# pH, pKa, pI and protein charge

25 Oct 2010

## What is acidity

Most people know that pH is a measure of acidity, but what is acidity? The Brønsted-Lowry theory tells us that an acid is a chemical that donates a proton (H⁺), while a base is a chemical that accepts a proton. The Lewis theory defines acids as chemicals that accept electrons, which is often, but not always, the same thing. For our purposes, acids are chemicals that can donate a proton, while bases (or alkaline substances) are chemicals that accept a proton.

An interesting result of this definition is that acidity is relative: whether a chemical is an acid or not depends on the other chemicals in the solution. If we add a chemical to water and it donates protons to water, then it is an acid. If we then add a chemical that is even better at donating protons to water, the first chemical will now accept protons from the second, thus acting as a base.

So what does pH measure? pH is a measure of the concentration of H⁺ in a solution (technically it's a measure of the activity of H⁺, but that is roughly the same thing under normal circumstances). More specifically pH is the negative logarithm (in base 10) of the molar concentration of H⁺. A pH of 7 therefore corresponds to 10-7 molar H⁺ (100 nM). The reason a negative logarithm is used is that it converts a huge range of concentrations into an easy range of positive numbers (although it is technically possible to have a negative pH: a 10 molar solution of H⁺ would have a pH of -1, and a pH of -3.6 has been reported, although this is due to the activity of the protons and not just their concentration).

## pI, pH and pKa

The kinetics of protons dissociating (and associating) from an acid, A, can be treated like any other reaction. The dissociation reaction is:

$\qquad\large{\text{HA} \leftrightharpoons \text{H}^+ + \text{A}^-}$

where HA is the acid (e.g. hydrochloric acid, HCl) and A⁻ is what is known as the conjugate base (Cl⁻ is the conjugate base of HCl). An association constant, $K_a$, can be defined just like any other equilibrium constant. This is the ratio of the concentration chemical species at equilibrium.

$\qquad\large{K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}}$

A strong acid is good at losing protons, so equilibrium will only be reached when [H⁺] and [A⁻] are high and [HA] is low. A strong acid, therefore has a large $K_a$. In the same way that pH is the negative logarithm of [H⁺], $pK_a$ is the negative logarithm of $K_a$:

$\qquad\large{pK_a = -log\left(\frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\right)}$

On a side note, an analogous equation can be derived for the dissociation of proton from a base:

$\qquad\large{\text{HB}^+ \leftrightharpoons \text{H}^+ + \text{B}}$

## Calculating the charge of a chemical

We can calculate the charge of a chemical (such as an amino acid) at a given pH by rearranging the $pK_a$ equation:

\qquad\begin{align} pK_a &= -log\left(\frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\right) \\ pK_a &= -log[\text{H}^+] - log[\text{A}^-] + log[\text{HA}] \\ pK_a &= pH - log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right) \\ log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right) &= pH - pK_a \\ \frac{[\text{A}^-]}{[\text{HA}]} &= 10^{(pH - pK_a)} \end{align}

This tells us the ratio of non-protonated to protonated acid species. To get the proportion of acid A that is protonated, we need to calculate the ratio of protonated acid to all acid species. If we set the amount of protonated acid, HA, to 1, then by the equation above, [A⁻] is equal to $10^{(pH - pK_a)}$ so the ratio is:

$\qquad\large{\frac{[\text{HA}]}{[\text{HA}] + [\text{A}^-]} = \frac{1}{1 + 10^{(pH - pK_a)}}}$

We can use this ratio to determine the pI of a protein.