Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems

The final publication is available at Springer via http://dx.doi.org/ 10.1007/s10543-016-0601-5.We investigate how to adapt the Q-Arnoldi method for the case of symmetric quadratic eigenvalue problems, that is, we are interested in computing a few eigenpairs of with M, C, K symmetric matrices. This problem has no particular structure, in the sense that eigenvalues can be complex or even defective. Still, symmetry of the matrices can be exploited to some extent. For this, we perform a symmetric linearization , where A, B are symmetric matrices but the pair (A, B) is indefinite and hence standard Lanczos methods are not applicable. We implement a symmetric-indefinite Lanczos method and enrich it with a thick-restart technique. This method uses pseudo inner products induced by matrix B for the orthogonalization of vectors (indefinite Gram-Schmidt). The projected problem is also an indefinite matrix pair. The next step is to write a specialized, memory-efficient version that exploits the block structure of A and B, referring only to the original problem matrices M, C, K as in the Q-Arnoldi method. This results in what we have called the Q-Lanczos method. Furthermore, we define a stabilized variant analog of the TOAR method. We show results obtained with parallel implementations in SLEPc.This work was supported by the Spanish Ministry of Economy and Competitiveness under Grant TIN2013-41049-P. Carmen Campos was supported by the Spanish Ministry of Education, Culture and Sport through an FPU Grant with reference AP2012-0608.Campos, C.; Román Moltó, JE. (2016). Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems. BIT Numerical Mathematics. 56(4):1213-1236. https://doi.org/10.1007/s10543-016-0601-5S12131236564Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. 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Four plant defensins from an indigenous South African Brassicaceae species display divergent activities against two test pathogens despite high sequence similarity in the encoding genes

<p>Abstract</p> <p>Background</p> <p>Plant defensins are an important component of the innate defence system of plants where they form protective antimicrobial barriers between tissue types of plant organs as well as around seeds. These peptides also have other activities that are important for agricultural applications as well as the medical sector. Amongst the numerous plant peptides isolated from a variety of plant species, a significant number of promising defensins have been isolated from Brassicaceae species. Here we report on the isolation and characterization of four defensins from <it>Heliophila coronopifolia</it>, a native South African Brassicaceae species.</p> <p>Results</p> <p>Four defensin genes (<it>Hc-AFP1</it>-<it>4) </it>were isolated with a homology based PCR strategy. Analysis of the deduced amino acid sequences showed that the peptides were 72% similar and grouped closest to defensins isolated from other Brassicaceae species. The Hc-AFP1 and 3 peptides shared high homology (94%) and formed a unique grouping in the Brassicaceae defensins, whereas Hc-AFP2 and 4 formed a second homology grouping with defensins from <it>Arabidopsis </it>and <it>Raphanus</it>. Homology modelling showed that the few amino acids that differed between the four peptides had an effect on the surface properties of the defensins, specifically in the alpha-helix and the loop connecting the second and third beta-strands. These areas are implicated in determining differential activities of defensins. Comparing the activities after recombinant production of the peptides, Hc-AFP2 and 4 had IC₅₀values of 5-20 μg ml^-1against two test pathogens, whereas Hc-AFP1 and 3 were less active. The activity against <it>Botrytis cinerea </it>was associated with membrane permeabilization, hyper-branching, biomass reduction and even lytic activity. In contrast, only Hc-AFP2 and 4 caused membrane permeabilization and severe hyper-branching against the wilting pathogen <it>Fusarium solani</it>, while Hc-AFP1 and 3 had a mild morphogenetic effect on the fungus, without any indication of membrane activity. The peptides have a tissue-specific expression pattern since differential gene expression was observed in the native host. <it>Hc-AFP1 </it>and <it>3 </it>expressed in mature leaves, stems and flowers, whereas <it>Hc-AFP2 </it>and <it>4 </it>exclusively expressed in seedpods and seeds.</p> <p>Conclusions</p> <p>Two novel Brassicaceae defensin sequences were isolated amongst a group of four defensin encoding genes from the indigenous South African plant <it>H. coronopifolia</it>. All four peptides were active against two test pathogens, but displayed differential activities and modes of action. The expression patterns of the peptide encoding genes suggest a role in protecting either vegetative or reproductive structures in the native host against pathogen attack, or roles in unknown developmental and physiological processes in these tissues, as was shown with other defensins.</p

Structure-Activity Determinants in Antifungal Plant Defensins MsDef1 and MtDef4 with Different Modes of Action against Fusarium graminearum

Plant defensins are small cysteine-rich antimicrobial proteins. Their three-dimensional structures are similar in that they consist of an α-helix and three anti-parallel β-strands stabilized by four disulfide bonds. Plant defensins MsDef1 and MtDef4 are potent inhibitors of the growth of several filamentous fungi including Fusarium graminearum. However, they differ markedly in their antifungal properties as well as modes of antifungal action. MsDef1 induces prolific hyperbranching of fungal hyphae, whereas MtDef4 does not. Both defensins contain a highly conserved γ-core motif (GXCX3–9C), a hallmark signature present in the disulfide-stabilized antimicrobial peptides, composed of β2 and β3 strands and the interposed loop. The γ-core motifs of these two defensins differ significantly in their primary amino acid sequences and in their net charge. In this study, we have found that the major determinants of the antifungal activity and morphogenicity of these defensins reside in their γ-core motifs. The MsDef1-γ4 variant in which the γ-core motif of MsDef1 was replaced by that of MtDef4 was almost as potent as MtDef4 and also failed to induce hyperbranching of fungal hyphae. Importantly, the γ-core motif of MtDef4 alone was capable of inhibiting fungal growth, but that of MsDef1 was not. The analysis of synthetic γ-core variants of MtDef4 indicated that the cationic and hydrophobic amino acids were important for antifungal activity. Both MsDef1 and MtDef4 induced plasma membrane permeabilization; however, kinetic studies revealed that MtDef4 was more efficient in permeabilizing fungal plasma membrane than MsDef1. Furthermore, the in vitro antifungal activity of MsDef1, MsDef1-γ4, MtDef4 and peptides derived from the γ-core motif of each defensin was not solely dependent on their ability to permeabilize the fungal plasma membrane. The data reported here indicate that the γ-core motif defines the unique antifungal properties of each defensin and may facilitate de novo design of more potent antifungal peptides