The box-ball systems are integrable cellular automata whose long-time behavior is characterized by soliton solutions, with rich connections to other integrable systems such as the Korteweg-de Vries equation. In this paper, we consider a multicolor box-ball system with two types of random initial configurations and obtain sharp scaling limits of the soliton lengths as the system size tends to infinity. We obtain a sharp scaling limit of soliton lengths that turns out to be more delicate than that in the single color case established in [LLP20]. A large part of our analysis is devoted to studying the associated carrier process, which is a multidimensional Markov chain on the orthant, whose excursions and running maxima are closely related to soliton lengths. We establish the sharp scaling of its ruin probabilities, Skorokhod decomposition, strong law of large numbers and weak diffusive scaling limit to a semimartingale reflecting Brownian motion with explicit parameters. We also establish and utilize complementary descriptions of the soliton lengths and numbers in terms of modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes.

The K-theoretic Schur P- and Q-functions $G\hspace {-0.2mm}P_\lambda $ and $G\hspace {-0.2mm}Q_\lambda $ may be concretely defined as weight-generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of stable Grothendieck polynomials and were introduced by Ikeda and Naruse for applications in geometry. Nakagawa and Naruse specified families of dual K-theoretic Schur P- and Q-functions $g\hspace {-0.1mm}p_\lambda $ and $g\hspace {-0.1mm}q_\lambda $ via a Cauchy identity involving $G\hspace {-0.2mm}P_\lambda $ and $G\hspace {-0.2mm}Q_\lambda $. They conjectured that the dual power series are weight-generating functions for certain shifted plane partitions. We prove this conjecture. We also derive a related generating function formula for the images of $g\hspace {-0.1mm}p_\lambda $ and $g\hspace {-0.1mm}q_\lambda $ under the $\omega $ involution of the ring of symmetric functions. This confirms a conjecture of Chiu and the second author. Using these results, we verify a conjecture of Ikeda and Naruse that the $G\hspace {-0.2mm}Q$-functions are a basis for a ring.

We enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of W-noncrossing partitions.

We have crystallized Drosophila melanogaster acetylcholinesterase and solved the structure of the native enzyme and of its complexes with two potent reversible inhibitors, 1,2,3,4-tetrahydro-N-(phenylmethyl)-9-acridinamine and 1,2,3,4-tetrahydro-N-(3-iodophenyl-methyl)-9-acridinamine—all three at 2.7 Å resolution. The refined structure of D. melanogaster acetylcholinesterase is similar to that of vertebrate acetylcholinesterases, for example, human, mouse, and fish, in its overall fold, charge distribution, and deep active-site gorge, but some of the surface loops deviate by up to 8 Å from their position in the vertebrate structures, and the C-terminal helix is shifted substantially. The active-site gorge of the insect enzyme is significantly narrower than that of Torpedo californica AChE, and its trajectory is shifted several angstroms. The volume of the lower part of the gorge of the insect enzyme is ∼50% of that of the vertebrate enzyme. Upon binding of either of the two inhibitors, nine aromatic side chains within the active-site gorge change their conformation so as to interact with the inhibitors. Some differences in activity and specificity between the insect and vertebrate enzymes can be explained by comparison of their three-dimensional structures.

We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every set. for example, is decidable by such machines, and the semi-decidable sets form a portion of the sets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators.

The complete amino acid sequence of the lectin KM+ from Artocarpus integrifolia (jackfruit), which contains 149 residues/mol, is reported and compared to those of other members of the Moraceae family, particularly that of jacalin, also from jackfruit, with which it shares 52% sequence identity. KM+ presents an acetyl-blocked N-terminus and is not posttranslationally modified by proteolytic cleavage as is the case for jacalin. Rather, it possesses a short, glycine-rich linker that unites the regions homologous to the α- and β-chains of jacalin. The results of homology modeling implicate the linker sequence in sterically impeding rotation of the side chain of Asp141 within the binding site pocket. As a consequence, the aspartic acid is locked into a conformation adequate only for the recognition of equatorial hydroxyl groups on the C4 epimeric center (α-d-mannose, α-d-glucose, and their derivatives). In contrast, the internal cleavage of the jacalin chain permits free rotation of the homologous aspartic acid, rendering it capable of accepting hydrogen bonds from both possible hydroxyl configurations on C4. We suggest that, together with direct recognition of epimeric hydroxyls and the steric exclusion of disfavored ligands, conformational restriction of the lectin should be considered to be a new mechanism by which selectivity may be built into carbohydrate binding sites. Jacalin and KM+ adopt the β-prism fold already observed in two unrelated protein families. Despite presenting little or no sequence similarity, an analysis of the β-prism reveals a canonical feature repeatedly present in all such structures, which is based on six largely hydrophobic residues within a β-hairpin containing two classic-type β-bulges. We suggest the term β-prism motif to describe this feature.