Have you ever asked what point a genetic sequence will go berserk and just stop working? The answer, it turns out, comes from a relatively simple math equation from an unexpected corner of pure mathematics: a surprising link just uncovered by a team of researchers shows that the sum-of-digits function can be used to predict when genetic mutations often lead to changes in function.
For every genotype roughly the DNA letters for a given gene there is a resultant phenotype: the final output, a new protein, or even a behavior. Many mutations are possible before a genotype is actually changed. This is one context in which mutational robustness is important to evolution. It allows organisms to go through a great deal of genetic “damage” without changing their basic nature.
“Part of what we’re trying to do is find a universal explanation for a lot of these trends which have been observed in nature,” said Vaibhav Mohanty, a theoretical physicist and doctoral and MD candidate at Harvard Medical School and MIT. Mohanty and his team came up with imagined genetic sequences as points in a hypercube. It is known that the hypercube represents an occupant of the same phenotype if it contains the genotypes with the same phenotype. The question was, what shape do these clusters form?
This is a rather surprising question that was studied in a very different context: number theory, a field devoted to understanding properties of positive integers. Average robustness of a phenotype with respect to mutations is defined using an adelic sum-of-digits function. In simple terms, the sum of the digits that the genotype refers to gives scientists a way to know the number of mutations a genotype can take before its phenotype will alter.
“Let’s say there are five genotypes that map to a particular phenotype,” Mohanty explained. “If you add up all the digits of those genotypes together all the t’s of x for each of the g’s that tells you the average number of mutations that those genotypes can sustain.” Well, turns out that those sums of digits, when graphed, result in a fractal curve called the blancmange curve.
On a fractal curve, says Mohanty, “if you zoom into the curve it looks exactly the same as if you were zoomed out, and you can continue to zoom in infinitely and infinitely and infinitely and it would be the same.” But there is more to the fractal nature; it reveals a deep insight into how much these biological systems can contain errors.
These are not just theoretical results; their “Understanding mutational robustness could eventually be important for preventing disease”. Pathogenic viruses and bacteria are evolving constantly and, as such, accumulate a huge number of neutral mutations. If scientists could find a way to stop these pathogens from picking up beneficial mutations in the vast sea of neutrals accumulated every evolutionary minute, then their ability to become ever more virulent or ever less susceptible to antibiotics could potentially be retarded.
“An understanding of mutational robustness may ultimately be of significant importance to disease prevention,” Mohanty says. If they understand ways to prevent viruses or bacteria from coming down on any of the many beneficial mutations their research has identified so far, they may be able to slow the spread of infections or the march toward antibiotic resistance.
Professor Ard Louis from the University of Oxford, who is also involved in the same study, commented, “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 know what the absolute maximal robustness possible would be, or if there even was a maximum”
The study from the team also further indicates that maximum robustness is linked to a function known as the Takagi, which is a peculiar fractal function that is continuous at every point but differentiable at no point. This peculiar function is sometimes referred to as the Blancmange because it bares a rather striking resemblance to the French dessert.
“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,” said Mohanty.
What’s beautiful about the discovery is not just its mathematical elegance but the illumination it’s cast on deep structures in our natural world. It is a vivid reminder of the interconnection between scientific disciplines. Could anyone have ever thought that a function from pure mathematics could explain the resilience of genetic sequences?
As Louis put it, “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. We believe that many intriguing new links between number theory and genetics will be found in the future.”
It is groundbreaking research, opening up absolutely new paths of investigation into how mathematics can help to demystify the complexities of life. This text-case example shows that curiosity-driven, directed basic science can work miracles of revealing the unexpected and give a twist to the way we think. Next time you consider genetics, remember: a tinge of this elegant math is within those DNA sequences, silently ruling over the very stuff that is life.
Relevant articles:
– Scientists uncover hidden math that governs genetic mutations, Live Science
– Scientists uncover a surprising connection between number theory and evolutionary genetics, ScienceDaily
– Scientists uncover a surprising connection between number theory and evolutionary genetics, University of Oxford
– Scientists uncover hidden math, Vocal