Because so many of us fail to perceive math's beauty, mathematicians find
great joy in it. However, nature is a beautiful place where one can witness
beauty derived from mathematical correlations.

If we can detect them, the seemingly limitless patterns seen in the natural
world are supported by numbers.

Fortunately for us, a disorganized group of scientists has recently
discovered another amazing relationship between mathematics and the natural
world: number theory, one of the purest branches of mathematics, and
genetics, the principles controlling the evolution of life on a molecular
level.

Despite its abstract nature, number theory is also a somewhat well-known
branch of mathematics for most of us. It includes the arithmetic operations
of addition, subtraction, division, and multiplication of integers, or whole
numbers, and their inverses.

One well-known example of a series where each number is the sum of the two
numbers before it is is the Fibonacci sequence. Patterns of it may be seen
in pinecones, pineapples, and sunflower seeds, among other natural
objects.

The principal author of the new study, mathematician Ard Louis of Oxford
University,
says, "The beauty of number theory lies not only in the abstract relationships
it uncovers between integers, but also in the deep mathematical structures
it illuminates in our natural world."

The genetic mistakes known as mutations, which gradually seep into an
organism's DNA and propel evolution, piqued the curiosity of Louis and his
colleagues.

While many mutations result in unanticipated benefits or diseases due to
single-letter changes in the DNA sequence, other mutations may not affect an
organism's phenotype—its outward characteristics—in any noticeable
way.

The latter are frequently called neutral mutations, and while they don't
seem to affect anything, they are signs of evolution in action. Over time,
mutations accumulate steadily and map the genetic links among creatures as
they gradually diverge from a common ancestor.

However, in order to maintain their distinctive phenotype, organisms
must be able to
withstand some mutations while the genetic lottery continues to produce
potentially favorable replacements.

Genetic variety is produced through this so-called mutational resilience,
which differs among species and is even seen in the proteins found inside of
cells.

Sixty-six percent of mutations are harmless and have no influence on the final structure of
the studied proteins, which can withstand about two-thirds of random
mistakes in their coding sequences.

"Evolution would not be possible without the extraordinarily high
phenotypic robustness that many biological systems exhibit,"
says
Louis.

"However, we were unsure of the maximum possible robustness or even if one
existed at all."

Louis and colleagues investigated the relationship between a genotype—a
distinct genetic sequence that corresponds to a particular phenotype or
trait—and
short RNA structures
and
protein folding.

For proteins, the structure is encoded by piecing together the building
pieces of the protein, which are spelled out in a brief DNA sequence.

RNA secondary structures, which are free-floating strands of genetic
information that aid in the construction of proteins, are smaller than
proteins.

Louis and associates conducted numerical simulations to calculate the
likelihood that nature would get near to the top boundaries of mutational
resilience.

They investigated the mathematically abstract aspects of how many genetic
variants map to a given phenotype without altering it, and they demonstrated
that naturally occurring proteins and RNA structures might in fact enhance
mutational resilience.

Furthermore, the maximal resilience was related to the
sum-of-digits fraction, a fundamental idea in number theory, and followed a self-repeating
fractal pattern known as a
Blancmange curve.

According to Vaibhav Mohanty of Harvard Medical School, "we found clear evidence in
the mapping from sequences to RNA secondary structures that nature in some
cases achieves the exact maximum robustness bound."

In a way, biology seems to be aware of the fractal sums-of-digits
function.

Once more, math seems to be a fundamental aspect of nature that provides
the physical universe order, even at the minuscule scale.

__The study has been published in the Journal of The Royal Society Interface.__