The power and scope of random mutations in evolution.

People who agree with the power of natural selection still tend to view evolution as far fetched because they doubt that random mutations can cause required effects. In this thread I'll try to demonstrate how.
I hope the ones reading this are aware of the structure of DNA and how mutations are basically copying errors, duplications or insertions that change the text of the genome which will look something like this. Let's take a look at the chances of a random mutation giving results.
Let's consider the SWS opsin gene in birds in which a mutation at position 268 will confer them with UV vision. 4 different orders of birds have evolved this ability seperately.
The per site rate of mutation is about 1 in 500,000,000 bases in DNA of most vertebrates.
This means that the exact change at position 268 in one copy of the bird SWS gene will be mutated on average, about once in every 500 million offspring. It has 2 copies of the gene, so this cuts the average to 1 in 250 million chicks. However there are 3 possible kinds of...
... mutations at this site. Only one of these can cause the switch to UV vision. So the required mutation will occur roughly one in 750 million (250×3) birds. Seems like a long shot right? Not really.
If we factor in the number of offspring produced per year, it can be 1 million to more than 20 million individuals. We will go with the conservative figure of 1 million. Divide this into the rate of one mutation per 750 million birds; the result is that the required mutation...
....will happen once in every 750 years. This is a blip in geological time scale. If we take a million years into account, this specific mutation will have occured 1200 times seperately. Got the idea? The orders of these birds have been evolving for 10s of millions of years.
Now what if the mutations doesn't need to be so exact? Many genes don't require such specific mutations for these changes and different mutations can confer the same change. One such example is the MC1R gene which are responsible for dark coloration in many animals.
There is at least 10 different mutations that can cause the same effect in the MC1R gene. With 10 target sites and the same mutation rate as earlier, what are the probabilities that black this dark causing mutation will arise? Let's take a look.
The chances go up by 10 times. The 1 in 750 million probability will go up by 1 in 75 million becoming dark in color. A species that produces 1 million offspring a year will give rise to a new dark variant in every 75 years.
A species that produces 7.5 million offsprings a year will give rise to the variant once in every 10 years. Even a very confined, small population of 75,000 will give a new variant every 1000 years. Not that surprising now when we factor in the pruning of natural selection.
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