My brine recipe calls for 1/2 cup sugar and 1/2 cup of salt per gallon of water. I want to add soy sauce for flavor.

I suspect a direct sodium swap will work well. My Diamond kosher salt has 280 mg sodium per 1/4 teaspoon, or 26880 mg per half cup. My lite soy sauce has 575 mg sodium per tablespoon, so would be just under 3 cups to get equivalent sodium. My regular soy sauce has 920 mg sodium per tablespoon, or just over 1 3/4 cups to get equivalent sodium.

This one is from Kikkoman, a blogger and America's Test Kitchen thrown in for good measure. The ratios are different than yours, but it gives you an idea. I've used this one a couple of times with good results.

Personally, I suggest you do this experiment yourself by scaling down.

There are 16 cups per gallon. To make a test batch, use 1 cup water and 1/16th the amount of sugar and salt. This works out to 2/3 tablespoon each of sugar and salt. 2/3 Tbs. is actually 2 teaspoons.

Now that you have your control sample, I suggest you make another batch with 1 cup of water and 2 teaspoons of sugar. Then start adding soy sauce from a graduated measuring cup, noting how much you've poured into the solution until you reach a similar salinity by taste.

This is really the most logical way to perform this conversion since different soy sauces have different salinity levels.