When low carb it's easier to get dehydrated if you're not careful, so keep that in mind.
There is contradictory data as to whether purines are the cause or not. I doubt humans are not adapted eat protein though. What is likely is that we should avoid:
Avoid caffeine (a purine and diuretic)
Alcohol cuts the kidney?s ability to rid the blood of excess purines by at least 50%.
High GI carbs: Insulin tells the kidneys to reabsorb uric acid into the blood instead of excreting it into the urine. Fructose appears to be especially bad. Protein, especially those rich in BCAAs, or hydrolyzed, would also spike insulin as well.
Things you can take:
Drink lots of water. Staying hydrated avoids raises in blood uric acid levels.
You can try alkalinizing your urine. http://www.ncbi.nlm.nih.gov/pubmed/20958145