An interdisciplinary team of mathematicians, engineers, physicists and medical scientists has uncovered an unexpected link between pure mathematics and genetics that has revealed important insights into the structure of neutral mutations and the evolution of organisms.
Number theory, the study of the properties of positive integers, is perhaps the purest form of mathematics. At first glance, it may seem too abstract to apply to the natural world. In fact, the influential American number theorist Leonard Dickson wrote: “Thank God number theory has not been tainted by any application.”
Yet number theory finds unexpected applications again and again 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 proven an unexpected link between number theory and evolutionary genetics.
Specifically, the team of researchers (from Oxford, Harvard, Cambridge, GUST, MIT, Imperial and Alan Turing Institute) discovered a deep link between the sum of digits function in number theory and a substantial amount in genetics. phenotype mutational robustness. This quality is defined as the average probability that a point mutation will not change a phenotype (a trait of an organism).
The discovery could have important implications for evolutionary genetics. Many genetic mutations are neutral, meaning they can accumulate slowly over time without affecting the viability of the phenotype. These neutral mutations cause genome sequences to change at a constant rate over time. Because this ratio is known, scientists can compare the percent difference in sequence between two organisms and figure out when their last common ancestor lived.
But the existence of these neutral mutations raised an important question: Which fraction of mutations in a sequence are neutral? This feature, called phenotype mutation robustness, defines the average amount of mutations that can occur across all sequences without affecting the phenotype.
We have known for some time that many biological systems exhibit remarkably high phenotype robustness, without which evolution would not be possible. But we didn’t even know what the absolute maximum robustness possible would be or if there was a maximum.“
Ard Louis, Study Lead Author and Professor, University of Oxford
This is exactly the question the team is answering. They proved that the maximum robustness is proportional to the logarithm of the fraction of all possible sequences mapped to a phenotype; The natural number in base k is n. For example, for base 10 n = 123, the sum of digits would be s10(123) = 1 + 2 + 3 = 6.
Another surprise was that the maximum robustness turned out to be related to the famous Tagaki function, a peculiar function that is continuous everywhere but cannot be differentiated anywhere. This fractal function is also called the blancmange curve because it resembles the French dessert.
First author Dr. Vaibhav Mohanty (Harvard School of Medicine) added: “The most surprising thing is that in mapping from sequences to RNA secondary structures, we find clear evidence that nature has in some cases reached its limit of maximum robustness precisely. As if biology knew the function of the sum of fractal digits.”
Professor Ard Louis added: “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. We believe many interesting new connections between number theory and genetics will be found in the future.”
Mohanty, V., take meat. (2023) Maximum mutation robustness in genotype-phenotype maps follows a self-similar blancmange-like curve. Journal of The Royal Society Interface. doi.org/10.1098/rsif.2023.0169.
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