There are several schools of thought on fasting. A popular rule of thumb is that one can consume up to 50 calories during a fasting period without breaking the fast. Here is a study citing that less than 50 calories of carbohydrates or protein will not create an insulin response that stops autophagy.

Glynn, E. L., Fry, C. S., Drummond, M. J., Timmerman, K. L., Dhanani, S., Volpi, E., & Rasmussen, B. B. (2010). Excess leucine intake enhances muscle anabolic signaling but not net protein anabolism in young men and women. The Journal of nutrition, 140(11), 1970-1976.