How to prepare media from salts

The composition of liquid growth media is frequently given in its ion composition in literature. The preparation of such media from salts diluted in aqua dest leads to the practical problem of how to combine available salts in a way best approximating the given composition. Let us call the sought grams of salt and the known molars of ion . Elementary chemistry allows us to compose a matrix , such that a typical element stands for the molar contribution of 1 gram of salt $j$ to ion $i$. If we want to prepare a litre of medium, the problem is thus to find a useful , given and . We can only hope for a unique solution if . Let us assume this to be the case. Let us first try to minimize the sum of the absolute errors. So, we are looking for , such that

(8.1)

is minimal for all possible choices for . In matrix notation we have is minimal. To find such a , we have to solve . In elements we have

(8.2)

In matrix notation this becomes

In terms of the latter notation, the solution is easily written down, if it exists:

(8.3)

So, we have to multiply the column vector to the left by the matrix . This matrix is known as the (left) generalized Moore inverse of the matrix . It is an inverse, because multiplication by its original results in the identity matrix, . It is generalized because originally, the inverse matrix is only defined for square matrices. It is a particular generalized inverse, because more ways exist to define an inverse matrix, namely the right generalized inverse. The left and the right inverse are identical for a symmetrical (and thus square) matrix only, if they exist. Usually, the ions differ widely in abundance. From a biological point of view, the way they play a role in supporting life can be widely different. Therefore it is questionable wether the loss function makes sense on physical grounds. A lot more elegant is to minimize the sum of relative errors rather than the absolute ones. The loss function now takes the form

In matrix notation we have

where . In elements, we have to solve

(8.5)

The solution, if it exists, is of course

(8.6)

Although this choice for will be satisfying in most practical cases, it is possible that one or more elements of are negative. Since it is far from easy to take a certain salt out of a solution, we might want to have non- negative solutions only. This has to be obtained by changing the salts to be use for the preparation of the medium. As an example, consider the preparation of artificial seawater, as far as the main ions are concerned, from a set of salts. Table 8.1 gives the relevant data. An "exact" solution is not possible, because the given combination of ions is not electrically neutral. The values has been axtracted from "the handbook", ignoring the many rare ions. We can conclude that, when both magnesiumchloride and magnesiumsulfate are available, the sum of relative errors can be made quite small. The option with magnesiumsulfate only, has to be preferred above that with magnesiumchloride only. In practice we should bother about how to prevent gyps to precipitate, in this case.

Table 8.1: The preparation of artificial seawater. The $i,j$-th element of the matrix is obtained by dividing the $i,j$-th element of the main table given below, by the molar concentration of ion $i$ of seawater (second column) and the molar weight of salt $j$ (second row). The last three rows consist of the best choice for salts using one or both magnesium salts in g/l. The first component indicates the sum of relative errors.