Parameterizing Binding, Complex Turnover, and Clearance

Why Parameterization and Units Matter

A TMDD model can have the right biological structure and still produce incorrect estimates if its parameters do not match the state definitions and units. A binding rate expressed per molar concentration cannot be inserted unchanged when concentrations are represented in nanomolar units. Likewise, systemic clearance \(CL\) is not itself the first-order rate constant for concentration decline: in a one-compartment model that rate is \(CL/V\).

This tutorial uses the full mass-action model with free drug concentration \(C\), free target concentration \(R\), and drug–target complex concentration \(B\). The species are expressed in compatible molar units, with one drug molecule binding one target site. If drug and target are initially recorded in different units, convert them to compatible units before applying these equations. The input \(u(t)\) is an amount rate into the compartment; a bolus dose can instead be represented as an instantaneous concentration change.

The corresponding full model is:

The association term removes free drug and free target while forming complex; dissociation returns both free species. Complex internalization removes \(B\), and systemic elimination removes free drug at rate \((CL/V)C\). In this model, systemic clearance acts on free drug, while drug in complex is lost through the complex-removal process. These are model assumptions, not universal biological facts; a different mechanism requires different equations.

Rate Constants, Turnover, and Affinity

The product \(k_{\mathrm{on}}CR\) must have concentration/time units. For example, if \(C\) and \(R\) are in nM and time is measured in hours, \(k_{\mathrm{on}}\) is in nM\(^{-1}\)h\(^{-1}\), and \(k_{\mathrm{on}}CR\) is in nM/h. Both \(k_{\mathrm{off}}B\) and \(k_{\mathrm{int}}B\) also have concentration/time units. This unit check is a quick way to detect a missing concentration conversion or an incorrectly entered rate constant.

For the reversible binding reaction by itself, the equilibrium dissociation constant is \(K_D=k_{\mathrm{off}}/k_{\mathrm{on}}\), with concentration units. This ratio describes the binding reaction’s equilibrium scale when internalization is not part of that reaction. It does not replace the separate rate constants in the full dynamic model: two parameter sets can share a ratio but imply different binding and dissociation time courses.

Target turnover can be parameterized directly by \(k_{\mathrm{syn}}\) and \(k_{\mathrm{deg}}\). A useful alternative uses a baseline free-target concentration \(R_0\) and \(k_{\mathrm{deg}}\), setting \(k_{\mathrm{syn}}=k_{\mathrm{deg}}R_0\). This identity follows when synthesis and first-order degradation balance in the absence of drug and complex. The next tutorial addresses the use of baseline target equilibrium in setting initial conditions; here, the identity is a parameterization relationship.

A target half-life \(t_{1/2,R}\) under first-order degradation is related to \(k_{\mathrm{deg}}\) by \(k_{\mathrm{deg}}=\ln(2)/t_{1/2,R}\). It is the half-life of free target under the specified degradation process, not necessarily the half-life of total target in a system where binding and complex internalization also affect target amounts.

Volume and Systemic Clearance

In the free-drug equation, \(CL\) and \(V\) enter through \(CL/V\). Since \(CL\) has units volume/time and \(V\) has units volume, \(CL/V\) has units time\(^{-1}\). For a one-compartment model with no input or binding, \(dC/dt=-(CL/V)C\), so concentration declines exponentially at rate \(CL/V\). This is why a fitted elimination rate \(k_{\mathrm{el}}\) can be translated to clearance by \(CL=k_{\mathrm{el}}V\), provided the model uses this same one-compartment elimination structure.

Changing Concentration Units Without Changing the Model

Changing concentration units changes the numerical values of some parameters, even though the physical process is unchanged. Suppose every concentration state is rescaled by a positive factor \(q\), so \(C'=qC\), \(R'=qR\), and \(B'=qB\), with time units unchanged. For example, conversion from mol/L to nmol/L uses \(q=10^9\). The derivatives must then satisfy \(dC'/dt=q\,dC/dt\), with the same relationship for \(R\) and \(B\).

The transformed parameters follow by requiring every term in each equation to scale by \(q\). In particular, \(k'_{\mathrm{on}}=k_{\mathrm{on}}/q\), because \(k'_{\mathrm{on}}C'R'=(k_{\mathrm{on}}/q)(qC)(qR)=qk_{\mathrm{on}}CR\). The first-order rates \(k_{\mathrm{off}}, k_{\mathrm{int}}, k_{\mathrm{deg}}\), and \(CL/V\) are unchanged. The synthesis rate becomes \(k'_{\mathrm{syn}}=qk_{\mathrm{syn}}\).

The input term must transform too. If \(a(t)=u(t)/V\) is the concentration-rate input, then \(a'(t)=q\,a(t)\). Equivalently, when \(V\) retains its numerical value in the same volume unit, the input amount rate must be expressed in the amount unit corresponding to the new concentration unit. Leaving \(u(t)/V\) numerically unchanged while scaling \(C\), \(R\), and \(B\) would not produce the same trajectories.

Practical Checks, Alternatives, and Limitations

Before fitting, write down the units of every state, parameter, and input. Then check each right-hand-side term for concentration/time units. A useful model comparison is between parameterizing elimination with \(CL\) and \(V\), or directly with \(k_{\mathrm{el}}=CL/V\). The former has a direct clearance interpretation and can be useful when volume is separately meaningful; the latter is a compact parameterization of concentration decline. If only \(CL/V\) is informed by the data, reporting separate \(CL\) and \(V\) estimates without adequate information about both can imply more precision than the model supports.

Similarly, target turnover can be fitted as \(k_{\mathrm{syn}}\) and \(k_{\mathrm{deg}}\), or expressed using \(R_0\) and \(k_{\mathrm{deg}}\) through \(k_{\mathrm{syn}}=k_{\mathrm{deg}}R_0\). The second form can make a baseline target scale easier to communicate; it does not add information by itself. Binding can be reported using \(k_{\mathrm{on}}\) and \(k_{\mathrm{off}}\), or using \(k_{\mathrm{on}}\) and their ratio \(K_D\), but the dynamic equations still require a consistent representation of the rate processes.

Common warning signs include a fitted \(k_{\mathrm{on}}\) changing by a factor of \(10^9\) after a concentration-unit conversion without the reciprocal conversion being applied, or \(k_{\mathrm{syn}}\) retaining the same numerical value even though its concentration unit changed. Inspect units and transformed equations before interpreting a striking parameter estimate. A poor fit can also result from an incorrect structural assumption—such as treating complex internalization as free-drug clearance—rather than from an optimizer failure.

• Wrong: “All concentrations were converted to nM, so I can keep \(k_{\mathrm{on}}\) and the input term \(u/V\) unchanged.” Correction: divide \(k_{\mathrm{on}}\) by the concentration scale factor and multiply the concentration-rate input by it; also scale synthesis and verify every equation term.
• Wrong: “Clearance is the rate constant in the concentration equation.” Correction: the rate constant is \(CL/V\); \(CL\) alone has units volume/time.
• Wrong: “The target half-life is always \(\ln(2)/k_{\mathrm{deg}}\) for total target.” Correction: that relationship describes first-order degradation of free target in this parameterization; binding and complex removal affect the broader system dynamics.

These parameterizations do not guarantee that every parameter can be estimated precisely from a given data set. The full model specifies the meaning and units of the parameters, but data informativeness and numerical behavior must still be assessed during fitting.