why infinite site assumption implies that convergent evolution not existing?

Question

[2001linana]

The infinite site assumption implies that, every genome position mutates at most once in the evolutionary history of a tumour. With this assumption, one could easily exclude convergent evolution. Here, """ Convergent evolution is a common theme in the evolution of animals. It occurs when two unrelated species independently evolve similar traits to cope with specific evolutionary challenges, like living in ice-cold water or eating ants. Sometimes convergent evolution is so powerful that creatures that began as entirely different animals start to look almost the same, as is the case with the skulls of the extinct marsupial Thylacine and the living Grey Wolf.""" Why is this? Could anyone please help to clarify more about it?

[2001linana]

Moreover, the infinite sites assumption, would preclude back mutations, why is this?

[Michael]

Because, if a locus e.g. gets a mutation from A -> C, to get A back would require a mutation at the same position. The ISA, if it holds, precludes a second mutation of the same site. I think you need to understand that ISA is just an abstraction to make models of site evolution simpler, and it is reasonable to assume that back-mutation of any site has very low probability. If you wanted to take back-mutations into account, every site could in addition to the observed state (A,C,G,T) be the consequence of infinitely many unobserved back and forward transitions, like A -> C -> A, A -> T -> A, A -> G -> A, A -> C -> G -> A, ... .

However, if one could show that joint probability for all chains of back-mutations of length i is in fact bounded by a very small quantity as i approaches infinity, then this quantity can be neglected. And indeed this is the case because the joint probability can be expressed as a geometric series which converges quickly for a given probability of a single point mutation (0< p <1).

Answer 1

[Michael]

I am not so sure that what you are trying to compare here are in fact comparable concepts. I think there are some illogical traits in what you write above.

1. a tumor is not a Grey Wolf, even if your assumption about the infinite site assumption (ISA) would hold for a bunch tumor cells, it doesn't necessarily follow that it holds for animals or other organisms.
2. What you propose about the ISA is an assumption. I propose there is likely nothing that can prevent a single genomic position to mutate twice if there is enough time, it is just that the probability is low given the life-time of a tumor.
3. Even if ISA holds, there are multiple loci in a genome that can mutate to lead the same phenotypic change. Simple example: loss of protein A leads to an advantageous trait, then this can happen in many ways, like insertion of a stop codon or a frameshift anywhere in it's coding region, deletion of it's promotor sequence, insertion of a mobile genetic element. Any of those thousands of possible variations would result in protein A being deleted leading to the same trait, thus convergent evolution.
4. One could say that tumor cells in fact have converging phenotypes, for example in multi-drug resistance, tumor cells tend to accumulate multiple (potentially different from other resistant tumor cells) mutations that lead to the same outcome, resistance against chemotherapy.
5. Convergent evolution is an observable fact, so it cannot be simply excluded based on an assumption.