The Mathematics of Evolution

by | Feb 1, 2019


Evolution is a tough concept to wrap our minds around. How is it possible that humans and other species of plants and animals have evolved to be such wonderous and complex organisms? Can this really have occurred through chance? The answer is yes, and this IFOD digs a bit into how evolution works and the mathematics of evolution. If you think this sounds long and boring, try out these music videos instead.

The drivers of evolution: (i) chance mutation of genes, (ii) selection for or against those mutations, and (iii) time (lots and lots of it!).

Chance:  Evolution occurs due to gene mutations.  A “mutation” does not connote a harmful change.  It just means that through chance gene duplication produced an error. Mutations happen randomly and occur in every human (and every other living thing).  The copying of DNA is a complex process and mistakes happen all the time in the replication of DNA.  Of the 7 billion DNA letters in the human genome, there are an estimated 175 new mutations in each human.  Most mutations do not have an effect because: “(i) many of the mutations occur in areas of DNA that are empty of useful information, or (ii) the mutation doesn’t change how a gene works, or (iii) the mutation doesn’t affect us because we carry two copies of many genes, or (iv) the mutation causes a change within an acceptable range of variation.”  But, over time, these mutations create variation in the size, shape, color and other physical properties.  It is this variation that drives evolution.

Selection:  Mutation generates random variation, but it is “adverse selection” that sorts out the winners and losers.  Selection occurs when a trait provides an advantage or disadvantage.  Biologists calculate the power of selection in terms of “selection coefficients.”  For example, if the presence of a trait conveys a small advantage to individuals such that they produce 101 viable offspring vs. 100 viable offspring for those lacking the trait then this particular trait has a 1 percent advantage or a selection coefficient of 0.01.  Coefficients work the opposite way as well, if a trait produces 99 viable offspring vs. 100 for those without the trait then the coefficient is -0.01.  The selection coefficients are a measure of fitness, which are relative, not absolute measures.  Mathematically, these selection coefficients really add up relatively quickly.  For instance, assume a trait has a positive selection coefficient of 0.01.  That positive trait would increase in frequency from just a few individuals per 1,000 to over 90% the entire population in just 3,000 generations.  Change the coefficient to 0.1, and that increase would take just 300 generations.  300 or 3,000 generations may sound like a long time, but biologically speaking, it is a very short time period.  Many life forms have generation times of a year or less. 

Time:  The Earth is approximately 4.5 billion years old. Life on Earth is thought to have begun about 3.5 to 4 billion years ago.  For us 50 years seems like a long time and 100 years is a very long time ago. 100 years ago was 1919 – WWI had just ended, only about 30% of houses had telephones and no homes had air conditioning. Donald Trump has not even been born yet! Thus, it is very hard for us to wrap our minds around how long 1,000 years is, or 10,000, or 100,000 or 500,000 years.  One million years or more is unfathomable. Tens and hundreds of thousands and even millions of generations of a species can occur over these vast time periods. Random mutation along with positive and negative selection is very powerful when very long periods of time is involved. A bit of mutation and many generations can create big changes.

Similarly, it is difficult to appreciate exponential growth over time. An example that Charles Darwin presented of the mathematics of reproduction over time is that if pairs of elephants produce six offspring during their 60 year lives, the original pair of elephants would produce 15 million offspring in 500 years without constraints on land or resources and no competition from other species. (I verified this in a simplified model in excel.) Crazy.

An interesting example of the mathematics of evolution concerns mice living along the coast of Maine is discussed in the fantastic book The Making of the Fittest: DNA and the Ultimate Forensic Record of Evolution, by Sean B. Carroll. The example relates to a species of mice changing over time from sandy colored to black due to a change in their environment. Study of the mice in the lab has determined where in their genome a mutation can occur that will cause the mouse to have black fur rather than sandy colored and determined that the frequency of mutation will on average produce one black mouse per every 25 million mice that are born. This is such a small probability it seems like there is no way a population of mice could actually change from sandy to black.  But hold on, the mathematics of evolution says otherwise.

Mice breed a lot, 2-3 litters per female per year and each litter has 2-5 babies. Thus, there are a lot of mice being born and a very short time between generations. For an initial population of 10,000 mice you get 25,000 babies born per year and assuming 1 black mouse per 25 million you’d have one black mouse every 1,000 years.  Thus, in one million years you will get 1,000 independently occurring black mice.  However, If you use a population of 100,000 mice you will hit a black mouse every 100 years.

This black mutation will spread if there is an advantage to having black fur.  For these mice in Maine, there is an advantage to being black because much of their living area is black volcanic rock.  For a population of 10,000 mice, the average time for a 0.001 selection coefficient to make more than 50% of the mice black is 19,800 generations (you’ll have to trust the math – I’ve seen it worked out in a proof – too long and boring to reproduce). It’s 1,980 generations for a 0.01 coefficient, 400 generations for 0.05 coefficient, 200 generations for a 0.1 coefficient and 100 generations for a 0.2 coefficient.  It has been calculated based on observations that the survival advantage of having black fur if you are a mouse living on black lava rock is a coefficient of somewhere north of 0.01.  So, even assuming one generation of mice per year, it would take less than 2,000 years for the mouse population to move from sandy colored to black. Using the 2-3 generation per year number it is less than one thousand years.  With respect to the actual mice in Maine, part of the population is sandy colored, and part of the population is black.  Study of the DNA record of these mice reveals that the population was originally sandy colored, and the mice are becoming black over time in accordance with the positive selection associated with being black.

Thus, what the study of sandy vs. black colored mice show is that a relatively rare mutation (1 in 25 million) and a slight survival advantage (0.01) will result in most of the population being black in just a few thousand years, and maybe much soon if the initial population is bigger.

I read recently that less than 50% of Americans believed in evolution.   That is understandable.  To look around at the wonderous complexity of life you would think that there is no way it could have happened by chance.  However, based on the frequency of random mutation, the power of selection, and time, mathematically, the mouse population changed color by chance.  The same is true for every aspect of every type of life that currently exists.  It is mathematically possible for all biological variation on Earth to have arisen by chance.  3.5 Billion years is a very very long time.

Source: In this IFOD I drew heavily upon the fantastic book The Making of the Fittest: DNA and the Ultimate Forensic Record of Evolution, by Sean B. Carroll. In the book, he goes into much greater depth as to the assumptions of the calculations as well as providing a few other real-world examples of the mathematics of evolution. The main point of the book is the discussion of the DNA record of evolution. No longer is just the fossil record needed to trace evolution. Fascinating.


  1. So: given the earth is about 4 billion years old and the universe is more than 10 billion years old. Are we not fools to not believe there is life in this universe that had an higher selection coefficient and more time to evolve than humans and would therefore be more advanced than us. It also seems given those advanced species that they may have intervened in the selection process of species on earth. One must consider the fact that we humans are now intervening in the evolutionary process of species on our own planet? The fact that random selection is a fact does not mean that design has not occurred.

  2. It would be interesting to watch you have a conversation with a person equally as knowledgeable and talkative as you.
    For about thirty minutes until my mind drifts off on pony’s and I fall asleep.


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