L-Glutamine is a naturally occurring nonessential neutral amino acid. Glutamine is utilized at high rates by leukocytes to provide energy. Glutamine supplementation is becoming increasingly popular among athletes because of its beneficial effect on glycogenesis .
In fasted state muscle protein break down occurs. One of more recent studies show that resistance-training to some extent reduces this protein catabolism, while anabolic response requires an intake of essential amino acids in the recovery period . This increase in amino acid uptake into muscles increases the rate of protein synthesis. While essential amino acid surplus seems to have beneficial effect, non-essential amino acid supplementation is unlikely to provide any additional benefit . However, oral glutamine supplementation has been shown to decrease protein degradation and increase protein synthesis [4,5]. In contrast, many studies found no beneficial effect of glutamine supplementation on glycogenesis after glycogen depleting workout .
In a study that investigated the effects of oral glutamine supplementation in combination with strength training, glutamine showed no effect on glycogen resynthesis, strength and lean tissue mass. The lack of effect may be due to the consumption of glutamine by many other tissues before reaching the skeletal muscles. One other reason my be that strength training is not stressful enough to benefit from glutamine supplementation 
It is controversial whether glutamine supplementation has the potential to improve exercise due to many contrasting studies.
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Hankard, Regis G., Morey W. Haymond, and Dominique Darmaun. “Effect of glutamine on leucine metabolism in humans.” American Journal of Physiology-Endocrinology And Metabolism 271.4 (1996): E748-E754.
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Candow, Darren G., et al. “Effect of glutamine supplementation combined with resistance training in young adults.” European journal of applied physiology 86.2 (2001): 142-149.