Mathematicians enjoy the beauty of mathematics that many of us never see. But nature is a wonderful world in which the beauty resulting from mathematical relationships can be observed.

The natural world offers seemingly endless patterns based on numbers – if we can recognize them.

Fortunately for us, a diverse team of researchers has just discovered another amazing connection between mathematics and nature; Between one of the purest forms of mathematics, number theory, and the mechanisms that govern the evolution of life at molecular scales, genetics.

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

The famous one Fibonacci sequence This is just one example, where each number in the sequence is the sum of the previous two numbers. Its patterns can be found throughout nature, in pinecones, 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 it illuminates in our natural world.” He explains Oxford mathematician Ard Lewis, lead author of the new study.

Of interest to Lewis and his colleagues were mutations, which are genetic errors that slip into an organism’s genome over time and lead to evolution.

Some mutations can be a single letter change in the genetic sequence that causes disease or produces some unexpected advantage, while other mutations can have no noticeable effect on an organism’s appearance, traits, or behaviors (phenotype).

The latter are sometimes referred to as neutral mutations, and although they have no noticeable effect, they are indicators of evolution in action. Mutations accumulate at a constant rate over time, shaping genetic relationships between organisms as they slowly diverge from a common ancestor.

However, organisms must be able to tolerate some mutations, to maintain their distinctive phenotype while the genetic lottery continues to distribute alternatives that Maybe yes, maybe not be useful.

That’s what it’s called Mutational robustness It generates genetic diversity, but it varies between species and can even be observed in proteins within cells.

The proteins studied can tolerate about two-thirds of random errors in their coding sequences 66% of mutations It is neutral and has no effect on its final appearance.

“We have known for some time that many biological systems exhibit remarkably high apparent robustness, without which evolution would not be possible.” He explains Louis.

“But we didn’t know what the absolute maximum power was possible, or whether there was a maximum.”

To investigate, Lewis and his colleagues looked to Protein folding And Small RNA structures As examples of how a unique genetic sequence, known as a genotype, assigns a specific phenotype or trait.

In the case of proteins, a short DNA sequence shows the protein’s building blocks, which, when put together, encode its shape.

Smaller than proteins are the secondary structures of ribonucleic acid (RNA); Free-floating strands of genetic code that help build proteins.

Lewis and his colleagues wondered how close nature was to reaching the upper limits of mutational strength, so they ran numerical simulations to calculate the probabilities.

They studied the abstract mathematical features of how many genetic variations map to a particular phenotype without changing it, and showed that mutational power can indeed be maximized in naturally occurring proteins and RNA structures.

Moreover, the maximum robustness follows a self-repeating fractal pattern called a Blancman curveand it was consistent with the basic concept of number theory, called Break the sum of numbers.

“We found clear evidence in mapping from sequences to RNA secondary structures that in some cases nature achieves a precise maximum in strength.” He says Vaibhav Mohanty, of Harvard Medical School.

“It’s as if biology knows about the function of the sum of rational numbers.”

Once again, mathematics seems to be a fundamental element of nature, giving structure to the physical world, even at microscopic levels.

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

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