Mass-Action TMDD State Equations

Why Write the Full State Equations?

When drug binds a target, the free-drug concentration can change for two different reasons: drug can be cleared from the system, or it can move temporarily into a bound complex. These processes are not interchangeable. A mechanistic model therefore tracks free drug, free target, and drug–target complex as separate states and accounts explicitly for each reaction that moves material between them.

The equations below describe a standard one-to-one mass-action TMDD model. They specify how the states change over time; they do not, by themselves, determine which concentrations an assay measures or how those measurements are modeled.

The standard model assumes a well-mixed compartment of fixed volume, one drug molecule binding one target molecule, and constant rate parameters over the modeled period. The association reaction follows mass action, so its rate is proportional to both free concentrations. Dissociation, internalization, free-target degradation, and systemic drug clearance are first-order in their respective states. Target production is zero-order. The model also assumes that internalization removes both drug and target in the complex, and that systemic clearance acts on free drug rather than directly on complex.

From Reactions to Differential Equations

Write each process as a reaction or input, then record its effect on the three states. The association flux is \(k_{\mathrm{on}}DR\), with concentration-per-time units. Dissociation returns one drug and one target to their free states, at flux \(k_{\mathrm{off}}C\). Internalization removes complex at flux \(k_{\mathrm{int}}C\). The remaining processes are production of free target, degradation of free target, and clearance of free drug.

Let \(u_D(t)\) denote an external drug input expressed as concentration per unit time. For an infusion into a fixed-volume compartment, this is the amount infusion rate divided by volume. A bolus is instead represented as a jump in \(D\), not as an ordinary finite rate acting over time.

Each term has a direct state-based interpretation. Association decreases both \(D\) and \(R\) and increases \(C\); dissociation does the reverse. Internalization decreases \(C\), while target production and degradation affect free \(R\), and systemic clearance decreases free \(D\). The model does not add an internalized state because internalized material is outside the three-state system being represented.

Worked Applications

A Mass-Balance Check

A useful derivation follows by adding the free-drug and complex equations. This sum tracks drug moiety in the modeled states: each molecule of \(D\) or \(C\) contains one drug molecule. Association and dissociation redistribute drug between the two states, so their terms must cancel in the sum.

The cancellation is a structural check: binding alone cannot create or destroy drug. Drug leaves the modeled free-plus-complex pool through systemic clearance of free drug and internalization of complex. In the first worked example, the derivative of this pool is \(-22+18.5=-3.5\) nM/h. The same result follows from \(-3.0-0.50=-3.5\) nM/h, exactly the free-drug clearance plus complex internalization losses.

A parallel check can be made for target moiety by adding the equations for \(R\) and \(C\). Binding and dissociation cancel there too, leaving \(d(R+C)/dt=k_{\mathrm{syn}}-k_{\mathrm{deg}}R-k_{\mathrm{int}}C\). This reflects the stated assumption that degradation applies to free target and internalization removes complex.

Assumptions, Diagnostics, and Alternatives

Before using these equations, check that state concentrations and parameter units are compatible. If \(D\), \(R\), and \(C\) are in nM and time is in hours, then \(k_{\mathrm{on}}\) must be in nM\(^{-1}\)h\(^{-1}\), while the first-order rate constants must be in h\(^{-1}\), and \(k_{\mathrm{syn}}\) in nM/h. Every term in each derivative must have concentration-per-time units. If amount states are used instead, the association expression and all input and clearance terms must be formulated consistently for that representation.

The equations also imply nonnegative forward fluxes when states and rate constants are nonnegative. At a zero boundary, the derivative should not point toward a negative concentration: for example, at \(D=0\), \(dD/dt=u_D+k_{\mathrm{off}}C\), which is nonnegative for nonnegative input and complex. At \(R=0\), \(dR/dt=k_{\mathrm{syn}}+k_{\mathrm{off}}C\), also nonnegative. At \(C=0\), \(dC/dt=k_{\mathrm{on}}DR\), nonnegative. A numerical solution that becomes substantially negative is therefore a warning about numerical integration, coding, or incompatible inputs—not a biological prediction.

Several additional checks help locate errors:

• Association must have a negative sign in both free-state equations and a positive sign in the complex equation.
• Dissociation must have a positive sign in both free-state equations and a negative sign in the complex equation.
• Internalization must reduce complex. Under the stated model, it does not return free drug or free target.
• Systemic clearance must reduce free drug, and target production must increase free target.
• The association and dissociation terms must cancel from both \(d(D+C)/dt\) and \(d(R+C)/dt\).

The full state model is appropriate when binding and complex turnover are represented as distinct processes. A simpler empirical concentration-dependent clearance model may be easier to fit, but it does not explicitly represent target or complex dynamics and should not be interpreted as if it did. A target turnover model without explicit complex formation can describe changing target abundance but omits the mass-action binding mechanism. Choosing among such formulations depends on the scientific question and the information in the data; the full equations do not guarantee that every rate parameter can be estimated precisely.