Cells contain small membrane-bound compartments called vesicles. These vesicles can be so small that you can’t even see them in your typical light microscope. They are used for all sorts of things, from transporting compounds around the cell to deconstructing compounds or digesting bacteria. Some of these digesting vesicles, called lysosomes, use acid (low pH) and enzymes to tear apart molecules or entire bacteria. Since acidity is expressed in pH, a measure of how many hydrogen ions (H⁺ or protons) are floating around in solution, I started thinking…

Molecules are usually talked about in numbers too large to count — millions, quadrillions, or 10²³ — but this is a small space, a really small space. In a space so incredibly small, how few free protons are there? (I can submit my own questions, right?)

In case you can’t tell from the question, this is going to get into some technical details pretty quickly, so if you have an aversion to mathematics, you may want to skip to the summary.

The vesicles in a cell can be as small as 20 nm across, which would have a volume of roughly 4.189×10^-21 L ((4/3)πr³). At pH 5, the proton concentration is 10 mM or 10^-5 M (pH=-log([H⁺])). This would mean that a 20 nm vesicle with a pH of 5 contains approximately 4.2×10^-26 mol of H⁺ or 0.026 free protons per vesicle (6.022×10²³ molecules/mol). Since you can’t have a fraction of a subatomic particle (yes, a proton can be divided into its three quarks, but they aren’t ever observed alone anyway), there must be something more to this simple definition of pH.

A common mistake is to forget that altering the proton concentration by a small amount (<10^-7 M) does not contribute significantly to the proton concentration of water at neutral pH (pH 7). Although we’re talking about very small numbers of protons in these vesicles, the concentration is still well above that cutoff at 10^-5 M. The other point to consider is that pH is a balance between the forces of acids and bases. Lowering the amount of basic ions (which is how a Lewis acid functions) relative to acidic ions would also make a solution more acidic. At pH 5, there should be 10000× more H⁺ than OH^– (water’s basic half). If the concentration of OH^– could be raised to 10^-3 M (~2 molecules/vesicle), then H⁺ would have to be 10 M (~20000/vesicle) to obtain the desired pH. At least now we would be talking about whole molecules, this would also require lowering the concentration of H₂O relative to H⁺ and OH^– can be lowered. This is where the dissociation constant of water becomes an issue. The typical concentration of dissociated water molecules is 10^-14 M, but if the environment was under such tight constraints that the normal equilibrium of H⁺ + OH^– ⇌ HOH was shifted to the left, then the previously described hypothesis may hold true. The dissociation constant of water is affected by temperature, but such dramatic alterations of the dissociative properties of water in a controlled microenvironment is quickly escaping out capacity, so we’re hopeful for a simpler solution.

Stepping back for a moment, even with the tiny size of a vesicle, there are even smaller spaces in which we assume there is a maintained pH. Within and around proteins, there are microenvironments containing only a handful (if you had very small hands) of atoms and molecules. How can you claim such a small space has any particular pH when only five molecules are present?!

tl;dr

If the amino acid residues of a protein are sufficient to maintain a pH of 5, then surely a 20nm wide vesicle can maintain a pH of 5 without the need of impossibly small numbers of free protons. Much more likely is that pH really has nothing to do with the actual number of acid or alkaline ions at a given time; instead these numbers represent averages. If you spend an hour a day in your car, on average, your car would contain 0.04 people. This would not sound right, and you’re more likely to say that, 1/24 of the time, there is a person in your car. A molecule is considered more acidic if it has a proton that spends more time away from the molecule as it spends bound to it. Now let’s pretend that you are that proton and your car is a side-chain on a protein where the proton spends a small portion of its time. To get .026 free protons, the proton could be bound 97.4% of the time and free the other 2.6%.

Sometimes things in science seem confusing or impossible simply because science is not described in the same way that we speak and communicate. Even when the answer seems hard to find, translating science into terms we can all understand makes some things so much simpler.

Note: The back-of-the-envelope calculations were verified using Wolfram|Alpha, although I welcome any corrections.