The Algebra of Evolution

In the deserts of Arizona, million-year-old black lava flows are inhabited by rock pocket mice. In this region, the mouse can be seen in two colors dark black and sandy-colored. The dark color mouse are found most often in black lava rocks. And white color mouse are found most often in sandy-colored habitat. Take a look at the image given below. Before the lava flows all the mouse were sandy-colored. A curious mind should ask couple of questions (1) how did some mice manage to change from sandy-colored to black (2) how did the mice organize itself according to its surroundings?


In order to answer these questions we need to open the book of evolution. Before proceeding further make sure that you read my earlier post which talks about the basics of DNA, genes, and proteins. For the mouse to change from sandy to black color three things needs to take place. They are (1) mutations (2) natural selection and (3) time. Let us look at each one of them in detail.

Mutations: In order to reproduce, organisms must make copies of their DNA. The copying of DNA is a complex biochemical processes. And mistakes happen during the copying process. These mistakes are mutations and they are the source of all the varieties (plants, bacteria, animals, and humans). There are several ways mutations can happen.

If we think of DNA as being like a written text, then the categories of mutations are just like the familiar kinds of word processing errors. The DNA of a given species ranges from millions to billions of permutations of the four letters A, C, G, and T. The most common mistake is the substitution of an incorrect letter—a typo. But there are many other kinds of events that also occur, such as deletions and insertions of blocks of letters. Copy and paste errors also occur; these result in duplications of text. Groups from just a few letters on up to entire genes, or large blocks of genes, are duplicated at a significant frequency. Blocks of DNA letters are also rearranged—by inversions and the breakage and joining of parts of text. As a result, in every new individual, there are some new mutations. – The Making of the Fittest

In a mouse there is a gene called MC1R and when mutated turns the color of the mouse to black. What is the probability of this gene to mutate? In a gene the place where mutation can occur is called as a site. On average a mutation can occur in 2 out of every billion site per mouse. There are 2 copies of MC1R gene. And each MC1R gene has 10 sites. This tells us that there is about a 1 in 25 million chance of a mouse having a black-causing mutation in the MC1R gene. This shows how accurate DNA copying is. But it’s not perfect.

Total no of mutations for very 1 billion site = 2
Total sites in MC1R gene that can be mutated  = 20 (2 copies * 10 sites)
No of mutations that can happen in MC1R gene  = [ 20 / 109 ] * 2
No of mutations that can happen in MC1R gene  = 0.00000004
One black mouse out of 25 million             = 1 (25 million * 0.00000004)

Imagine that there are 5,000 white female mice in the population and each one of them is capable of producing 5 offsprings. This means that every year there will be 25,000 offsprings and in 1,000 years a group of white female mice would have produced one black mouse just by random chance. The lava in Arizona was formed over 1.7 million years and during this period the mutation would have produced 1,700 black mice. Our evolutionary clock is 3 billion years old. So don’t be surprised of 1.7 million years as it represents only 0.056% of the total evolutionary time. Mutation is a random chance occurence. After creating a variation it hands over the job to natural selection.

Natural Selection and Time: Many think that the entire process of evolution is blind. But that’s not correct. The mutational process is blind, natural selection is not. If the black mouse was born around black lava rocks then the predators like owls and snakes will not be able to spot it easily. So this trait (black color) gives it survival and reproduction advantage. On the other hand if it was born around sandy environment then it would have been killed before passing its traits to its offsprings. Thus the process of natural selection acts as a cop to favor mutations which suits the environment. In order to understand the power of natural selection and time let us jump into the field of finance.

Imagine Peter and Paul both invested $1,000. Peter is compounding his money at 1% and Paul at 6%. For the first few years their accounts will differ by a little. But over time the difference will be huge. The rate of difference of just 5% over long periods of time gave Paul 30x more money than Peter.


Using the same idea let us assume that the black mice has a survival advantage in black lava rocks over white mice by 1.9%. Survival advantage in natural selection is akin to interest rate in finance. Since mutation has already produced many black mice, let us assume that there are 10,000 white mice and 8 black mice. At the start the population will have 99.92% white mice and 0.08% black mice. After 500 years with black mice growing at 2% and white mice growing at 0.1% the population will have 90.64% black mice and and 9.36% white mice.


Adding the mutation and selection time, the entire population of white mice almost turned into black in 1.75 million years. It’s not the mice but the agents (owls and snakes) of natural selection which organized the mice to suit the surroundings. A human being looking at the current situation will incorrectly conclude the orderliness of the world. One cannot blame his incorrect conclusion as our average life time is around 60 years. This is a tiny number when compared to 1.75 million years. Take a look at the excellent video which explains what I wrote so far.

The algebra of evolution is simple. It uses probability (random chance) during the mutation phase. This results in creating a new trait or removing an existing trait. If that trait gives a survival advantage (read it as interest rate) then natural selection ensures that the species population spreads. Natural selection uses compounding to do its work. Evolution ran the compounding engine for 3 billion years. That’s an insane amount of time. And evolution was run in parallel in several disconnected parts of the world (99.99% of evolution time didn’t have boats, ships, flights, and humans) under varying conditions. The result is a lollapalooza. Long before DNA was discovered, Charles Darwin was able to see the movie of evolution. It’s fascinating to read what he wrote in 1859.

The chief cause of our natural unwillingness to admit that one species has given birth to other and distinct species, is that we are always slow in admitting any great change of which we do not see the steps. The difficulty is the same as that felt by so many geologists when Lyell first insisted that long lines of inland cliffs had been formed, and great valleys excavated, by the slow action of the coast waves. The mind cannot possibly grasp the full meaning of the term of a hundred million years; it cannot add up and perceive the full effects of many slight variations, accumulated during an almost infinite number of generations.On the Origin of Species

8 thoughts on “The Algebra of Evolution

  1. Simple concept. But in your words, it gets even simpler 🙂
    The video did not address one particular aspect in detail.
    it appeared as if these black rats did not develop black fur on the underbelly.
    Natural selection (i.e, predators) is indifferent to the color of the fur on the underbelly.
    that being the case, i’d have expected that those mutations (the ones that developed black underbelly) also would have thrived equally well, unless ofcourse there were other forces.
    For someone looking at this after a million years, it appears as if there is another force acting here – i.e, the process of mutation is somehow self-optimizing!

    • Raghav,

      Thanks. You are absolutely right. If the natural selection is neutral for black and white belly (which it is) then you will see both kinds survive. The only question then is how lucky the mutation was between white and black color on the belly.

      If you haven’t read this book then you should. The author did a fantastic job in explaining the concepts of evolution in layman terms.


  2. I know this is an emotionally charged question but is there relevant genetic math that explains the vastly different races of humans that evolved in just 80,000 years? It would appear climate played the primary role in selection initially but given that travel was incorporated into the process, are there mathematical renderings that reflect human genetic distribution? Is the science solid or are there unknowns i.e. aspects that we cannot quantify or explain?

  3. Just have to say that although the discussion of the evolutionary concept in the piece is very good, your choice of mathematical terminology (“algebra”) is a very inaccurate description of the mathematics you are using in your examples. Yes there is technically some algebra, if you look hard enough, but most of it has nothing to do with algebra – probability, exponential functions, etc. – which are not “algebra”.

    It’s a very good piece, but I’m just saying that you might want to consult with a mathematician before using the term like that again.

    • Andrew,

      Fair point. May be Mathematics of evolution might suit this post better.


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