Unbound Intracellular Concentrations in Transport Models

Why Total and Unbound Hepatocyte Concentrations Differ

Hepatocyte assays commonly report drug associated with cells, which includes both unbound drug and drug bound to intracellular proteins, lipids, and other sites. A transporter or enzyme may respond to the unbound concentration available at its binding site rather than to this total concentration. Treating the measured total concentration as the concentration driving every process can therefore distort predicted fluxes, even when the total amount in the cell is measured accurately.

The distinction matters in both steady-state calculations and time courses. Binding changes how much drug is present at a given unbound concentration. If binding equilibrates rapidly, it also changes how quickly the unbound concentration responds to a net influx or loss. The transporter and enzyme rate laws themselves need not change: what changes is the concentration supplied to those laws and the relationship between unbound concentration and total intracellular drug.

Definition: Let \(C_{h,\mathrm{tot}}\) denote total intracellular hepatocyte concentration and \(C_{h,u}\) denote unbound intracellular concentration, both expressed per unit hepatocyte volume. The intracellular unbound fraction is \(f_{u,h}=C_{h,u}/C_{h,\mathrm{tot}}\), when the ratio is defined at the concentration of interest. Thus \(C_{h,u}=f_{u,h}C_{h,\mathrm{tot}}\). This fraction is not necessarily constant: it is constant under linear binding, but can vary with concentration under saturable binding.

In this tutorial, transporter and enzyme rates are expressed as concentration rates per unit hepatocyte volume, such as \(\mu\mathrm{M}/\mathrm{min}\). The basic free-drug model assumes that intracellular binding is at rapid equilibrium and that the unbound intracellular concentration is the concentration relevant to the transporter or enzyme being modeled. These are modeling assumptions, not universal properties of every transporter, enzyme, or assay.

Linear Binding: A Constant Intracellular Unbound Fraction

Under linear, nonsaturable binding, the bound concentration is proportional to the unbound concentration. If \(C_{h,\mathrm{bound}}=K C_{h,u}\), where \(K\) is a nonnegative dimensionless binding ratio, then total concentration is \(C_{h,\mathrm{tot}}=C_{h,u}+C_{h,\mathrm{bound}}=(1+K)C_{h,u}\). Therefore \(f_{u,h}=1/(1+K)\), and the conversion between total and unbound concentration is linear.

C_{h,\mathrm{tot}}=\frac{C_{h,u}}{f_{u,h}}, \qquad C_{h,u}=f_{u,h}C_{h,\mathrm{tot}}, \qquad 0<f_{u,h}\leq 1.

A low \(f_{u,h}\) means that a relatively large total cellular concentration can correspond to a modest unbound concentration. The fraction is specific to the drug, the cellular preparation, and the conditions under which binding is measured. For the simple conversion above, it is also assumed not to change over the concentration range being modeled.

Worked Example: Converting Total Concentration to Unbound Concentration

Suppose a hypothetical hepatocyte experiment reports a total concentration of \(25\ \mu\mathrm{M}\), and an independent binding measurement supports a constant \(f_{u,h}=0.080\) over this range. What unbound concentration should be used in a free-drug transporter rate law?

Apply the linear-binding relationship:

C_{h,u}=f_{u,h}C_{h,\mathrm{tot}} =(0.080)(25\ \mu\mathrm{M}) =2.00\ \mu\mathrm{M}.

Under the stated assumptions, \(2.00\ \mu\mathrm{M}\), not \(25\ \mu\mathrm{M}\), is the concentration supplied to the rate law. The remaining \(23.0\ \mu\mathrm{M}\) is bound. This calculation does not by itself establish that a specific transporter senses only the unbound pool; that is an additional mechanistic assumption.

Using Unbound Concentration in Transporter and Enzyme Rates

For a transporter or enzyme represented by a Michaelis–Menten rate law, the free-drug model evaluates the rate at \(C_{h,u}\). For example, an intracellular loss process \(j\) may be written with maximum rate \(V_{\max,j}\) and Michaelis constant \(K_{m,j}\). Both parameters must use a consistent concentration basis: \(V_{\max,j}\) is a concentration rate and \(K_{m,j}\) is a concentration. When the law is written in terms of unbound drug, \(K_{m,j}\) is interpreted on that unbound concentration basis.

v_j(C_{h,u})= \frac{V_{\max,j}C_{h,u}}{K_{m,j}+C_{h,u}}.

Uptake from outside the cell is often parameterized using an extracellular unbound concentration, whereas intracellular efflux and intracellular metabolism may be parameterized using \(C_{h,u}\). The driving concentration must be specified separately for each process; replacing every concentration in a model with one intracellular value would not be appropriate. In particular, the choice of free versus total concentration in an empirical rate law should match the concentration basis used to estimate its parameters.

Worked Example: Calculating Efflux and Metabolic Loss From Free Drug

Consider hypothetical intracellular efflux and metabolic-loss processes. Suppose \(C_{h,\mathrm{tot}}=125\ \mu\mathrm{M}\) and \(f_{u,h}=0.080\), so that \(C_{h,u}=10.0\ \mu\mathrm{M}\). Let the efflux parameters be \(V_{\max,\mathrm{eff}}=8.00\ \mu\mathrm{M}/\mathrm{min}\) and \(K_{m,\mathrm{eff}}=2.00\ \mu\mathrm{M}\). Let the metabolism parameters be \(V_{\max,\mathrm{met}}=6.00\ \mu\mathrm{M}/\mathrm{min}\) and \(K_{m,\mathrm{met}}=4.00\ \mu\mathrm{M}\). Assume both processes follow the free-drug rate law above.

The efflux rate is

v_{\mathrm{eff}}= \frac{(8.00\ \mu\mathrm{M}/\mathrm{min})(10.0\ \mu\mathrm{M})} {2.00\ \mu\mathrm{M}+10.0\ \mu\mathrm{M}} =6.6667\ \mu\mathrm{M}/\mathrm{min}.

The metabolic rate is

v_{\mathrm{met}}= \frac{(6.00\ \mu\mathrm{M}/\mathrm{min})(10.0\ \mu\mathrm{M})} {4.00\ \mu\mathrm{M}+10.0\ \mu\mathrm{M}} =4.2857\ \mu\mathrm{M}/\mathrm{min}.

Adding the unrounded component rates gives a combined loss of \(6.6667+4.2857=10.9524\ \mu\mathrm{M}/\mathrm{min}\), or approximately \(10.95\ \mu\mathrm{M}/\mathrm{min}\). The rates depend on \(C_{h,u}=10.0\ \mu\mathrm{M}\), not directly on the total concentration of \(125\ \mu\mathrm{M}\). Substituting total concentration into these same rate laws would be inconsistent with the stated free-drug assumption and would produce different rates.

How Binding Changes a Transient

The effect of binding on a time course follows from mass balance. Let \(r_{\mathrm{net}}\) be the net rate of change of total intracellular concentration: influx minus all efflux and metabolic losses, with each term expressed on the same cellular-volume basis. Then \(dC_{h,\mathrm{tot}}/dt=r_{\mathrm{net}}\). If binding equilibrates rapidly and \(f_{u,h}\) is constant, \(C_{h,\mathrm{tot}}=C_{h,u}/f_{u,h}\). Differentiating this relation gives the unbound-concentration equation.

\frac{dC_{h,\mathrm{tot}}}{dt} =\frac{1}{f_{u,h}}\frac{dC_{h,u}}{dt} =r_{\mathrm{net}}, \qquad \frac{dC_{h,u}}{dt}=f_{u,h}r_{\mathrm{net}}.

Thus, at a given net rate of total accumulation, a smaller constant unbound fraction produces a smaller instantaneous change in unbound concentration. This is a buffering effect. Binding does not add a separate transporter or enzyme flux in this equilibrium model; it changes the relationship between the net flux and the unbound concentration response. At steady state, \(r_{\mathrm{net}}=0\), so the steady-state balance among fluxes is still determined by the flux laws. Binding affects the total concentration corresponding to that balance.

Worked Example: Binding Slows the Rise in Unbound Concentration

Consider a hypothetical cell with constant input \(I=2.00\ \mu\mathrm{M}/\mathrm{min}\) and a first-order intracellular loss \(v_{\mathrm{loss}}=kC_{h,u}\), where \(k=0.500\ \mathrm{min}^{-1}\). Assume rapid linear binding, constant input, and an initial unbound concentration of zero. Compare \(f_{u,h}=0.100\) with no binding, represented by \(f_{u,h}=1.00\).

The total-concentration balance is \(dC_{h,\mathrm{tot}}/dt=I-kC_{h,u}\). Using the constant-fraction relation gives

\frac{dC_{h,u}}{dt} =f_{u,h}(I-kC_{h,u}).

The steady-state unbound concentration is obtained by setting the derivative to zero: \(I-kC_{h,u}=0\), so \(C_{h,u,\mathrm{ss}}=I/k=(2.00\ \mu\mathrm{M}/\mathrm{min})/(0.500\ \mathrm{min}^{-1})=4.00\ \mu\mathrm{M}\) for either binding condition. Solving the linear differential equation from \(C_{h,u}(0)=0\) gives

C_{h,u}(t)=\frac{I}{k}\left(1-e^{-f_{u,h}kt}\right).

At \(t=10.0\ \mathrm{min}\) with \(f_{u,h}=0.100\), \(C_{h,u}=4.00(1-e^{-0.500})=1.57\ \mu\mathrm{M}\). The corresponding total concentration is \(1.57/0.100=15.7\ \mu\mathrm{M}\). With no binding, \(C_{h,u}=4.00(1-e^{-5.00})=3.97\ \mu\mathrm{M}\), and total concentration equals unbound concentration. Both cases approach the same \(4.00\ \mu\mathrm{M}\) unbound steady state, but the bound case approaches it more slowly and accumulates more total drug.

Saturable Binding and the General Buffering Relationship

A constant unbound fraction is not suitable when binding sites can become saturated. One simple equilibrium model has a single class of binding sites with total binding capacity \(B_{\max}\) and dissociation constant \(K_d\). Here \(B_{\max}\), \(K_d\), and both concentrations use the same concentration units. Bound concentration is \(B_{\max}C_{h,u}/(K_d+C_{h,u})\), so total concentration is the sum of free and bound drug.

C_{h,\mathrm{tot}} =C_{h,u}+\frac{B_{\max}C_{h,u}}{K_d+C_{h,u}}.

For a given total concentration, this equation can be solved for \(C_{h,u}\), choosing the nonnegative solution. In contrast to linear binding, the ratio \(C_{h,u}/C_{h,\mathrm{tot}}\) varies with concentration. Differentiating the total concentration with respect to the unbound concentration gives a local binding-capacity factor. Combining this derivative with the mass balance yields the transient relation below.

\frac{dC_{h,\mathrm{tot}}}{dC_{h,u}} =1+\frac{B_{\max}K_d}{(K_d+C_{h,u})^2}, \qquad \frac{dC_{h,u}}{dt} = \frac{r_{\mathrm{net}}} {1+B_{\max}K_d/(K_d+C_{h,u})^2}.

The denominator is at least one, so rapid equilibrium binding buffers the unbound response to a given net flux. At low unbound concentrations relative to \(K_d\), many sites are available and the binding-capacity factor can be large. As sites saturate, the factor approaches one, and additional drug produces a larger increase in unbound concentration per increment of total concentration.

Worked Example: Local Buffering With Saturable Binding

Suppose a hypothetical hepatocyte has \(B_{\max}=20.0\ \mu\mathrm{M}\), \(K_d=1.00\ \mu\mathrm{M}\), and \(C_{h,u}=2.00\ \mu\mathrm{M}\). Calculate total concentration and the effect of a net accumulation rate of \(1.00\ \mu\mathrm{M}/\mathrm{min}\) on the instantaneous unbound-concentration rate.

First, total concentration is

C_{h,\mathrm{tot}} =2.00+\frac{(20.0)(2.00)}{1.00+2.00} =2.00+13.3333 =15.3333\ \mu\mathrm{M} \approx 15.3\ \mu\mathrm{M}.

Next, the local buffering factor is

1+\frac{B_{\max}K_d}{(K_d+C_{h,u})^2} =1+\frac{(20.0)(1.00)}{(1.00+2.00)^2} =1+\frac{20.0}{9.00} =3.2222.

Therefore, for the stated net accumulation rate,

\frac{dC_{h,u}}{dt} =\frac{1.00\ \mu\mathrm{M}/\mathrm{min}}{3.2222} =0.3103\ \mu\mathrm{M}/\mathrm{min}.

The rate is local to the stated concentration: as \(C_{h,u}\) changes, the buffering factor changes too. The calculation assumes binding equilibrates quickly enough for the equilibrium relationship to hold throughout the transient.

Assumptions, Diagnostics, and Alternatives

The free-drug treatment is useful only when its assumptions are defensible. It assumes that the relevant intracellular free concentration is represented by a well-mixed hepatocyte concentration, that binding is either adequately described by the chosen equilibrium model or by a measured constant fraction, and that the transporter or enzyme rate law is calibrated on the same concentration basis used in the model. Rapid equilibrium binding is a separate assumption from free-drug-dependent flux: a process may depend on unbound drug even when binding is too slow to remain at equilibrium.

Several observations can signal a mismatch. If the apparent relationship between total concentration and flux changes with concentration, a constant \(f_{u,h}\) may be inadequate. If estimates of \(f_{u,h}\) depend strongly on assay duration, the preparation may not have reached binding equilibrium. Unexplained differences between predicted and measured accumulation can also arise from cell damage, nonspecific adsorption, unmeasured compartments, or uncertainty in the cellular-volume normalization. These diagnostics do not identify a unique cause, but they are reasons to check the assumptions before changing transporter capacity or enzyme parameters.

Approach	What it represents	Main caution
Constant \(f_{u,h}\)	Linear, rapid-equilibrium binding over the modeled range	Can misrepresent concentration-dependent saturation
Saturable equilibrium binding	Finite binding capacity with rapid equilibration	Requires binding parameters and a valid single-pool approximation
Explicit kinetic binding	Binding and unbinding with their own time scales	Needs kinetic data and additional states; equilibrium conversion is not sufficient
Empirical total-concentration rate law	A rate fitted directly against measured total concentration	May be condition-specific and should not be interpreted automatically as a free-drug mechanism

An intracellular unbound fraction is also distinct from \(K_{p,uu}\), the ratio of unbound liver concentration to unbound plasma concentration at a specified condition. The fraction \(f_{u,h}\) converts total hepatocyte concentration to unbound concentration within cells; \(K_{p,uu}\) compares unbound concentrations across liver and plasma. Neither quantity can be substituted for the other without an additional model linking the compartments.

Common pitfall: A modeler may argue that a measured total hepatocyte concentration of \(20\ \mu\mathrm{M}\) means an enzyme experiences \(20\ \mu\mathrm{M}\), and then use that value in a free-drug Michaelis–Menten law. The correction is to convert total to unbound concentration using a justified binding relationship, then evaluate the law on the concentration basis on which its parameters are defined. If that basis is unknown, the free-drug interpretation is not established merely by having a total-concentration measurement.

The most important limitation is that a single intracellular concentration may not capture spatial gradients, membrane-associated pools, or transporter-specific access to drug. A transporter can have an empirical rate relationship that does not reduce to the well-mixed free-drug expression. Binding measurements also carry uncertainty, and a wrong binding model can shift apparent kinetic parameters. Consequently, \(C_{h,u}\) is a mechanistic concentration choice to justify and test, not a guarantee that every measured flux will be predicted correctly.

Check Your Understanding

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

If \(C_{h,\mathrm{tot}}=40.0\ \mu\mathrm{M}\) and constant \(f_{u,h}=0.050\), what is \(C_{h,u}\)?
With linear rapid-equilibrium binding, does changing \(f_{u,h}\) necessarily change the steady-state unbound concentration when input and free-drug loss laws are unchanged? Explain.
For saturable binding, what happens to the binding-capacity factor as \(C_{h,u}\) becomes much larger than \(K_d\)?
Is \(f_{u,h}\) interchangeable with \(K_{p,uu}\)? State what each quantity relates.

Solutions. (1) \(C_{h,u}=(0.050)(40.0)=2.00\ \mu\mathrm{M}\). (2) No. At steady state the net flux is zero, and the balance among input and losses is determined by their unchanged free-drug laws; \(f_{u,h}\) changes the corresponding total concentration and transient response. (3) The factor \(1+B_{\max}K_d/(K_d+C_{h,u})^2\) approaches one as \(C_{h,u}\) becomes large relative to \(K_d\), because the binding sites are becoming saturated. (4) No. \(f_{u,h}\) is the ratio of unbound to total intracellular concentration, while \(K_{p,uu}\) is a ratio comparing unbound liver and unbound plasma concentrations.