Long and short paths in uniform random recursive dags

In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \sigma such that S_n/log(n) tends to \sigma in probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to \sigma in probability.Comment: 16 page

Depsipeptide substrates for sortase-mediated N-terminal protein ligation

Technologies that allow the efficient chemical modification of proteins under mild conditions are widely sought after. Sortase-mediated peptide ligation provides a strategy for modifying the N or C terminus of proteins. This protocol describes the use of depsipeptide substrates (containing an ester linkage) with sortase A (SrtA) to completely modify proteins carrying a single N-terminal glycine residue under mild conditions in 4–6 h. The SrtA-mediated ligation reaction is reversible, so most labeling protocols that use this enzyme require a large excess of both substrate and sortase to produce high yields of ligation product. In contrast, switching to depsipeptide substrates effectively renders the reaction irreversible, allowing complete labeling of proteins with a small excess of substrate and catalytic quantities of sortase. Herein we describe the synthesis of depsipeptide substrates that contain an ester linkage between a threonine and glycolic acid residue and an N-terminal FITC fluorophore appended via a thiourea linkage. The synthesis of the depsipeptide substrate typically takes 2–3 d