Here is how I would do it:
Let the number of proteins of a certain type (p) by a strain of virus (v) be given by
Ln(p) = b0 + b1(v) + b1b2(v,p) + e
where b1(v) can vary between strains and b1b2(v,p) regulates excess production of a certain type.
Now the null hypothesis is b1b2(v,p) are jointly zero.
Now if you have the protein volumes, this can be estimated directly, though you will need to define one protein as the default in order to avoid collinearity. If you only have the shares, then you will need to estimate it via maximum likelihood as your shares equation will have multiple error terms. You can then use Wilk's theorem for significance testing. But it will be a major pain in the ass as the GRG problem will have a stupidly large number of unknowns.