# DNA sequence evolution

## Homework Statement

A strand of length L begins life as all A's. Assume that each letter evolves independent of all the rest until today, 1000 generations later. Within each generation there is a $\mu$ probability that the letter mutates to either C, G, T. Finally, assume that once a letter mutates that it cannot mutate again.
Calculate the number of A's as a function of $\mu$. Then equate this expectation to $N_A$ and write down a function for $\mu$ in terms of$N_A$.

## The Attempt at a Solution

So, I have 1000 generations where each A has the possibility to mutate to something else with probability $\mu$. The first generation the total number of A's is $N_A=L$. The second generation we must multiply each A by the mutation probability. Since there is L A's we will get: $N_A=\mu L$. The third generation occurs and we have to multiply the current number of A's by $\mu$ again. Which gives us $N_A=\mu \mu L$. Taking this to 1000 generations we'd have $N_A= \mu^{1000-1} L$ which doesn't really seem likely at all.

Any suggestions, or is this correct?

Related Precalculus Mathematics Homework Help News on Phys.org
What your solution is working toward is the number of non-A's in a given generation. What you want is to apply the opposite probability, the probability of not mutating.

For example, think if the probability was 1% to mutate. After the 1st generation, you would expect .99L A genes and .01L non-A genes. if you just took NA = μL, you would effectively be saying that NA in the first generation is (.01)L which would actually be the Nnot-A

So, your concept of multiplying the probability successively is correct, but you just need to use the right probability.

Last edited:
You are right. I had a feeling that I was getting the opposite result that I was meaning to get. I should have realized I had them mixed up! Thanks!

No problem!

And one more thing, if the second generation is P2L, and the third is P3L. Wouldn't the 1000th be P1000L? Just wondering since you put P1000-1L
(assuming P is the corrected probability)

Edit: nevermind, the first generation is just L, haha 