= Membrane topology
{wiki}
<Membrane topology> specifies the sidedness of a membrane-associated <polypeptide chain>: where its <N-terminus>, <C-terminus> and intervening <protein domains> lie relative to the <lipid bilayer>. Side-specific placement is essential to interpreting a <cell-surface receptor>.<Molecular Biology of the Cell, 4th edition by Bruce Alberts and others/Membrane Proteins>[Molecular Biology of the Cell, 4th edition by Bruce Alberts and others; Membrane Proteins Can Be Associated with the Lipid Bilayer in Various Ways: aqueous domains on either side and receptor orientation]{ref}
For an original worked example, stipulate a continuous <polypeptide chain> with its <N-terminus> outside a <biological cell>, exactly three complete crossings of the <cell membrane>, and no additional excursions into the <lipid bilayer>. Its segments lie, in order, outside, inside, outside, inside. A <protein domain> between the first and second crossings faces the <cytosol>; one between the second and third faces the exterior; the <C-terminus> faces the <cytosol>. Counting crossings determines these placements without specifying an atomic <protein structure>.
The proof is a side-switching argument. Assign exterior the sign $+1$ and interior $-1$. Each complete crossing multiplies the sign by $-1$, so after $m$ crossings,
$$
s_m=(-1)^m s_0.
$$
An odd number puts the termini on opposite sides, an even number on the same side. This assumes complete crossings of one boundary; a chain that enters and returns to the same side without crossing must not be counted as a complete crossing. Nor does the rule determine which side the <N-terminus> starts on.
As a constructed accessibility test, put an <epitope> on the outside segment between crossings two and three, and another on the inside segment between crossings one and two. Stipulate an intact <cell membrane> that excludes the test <antibody> and accessible <epitopes> that bind it. Applying the <antibody> outside can label only the first <epitope>. If both become accessible after opening the boundary, the difference tests sidedness under those assumptions. Failure to label an <epitope> alone is insufficient: it could also be inaccessible because of <protein folding> or poor <antibody affinity>. This is a proposed reasoning exercise, separate from the historical mixing experiment below.
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