The final feature that needs to be addressed is the probability of mutations cumulating to produce new organs and species.

This question lies at the heart of many people’s objections to evolutionary ideas. They cannot envisage how infinitesimal changes, each invisible to the eye, can add up to major changes. That is because they tend to think that the two foundations for this to occur (the occurrence of successful mutations and the mutations then spreading throughout the population) are both highly unlikely, and so that the chance of a whole sequence of such processes occurring must be infinitesimally small.
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In the previous post, we saw that if we start with a trait that is present in just 0.1% of the population (i.e., f=0.001), and if this has a small selection advantage of size s=0.01, this will grow to 99.9% (F=0.999) in just under 1,400 generations, which is a very short time on the geological scale.

But in a population of one million, an initial fraction of f=0.001 means that we are starting with about 1000 organisms having the favorable mutation. But it could be argued that new mutations usually start with just a single new kind of organism being produced in one single organism. How does that affect the calculation?

Suppose that you have a population of organisms of size N and they all start out having the same gene at a particular position (called the ‘locus’) on one of the chromosomes that make up the DNA. Now suppose a random mutation occurs in just one organism, the way that it was described in an earlier post in this series describing the shift from violet to UV sensitive sight in some birds. Most of the time, even a favorable mutation will disappear because of random chance because (say) that mutated organism died before it produced any offspring or it did produce a few and that particular gene was not inherited. But on occasion that mutation will spread. How likely is it that such a single mutation will spread to every single organism (i.e., become ‘fixed’ in the population)?
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Of the three stages of natural selection outlined before, the only one that occurs purely by chance is the first one, that of the occurrence of mutations. I discussed how although the chances of producing a favorable mutation by changes in any individual site in the DNA (called ‘point mutations’) on an individual member of the species is very small, when the number of individuals in a species and the long times available for the changes to occur are factored into the calculation, the result is that such mutations are not only likely, they are almost inevitable to occur and furthermore are likely to occur many times.
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In the previous post, I discussed the puzzle posed by a naïve understanding of Mendelian genetics, which was that one might expect that organisms that displayed recessive gene traits would slowly disappear in a population while those with dominant gene traits would grow in number. But if that were true that would prevent new mutations from gaining a foothold in the population and growing in number, if it happened to be a recessive trait.

The crucial work that formed the breakthrough that revived the theory of natural selection was done in 1908 by G. H. Hardy (a Cambridge University mathematician and author of a fascinating book A Mathematician’s Apology) and Wilhelm Weinberg (a German physician), working independently. What is nice is that the result is quite simple to derive, and surprising.
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The joining of Darwin’s theory of natural selection with the Mendelian theory of genetics is one of the great triumphs of biology, now called somewhat grandly the ‘neo-Darwinian synthesis’. It forms the basis of all modern biology, and was strengthened by the discovery of DNA as the structure of genetic information and which explained how Mendelian genetics worked on a microscopic scale. The modern ability to map out the entire genome of humans and other species has produced overwhelming evidence in support of Darwin’s theory of how organisms evolve and branch out into different forms. The rough tree of life that Darwin sketched out in his book based on the anatomy of biological species has now been made more precise and detailed by the mapping of the DNA of species, showing ever more clearly how species are related to one another and when they separated from a common ancestor.
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In Darwin’s own time, there was a three-way dispute concerning the theory of evolution. Strange as it may sound these days in the US where so many question whether evolution even occurs at all, the idea that evolution had occurred and new species were being created and old ones dying out was not such a major problem in the mid-to-late 19th century. Elite opinion of that time had been exposed to that idea and had accepted it even before Darwin because of all the fossil records that were being discovered all the time. Even Darwin’s own grandfather Erasmus Darwin, a freethinker, had around 1795 published a book Zoonomia that had floated the idea that species had evolved, but he used a Lamarckian model. What religious people mostly shied away from was the idea that human beings were also part of the evolutionary process and shared common ancestors with other species, a reluctance that still persists.
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One of the challenges faced by Darwin was whether the rate at which mutations creating new favorable varieties would occur was sufficiently rapid for his purposes. Since during his time the laws of inheritance were not known and neither was the mathematics involved, advocates of natural selection had to assume that things would work out eventually.

In his excellent book The Making of the Fittest (2006), Sean B. Carroll demystifies the various numbers and calculations involved in natural selection using our current knowledge.

Recall from the previous post in this series that DNA is made up of a string of bases A, C, T, and G. New genetic information is created when there is a change in the DNA and the most basic (but not the only) way that this can occur is by mutations acting at the level of a single base site in the DNA, changing one of the bases A, C, T, and G to a different one.
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In order to understand how inheritance works and the mathematics involved, it may be helpful to have a quick summary of some basic facts about genetics (a little simplified), using the human genome for concreteness.

All the genetic information in our bodies is found in the DNA, whose famous double helix structure was discovered in 1953. Thanks to the Human Genome Project, we now have a complete map of the DNA of humans, called the human genome, and know that it consists of a sequence of 3.1647 billion sites arranged in a row, each site containing one of four complex molecules (called bases) labeled A, C, T and G. It is this long arrangement of the four bases that define each of us genetically. Almost 99.9% of the arrangement of these bases is identical in all humans, and about 98% is identical between chimpanzees and us.
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There are three mathematical ideas that one needs to come to terms with in order to get the full flavor of how natural selection works.

1. One is the rate at which favorable mutations occur in organisms. These do occur by chance and the question is whether the frequency of such occurrences is sufficient to explain evolution.
2. The second is the rate at which favorable mutations become more numerous in the population. It is not enough to produce a single favorable organism. The population of varieties with advantageous properties has to eventually grow to sufficiently high numbers that it dominates the population and can form the basis for yet further mutations.
3. The third is whether the rate at which repeated small and favorable mutations build on each other is sufficient to produce major changes in complex systems (the eye, ear, and other organs for example) and even entirely new species.

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