2. Native ATCase is a dodecamer (R₆C₆) composed of six regulatory (R) subunits and six catalytic (C) subunits. The researchers in Dr. Schachman's laboratory wanted to figure out the quaternary structure of the complex by distinguishing between stronger and weaker subunit interactions.  Under mild conditions they were able to partially dissociate the complex into smaller aggregates containing only R or C subunits.  They could separate the two classes of subunit aggregates and then recombined them to reform active dodecameric enzyme.  In addition, they reacted the native enzyme with succinic anhydride which succinylates the epsilon-amino groups of lysine residues and creates a modified but still active enzyme (R'₆C'₆) with a substantial negative charge.  By dissociating, isolating, and recombining various combinations of modified and unmodified subunit complexes, they generated various hybrid dodecameric enzymes and analyzed them by gel electrophoresis (not SDS-PAGE) as diagrammed below. The dark spots represent ATCase activity in the gel.  The parentheses indicate combinations of native and modified subunits summing to 6.

b. Assume that the recombination of modified and unmodified subunits occurs randomly.  What are the relative amounts of ATCase in each of the electrophoretic bands for (RR')₆C and R₆(CC')₆ above?  Assume a binomial distribution and that prior to electrophoresis the native and modified subunit aggregates were present in equal amounts. e.g. [R_n] = [R'_n].

c. Constraints of symmetry restrict the possible arrangements R₆C₆ aggregates. X-ray crystallography of the enzyme shows that each subunit is isomorphous with the other subunits of its type. i.e. they are indistinguishable from each other since they fit into the complex structurally identical ways. Draw a representation of E.coli ATCase which satisfies the X-ray observations and is consistent with the electrophoretic observations .