Fundamentally, a carcinogen is almost always a type of mutagen. A mutagen has the general ability to mutate, or change, DNA by a variety of means ranging from alteration of chemical bonds to outright destruction of structural components. A carcinogen is something that does that in a way that promotes development of cancer in a statistically significant way. Not all carcinogens are mutagens, but a majority are.
Determining carcinogenic potential in the lab is difficult. Generally it culminates in exposing animals to the substance, and seeing if they develop cancer in an experimentally repeatable way.
Mutagenic potential can be more easily tested, and can serve as a good indicator of whether a substance is likely to be carcinogenic.
It’s called the [Ames Test](https://en.m.wikipedia.org/wiki/Ames_test). What you do is take a sample of fast-growing bacteria that have been genetically modified to be unable to produce a critical amino acid- histidine. Grow them on a sample of histidine until they’re fat and happy, expose them to your suspected mutagen, and then transfer them to a growth medium containing no histidine.
What this does is put immense selection pressure on the bacteria. They need to essentially evolve their way back into making their own histidine, and in theory, the gene-scrambling substance you exposed them to will set them down the mutation path until they either figure out how to make histidine or die.
It’s a crude but generally effective test that can give you valuable data for how strongly mutagenic your substance is, and indicate potential carcinogenic potential.
Say a saboteur breaks into a fully automated factory, finds the computer that tells the factory what to do and what to produce and changes those instructions in a way that is not picked up by quality control. Obviously, the factory now goes haywire, starts overproducing, underproducing, producing garbage, using too much energy, etc.
This is essentially what a carcinogen is, and how it operates. Scientists can figure out what a carcinogen is by looking at the effects of exposure to a substance and the cancer risk in subjects exposed to said substances. For example, in people who smoke cigarettes, there is a known increase in lung cancer rates compared to non-smokers. Therefore, it is reasonable to conclude that cigarette smoke is a carcinogen. Scientists can also perform animal studies, by deliberately exposing an animal (usually a rat) to a suspected carcinogen, and checking if the rat develops cancer.
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