Mathematicians delight in the beauty of mathematics that most of us do not see. But nature is a wonderful place to see the beauty born of mathematical relationships.

The natural world offers seemingly endless patterns supported by numbers – if we can understand them.

Fortunately for us, a motley crew of researchers recently uncovered another surprising connection between math and nature; among the most refined methods of mathematics, number theory, and the mechanisms governing the evolution of life on molecular scales, genetics.

Abstract as it may be, number theory may be one of the most familiar forms of mathematics for most of us. It includes multiplication, subtraction, division, and addition (arithmetic operations) of integers, or whole numbers and their negative counterparts.

The famous Fibonacci sequence is one example, where each number in the sequence is the sum of the previous two. Patterns of it can be found all over nature, in pineapples, pineapples, and sunflower seeds.

“The beauty of number theory lies not only in the abstract relationships it reveals between integers, but also in the deep mathematical structures that illuminate our natural world,” explains Oxford University mathematician Ard Louis, senior author of the new study.

Of interest to Louis and his colleagues were mutations, genetic errors that slip into an organism’s genome over time and drive evolution.

Some mutations may be single-letter changes in the gene sequence that cause a disease or produce some unexpected benefit, while other mutations may not have a noticeable effect on appearance, symptoms, or behavior (its phenotype).

The latter are sometimes called neutral variables, and although they have no apparent effect, they are indicators of evolution at work. Mutations accumulate at a steady rate over time, charting the genetic relationships between organisms as they slowly descend from a common ancestor.

Organisms must be able to tolerate certain changes however, to maintain their phenotype while the genetic lottery runs to deal with what may or may not be useful.

This so-called mutational robustness creates genetic diversity, but it varies between species, and can also be seen in proteins within cells.

The proteins studied can tolerate about two-thirds of random errors in their sequence codes, which means that 66 percent of mutations are neutral and have no effect on their final shape.

“We’ve known for a long time that many biological systems show a high robustness of the phenotype, unless evolution is impossible,” explains Louis.

“But we didn’t know what the maximum potential strength would be, or if there was a maximum.”

To investigate, Louis and colleagues looked at folding proteins and small structures of RNA as examples of how a unique genetic sequence, known as a genotype, maps to a phenotype or a specific trait.

In the case of proteins, a short sequence of DNA defines the building blocks of a protein, which when put together, encodes its shape.

Smaller than proteins are the secondary structures of RNA; free-floating strands of genetic code that help build proteins.

Louis and his colleagues wondered how close nature is to the upper limits of evolutionary strength, so they ran numerical simulations to calculate the probability.

They studied abstract mathematical properties of how many genetic mutations can occur in a given map without changing it, and showed the robustness of mutation can be realistically extended to naturally occurring proteins and RNA structures.

Moreover, the peak strength followed a repeating fractal pattern called the Blancmange curve, and was aligned with a basic concept of number theory, called the sum-of-digits fraction.

“We found clear evidence in the mapping from sequence to RNA secondary structure that nature in some cases achieves high fitness,” said Vaibhav Mohanty, of Harvard Medical School.

“It’s like biology knows about the fractal sums-of-digits function.”

Mathematics also appears to be an important part of nature that gives structure to the physical world, even at microscopic levels.

The study was published in *Journal of the Royal Society Interface.*

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