= Gamma-chain pairing bounds from bulk composition
\b[Gamma-chain pairing bounds from bulk composition] describe what a measured <G-gamma globin>/<A-gamma globin> chain ratio can establish about the assemblies containing those chains. Assume every measured gamma chain belongs to a standard <fetal hemoglobin> tetramer containing exactly two gamma chains. A bulk chain fraction then constrains, but generally does not determine, how the two forms are paired. The molecular distinction between the forms is experimentally established by <glycine> versus <alanine> at residue 136.<Evidence for multiple structural genes for the gamma chain of human fetal hemoglobin by W. A. Schroeder and others>[Page 537, opening paragraph]{ref}
Let $p$ be the fraction of gamma chains that are G-gamma, and let $x,y,z$ be the fractions of tetramers containing respectively two G-gamma chains, one of each form, and two A-gamma chains. Counting assemblies and chains gives
$$
x+y+z=1,\qquad 2x+y=2p.
$$
Hence $y=2p-2x$ and $z=1-2p+x$. Requiring all three fractions to be nonnegative gives the sharp bounds
$$
\boxed{\max(0,2p-1)\leq x\leq p}.
$$
Every value in this interval yields nonnegative $x,y,z$ with the required bulk composition; a finite population additionally requires integral assembly counts.
For an original example, 100 tetramers containing 120 G-gamma and 80 A-gamma chains have $p=0.60$. The endpoint compositions $(x,y,z)=(0.20,0.80,0)$ and $(0.60,0,0.40)$ give the same chain counts. Thus the fraction of two-G-gamma tetramers can range from $20\%$ to $60\%$ despite an exactly known bulk composition.
Independent random placement is an additional model. It selects the <binomial distribution> values
$$
x=p^2,\qquad y=2p(1-p),\qquad z=(1-p)^2.
$$
At $p=0.60$, these are $0.36,0.48,0.16$, one feasible point rather than a consequence of the measured chain ratio. The inequalities $p^2\leq p$ and $p^2-(2p-1)=(1-p)^2\geq0$ show why the random-pairing value always lies within the bounds.
The argument does not infer actual biological pairing preferences. It identifies the information lost when an assay pools chains from many <protein complexes>. Free chains, other assemblies or unequal analytical responses would require a different input model. A change in bulk sequence proportions is therefore not automatically a measured change in every tetramer's composition or <oxygen> binding.
\Image[/gamma-chain-pairing-feasible-region.png]
{title=Bulk gamma-chain composition bounds but does not determine pairing}
{description=Original stoichiometric feasible region for tetramers containing two gamma chains. The shaded band contains every allowed two-G-gamma assembly fraction at each bulk G-gamma chain fraction. Independent placement gives the dashed curve, one extra model within the band. The highlighted p=0.60 example is constructed, not measured biological pairing.}
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