Unbound Inhibitor Concentrations at the Enzyme Site

Why the Concentration at the Enzyme Matters

An inhibition model compares inhibitor exposure with an inhibition constant. That comparison is meaningful only when the inhibitor concentration represents the exposure relevant to the enzyme and is on a compatible basis with the constant. Total plasma concentration, unbound plasma concentration, hepatic-inlet concentration, and intracellular unbound concentration are not interchangeable. They can differ substantially even at the same time point.

For example, plasma proteins may bind much of a circulating inhibitor, leaving a smaller unbound concentration available to enter tissues. Blood arriving at the liver may also contain inhibitor absorbed from the gut through the portal vein, so its concentration can differ from systemic arterial plasma. Within hepatocytes, transport and cellular partitioning may cause unbound concentration to differ from unbound plasma. The appropriate choice depends on the enzyme’s location and on how the inhibition constant was obtained.

Definition: \(C_{p,\mathrm{tot}}\) is the total inhibitor concentration in plasma, including bound and unbound inhibitor. \(C_{u,p}\) is the unbound inhibitor concentration in plasma. \(C_{H,\mathrm{in}}\) is the inhibitor concentration entering the liver, defined on a stated blood or plasma basis. \(C_{u,\mathrm{site}}\) is the unbound inhibitor concentration at the enzyme site. These quantities may be equal under particular assumptions, but they are not generally equal by definition.

From Total Plasma to Unbound Plasma

The plasma unbound fraction, \(f_{u,p}\), is the fraction of total plasma inhibitor that is unbound. If binding is at equilibrium, is linear over the concentration range of interest, and the measured total and unbound concentrations refer to the same time and plasma sample, then:

C_{u,p}=f_{u,p}C_{p,\mathrm{tot}}

This conversion is often the first step in selecting an inhibition concentration. It does not establish that unbound plasma is the enzyme-site concentration. It only estimates the unbound concentration in plasma. In particular, a low \(f_{u,p}\) makes total plasma concentration much larger than unbound plasma concentration, and substituting total concentration directly into a model that expects free inhibitor can overstate inhibition.

Worked Example: Converting Total Plasma to Unbound Plasma

Suppose a hypothetical inhibitor has a total plasma concentration of \(12.0\ \mu\mathrm{M}\) at a specified time and a plasma unbound fraction of 0.080. Assume binding is at equilibrium and the unbound fraction is applicable at this concentration.

Calculate unbound plasma concentration.

C_{u,p}=f_{u,p}C_{p,\mathrm{tot}} =(0.080)(12.0\ \mu\mathrm{M}) =0.960\ \mu\mathrm{M}

If an enzyme model uses an unbound inhibition constant \(K_i=0.240\ \mu\mathrm{M}\), the competitive inhibition factor is \(1+C_{u,p}/K_i=1+0.960/0.240=5.00\). This calculation assumes unbound plasma is the appropriate surrogate for inhibitor at the enzyme and that the \(K_i\) is expressed on a free-inhibitor basis. Neither assumption follows from the plasma conversion itself.

Using \(12.0\ \mu\mathrm{M}\) instead would give \(1+12.0/0.240=51.0\), a much larger factor. That result would not be appropriate for this free-concentration model merely because total concentration is more readily measured.

Hepatic-Inlet Concentration Is an Inflow Quantity

For a liver enzyme, systemic plasma concentration may not represent the inhibitor concentration entering the liver. Blood reaching the liver includes systemic arterial blood and portal venous blood. The portal contribution can be especially relevant when inhibitor is absorbed from the gastrointestinal tract. A hepatic-inlet concentration is therefore a flow-weighted concentration of the inflows, not simply another name for systemic plasma concentration.

When hepatic blood flow and the concentrations in each inflow are available on a whole-blood basis, a two-inflow calculation is:

C_{H,\mathrm{in},b} = \frac{Q_{\mathrm{art}}C_{\mathrm{art},b} +Q_{\mathrm{port}}C_{\mathrm{port},b}} {Q_{\mathrm{art}}+Q_{\mathrm{port}}}

Here \(Q_{\mathrm{art}}\) and \(Q_{\mathrm{port}}\) are arterial and portal blood flows, and the subscript \(b\) denotes whole-blood concentration. Use concentrations and flows on a consistent basis. If the inflow concentrations are measured in plasma, convert them to blood concentrations using the blood-to-plasma ratio \(R_b=C_b/C_p\), when that ratio is applicable. If \(R_b\) is the same for both inflows, the flow-weighted plasma-equivalent concentration can be calculated directly from the plasma concentrations. After obtaining an inlet concentration, any conversion to unbound concentration requires an appropriate binding assumption.

Worked Example: Flow-Weighting the Hepatic Inflows

Consider hypothetical arterial and portal plasma concentrations of \(0.400\ \mu\mathrm{M}\) and \(2.00\ \mu\mathrm{M}\), respectively. Let arterial and portal blood flows be \(0.800\ \mathrm{L/min}\) and \(1.20\ \mathrm{L/min}\), and assume \(R_b=0.800\) in both inflows.

First convert each concentration to whole blood: \(C_{\mathrm{art},b}=0.400/0.800=0.500\ \mu\mathrm{M}\) and \(C_{\mathrm{port},b}=2.00/0.800=2.50\ \mu\mathrm{M}\). Then flow-weight:

C_{H,\mathrm{in},b} = \frac{(0.800)(0.500)+(1.20)(2.50)} {0.800+1.20} = \frac{3.40}{2.00} =1.70\ \mu\mathrm{M}

Converting back to a plasma-equivalent inlet concentration gives \(C_{H,\mathrm{in},p}=R_bC_{H,\mathrm{in},b}=(0.800)(1.70)=1.36\ \mu\mathrm{M}\). If the relevant unbound fraction is \(0.100\), and binding is assumed to equilibrate consistently at the inlet, the corresponding unbound plasma-equivalent concentration is \(0.136\ \mu\mathrm{M}\). The calculation uses the portal contribution rather than assuming the arterial value alone represents liver inflow. It does not, by itself, establish the concentration inside hepatocytes.

From Plasma Exposure to the Enzyme Site

For an intracellular hepatic enzyme, unbound plasma can be a useful starting point, but an additional tissue-distribution relationship may be needed. Define the unbound liver-to-plasma partition coefficient \(K_{p,uu,\mathrm{liver}}\) as the ratio of unbound intracellular liver concentration to unbound plasma concentration at the relevant equilibrium or quasi-equilibrium condition:

K_{p,uu,\mathrm{liver}} = \frac{C_{u,\mathrm{liver}}}{C_{u,p}}

If the enzyme is exposed to the intracellular unbound concentration represented by this ratio, then \(C_{u,\mathrm{site}}\) can be approximated by \(C_{u,\mathrm{liver}}\), and:

C_{u,\mathrm{site}} \approx K_{p,uu,\mathrm{liver}}C_{u,p} = K_{p,uu,\mathrm{liver}}f_{u,p}C_{p,\mathrm{tot}}

This relationship assumes that the partition coefficient applies at the time and concentration range being modeled, that the measured liver concentration represents the compartment containing the enzyme, and that the relevant species is the unbound parent inhibitor. It is not a general identity. Active uptake or efflux, intracellular binding, ion trapping, pH gradients, and slow equilibration can all make the relationship more complicated. A total liver-to-plasma ratio is not a substitute for \(K_{p,uu,\mathrm{liver}}\): total tissue concentration can be high because of binding or sequestration without high unbound concentration at the enzyme.

Worked Example: Estimating Intracellular Unbound Concentration

Suppose a hypothetical inhibitor has \(C_{p,\mathrm{tot}}=5.00\ \mu\mathrm{M}\) and \(f_{u,p}=0.040\). An independently estimated \(K_{p,uu,\mathrm{liver}}\) is 3.00. Assume these values apply at the time of interest, the enzyme is in the compartment represented by the partition coefficient, and the parent inhibitor is at distribution equilibrium.

First, \(C_{u,p}=(0.040)(5.00)=0.200\ \mu\mathrm{M}\). Then:

C_{u,\mathrm{site}} \approx (3.00)(0.200\ \mu\mathrm{M}) =0.600\ \mu\mathrm{M}

For a competitive model with a matched unbound \(K_i=0.300\ \mu\mathrm{M}\), the inhibition factor is \(1+0.600/0.300=3.00\). Using unbound plasma instead would give \(1+0.200/0.300=1.67\), while using total plasma would give \(1+5.00/0.300=17.7\). These differences show why the assumed enzyme-site exposure must be stated. They do not demonstrate that the partition coefficient or the competitive model is correct.

Matching the Concentration to the Inhibition Constant

The concentration in the numerator of an inhibition ratio must be compatible with the \(K_i\) in its denominator. In the simple competitive model, the factor \(1+C_{u,\mathrm{site}}/K_i\) presumes that \(K_i\) and \(C_{u,\mathrm{site}}\) use the same free-inhibitor basis and refer to the relevant inhibition process. The mixed and noncompetitive models likewise require the concentration used for each binding interaction to be matched to the corresponding inhibition constant. The model form distinguishes binding modes; it does not remove the need to define exposure.

An in vitro inhibition constant may be reported using nominal inhibitor concentration in the assay rather than measured free concentration. If the unbound fraction in the incubation, \(f_{u,\mathrm{inc}}\), is known and binding is linear, at equilibrium, and adequately described by that fraction, then \(C_{u,\mathrm{inc}}=f_{u,\mathrm{inc}}C_{\mathrm{nom}}\). Under those conditions, a nominal-concentration constant \(K_{i,\mathrm{nom}}\) corresponds to an unbound constant \(K_{i,u}=f_{u,\mathrm{inc}}K_{i,\mathrm{nom}}\). This correction should not be applied automatically: it depends on the assay’s binding conditions and on the reported constant’s definition. The essential check is to establish whether both the model concentration and the constant are nominal or unbound, and to keep their basis consistent.

Derivation: Suppose the model’s inhibition ratio is \(C_{u,\mathrm{site}}/K_{i,u}\), and \(C_{u,\mathrm{site}}=K_{p,uu,\mathrm{liver}}f_{u,p}C_{p,\mathrm{tot}}\). Substitution gives:

\frac{C_{u,\mathrm{site}}}{K_{i,u}} = \frac{K_{p,uu,\mathrm{liver}}f_{u,p}C_{p,\mathrm{tot}}}{K_{i,u}}

If the concentration basis or partition assumptions are wrong, the ratio is wrong even if the inhibition equation is algebraically correct. This is why conversion and site selection precede interpretation of the model output.

Choosing Among Concentration Approaches

Approach	When it may be appropriate	Main limitation
Total plasma concentration	When the model constant and intended empirical predictor are defined on a total-plasma basis.	Does not account for plasma binding or tissue access; can greatly exceed free exposure.
Unbound plasma concentration	When plasma free concentration is the intended exposure or is a justified surrogate for the enzyme site.	May not represent hepatic inflow or intracellular unbound exposure.
Hepatic-inlet concentration	When liver inflow is the relevant exposure boundary and arterial and portal contributions are represented.	Is not intracellular concentration; depends on consistent flow and blood/plasma inputs.
Intracellular unbound concentration	When a liver-cell enzyme is the target and an appropriate unbound partition or distribution model is available.	May be uncertain or time-varying; bulk tissue measurements may not reflect the enzyme microenvironment.

These approaches are alternatives only when they represent competing exposure assumptions for the same model. In a mechanistic PBPK model, they can instead be successive steps: systemic and portal inflows determine liver entry, while cellular distribution determines exposure at an intracellular enzyme. Use the simplest approach that is supported by the data and appropriate to the question, and report its assumptions. More detailed modeling is not automatically more reliable if the transport, binding, or partition parameters are poorly known.

Diagnostics and Common Failure Modes

Check concentration calculations and assumptions before interpreting a predicted change in metabolism. Verify that concentrations and partition coefficients refer to the same analyte, time, units, and matrix; confirm whether a reported \(K_i\) is based on nominal or unbound inhibitor; and inspect whether the assumed unbound fractions are applicable over the exposure range. Large discrepancies between total and unbound concentration, or between systemic and portal concentrations, are reasons to check the basis of the model rather than to select whichever value produces a preferred prediction.

Using total plasma with an unbound \(K_i\). This mixes concentration bases and can inflate the inhibition ratio. Convert total plasma to unbound concentration and then address whether plasma free concentration represents the enzyme site.
Treating hepatic inlet as intracellular exposure. Inlet concentration describes blood entering the liver. Apply an appropriate distribution relationship before calling it the concentration at an intracellular enzyme.
Using total liver concentration as free site concentration. Tissue binding or sequestration can increase total tissue concentration without increasing unbound inhibitor at the enzyme. Use a defensible unbound partition estimate or a model that represents the relevant distribution.
Assuming equilibrium without evidence. Slow transport, active efflux, pH-dependent distribution, or rapidly changing concentrations can make an equilibrium partition ratio inappropriate. Time courses or a dynamic tissue model may be needed.
Ignoring the enzyme’s location. An enzyme in cytosol, a membrane, or another subcellular compartment may not experience the same local concentration. A bulk hepatocyte average is only a surrogate when that correspondence is justified.

A useful diagnostic is to calculate the implied concentration at each step—total plasma, unbound plasma, inlet, and site—and check whether each transition is supported by a measured fraction, flow, or partition parameter. If the prediction changes greatly under plausible alternative assumptions, present that sensitivity rather than treating one uncertain conversion as established fact.

Key takeaway: Select inhibitor exposure according to the enzyme location, convert concentrations on a consistent blood, plasma, or unbound basis, and match that basis to the inhibition constant. Total plasma concentration is not automatically the concentration at the enzyme.

Check Your Understanding

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

A hypothetical inhibitor has \(C_{p,\mathrm{tot}}=8.00\ \mu\mathrm{M}\) and \(f_{u,p}=0.050\). What is \(C_{u,p}\)?
Why can a high hepatic-inlet concentration not automatically be used as the concentration at a hepatocyte enzyme?
If \(C_{u,p}=0.250\ \mu\mathrm{M}\) and \(K_{p,uu,\mathrm{liver}}=2.00\), what site concentration is estimated under the equilibrium partition assumption?
An in vitro \(K_i\) is reported on a nominal basis, but the PBPK model uses free inhibitor concentration. What must be checked before using the reported \(K_i\)?

Solutions. (1) \(C_{u,p}=(0.050)(8.00)=0.400\ \mu\mathrm{M}\). (2) Hepatic inlet describes blood entering the liver; distribution across the liver and into the enzyme’s cellular compartment is a separate step. (3) \(C_{u,\mathrm{site}}\approx(2.00)(0.250)=0.500\ \mu\mathrm{M}\), provided the partition coefficient applies and equilibrium is reasonable. (4) Check the assay’s unbound fraction and conditions, and whether a justified conversion from nominal to unbound \(K_i\) is possible; do not combine a nominal constant with a free concentration without reconciling their bases.