The influential American number theorist Leonard Dickson wrote 'Thank God that number theory is unsullied by any application.'
And yet, again and again, number theory finds unexpected applications in science and engineering, from leaf angles that (almost) universally follow the Fibonacci sequence, to modern encryption techniques based on factoring prime numbers.
Now, researchers have demonstrated an unexpected link between number theory and evolutionary genetics.
Specifically, the team of researchers - from Oxford, Harvard, Cambridge, GUST, MIT, Imperial, and the Alan Turing Institute - have discovered a deep connection between the sums-of-digits function of number theory and a key quantity in genetics: #phenotype #mutational #robustness.
This quality is defined as the average probability that a point mutation does not change a phenotype (a characteristic of an organism).
They proved that the maximum robustness is proportional to the logarithm of the fraction of all possible sequences that map to a phenotype, with a correction which is given by the sums of digits function sk(n), defined as the sum of the digits of a natural number n in base k. For example, for n = 123 in base 10, the digit sum would be s10(123) = 1 + 2 + 3 = 6.
Another surprise was that the maximum robustness also turns out to be related to the famous Tagaki function, a bizarre function that is continuous everywhere, but differentiable nowhere.
First author Dr. Vaibhav Mohanty (Harvard Medical School) added: 'What is most surprising is that we found clear evidence in the mapping from sequences to RNA secondary structures that nature in some cases achieves the exact maximum robustness bound. It's as if biology knows about the fractal sums-of-digits function.'
https://www.news-medical.net/news/20230808/Unexpected-link-between-pure-mathematics-and-genetics-discovered.aspx