Thoughts on serial dilutions

An adventurer has been bitten by a poisonous creature and has sought refuge in a shack. The adventurer is too weak to move from the shack and will die unless an anti-toxoid is administered. Luckily for the adventurer, the shack seems to be someone’s old abandoned home lab. One of the vials in the lab is precisely the quantity of anti-toxoid that the adventurer needs. Unfortunately, the anti-toxoid is dissolved together with a chemical preservative which the adventurer is highly allergic to. This preservative will kill the adventurer unless its levels are reduced by at least 150 times.

Besides 5 ml of anti-toxoid, the adventurer has also found reels of dialysis tubing and 25 ml of water. The adventurer thinks to himself: “I will arbitrarily divide the 25 ml of water into r portions and then use these portions to serially dialyze the anti-toxoid. If r is large enough, I will surely bring the lethal chemical preservative down by at least 150 times.” Is the adventurer correct?

Let us generalize with a starting sample volume Vs and a fixed dialysis buffer volume V_d with which to perform dialysis. If V_s is dialyzed against V_d, we get a dilution factor D where

$D=\frac{V_s + V_d}{V_s}$

If we split the dialysis buffer into r portions of V_d/r each, we will get:

V_s = sample volume

V_d = total volume of dialysis buffer

r = number of dialysis rounds each performed with V_d/r volume of dialysis buffer

$D=\left(\frac{V_s+\frac{V_d}{r}}{V_s}\right)^r$

Does our dilution factor improve indefinitely with increasing r or is there a theoretical limit? We define the ratio

$N=\frac{V_d}{V_s}$ giving us $D=(1+\frac{N}{r})^r$

We notice at this point that the above equation is similar to the derivation of the mathematical constant e. Consistent with intuition, the theoretical limit as r goes to infinity is given by

$\lim_{r \to \infty}(1+\frac{N}{r})^r=e^N}$

In short, most unfortunately for the adventurer, the best dilution factor he can hope for is e⁵ which works out to ~148.4, just below the 150-fold dilution required and this is assuming that he can infinitely divide his water for dialysis.

Addendum: My wife was upset at the implied demise of the adventurer. What she did not know was that a magical unicorn teleported from a parallel universe and used its healing spittle to save the adventurer.

Thoughts on serial dilutions

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