The Ames test

The Ames test is a bioassay for testing the mutagenic potential of chemicals on the basis of the frequency of backward mutations in histidine auxotrophic mutants of the pathogenic bacterium Salmonella typhimurium (i.e. these bacteria have been mutated such that they are no longer able to synthesize the amino acid histidine themselves, so they need it in the medium). At the start of the experiment, a certain amount of (primairy) histidine auxotrophs are inoculated on an agar plate containing a rich medium with, however, a little bit of histidine and the chemical to be tested. (Usually, there are several plates with different concentrations of chemical.) The incolated bacteria grow and divide, until the histidine is exhausted. So each produces a micro-colony, usually consisting of some 100 bacteria. If one or several bacteria have been permanently converted to the prototrophic state, however, its colony will continue to grow and will become visible with the naked eye. Its is then called a revertant colony. The growing state seems to be a condition for the expression of the backward mutation. So, no visible mutations occur after colony growth ceased due to histidine limitation. For mutagenic chemicals, the number of revertant colonies generally increases linearly with its concentration, at a rate that is taken to be proportional to its mutagenic potential. Let us study the quatitative aspect. To derive an expression for the number of revertant colonies, we first need the fixation probability , which is the probability that a prototrophic cell of age a at time zero will grow into a revertant colony after the moment the histidine is exhausted. Generally, this fixation probability is less than 1 because prototrophic cells may become extinct due to conversion to multiple auxotrophy. This extinction probability is easily found from the probability generating function, , of the number of prototrophic cells arising from a prototrophic cell after division, i.e. , where and is the conversion probability from a prototrophic cell into a multiple auxotrophic one in a period $d$, i.e. the division interval. The extinction probability , of prototrophic cells starting with a prototrophic one of age $d$ is given by . If the probability rate for conversion of a multiple auxotrophy to a primairy auxotrophy or prototrophy is neglibibly small, and if the conversion probability rates from primary auxotrophy and prototrophy to multiple auxotrophy is equal and constant at value , say, we have . This leads to . For an arbitrary initial age , we should account for the probability of conversion to multiple auxotrophy before the moment of first division, so

			(5.1)

where the + sign indicates the maximum of zero and the term within the brackets. Suppose that we inoculate with cells (usually some ) and that the amount of histidine is just enough for the formation of cells on the plate. Suppose further that the division interval $d$ is constant. The number of cells on the plate then grows exponentially in time, i.e. , as long as the histidine is not limiting. This occurs at time $T$, found from , giving . In writing an expression for the probability on a revertant colony, we have to realize that the inoculated cells as well as the cells present at $T$, have not been exposed to the chemical during their full life cycle. To deal with this fact, we partition $T$ as , where and . Suppose that each inoculated cell has a probability of being primary auxotrophic and of being prototrophic. If the probability rate of conversion from primary auxotrophy to prototrophy is small and constant at value $\alpha$, say, the probability, , that an inoculated cell of age will grow into a revertant colony is given by

Suppose that we start with the stable age distribution for the inoculated cells, i.e. . The probability that a randomly chosen cell will then grow into a revertant colony is after some calculations found to be

which reduces for negligibly small to

The number of revertant colonies will be binomially distributed, but in view of the small number of revertant colonies with respect to the number of inoculated cells, it is safe to use the approximation with the Poisson distribution with parameter . If each molecule of the chemical has an equal (and small) probability of causing a mutation, a will be proportional to the concentration. One can use likelihood based linear Poisson regression, to obtain the propartionality constant from the number of revertant colonies at several concentrations of test compound. In conclusion, (5.3) confirms the empirical observation that the mean number of revertant colonies depends linearly on the concentration of chemical. However, it also reveals that a number of other factors contribute as well, like the age distribution and the amount of inoculated cells, the amount of histidine supplied and the division interval (and so the composition of the medium). In practice these problems are "solved" by standardization. Additional complications arise if the chemical also affects growth (so prolonging the period of exposure), particularly if the agar contains slightly mutagenic compounds (due to autoclavation, and referred to as "background" mutations).