Degenerations and representations of twisted Shibukawa-Ueno R-operators

We study degenerations of the Belavin R-matrices via the infinite dimensional operators defined by Shibukawa-Ueno. We define a two-parameter family of generalizations of the Shibukawa-Ueno R-operators. These operators have finite dimensional representations which include Belavin's R-matrices in the elliptic case, a two-parameter family of twisted affinized Cremmer-Gervais R-matrices in the trigonometric case, and a two-parameter family of twisted (affinized) generalized Jordanian R-matrices in the rational case. We find finite dimensional representations which are compatible with the elliptic to trigonometric and rational degeneration. We further show that certain members of the elliptic family of operators have no finite dimensional representations. These R-operators unify and generalize earlier constructions of Felder and Pasquier, Ding and Hodges, and the authors, and illuminate the extent to which the Cremmer-Gervais R-matrices (and their rational forms) are degenerations of Belavin's R-matrix

Structure-Guided Recombination Creates an Artificial Family of Cytochromes P450

Creating artificial protein families affords new opportunities to explore the determinants of structure and biological function free from many of the constraints of natural selection. We have created an artificial family comprising ~3,000 P450 heme proteins that correctly fold and incorporate a heme cofactor by recombining three cytochromes P450 at seven crossover locations chosen to minimize structural disruption. Members of this protein family differ from any known sequence at an average of 72 and by as many as 109 amino acids. Most (>73%) of the properly folded chimeric P450 heme proteins are catalytically active peroxygenases; some are more thermostable than the parent proteins. A multiple sequence alignment of 955 chimeras, including both folded and not, is a valuable resource for sequence-structure-function studies. Logistic regression analysis of the multiple sequence alignment identifies key structural contributions to cytochrome P450 heme incorporation and peroxygenase activity and suggests possible structural differences between parents CYP102A1 and CYP102A2

Cremmer-Gervais r-matrices and the Cherednik Algebras of type GL2

We give an intepretation of the Cremmer-Gervais r-matrices for sl(n) in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL2 on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer-Gervais quantization in terms of the corresponding double affine Hecke algebra.Comment: 6 page

Semi-classical twists for sl(3) and sl(4) boundary r-matrices of Cremmer-Gervais type

We obtain explicit formulas for the semi-classical twists deforming the coalgebraic structure of U(sl(3)) and U(sl(4)). In rank 2 and 3 the corresponding universal R-matrices quantize the boundary r-matrices of Cremmer-Gervais type defining Lie Frobenius structures on the maximal parabolic subalgebras in sl(n)

SCHEMA Recombination of a Fungal Cellulase Uncovers a Single Mutation That Contributes Markedly to Stability

A quantitative linear model accurately (R^2 = 0.88) describes the thermostabilities of 54 characterized members of a family of fungal cellobiohydrolase class II (CBH II) cellulase chimeras made by SCHEMA recombination of three fungal enzymes, demonstrating that the contributions of SCHEMA sequence blocks to stability are predominantly additive. Thirty-one of 31 predicted thermostable CBH II chimeras have thermal inactivation temperatures higher than the most thermostable parent CBH II, from Humicola insolens, and the model predicts that hundreds more CBH II chimeras share this superior thermostability. Eight of eight thermostable chimeras assayed hydrolyze the solid cellulosic substrate Avicel at temperatures at least 5 °C above the most stable parent, and seven of these showed superior activity in 16-h Avicel hydrolysis assays. The sequence-stability model identified a single block of sequence that adds 8.5 °C to chimera thermostability. Mutating individual residues in this block identified the C313S substitution as responsible for the entire thermostabilizing effect. Introducing this mutation into the two recombination parent CBH IIs not featuring it (Hypocrea jecorina and H. insolens) decreased inactivation, increased maximum Avicel hydrolysis temperature, and improved long time hydrolysis performance. This mutation also stabilized and improved Avicel hydrolysis by Phanerochaete chrysosporium CBH II, which is only 55–56% identical to recombination parent CBH IIs. Furthermore, the C313S mutation increased total H. jecorina CBH II activity secreted by the Saccharomyces cerevisiae expression host more than 10-fold. Our results show that SCHEMA structure-guided recombination enables quantitative prediction of cellulase chimera thermostability and efficient identification of stabilizing mutations

Optimal Compost Rates for Organic Crop Production Based on a Decay Series

One of the more challenging aspects of organic farming is the development of an appropriate fertility plan, which may include crop rotation, cover crops, and/or soil amendments. When fertility is maintained by applying manure and/or compost, a pressing question is how much should be used. A framework was developed to address this question based on the idea of a decay series, which is a sequence of numbers quantifying the effects of compost on crop yield over a multi-year period. Prior research has focused on decay series expressed in nitrogen fertilizer equivalents. Given this information, I show how to calculate what manure/compost rates are needed to meet the nitrogen targets in a multi-crop rotation. Analogous results are presented for when the objective is profit rather than yield maximization. The planning framework is then generalized to include decay series where the carryover effects of manure/compost are measured, not against nitrogen fertilizer, but against new applications of the amendment. This change of basis, from nitrogen fertilizer equivalents to manure/compost equivalents, allows for field research on organically certified land and quantifies non-nutritive effects in a more meaningful way. Two case studies are presented to illustrate how this new type of decay series may be estimated and used to optimize crop production. By using data from a continuous corn (Zea mays L.) system amended with cattle manure slurry, the case study in estimation explores the methodological challenges that arise when the yield response to nitrogen fertilizer is not available as a benchmark. The case study in optimization looks at profit-maximizing compost rates for dryland, organic wheat (Triticum aestivum L.) in northern Utah