Testing the difference between two distributions

I have the following experimental setup: Protein A is capable of cutting protein B in small fragments. The small fragments are identified and the nature of the last amino acid in each fragment is counted. Thus, in one experiment it is possible to detect all 20 amino acids but with a different total count. The total count depends on the nature of Protein A and the conditions of the experiment. At the end, for the two conditions tested I end up with a table like this:

Amino-acid  Exp1   Exp2
A            0      3
R            20     12
G            10     15
H            14     22
E             5      0


with entries for all 20 amino acids and I also know the total number of fragments from Protein B that were identified in each condition.

The question I need to answer is: Are the amino acids total counts significantly different under the two experimental conditions?

First I thought to use a chi-square test since with the chi-square test I can take into account the different number of fragments that were identified in the two conditions. But inevitably I will end up with expected values being 0 and thus I cannot use the chi-square test.

What test can be used in this case?

• A chi-squared test allows you to compare an observed distribution with an expected distribution, but it is not clear that you have an expected distribution and so that test might be inappropriate even if you did not have zero expectations or counts. Is there a condition under which the distribution of amino-acid counts is characterised well enough to approximate a theoretical distribution? If not then I don't think chi-squared will be your friend. – Michael Lew Jul 2 '18 at 21:32
• Seems to me that you need to know how much the counts vary between runs of the experiment in each condition because your question of interest is something like "do the results between Exp1 and Exp2 differ by more than would be expected when the conditions of the experiments did not differ in any functionally important way?" You will need to know how consistent the results are across experiments where the conditions are fixed. – Michael Lew Jul 2 '18 at 21:35
• @Michael A $k\times 2$ chi-square (homogeneity of proportions test) is possible; as OP says the issue will be with some categories with very small expected (zero if both counts were 0). – Glen_b Jul 3 '18 at 3:10