Scientists have discovered that a key function from a “pure” branch of mathematics can predict how often genetic mutations lead to changes in function.
These rules, laid down through the so-called sum function of numbers, also govern some aspects of protein folding, computer coding and certain magnetic states in physics.
said lead study author Vibhav Mohanty, a theoretical physicist, doctoral candidate and MD at Harvard Medical School and MIT.
For every genotype — the DNA letters of a particular gene — there is a phenotype, or end result: a new protein, or even behavior in the case of a gene that regulates another set of genes. A genotype can acquire a number of mutations before its phenotype changes; This accumulation of neutral mutations is a major way in which evolution proceeds.
“We want to understand, how robust is the actual phenotype of the mutations?” Mohanty said. “It turned out that this strength was too high.” In other words, a lot of the “characters,” or base pairs that make up the DNA code, can change before the output does.
Since this strength is seen not only in genetics but also in fields such as physics and computer science, Mohani and his colleagues suspected that its roots might lie in the underlying mathematics of potential sequences. They envisioned these possible sequences as a multi-dimensional cube, known as a hypercube, with every point on this impossible cube visualized as a possible genotype. Mohany said genotypes with the same phenotype should eventually cluster together. The question was, what shape would these clusters form?
Turns out, the answer is found in number theory, the field of mathematics concerned with the properties of positive integers. The average phenotypic robustness of mutants has been shown to be determined by what is called a sum-of-numbers function. This means that by adding the numbers representing each genotype on the cube, you can arrive at the average robustness of the genotype.
“Let’s say there are five genotypes that correlate with a particular phenotype,” Mohanty said. So, for example, five DNA sequences, each with a different mutation, but all still code for the same protein.
The researchers found that adding the numbers used to represent these five sequences gives you the average number of mutations those genotypes can have before their phenotypes change.
This led to the second interesting discovery: these numbers from the graph, formed the so-called planmange curve, a fractal curve named after a French pudding (which looks like a fancy molded pudding).
On a fractal curve, Mohany said, “If you zoom in on the curve, it looks just like if it’s been zoomed out, and you can keep zooming in infinitely, infinitely, infinitely, and it’ll be the same.”
Mohani said these results revealed some interesting secrets about error correction. For example, the natural systems the researchers studied tend to handle errors differently than humans do when setting up data storage, such as in digital messages or on CDs or DVDs. In these technological examples, all errors are treated equally, while biological systems tend to protect certain sequences more than others.
This isn’t surprising for genetic sequences, Muhany said, as there may be many linchpin sequences and then others more peripheral to the main genetic function.
Understanding the dynamics of these neutral mutations may ultimately be important for disease prevention, Mohany said. Viruses and bacteria evolve rapidly, accumulating many neutralizing mutations in the process. If there is a way to prevent these pathogens from landing the needle mutation in the beneficial haystack among all the chaff, researchers may be able to impede the ability of pathogens to become more infectious or resistant to antibiotics, for example.
The researchers published their findings on July 26 in the Interface Journal of the Royal Society.
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