Why Separate States and Processes?
In target-mediated drug disposition (TMDD), drug molecules can bind to a target, and the resulting complex can be removed through processes different from those acting on free drug or free target. Describing all material with one “drug concentration” can therefore hide important mechanisms. A full TMDD model keeps track of free drug, free target, and drug–target complex as separate states, and assigns rates to the processes that transfer material among them or remove it from the modeled compartment.
This tutorial defines those states and elementary rates. It does not yet combine them into differential equations for net state changes. That distinction is useful: an elementary process rate describes one mechanism at an instant, whereas a state’s net change depends on all processes adding to or removing material from that state.
The usual minimal model assumes the three states occupy the same well-mixed compartment, concentrations are expressed on a compatible basis, binding follows mass-action kinetics, and the rate constants are fixed over the interval considered. The one-to-one stoichiometry means each binding event removes one molecule of free drug and one molecule of free target from their free pools and forms one complex. Other model structures may be needed for multiple binding sites, spatially distinct pools, or more elaborate target processing.
Elementary Processes and Their Units
The binding association rate constant is \(k_{\mathrm{on}}\), with units of concentration\(^{-1}\) time\(^{-1}\). Under the mass-action assumption, the instantaneous rate of complex formation is \(k_{\mathrm{on}}DR\), with units of concentration per time. The dissociation rate constant \(k_{\mathrm{off}}\) has units of time\(^{-1}\); the instantaneous rate of complex dissociation is \(k_{\mathrm{off}}C\). Dissociation forms free drug and free target in a one-to-one event.
Free drug may also be eliminated from the modeled compartment independently of target binding. In the minimal model, this is represented as first-order elimination with rate constant \(k_{\mathrm{el}}\) (time\(^{-1}\)); its instantaneous rate is \(k_{\mathrm{el}}D\). This term describes loss of free drug, not loss of complex. If drug elimination is better represented by another mechanism or compartment structure, the process description must be changed accordingly.
Free target is supplied by synthesis at rate \(k_{\mathrm{syn}}\), typically expressed as concentration per time, and removed by first-order degradation at rate \(k_{\mathrm{deg}}R\), where \(k_{\mathrm{deg}}\) has units of time\(^{-1}\). The synthesis rate is an input to the free-target pool. In this minimal structure, \(k_{\mathrm{deg}}R\) applies to free target; it is not the rate of complex internalization.
The complex can be removed from the modeled compartment by internalization, represented by \(k_{\mathrm{int}}C\), where \(k_{\mathrm{int}}\) has units of time\(^{-1}\). Internalization removes complex from the extracellular or otherwise modeled complex pool. What happens after internalization—degradation, recycling, or release—is a biological detail that may require additional states and processes. Unless such processes are explicitly added, they are not separately represented by the minimal model.
| Process | Instantaneous rate | Units of rate constant | Directly acts on |
|---|---|---|---|
| Binding | \(k_{\mathrm{on}}DR\) | concentration\(^{-1}\) time\(^{-1}\) | Free drug and free target forming complex |
| Dissociation | \(k_{\mathrm{off}}C\) | time\(^{-1}\) | Complex forming free drug and free target |
| Free-drug elimination | \(k_{\mathrm{el}}D\) | time\(^{-1}\) | Free drug |
| Free-target degradation | \(k_{\mathrm{deg}}R\) | time\(^{-1}\) | Free target |
| Complex internalization | \(k_{\mathrm{int}}C\) | time\(^{-1}\) | Drug–target complex |
| Target synthesis | \(k_{\mathrm{syn}}\) | concentration per time | Free-target pool |
A unit check follows directly from the rate definitions. For binding, multiplying \(k_{\mathrm{on}}\), \(D\), and \(R\) gives concentration\(^{-1}\) time\(^{-1}\) × concentration × concentration, or concentration per time. For dissociation, elimination, degradation, and internalization, multiplying a first-order rate constant by a concentration also gives concentration per time. Synthesis already has those units. This check can expose a missing concentration factor or a rate constant entered with incompatible units before fitting or simulation.
Calculating Process Rates
Worked Example: Binding and Dissociation at One Instant
Suppose, hypothetically, that \(D=2.00\) nM and \(R=0.500\) nM, with \(k_{\mathrm{on}}=0.0200\) nM\(^{-1}\)h\(^{-1}\). At the same instant, let \(C=0.300\) nM and \(k_{\mathrm{off}}=0.100\) h\(^{-1}\). Assume one-to-one mass-action binding in a common, well-mixed compartment, with the stated concentrations on the same basis.
The binding rate is
The dissociation rate is
At this instant, dissociation produces complex at a rate of \(0.0300\) nM/h, while binding forms it at \(0.0200\) nM/h. These are rates of two separate processes, not a complete calculation of the net change in \(C\); other processes, such as internalization, can also remove complex. The calculation is valid under the stipulated mass-action and shared-compartment assumptions.
Worked Example: Target Synthesis and Free-Target Degradation
Consider hypothetical target turnover with \(k_{\mathrm{syn}}=0.0400\) nM/h, \(k_{\mathrm{deg}}=0.200\) h\(^{-1}\), and a current free-target concentration \(R=0.200\) nM. Assume synthesis is constant and free-target degradation is first order.
The synthesis rate is \(0.0400\) nM/h by definition. The free-target degradation rate is
Thus, at this concentration, synthesis supplies free target at \(0.0400\) nM/h and degradation removes it at \(0.0400\) nM/h. If no other process acted on free target, these two rates would balance at that instant. In the presence of drug, binding and dissociation also affect the free-target pool, so equality of synthesis and degradation alone does not establish that the target state is at steady state.
Worked Example: Internalization Versus Dissociation
Suppose a hypothetical system has \(C=0.400\) nM, \(k_{\mathrm{int}}=0.350\) h\(^{-1}\), and \(k_{\mathrm{off}}=0.150\) h\(^{-1}\). Assume both rates act as first-order processes on the same complex pool, and there are no additional complex-removal pathways in this calculation.
The internalization rate is
The dissociation rate is
Internalization removes complex from the modeled pool more rapidly than dissociation at this instant. The difference between the two rates is \(0.0800\) nM/h, but it is not the net complex change unless every other process affecting \(C\), including binding, is also accounted for.
From Process Rates to State Changes
The reason to define elementary rates before writing state equations is stoichiometry: each process has a particular direction and acts on particular states. Binding consumes free drug and free target while forming complex; dissociation does the reverse; free-drug elimination removes free drug; free-target degradation removes free target; and internalization removes complex from the modeled pool. Synthesis supplies free target. A complete balance must assign each of these contributions to every state it affects, with the appropriate sign.
For example, the same binding rate appears as a loss process for both \(D\) and \(R\), and as a formation process for \(C\). It should not be counted as three unrelated binding rates. This follows from one-to-one binding stoichiometry: each binding event changes the amount of each of these species by one molecule. The full mass-balance equations use these signed contributions to calculate net rates of change; deriving those equations is a separate step.
An equilibrium occupancy summary answers a different question from these kinetic rates. Under one-to-one binding, the fraction of target that is bound at an instant can be expressed using the free and complex target pools, but that fraction alone does not specify how rapidly binding, dissociation, or internalization occurs. Two systems can have the same occupancy and different process rates if their concentrations or kinetic rate constants differ.
Assumptions, Diagnostics, and Limitations
The process definitions are only as suitable as their assumptions. Mass-action binding presumes that encounter and complex formation can be represented by the product \(DR\), with an appropriate rate constant. First-order removal presumes that the rate is proportional to the relevant free-drug, free-target, or complex concentration. A constant synthesis rate presumes target input does not vary with time or state. If those assumptions fail, the rate laws may need to be replaced rather than merely re-estimated.
In model development, check that every rate has the expected units, that parameters are nonnegative, and that each process is assigned to the correct state. In data and simulation diagnostics, investigate systematic concentration-time residual patterns, implausible or boundary parameter estimates, and predictions that conflict with known target turnover or complex processing. Such patterns can indicate misspecified rate laws, insufficient information to distinguish processes, or a mismatch between the modeled compartment and the measurements. They do not, by themselves, identify which mechanism is wrong.
The full state description is more mechanistic than a simple occupancy-only description because it distinguishes free and bound pools and their rates of exchange and removal. That added detail also requires more information to estimate reliably. Sparse observations may not distinguish \(k_{\mathrm{on}}\), \(k_{\mathrm{off}}\), synthesis, degradation, and internalization. Simplified or reduced models may be useful when their assumptions are supported by the question and data, but they do not preserve every process as an independently represented state transition. The full model also does not automatically represent distribution among tissues, target recycling, multiple target pools, or intracellular drug action; those require explicit extensions.
Common Mistake: Treating Occupancy as a Rate
A common error is to report, for example, “30% occupancy per hour” after calculating a complex concentration or a binding rate. Occupancy is a fraction of target bound at a particular time; a process rate has units of concentration per time (or, for a fractional rate, inverse time). They are not interchangeable. Another related error is to assume that equal binding and dissociation rates imply no change in complex when internalization is also removing complex. The correction is to name the quantity, retain its units, and account for every process that affects the state before making a statement about its net change.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- With \(D=1.50\) nM, \(R=0.400\) nM, and \(k_{\mathrm{on}}=0.0500\) nM\(^{-1}\)h\(^{-1}\), what is the binding rate?
- If \(C=0.250\) nM and \(k_{\mathrm{off}}=0.0800\) h\(^{-1}\), what is the dissociation rate?
- Why does equality between target synthesis and free-target degradation not necessarily establish a steady free-target concentration in a drug-treated system?
- A model predicts a complex internalization rate of \(0.100\) nM/h. Is this the net rate of change in complex? Explain.
Solutions. (1) \(k_{\mathrm{on}}DR=(0.0500)(1.50)(0.400)=0.0300\) nM/h; the units reduce to concentration per time. (2) \(k_{\mathrm{off}}C=(0.0800\ \mathrm{h}^{-1})(0.250\ \mathrm{nM})=0.0200\) nM/h. (3) Binding and dissociation also alter free target, so synthesis and degradation are not its only processes. (4) No. It is the rate of one complex-removal process; binding, dissociation, and any other modeled complex processes must also be included to determine the net change.