Maths and Evolutionary Biology

Maths and Evolutionary Biology

Mathematics is often utilised across many fields – lets look at an example from biology, evolutionary biology and paleontology, in trying to understand the development of homo-sapiens.  We can start with a large data set which gives us the data for mammal body mass and brain size in grams (downloaded from here).  I then tidied up this to remove the rows with NA (how to: here).  This gave me a data set with more than 5000 rows with mammal data.  I then found the log of these values and then copied this data into Desmos.  This gave the following graph:

I then used the regression tool to find the linear regression for this data (purple line).  The dotted red lines were plotted by finding the standard deviation for the log(Brain Mass) data and then adjusting the regression line by +- 0.5 of this standard deviation.  We can see that this then captures the majority of the data set nicely.  We can notice that we get some negative values – but this is OK – as if an animal has a small body mass (say 0.1) then the log of this value (log0.1 = -1) will return a negative value.

Analysis

We can see that modern humans have a relatively large brain size compared with body mass over what is expected for mammals.  We can also see that whales (blue whale shown) have a much smaller brain size to mass than expected.

Finding the equation linking body mass and brain mass.

Log plots are nice ways of finding equations for non linear data.  We start by noticing that there is a linear trend of the form y = mx+c, but in this case the x and y axis are in a log scale.  We have the definitions:

And then use the Desmos regression line to give:

We then use the laws of logs to rearrange:

This then gives us an equation linking brain mass and body mass in mammals.

Evolutionary biology and paleontology

I then looked up some data for earlier hominid data and then also plotted this.  We can see that the looks like another linear trend.  When plotting this with the Homo Habilis data the linear regression is only moderately correlated – but if we look at the data for the Chimpanzee (using them as a shortcut for a prediction of our common ancestor), Australopithecus, Homo Erectus and Homo Sapiens (us) we get the following:

This has a correlation coefficient of 0.977 and represents very clearly the evolutionary changes in relative brain size in hominid development.  That Homo Habilis is not closer to this line could (potentially) be evidence that the current estimates for this species are not accurate enough.  The body and brain mass for Homo Habilus has been calculated from extremely limited fossil data.  So in this case maths can be used to highlight an inconsistency in the paleontological record.