samples such as 5, 10, 20, and 30 minutes. The exact schedule should also respect the assay's practical time resolution and measurement variability.

Diagnostics, Alternatives, and Limitations

Before collecting data, compare the proposed times with the plausible range of time constants, not just a single nominal estimate. If the fastest expected time constant is shorter than the interval between the first few samples, the rapid change may be unresolved. If the observation period is much shorter than the slowest expected time constant, the slow process may appear almost flat. A design that has many samples clustered in one narrow time window can have the same problem even when its total sample count is large.

After data collection, inspect whether the measurements span the expected transitions. A response that has already reached its apparent plateau at the first post-baseline point indicates that the schedule may be too late to characterize an expected fast phase. A continued trend at the last sample indicates that the observation window may be too short to characterize a slow phase or equilibrium. These patterns do not prove a particular mechanism; assay limits, time-varying concentration, delayed biomarkers, and model mismatch can produce similar appearances.

Design approach	Strength	Potential weakness
Uniform spacing	Simple to schedule and interpret across a fixed observation period	May waste samples in slowly changing regions while missing a rapid early transition
Logarithmic spacing	Covers a broad range of times with relatively few samples	May leave gaps near a time scale of particular interest
Phase-targeted spacing	Concentrates observations before and around expected fast and slow changes	Depends on having plausible prior ranges for the rates

A proposed schedule should also be checked against experimental timing error, the assay's lower and upper measurement limits, and the feasibility of sampling at very early times. If dissociation is the objective, washout must actually reduce free ligand enough that rebinding is negligible; otherwise the simple dissociation time constant may not describe the observed decline. If receptor turnover, internalization, or recycling occurs on a relevant time scale, occupancy dynamics may not be isolated by the simple model. The time-constant calculations are therefore planning tools conditional on the model assumptions, not substitutes for checking whether those assumptions are credible.

The main alternative is not a different formula but a different allocation of observations: uniform spacing may be adequate for a narrow, well-known time scale, while logarithmic or phase-targeted spacing is often more robust when rates are uncertain or span a wide range. When uncertainty is substantial, a pilot time course can identify whether the main transition occurs earlier or later than expected, followed by a revised schedule. This tutorial concerns choosing observation times; estimating binding-rate parameters from the resulting time course requires additional analysis.

Common Mistakes

Using one time constant for both phases. Association depends on \(k_{\mathrm{on}}C+k_{\mathrm{off}}\), whereas dissociation after ligand removal depends on \(k_{\mathrm{off}}\). Calculate each for its own conditions.
Starting too late. If early samples occur after several fast time constants, they cannot reveal the timing of the fast change. Add feasible samples before and around the expected rapid transition.
Stopping at one time constant. A single time constant corresponds to about 63.2% of a single-exponential change, not completion. Continue later when the plateau or residual decline matters.
Defining a finite time constant for a zero rate. If the relevant rate is zero, its reciprocal is undefined and the process has no finite time scale under the model.
Treating a two-phase signal as proof of two binding rates. Multiple apparent phases can have different explanations. Sampling can reveal phases, but the curve alone does not establish their biological cause.

Key takeaway: Calculate the association and dissociation time constants under the conditions actually used, then place observations before and around those time scales and extend the schedule far enough to observe slower changes. A schedule that never samples a phase cannot resolve it, regardless of the total number of later measurements.

Check Your Understanding

Use the stated equations and show how each proposed schedule relates to its expected time constant.

At constant concentration, \(k_{\mathrm{on}}C=0.300\ \mathrm{h}^{-1}\) and \(k_{\mathrm{off}}=0.200\ \mathrm{h}^{-1}\). What is \(\tau_{\mathrm{assoc}}\), and about what fraction of the total association change has occurred by that time?
A dissociation experiment has \(k_{\mathrm{off}}=0.0500\ \mathrm{min}^{-1}\). What fraction of the starting occupancy remains after 60.0 minutes?
A proposed association schedule has its first post-baseline sample at 20 minutes, while the fastest expected time constant is 2 minutes. What is the design concern?
Why can observations only at 0 and 30 minutes fail to reveal a fast phase with a time constant of 0.5 minutes?
If \(k_{\mathrm{off}}=0\), should \(\tau_{\mathrm{diss}}\) be reported as zero, infinity, or a finite positive number? Explain.

Solutions. 1. \(k_{\mathrm{obs}}=0.300+0.200=0.500\ \mathrm{h}^{-1}\), so \(\tau_{\mathrm{assoc}}=1/0.500=2.00\ \mathrm{h}\). By one time constant, \(1-e^{-1}=0.632\), or 63.2%, of the total change has occurred. 2. The remaining fraction is \(e^{-k_{\mathrm{off}}t}=e^{-(0.0500)(60.0)}=e^{-3}=0.0498\), or 4.98% of starting occupancy. 3. At 20 minutes, ten time constants have elapsed; the rapid change would be essentially complete before the first post-baseline observation, so earlier samples are needed to characterize it. 4. Thirty minutes is 60 fast time constants, so the fast component would be effectively complete by the 30-minute measurement; with no intermediate observations its timing is unresolved. 5. The time constant is not finite: \(1/k_{\mathrm{off}}\) is undefined at zero, and the model predicts no dissociation change on a finite time scale.