Technically, taurine is not an amino acid, but rather an amino sulfonic acid. But it is often referred to as an amino acid, even in scientific literature.
Taurine is not an essential nutrient; in other words, the human body makes its own taurine. Cats, on the other hand, are not able to make taurine and it must be supplied by the diet in order to keep their retinas healthy.
Taurine is made by the body from cysteine, which is a protein amino acid. If you eat the recommended amounts of protein, you should be getting enough cysteine to provide enough taurine.
Taurine is not found in plant foods. Non-vegetarians typically eat 40 - 70 mg of taurine per day (1). Vegans have been shown to have lower blood levels of taurine (3). It is not known whether this compromises health in any way, but very few vegans supplement with taurine, including healthy teenagers who have been vegan from birth.
Carnitine is a non-essential amino acid found primarily in animal products. If you are eating enough protein, your body should make what you need. While there is no reason for most vegetarians or vegans to be concerned with carnitine, there have been cases of vegans who do not thrive unless they are taking carnitine supplements.
A carnitine metabolic problem has been linked to migraines. If you are a vegan who started getting migraines after becoming vegan, you might consider talking to your health professional about carnitine supplementation. The average person consumes 100 - 300 mg of carnitine per day (2).
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3. Laidlaw SA, Shultz TD, Cecchino JT, Kopple JD. Plasma and urine taurine levels in vegans. Am J Clin Nutr. 1988 Apr;47(4):660-3.