Is it possible to determine if a certain specific mutation had a spontaneous origin (for example from a mistake of the DNA polymerase) as opposed to an induced origin (for example, from some genotoxic agent)?
Here's a quick answer; hopefully someone will give a more complete one. But meanwhile you've got something.
Some genotoxic agents have predictable results. For example, they cause Gs to substitute for Cs, or they cause mutations at specific spots on the genome. So if you had a bunch of mutations in a single cell (or collection of related cells) that matched the modus operandus of a known genotoxic agent, then it would be reasonable to conclude that the mutations were induced. But not all genotoxic agents are like that. And if you only have 1 or 2 mutations, that's not enough to prove they were induced. Also, of course, it depends on your sample size and your controls.
The above answer is mostly right. In the lab one can show that induced mutations are caused by exposure, compared to unexposed controls (e.g. in bacteria). The induced mutations will have a frequency, position, and type which can be typified. As a group, theyc ompris a "spectrum" of mutations. In a real case problem, such as humans exposed to diesel exhaust, the number of mutations and their trype are usually going to be too few to have a"spectrum". Naturally occurring (a serious misnomer in its own right) could also occur in those same positions and types. The usual language is that the observed mutations are consistent with an exposure, but does not prove the exposure caused them.