There are several versions of the insulin hypothesis of obesity, but the versions that are most visible to the public generally state that elevated circulating insulin (whether acute or chronic) increases body fatness. Some versions invoke insulin's effects on fat tissue, others its effects in the brain. This idea has been used to explain why low-carbohydrate and low-glycemic-index diets can lead to weight loss (although frankly, glycemic index per se doesn't seem to have much if any impact on body weight in controlled trials).
I have explained in various posts why this idea does not appear to be correct (1, 2, 3), and why, after extensive research, the insulin hypothesis of obesity lost steam by the late 1980s. However, I recently came across two experiments that tested the hypothesis as directly as it can be tested-- by chronically increasing circulating insulin in animals and measuring food intake and body weight and/or body fatness. If the hypothesis is correct, these animals should gain fat, and perhaps eat more as well.
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