Bullet point summary of the full paper
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"The consumption of foods prepared at high temperatures has been associated with numerous health risks."
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In previous studies, high heat has specifically been examined with its ability to transform harmless molecules into small-molecule metabolites that can interact with DNA such as "polycyclic aromatic hydrocarbons (PAHs) and heterocyclic amines (HCAs)... aldehydes, acrylamide, and N-nitroso compounds"
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When such species interact with DNA, "mutations when replication specificity is altered by modified nucleobases and in genotoxicity and chromosomal rearrangements when strand breaks occur during repair"
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The amount of DNA seems to be reflective of how many of these small metabolites are formed. For example, in animal products, often associated with health concerns, we see high amounts of DNA content (cows = 5.3-19.5g/kg, pigs = 6.9 - 21.2g/kg). But in plants, we see a far lower amounts with grains, starches, fruits, and legumes the lowest (wheat = 0.6g/kg, lentils = 0.7-0.8g/kg, potatoes = 1g/kg, avocado = 0.6g/kg), and vegetables being higher (spinach = 2.6g/kg, cauliflower = 2.8g/kg, broccoli = 5.1g/kg)
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This study notes that the link between small-molecule agents and health concerns are not yet proven. They primarily focused on examining connection between individual parts of said DNA hypothesis.
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They note that if their findings continue to be held up as true, frying/grilling/smoking are probably the worst, roasting being similarly bad, and boiling being the least harmful. With raw and non-heat processed foods being the best.
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The authors note they are not yet clear on why many plant foods seem to have protective effects in other studies for cancer, but they do write "starch may contribute to some protection against reactive oxygen species" which is common in many unprocessed plant foods like legumes and tubers.
TL;DR: Food products still contain DNA from their sources. When consumed, the DNA must be broken down and utilized by our digestive system. When exposed to high heat, DNA from foods may break down into harmful molecules. When we consume said DNA in our food, it can be harmful to our existing DNA causing deleterious mutations. Some foods have less DNA content and speculated protective compounds such as starch which may prevent some of damage to us including many plant foods, but specifically tubers, legumes, fruits, vegetables, and grains. The study reinforces previous studies showing diets high in said protective foods and low in animal products may result in positive health impacts including lower rates of cancer.