Let C(n) denote the maximum number of induced copies of 5-cycles in graphs on n vertices. For n large enough, we show thatand a, b, c, d, e are as equal as possible.Moreover, if n is a power of 5, we show that the unique graph on n vertices maximizing the number of induced 5-cycles is an iterated blow-up of a 5-cycle.The proof uses flag algebra computations and stability methods.
De novo purine nucleotide synthesis is regulated, at least in part, by end-product inhibition of glutamine PRPP amidotransferase. An important feature of this inhibition is the fact that certain synergistic nucleotide pairs give more than additive inhibition. The physiological importance of synergism is in amplifying regulation by the adenine and guanine nucleotide end products of de novo synthesis. Using a new method to quantitate synergism, ADP plus GMP were confirmed [Meyer, E., and Switzer, R. L. (1978) J. Biol. Chem. 254, 5397-5402] to give strong synergistic inhibition of Bacillus subtilis glutamine PRPP amidotransferase. An X-ray structure of the ternary enzyme.ADP.GMP complex established that ADP binds to the allosteric A site and GMP to the catalytic C site. GMP increased the binding affinity of ADP for the A site by approximately 20-fold. Synergism results from a specific nucleotide-nucleotide interaction that is dependent upon a nucleoside diphosphate in the A site and a nucleoside monophosphate in the C site. Furthermore, synergism is enhanced by the competition between nucleotide inhibitor and PRPP substrate for the C site. Purine base specificity results from a backbone carbonyl interaction of Lys305' with the 6-NH2 group of adenine in the A site and a Ser347 Ogamma interaction with the 2-NH2 group of guanine in the C site. Steric considerations favor binding of the nucleoside diphosphate to the A site. Site-directed replacements of key residues increased the nucleotide concentrations needed for 50% inhibition and in some cases perturbed synergism. Mutations in either of the nucleotide sites perturbed function at both sites, supporting the important role of synergism.
We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least. Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colorings of complete graphs with two colors, where the number of monochromatic K 4 's is minimized. We show that all the extremal colorings must contain monochromatic K 4 's only in one of the two colors. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.
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