From Property Values to a Usable Input Set
A PBPK input table can contain individually plausible values that do not describe the same chemical conditions. A pKa must be interpreted with its ionization convention; a solubility needs a concentration basis, pH, and solid form; and a logD value is meaningful only with its measurement pH and method. The practical task is therefore not to collect three numbers, but to assemble and audit a set whose units, conditions, and assumptions are explicit.
This tutorial works through one hypothetical monoprotic weak acid and one hypothetical monoprotic weak base. The calculations use equilibrium, ideal-solution relationships to check the inputs. They are screening calculations, not substitutes for measured values or for a more detailed model when the assumptions fail.
A Consistent Calculation Sequence
Start by recording molecular weight (MW) in g/mol, pKa, logP, intrinsic solubility \(S_0\), the pH of interest, and the measurement conditions for each reported value. In the calculations below, logP is the base-10 logarithm of the neutral species' octanol-to-water partition coefficient. LogD is the corresponding distribution measure for all dissolved forms at a specified pH. A reported experimental logD should retain its pH and assay conditions; it is not a universal constant for an ionizable drug.
For a monoprotic acid, the ratio of ionized to neutral concentration is \(10^{pH-pKa}\). For a base whose pKa is that of \(BH^+\), the ratio of ionized to neutral concentration is \(10^{pKa-pH}\). These ratios give the neutral fraction by dividing the neutral concentration by the sum of neutral and ionized concentrations. The complementary fraction is ionized.
The solubility expressions assume a fixed pH, equilibrium with the neutral solid, ideal concentration-based behavior, and no additional solubility contributions from salts, complexes, aggregates, or other solid phases. They describe total dissolved concentration, not merely the concentration of the neutral species.
Worked Assessments
Worked Example: Auditing a Hypothetical Weak Acid
Consider a hypothetical weak acid with MW \(300\ \mathrm{g/mol}\), pKa \(4.50\), logP \(2.00\), and reported intrinsic solubility \(0.0200\ \mathrm{mg/mL}\). Assess its ionization, approximate logD, and equilibrium solubility at pH \(6.50\). Assume the solid is the neutral acid and that ionized-species partitioning into octanol is negligible.
Convert \(S_0\) to molar units. Since \(0.0200\ \mathrm{mg/mL}=20.0\ \mathrm{mg/L}=0.0200\ \mathrm{g/L}\),
Calculate the fractions. Here \(pH-pKa=6.50-4.50=2.00\), so the ionized-to-neutral ratio is \(10^2=100\). Thus
At this pH, the calculation predicts about \(0.9901\), or \(99.01\%\), ionized drug in the dissolved acid-base pair.
Estimate logD under the stated partition assumption.
This is a model-based estimate near zero, not a replacement for a measured logD. It follows from treating neutral acid as the only species with appreciable octanol partitioning.
Calculate total equilibrium solubility.
On a molar basis, \(2.02\ \mathrm{g/L}/(300\ \mathrm{g/mol})=0.00673\ \mathrm{mol/L}=6.73\ \mathrm{mmol/L}\). The result is conditional on the stated solid-form and ideal-equilibrium assumptions.
Worked Example: Auditing a Hypothetical Weak Base
A hypothetical weak base has MW \(250\ \mathrm{g/mol}\), pKa \(8.00\) for \(BH^+\), logP \(1.50\), and \(S_0=0.0500\ \mathrm{mg/mL}\). Calculate the neutral fraction, approximate logD, and total equilibrium solubility at pH \(7.00\). Assume the neutral free base is the solid in equilibrium and ionized-species octanol partitioning is negligible.
Convert intrinsic solubility. \(0.0500\ \mathrm{mg/mL}=50.0\ \mathrm{mg/L}=0.0500\ \mathrm{g/L}\), so
Calculate the acid-base ratio and fractions. Since \(pKa-pH=8.00-7.00=1.00\), the ionized-to-neutral ratio is \(10\). Therefore
Estimate logD.
Calculate total solubility.
The molar result is \(0.550\ \mathrm{g/L}/(250\ \mathrm{g/mol})=0.00220\ \mathrm{mol/L}=2.20\ \mathrm{mmol/L}\). A pH below the conjugate-acid pKa favors \(BH^+\), so the calculated dissolved total exceeds the intrinsic solubility of neutral \(B\).
Worked Example: Checking a Reported Solubility Against the Input Set
For the hypothetical acid in the first example, suppose a separate equilibrium assay reports total solubility \(1.80\ \mathrm{mg/mL}\) at pH \(6.50\). The acid pKa and \(S_0\) are as previously stated. Check whether this is exactly consistent with the ideal equation, and quantify the discrepancy without assuming that either measurement is wrong.
The ideal prediction at that pH is \(0.0200(1+10^{6.50-4.50})=2.02\ \mathrm{mg/mL}\). The reported-to-predicted ratio and relative difference are
Thus the assay result is \(10.89\%\) below the ideal prediction; it is not an exact match. As a diagnostic, the \(S_0\) that would reproduce the reported result with the same pKa and ideal model is
That inferred value is about \(10.89\%\) below the reported \(0.0200\ \mathrm{mg/mL}\). The calculation identifies the size of the discrepancy, but does not establish its cause. Assay variability, a different solid form, pH or unit-basis differences, nonideal behavior, or failure to reach equilibrium are possible issues to investigate. Whether the difference is acceptable depends on the measurement uncertainty and intended use.
Why the Relationships Hold—and When to Audit Them
The core calculations follow directly from the equilibrium concentration ratios. For an acid, the Henderson–Hasselbalch relationship gives \(pH=pKa+\log_{10}([A^-]/[HA])\), so \([A^-]/[HA]=10^{pH-pKa}\). Dividing each species concentration by their sum yields the neutral fraction. For a base, apply the same relationship to \(BH^+\rightleftharpoons H^++B\): \([B]/[BH^+]=10^{pH-pKa}\), or equivalently \([BH^+]/[B]=10^{pKa-pH}\). The neutral fraction then has the base form used above.
For solubility, assume the neutral dissolved concentration at saturation equals \(S_0\). For an acid, the ionized concentration at that pH is \(S_0\,10^{pH-pKa}\); adding it to the neutral concentration gives \(S_0(1+10^{pH-pKa})\). For a base, the ionized concentration is \(S_0\,10^{pKa-pH}\), giving \(S_0(1+10^{pKa-pH})\). This derivation also shows why pH must be specified and why the equation can fail if the solid phase or solution chemistry changes.
For the logD approximation, if only the neutral form partitions into octanol, the distribution coefficient is approximately the neutral partition coefficient multiplied by the fraction of total dissolved drug that is neutral: \(D\approx P f_{\mathrm{neutral}}\). Taking base-10 logarithms gives \(\log D\approx\log P+\log_{10}(f_{\mathrm{neutral}})\). If ionized species partition appreciably, or if the assay has additional complications, this simplification is not adequate; use a compatible measured logD or a model that represents those contributions.
Write every concentration with its unit and basis (for example, mg/mL of compound versus salt, or mmol/L), and record assay pH, temperature, buffer, and solid form when available.
Confirm molecular weight and whether the pKa belongs to the acid or to the conjugate acid of a base. A reversed pKa convention reverses the predicted pH dependence.
Compute pH-specific fractions, then compare an approximate logD or ideal solubility prediction only with data measured on a compatible basis and at the relevant pH.
Keep measured inputs distinct from derived values. Investigate mismatches through conditions and uncertainty before changing a parameter to force agreement.
A particularly common mistake is to use the acid formula for a base because both compounds have a pKa. For a weak base, lowering pH increases the fraction of protonated, ionized drug; the base equation therefore has \(pKa-pH\) in its exponent for the ionized-to-neutral ratio. Another mistake is to compare a salt-basis mass concentration directly with a free-drug-basis input. Convert using the molecular weight and chemical form that the reported concentration actually represents.
The equations are useful as consistency checks, not as universal corrections. Real assays may involve nonideal activities, multiple solid forms, precipitation, supersaturation, aggregation, degradation, or buffer-dependent effects. When those conditions matter, the alternatives are to prioritize an experimentally measured value under relevant conditions, characterize the uncertainty, or use a more detailed chemical or experimental model. Do not infer tissue partition coefficients from these calculations: logD describes an octanol-water distribution context, not a tissue-to-plasma partition coefficient.
Check Your Understanding
Use the equations and audit principles above. Answers are provided after the questions.
- A hypothetical weak acid has pKa 5.00. At pH 5.00, what are its neutral and ionized fractions?
- A weak base has conjugate-acid pKa 7.50. At pH 6.50, what is the ionized-to-neutral ratio, and which form predominates?
- A hypothetical compound has MW \(200\ \mathrm{g/mol}\) and solubility \(0.040\ \mathrm{mg/mL}\). Convert this concentration to mmol/L.
- For a weak acid with \(S_0=0.010\ \mathrm{mg/mL}\), pKa 4.00, and pH 5.00, calculate ideal total equilibrium solubility.
- Why is an experimental logD value without a stated pH insufficient to compare directly with a pH-specific estimate for an ionizable drug?
Solutions. (1) The acid ratio \(10^{pH-pKa}=1\), so each fraction is \(1/(1+1)=0.500\). (2) The ratio is \(10^{7.50-6.50}=10\); \(BH^+\) is the predominant, ionized form, with fraction \(10/11=0.9091\). (3) \(0.040\ \mathrm{mg/mL}=0.040\ \mathrm{g/L}\); dividing by \(200\ \mathrm{g/mol}\) gives \(0.000200\ \mathrm{mol/L}=0.200\ \mathrm{mmol/L}\). (4) \(S_{\mathrm{total}}=0.010(1+10^{5.00-4.00})=0.110\ \mathrm{mg/mL}\), conditional on the ideal neutral-solid assumptions. (5) Ionization, and therefore the distribution of total drug between phases, changes with pH. Without the measurement pH and compatible assay conditions, the reported value cannot be compared unambiguously with a pH-specific estimate.