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Research article Special Issues

Uncertain random problem for multistage switched systems

  • Received: 21 March 2023 Revised: 18 June 2023 Accepted: 25 June 2023 Published: 18 July 2023
  • MSC : 93C55, 49L20

  • Optimal control problems for switched systems how best to switch between different subsystems. In this paper, two kinds of linear quadratic optimal control problems for multistage switched systems composing of both randomness and uncertainty are studied. Chance theory brings us a useful tool to deal with this indeterminacy. Based on chance theory and Bellman's principle, the analytical expressions are derived for calculating both the optimal control input and the optimal switching control law. Optimal control is implemented by genetic algorithm instead of enumerating all the elements of a series of sets whose size grows exponentially. Finally, the results of numerical examples are provided to illustrate the effectiveness of the proposed method.

    Citation: Guangyang Liu, Yang Chang, Hongyan Yan. Uncertain random problem for multistage switched systems[J]. AIMS Mathematics, 2023, 8(10): 22789-22807. doi: 10.3934/math.20231161

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  • Optimal control problems for switched systems how best to switch between different subsystems. In this paper, two kinds of linear quadratic optimal control problems for multistage switched systems composing of both randomness and uncertainty are studied. Chance theory brings us a useful tool to deal with this indeterminacy. Based on chance theory and Bellman's principle, the analytical expressions are derived for calculating both the optimal control input and the optimal switching control law. Optimal control is implemented by genetic algorithm instead of enumerating all the elements of a series of sets whose size grows exponentially. Finally, the results of numerical examples are provided to illustrate the effectiveness of the proposed method.



    Fungal infections are responsible for significant reductions in agricultural crop yields and post-harvest product losses on a global scale [1]. Approximately $220 billion is wasted annually in the worldwide economy due to fungal diseases (FAO, 2019). In order to mitigate such losses, farmers employ multiple fungicides, a practice that proves to be ineffective due to the rapid development of disease resistance to these chemicals. Furthermore, this method presents substantial hazards to both human well-being and the ecosystem. In contrast, genetic approaches, such as the integration of resistant genes, are more reliable and enduring. Nevertheless, the existence of specific genes that confer resistance in host plants can exert substantial selective pressure on pathogens, leading to rapid evolutionary adaptations that often give rise to new races capable of evading host resistance, as observed in numerous interactions between plants and pathogens [2],[3]. Hence, to attain enhanced and long-lasting protection against fungal diseases, it is imperative to possess a profound comprehension of the pathogenicity factors secreted by the pathogen and the subsequent resistance responses of the plant [4].

    To effectively invade the host plant, plant-pathogenic fungi release several proteins to overcome various host protection barriers. These secretory proteins contain several enzymes that break down the cell wall, proteases, and small secretory proteins known as "effectors". Multiple investigations on plant–microbe interactions have demonstrated that effectors significantly influence the pathogenicity of numerous phytopathogens [5],[6]. Effectors are small proteins that pathogens release into both the extracellular and intracellular regions of host plants with the purpose of modifying specific targets inside the host. They exhibit distinct attributes, including small size (50–300 amino acids), high cysteine content, possession of an N-terminal signal peptide, absence of transmembrane structural domains, glycosylphosphatidylinositol (GPI) anchor sites, and subcellular localization signals for chloroplasts or other intracellular organelles [7]. These inherent traits allow scientists to predict effectors from the genomes of numerous phytopathogens. For instance, in a recent study, the draft genome was utilized to forecast the potential effectors of the citrus pathogen, Elsinoe fawcettii [8][13].

    Avocado scab, a prevalent disease in avocado-producing regions with warm and humid climates, is mostly caused by the fungus E. perseae [14],[15]. The initial symptoms manifest as distinct spots on the leaves along the midrib, which have the potential to combine and form star-shaped patterns. As the disease advances, the leaves undergo distortion and stunting [16]. The symptoms of avocado scab on fruits manifest as blackish/brownish oval and elevated cork scab formations that are dispersed on the fruit's skin [16],[17]. Even though it does not directly damage the pulp, it negatively affects the quality of the fruit and makes it inappropriate for the global market. This reduces the production value by up to 60%. To date, avocado disease has been reported in the USA and many countries in Africa and Asia [18].

    Current disease management strategies for avocado scab are generally based on a combination of cultural and chemical methods. A crucial cultural practice involves the elimination and destruction of affected plant material to avert the dissemination of the disease [19]. In the chemical method, many fungicides, including benomyl, azoxystrobin, and pyraclostrobin, have shown efficiency in controlling the disease's spread [20],[21]. Nonetheless, overdependence on chemical pesticides may result in environmental issues and the emergence of fungicide resistance, hence requiring the investigation of alternate approaches. Biocontrol techniques remain inadequately researched, despite investigations conducted to manage other avocado pathogens, such as Phytophthora cinnamomi [22],[23]. Currently, there has been no molecular analysis, including gene expression studies, concerning the E. perseae–avocado interaction, conducted to identify the pathogenicity factors of E. perseae that increase its virulence on avocado.

    To effectively mitigate the economic impact of avocado scab, a thorough comprehension of the molecular mechanisms underlying the pathogenicity of E. perseae is necessary [24],[25]. Although the draft genome of E. perseae is publicly available, there has been a dearth of comprehensive investigation about the exact characteristics and functionalities of its secretory proteins [24]. This study utilized a bioinformatics methodology to systematically predict secretory proteins in the genome of E. perseae. It provides valuable insights into numerous potential virulence factors of E. perseae, including cell wall–degrading enzymes, proteases, pathogenicity-related proteins, and potential effector proteins. This will serve as a great resource for future investigations into the molecular interaction between E. perseae and avocado.

    The E. perseae draft genome sequence (NCBI accession: GCA_029448695.1) was obtained from NCBI, and gene models were predicted in the current study using the GenSAS server [24],[25]. In this study, both homology and de novo methods were employed to predict gene models in the genome of E. perseae. In the homology-based prediction, the repeat-masked assembly was subjected to a BlastN search against expressed sequence tag (EST) sequences of the closely related species, E. ampelina, using an e-value cutoff of 1 × 10−5 [26],[27]. The de novo approach employed two ab initio gene prediction tools, namely Augustus and GeneMarkES [28],[29]. Ultimately, the gene prediction information was consolidated using EvidenceModeler to produce a non-redundant gene set [30]. The comparative analysis utilized genome data from six closely related species, viz., E. necatrix, E. batatas, E. arachidis, E. fawcettii, E. ampelina, and E. australis (Table 1) [31][35].

    The previously outlined pipeline was employed to predict the fungal secretome [36]. To identify the sequences with signal peptide, SignalP (version 6.0) was employed in conjunction with the Phobius server [37],[38]. The sequences that were identified as having a signal peptide by both systems were chosen for additional screening. The DeepTMHMM server was employed to eliminate the transmembrane proteins [39]. The proteins targeting the endoplasmic reticulum (ER) were eliminated by scanning them for PS00014 ER motif retention using the Prosite database and the ScanProsite web service [40]. The proteins targeting various organelles such as mitochondria and chloroplast were predicted using the TargetP and WoLF PSORT systems [41],[42]. The sequences containing glycophosphatidylinositol (GPI) anchor motifs were identified using NetGPI (version 1.1) [43].

    The refined secretome was subjected to scanning against NCBI, InterPro, and PFAM databases to get functional annotations for the predicted proteins [44],[45]. The annotation of carbohydrate-active enzymes was retrieved using the CAZy database and dbCAN web server [46],[47]. The effector prediction was performed using EffectorP CAZy (version 3.0) software, in conjunction with manual examination [48]. Furthermore, the BlastP algorithm with a significance threshold of E value lower than 1 × 10−10 was employed to query the pathogen–host interaction database (PHI database) in order to identify any resemblances to established effectors and pathogenicity factors [49]. The identification of proteolytic enzymes was accomplished by a BlastP search conducted on the MEROPS database [50]. Orthologue analysis was conducted using Orthovenn3 web server [51]. To infer the phylogenetic relationship among Elsinoe species, 50 shared orthologs were selected randomly and a concatenated alignment was made. The relationship was constructed by MEGA11 using the maximum likelihood method and JTT matrix-based model (based on 1000 bootstrap replications).

    The draft genome sequence of E. perseae (NCBI accession: GCA_029448695.1) was utilized to identify the secretome [24]. Furthermore, for the comparative study, the genome data of six Elsinoe species, namely E. necatrix, E. batatas, E. arachidis, E. fawcettii, E. ampelina, and E. australis, was chosen (Table 1) [31][35]. The gene models for E. perseae, E. necatrix, E. batatas, and E. arachidis were predicted in the current study using the GenSAS server due to the lack of publicly available annotations (Table 1, Table S1). Gene models for the remaining three species were acquired from publicly available databases (Table 1). In total, 9236 proteins were predicted from the E. perseae genome, which were subjected to secretome prediction using the methodology depicted in Figure 1. Of the 9236 proteins, an N-terminal secretory signal sequence was identified in 932 proteins. Out of these, 732 proteins, lacking any transmembrane domain, were chosen and examined for an ER-targeting signal in order to eliminate the proteins that reside in the endoplasmic reticulum. Of the 732 proteins, a subset of 12 were identified to possess the PS00014 ER motif and were therefore removed from subsequent research. Through the analysis of TargetP and WoLF PSORT, it was projected that the remaining 720 proteins were localized in the extracellular space. Out of the total 720 proteins, 66 proteins were identified as having GPI-anchor motifs using NetGPI (version 1.1). These proteins are most likely found on the surface rather than being secreted, and hence they were not included in the analysis. As a result, a list of 654 "refined secretome" was obtained, representing 7.1% of the entire predicted proteome of E. perseae (Figure 1, Table 1).

    Table 1.  List of Elsinoe species included in this study.
    Species Genome size (Mb) GenBank assembly Gene models Secretome
    E. perseae 23.5 GCA_029448695.1* 9236 654
    E. necatrix 25.5 GCA_033846785.1* 8501 619
    E. batatas 26.5 GCA_017309325.2* 8783 569
    E. arachidis 33.2 GCA_013372555.1* 9754 781
    E. fawcettii 26.3 GCA_007556565.1 10264 658
    E. ampelina 28.3 GCA_005959805.1 10209 621
    E. australis 23.8 GCA_007556505.1 9223 676

    *Gene models predicted in this study were used for the analysis.

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    Figure 1.  Pipeline for the identification and functional characterization of the secretome of Elsinoe perseae. CAZyme: carbohydrate-active enzymes; PHI: pathogen–host interactions database. Tools and the number of filtered proteins in each step are shown in the respective boxes.

    The refined secretome of E. perseae varied in length, ranging from 55 to 2410 amino acids (aa). Among these proteins, 50.3% (330) had a length of 55–399 aa, indicating an abundance of small secretory proteins in the secretome of E. perseae (Figure S1). Secretory proteins exhibited a molecular weight (MW) range of 5.9–248 kDa. The majority of these proteins (61.3%) fell between the 5.9 and 49.9 kDa range (Figure S1B). Furthermore, the theoretical isoelectric point (pI) of the secretory proteins varied between 3.45 and 11.89. The majority (53.3%) of these proteins had a pI ranging from 4 to 5.9 (Figure S1C). The domain analysis identified a minimum of one functional domain in 445 proteins. The most enriched domains were PAN_4 (PF14295), PAN_1 (PF00024), WSC (PF01822), LysM (PF01476), and FAD binding (PF00890) (Figure S3). From the total secreted proteins, 358 (55%) proteins were assigned at least one gene ontology (GO) term based on sequence homology. Based on the gene ontology terms, these proteins were categorized into three categories: biological process (251 proteins, 38.3%), molecular function (301 proteins, 46%), and cellular components (197 proteins, 30.1%) (Figure 2). The gene ontology terms that are most enriched under biological processes are carbohydrate metabolic process (GO:0005975), proteolysis (GO:0006508), polysaccharide catabolic process (GO:0000272), cellulose catabolic process (GO:0030245), and others (Figure 2). The molecular function category prominently encompasses several activities such as hydrolase activity (GO:0004553), serine-type endopeptidase activity (GO:0004252), cellulase activity (GO:0008810), or oxidoreductase activity (GO:0016614) (Figure 2). The cellular component comprises the extracellular region (GO:0005576), cell wall (GO:0005618), and membrane (GO:0016020) (not shown here).

    Figure 2.  Gene ontology of the secretome of Elsinoe perseae. GO terms assigned to molecular function and biological processes are listed.
    Figure 3.  Comparative secretome and phylogenetic analysis of the secretome of seven Elsinoe species. A. Orthologue clusters. Clusters of orthologous genes were detected utilizing Orthovenn3 and displayed utilizing UpsetR. For the analysis, the e-value cutoff of 1 × 10−10 was utilized. Blue bars represent the number of clusters shared by Elsinoe species, represented by numbers shown above the bar. Line and dot connections between species that share orthologs. B. Phylogenetic relationship of seven species of Elsinoe. A concatenated alignment of fifty shared orthologue proteins was used to infer the phylogenetic tree of seven Elsinoe species. The relationship was constructed using the maximum likelihood method and JTT+CAT matrix-based model.

    For the comparative analysis, the secretome of six closely related species of Elsinoe, as shown in Table 1, was identified using the methodology described above. The results showed that the secretome of E. perseae was comparable to that of closely related species, of which the closest was E. arachidis (781) and the furthest was E. batatas (569) (Table 1). The secretome of all species analyzed displayed the same pattern in terms of length, PI, and MW distribution (Figure S1A–C). The comparative orthologue analysis revealed that the secretome of all species of Elsinoe shared 152 clusters (Figure 3A). Among these clusters, 140 were identified as single-copy gene clusters. The analysis also identified singletons (proteins that do not form any clusters) in each species, which varied among species. E. australis had the maximum number of singletons (112), while E. batatas had the lowest (38) (Table S2). A total of 76 singletons were identified for E. perseae, indicating that 88.3% of the secretory proteins of E. perseae have orthologues in other species. A phylogenetic tree was constructed to elucidate the relationship between Elsinoe species using a composite alignment of 50 single-copy orthologue proteins (Figure 3B). Within the tree, E. fawcettii and E. arachidis showed a close relation with E. necatrix and E. batatas, respectively (Figure 3B). E. perseae constituted a distinct clade, indicating their monophyletic origin (Figure 3B), as previously shown [52].

    Table 2.  Cell wall–degrading enzymes in E. perseae.
    CAZy family Protein id PFAM id Enzyme name Substrate
    AA7 Ep.00g028650.m01 PF01565.18 Glucooligosaccharide oxidase Cellobiose
    AA7 Ep.00g057050.m01 PF01565.18 Glucooligosaccharide oxidase Cellobiose
    AA7 Ep.00g013270.m01 PF01565.18 Glucooligosaccharide oxidase Cellobiose
    AA7 Ep.00g035950.m01 PF01565.18 Glucooligosaccharide oxidase Cellobiose
    AA7 Ep.00g003880.m01 PF01565.18 Glucooligosaccharide oxidase Cellobiose
    AA7 Ep.00g009110.m01 PF01565.18 Glucooligosaccharide oxidase Cellobiose
    AA7 Ep.00g029870.m01 PF01565.18 Glucooligosaccharide oxidase Cellobiose
    AA7 Ep.00g005960.m01 PF01565.18 Glucooligosaccharide oxidase Cellobiose
    AA8 Ep.00g020830.m01 PF16010 Cellobiose dehydrogenase Cellobiose
    AA8 Ep.00g040980.m01 PF16010 Cellobiose dehydrogenase Cellobiose
    AA8 Ep.00g003740.m01 PF16010 Cellobiose dehydrogenase Cellobiose
    GH31 Ep.00g091100.m01 PF13802.1 α-glucosidase Cellobiose
    AA3 Ep.00g035640.m01 PF00732 Cellobiose dehydrogenases Cellulose
    AA3 Ep.00g090240.m01 PF00732 Cellobiose dehydrogenases Cellulose
    AA3 Ep.00g035790.m01 PF00732 Cellobiose dehydrogenases Cellulose
    AA9 Ep.00g003730.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g012020.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g019760.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g021550.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g027070.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g028770.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g030160.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g056790.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g058480.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g061930.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g067810.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g072230.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g078850.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g083810.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA9 Ep.00g089020.m01 PF03443 Lytic cellulose monooxygenase Cellulose
    AA16 Ep.00g004440.m01 PF03067 Lytic polysaccharide mono-oxygenase Cellulose
    AA16 Ep.00g076350.m01 PF03067 Lytic polysaccharide mono-oxygenase Cellulose
    AA16 Ep.00g091200.m01 PF03067 Lytic polysaccharide mono-oxygenase Cellulose
    GH1 Ep.00g078300.m01 PF00232.13 β-glucosidase Cellulose
    GH1 Ep.00g087300.m01 PF00232.13 β-glucosidase Cellulose
    GH3 Ep.00g040790.m01 PF00933.16 β-glucosidase Cellulose
    GH3 Ep.00g033660.m01 PF00933.16 β-glucosidase Cellulose
    GH3 Ep.00g062370.m01 PF00933.16 β-glucosidase Cellulose
    GH3 Ep.00g016320.m01 PF00933.16 β-glucosidase Cellulose
    GH3 Ep.00g005830.m01 PF00933.16 β-glucosidase Cellulose
    GH3 Ep.00g060380.m01 PF00933.16 β-glucosidase Cellulose
    GH3 Ep.00g019010.m01 PF00933.16 β-glucosidase Cellulose
    GH3 Ep.00g047030.m01 PF11220.3 β-glucosidase Cellulose
    GH39 Ep.00g072240.m01 PF01229.12 β-glucosidase Cellulose
    GH131 Ep.00g013780.m01 PF14099.1 β-glucosidase Cellulose
    GH128 Ep.00g075030.m01 PF18271.4 β-glucosidase Cellulose
    CE5 Ep.00g091760.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g080690.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g055890.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g028100.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g075580.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g001590.m01 PF04131.9 Cutinase Cutin
    CE5 Ep.00g021220.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g092920.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g052640.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g054680.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g024250.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g078410.m01 PF01083.17 Cutinase Cutin
    CE5 Ep.00g059400.m01 PF01083.17 Cutinase Cutin
    GH43 Ep.00g033080.m01 PF04616.9 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH43 Ep.00g079770.m01 PF04616.9 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH43 Ep.00g048420.m01 PF04616.9 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH43 Ep.00g013710.m01 PF04616.9 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH43 Ep.00g006660.m01 PF04616.9 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH43 Ep.00g014420.m01 PF00251.15 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH43 Ep.00g057540.m01 PF04616.9 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH43 Ep.00g033800.m01 PF04616.9 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH43 Ep.00g072460.m01 PF04616.9 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH43 Ep.00g017360.m01 PF04616.9 Endo-α-1,5-L-arabinanase Hemicellulose (arbinan)
    GH51 Ep.00g049020.m01 PF06964.7 α-L-arabinofuranosidase Hemicellulose (arbinan)
    GH54 Ep.00g038770.m01 PF09206.6 α-L-arabinofuranosidase Hemicellulose (arbinan)
    GH62 Ep.00g082120.m01 PF03664.8 α-L-arabinofuranosidase Hemicellulose (arbinan)
    GH10 Ep.00g006060.m01 PF00331.15 Endo-β-1,4-xylanase Hemicellulose (arbinan)
    GH10 Ep.00g084100.m01 PF12915.2 Endo-β-1,4-xylanase Hemicellulose (arbinan)
    GH10 Ep.00g062650.m01 PF00331.15 Endo-β-1,4-xylanase Hemicellulose (arbinan)
    GH11 Ep.00g028410.m01 PF00457.12 Endo-β-1,4-xylanase Hemicellulose (arbinan)
    GH11 Ep.00g025810.m01 PF00457.12 Endo-β-1,4-xylanase Hemicellulose (arbinan)
    GH12 Ep.00g082310.m01 PF01670.11 endo-xyloglucanase Hemicellulose (arbinan)
    GH12 Ep.00g067140.m01 PF01670.11 endo-xyloglucanase Hemicellulose (arbinan)
    CE1 Ep.00g065300.m01 PF00024.21 Acetylxylan esterase Hemicellulose (arbinan)
    CE1 Ep.00g014390.m01 PF00756.15 Acetylxylan esterase Hemicellulose (arbinan)
    CE2 Ep.00g034640.m01 PF13472.1 Acetylxylan esterase Hemicellulose (arbinan)
    GH93 Ep.00g013140.m01 PF13088.1 Exo-α-1,5-L-arabinofuranosidase Hemicellulose (arabinanose)
    GH142 Ep.00g065040.m01 PF06202.9 β-L-arabinofuranosidase Hemicellulose (arabinanose)
    GH146 Ep.00g090900.m01 PF07944.7 β-L-arabinofuranosidase Hemicellulose (arabinanose)
    GH146 Ep.00g035610.m01 PF07944.7 β-L-arabinofuranosidase Hemicellulose (arabinanose)
    AA5 Ep.00g066690.m01 PF07250 Galactose oxidase Hemicellulose (galactose)
    GH27 Ep.00g009190.m01 PF02065.13 α-galactosidase Hemicellulose (galactose)
    GH53 Ep.00g049270.m01 PF07745.8 endo-β-1,4-galactanase Hemicellulose (galactose)
    GH53 Ep.00g023560.m01 PF07745.8 endo-β-1,4-galactanase Hemicellulose (galactose)
    GH53 Ep.00g055170.m01 PF07745.8 endo-β-1,4-galactanase Hemicellulose (galactose)
    GH95 Ep.00g074320.m01 PF14498.1 α-L-galactosidase Hemicellulose (galactose)
    GH114 Ep.00g062230.m01 PF03537.8 endo-α-1,4-galactosaminidase Hemicellulose (galactose)
    GH115 Ep.00g071920.m01 PF03648.9 Xylan α-1,2-(4-O-methyl)-glucuronidase Hemicellulose (galactose)
    GH135 Ep.00g063890.m01 PF12138.3 endo-α-1,4-N-acetylgalactosaminidase Hemicellulose (galactose)
    GH6 Ep.00g027000.m01 PF00734.13 Endo-β-1,4-glucanase Hemicellulose (glucans)
    GH6 Ep.00g037220.m01 PF01341.12 Endo-β-1,4-glucanase Hemicellulose (glucans)
    GH7 Ep.00g074170.m01 PF00840.15 Endo-β-1,4-glucanase Hemicellulose (glucans)
    GH7 Ep.00g007280.m01 PF00840.15 Endo-β-1,4-glucanase Hemicellulose (glucans)
    GH16 Ep.00g045690.m01 PF00722.16 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH16 Ep.00g063100.m01 PF00722.16 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH16 Ep.00g078950.m01 PF00722.16 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH16 Ep.00g069650.m01 PF00722.16 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH16 Ep.00g032460.m01 PF00722.16 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH16 Ep.00g058160.m01 PF00722.16 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH17 Ep.00g008060.m01 PF00332.13 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH17 Ep.00g053040.m01 PF00332.13 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH17 Ep.00g065200.m01 PF00332.13 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH17 Ep.00g039010.m01 PF00332.13 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH45 Ep.00g068830.m01 PF02015.11 Endo-β-1,4-glucanase Hemicellulose (glucans)
    GH64 Ep.00g003160.m01 PF16483.8 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH81 Ep.00g082870.m01 PF10243.4 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH128 Ep.00g084710.m01 PF11790.3 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH128 Ep.00g012370.m01 PF11790.3 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH128 Ep.00g044710.m01 PF11790.3 Endo-β-1,3-glucanase Hemicellulose (glucans)
    GH132 Ep.00g092560.m01 PF03856.8 Exo-β-1,3-glucanase Hemicellulose (glucans)
    GH47 Ep.00g041820.m01 PF01532.15 Mannosyl-oligosaccharide α-1,2-mannosidase Hemicellulose (mannose)
    GH47 Ep.00g081180.m01 PF01532.15 Mannosyl-oligosaccharide α-1,2-mannosidase Hemicellulose (mannose)
    GH76 Ep.00g040900.m01 PF03663.9 α-1,6-mannanase Hemicellulose (mannose)
    GH76 Ep.00g079360.m01 PF03663.9 α-1,6-mannanase Hemicellulose (mannose)
    GH78 Ep.00g042190.m01 PF05592.6 α-L-rhamnosidase Hemicellulose (mannose)
    GH92 Ep.00g048360.m01 PF07971.7 α-1,4-mannosidase Hemicellulose (mannose)
    GH125 Ep.00g081770.m01 PF06824.6 exo-α-1,6-mannosidase Hemicellulose (mannose)
    GH125 Ep.00g060470.m01 PF06824.6 exo-α-1,6-mannosidase Hemicellulose (mannose)
    GH67 Ep.00g086010.m01 PF03648.9 Xylan α-1,2-(4-O-methyl)-glucuronidase Hemicellulose (uronic acids)
    GH79 Ep.00g044850.m01 PF06989.7 β-glucuronidase Hemicellulose (uronic acids)
    GH105 Ep.00g036690.m01 PF07470.8 D-4,5-unsaturated α-galacturonidase Hemicellulose (uronic acids)
    GH20 Ep.00g004780.m01 PF02838.10 β-N-acetylhexosaminidase Hexosamine
    AA1 Ep.00g003340.m01 PF07732 Multicopper oxidase Lignin
    AA1 Ep.00g086040.m01 PF07732 Multicopper oxidase Lignin
    AA1 Ep.00g062170.m01 PF07732 Multicopper oxidase Lignin
    AA1 Ep.00g032930.m01 PF07732 Multicopper oxidase Lignin
    AA2 Ep.00g062180.m01 PF00141 Peroxidase Lignin
    AA3 Ep.00g031600.m01 PF00732 Aryl alcohol oxidase Lignin
    AA3 Ep.00g041430.m01 PF00732 Aryl alcohol oxidase Lignin
    AA3 Ep.00g005910.m01 PF00732 Aryl alcohol oxidase Lignin
    AA3 Ep.00g046410.m01 PF00732 Aryl alcohol oxidase Lignin
    AA3 Ep.00g082320.m01 PF00732 Aryl alcohol oxidase Lignin
    AA3 Ep.00g046440.m01 PF00732 Aryl alcohol oxidase Lignin
    AA3 Ep.00g035860.m01 PF00732 Aryl alcohol oxidase Lignin
    AA3 Ep.00g041340.m01 PF00732 Aryl alcohol oxidase Lignin
    CE8 Ep.00g015010.m01 PF01095.14 Pectin methylesterase Pectin
    CE8 Ep.00g028320.m01 PF01095.14 Pectin methylesterase Pectin
    CE8 Ep.00g001980.m01 PF01095.14 Pectin methylesterase Pectin
    CE8 Ep.00g004170.m01 PF01095.14 Pectin methylesterase Pectin
    GH2 Ep.00g045820.m01 PF02837.13 β-galactosidase Pectin
    GH2 Ep.00g075560.m01 PF02837.13 β-galactosidase Pectin
    GH5 Ep.00g085260.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH5 Ep.00g060670.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH5 Ep.00g031200.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH5 Ep.00g037240.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH5 Ep.00g005540.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH5 Ep.00g074330.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH5 Ep.00g014300.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH5 Ep.00g081230.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH5 Ep.00g072670.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH5 Ep.00g065870.m01 PF00150.13 Endo-β-1,6-galactanase Pectin
    GH28 Ep.00g048380.m01 PF00295.12 Endo-polygalacturonase Pectin
    GH28 Ep.00g009600.m01 PF00295.12 Endo-polygalacturonase Pectin
    GH28 Ep.00g074850.m01 PF00295.12 Endo-polygalacturonase Pectin
    GH28 Ep.00g084490.m01 PF00295.12 Endo-polygalacturonase Pectin
    GH28 Ep.00g079230.m01 PF00295.12 Endo-polygalacturonase Pectin
    GH28 Ep.00g026980.m01 PF00295.12 Endo-polygalacturonase Pectin
    GH28 Ep.00g021250.m01 PF00295.12 Endo-polygalacturonase Pectin
    GH35 Ep.00g023860.m01 PF01301.14 β-galactosidase Pectin
    GH35 Ep.00g035150.m01 PF01301.14 β-galactosidase Pectin
    PL1 Ep.00g028120.m01 PF00544.14 Pectate lyase Pectin
    PL1 Ep.00g092740.m01 PF00544.14 Pectate lyase Pectin
    PL1 Ep.00g073980.m01 PF00544.14 Pectate lyase Pectin
    PL1 Ep.00g037200.m01 PF00544.14 Pectate lyase Pectin
    PL3 Ep.00g027020.m01 PF03211.8 Pectate lyase Pectin
    PL3 Ep.00g057340.m01 PF12708.2 Pectate lyase Pectin
    PL3 Ep.00g038030.m01 PF12708.2 Pectate lyase Pectin
    PL3 Ep.00g050710.m01 PF12708.2 Pectate lyase Pectin
    PL3 Ep.00g036650.m01 PF03211.8 Pectate lyase Pectin
    PL3 Ep.00g026880.m01 PF03211.8 Pectate lyase Pectin
    PL3 Ep.00g005980.m01 PF03211.8 Pectate lyase Pectin
    PL4 Ep.00g057920.m01 PF14686.1 Rhamnogalacturonan lyase Pectin
    PL4 Ep.00g092400.m01 PF14686.1 Rhamnogalacturonan lyase Pectin
    PL4 Ep.00g011470.m01 PF14686.1 Rhamnogalacturonan lyase Pectin
    PL4 Ep.00g070450.m01 PF14686.1 Rhamnogalacturonan lyase Pectin
    GH13 Ep.00g046360.m01 PF00128.19 Alpha amylase Starch
    GH13 Ep.00g010930.m01 PF00128.19 Alpha amylase Starch
    GH15 Ep.00g046350.m01 PF00723.16 Glucoamylase Starch

     | Show Table
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    Carbohydrate-active enzymes, known as CAZymes, are a broad category of enzymes that participate in the synthesis and degradation of carbohydrates and glycoconjugates [53],[54]. These enzymes are classified into six categories: glycoside hydrolases (GH), polysaccharide lyases (PL), carbohydrate esterases (CE), auxiliary activity (AA), glycosyltransferases (GT), and carbohydrate-binding modules (CBM) [46]. To determine the CAZymes in E. perseae and its closely related species, data from multiple sources was used, such as the blast description, gene ontology, EC number, PFAM domain, and the annotation results from the CAZy database [46]. A total of 190 CAZymes were detected in the secretome of E. perseae, as shown in Figure 4, Table S3, and Table S4. The CAZymes constituted 29.3% of the secretome of E. perseae, which was distributed in 67 CAZymes families (Table S3, Table S4). The number of CAZymes among the closely related species range from 178 to 240, with the lowest count observed in E. batatas and the greatest count observed in E. australis (Figure 4). Glycosidase hydrolase was found to be prominent among CAZymes in all species (Figure 4). In total, 78 CAZymes families were identified among all the seven species of Elsinoe, of which 51 were shared by all species. These families include 8 AA, 6 CE, 33 GH, 2 GT, and 3 PL families (Tables S3 and S4). Certain CAZymes exhibited restricted distribution, such as GH72 and GH81, which were exclusively detected in E. batatas and E. perseae, respectively (Tables S3 and S4). GH132 is exclusively present in E. perseae and E. australis; GH134 is exclusively found in E. arachidis and E. batatas; and GH135 is exclusively present in E. perseae and E. arachidis (Tables S3 and S4).

    Figure 4.  Distribution of CAZymes in seven species of Elsinoe. The numbers above the bar indicate the number of proteins in each group.

    The cell wall of the avocado is composed of various polysaccharides, including cellulose, hemicellulose, and pectin, which provide structural integrity and protection to the cells [55],[56]. To overcome these physical barriers, E. persea may secrete cell wall–degrading enzymes, such as cellulases and pectinases, to weaken and disrupt the structural integrity of the cell wall components. With the exception of the GT family, the GH, CE, PL, and AA families of CAZymes are classified as cell wall–degrading enzymes due to their involvement in the decomposition of several plant cell wall components [57],[58]. Cellulose is made up of a straight chain of numerous β-linked D-glucose units. The enzymes responsible for breaking down cellulose include exo-β-1,4-glucanases, endo-β-1,4-glucanases, β-1,4-glucosidases, cellobiose dehydrogenase, and lytic cellulose monooxygenase [59][61]. Out of the 190 CAZymes of E. perseae, 46 were identified as having the ability to break down cellulose. These include 8 glucooligosaccharide oxidases, 13 β-glucosidases, 1 α-glucosidase, 6 cellobiose dehydrogenases, and 18 lytic cellulose monooxygenases (Table 2). One α-glucosidase was identified within the GH31 family, while eight glucooligosaccharide oxidases were only discovered in the AA7 families (Table 2). A total of 18 lytic cellulose monooxygenases were identified in the AA9 (15) and AA16 (3) families, along with six cellobiose dehydrogenase enzymes in the AA3 (3) and AA8 (3) families (Table 2). Thirteen β-glucosidase enzymes were classified into four CAZymes families: GH1 (2), GH3 (8), GH39 (1), and GH131 (2) (Table 2).

    Hemicellulose is a significant constituent of the plant cell wall, comprising xyloglucans, xylans, mannans, glucomannans, and beta-(1-->3,1-->4)-glucans [62]. The primary enzymes responsible for breaking down hemicellulose are L-arabinanases, D-galactanases, D-mannanases, and D-xylanases [63]. In addition, numerous fungal species have been found to possess endo-β-1,4-glucanase that exhibit xyloglucanase activity [64]. Among the 190 CAZymes, 64 proteins were found to have the capability of degrading hemicellulose (Table 2). This group consisted of 60 GH proteins, 3 CE proteins, and 1 AA protein. The GH group consisted of 10 members from GH43, 6 members from GH16, 4 members from GH17, 3 members each from GH10, GH53, and GH128, 2 members each from GH11, GH12, GH47, GH76, GH125, and GH146, and one member each from 19 GH families (GH105, GH114, GH115, GH132, GH135, GH142, GH20, GH27, GH51, GH54, GH62, GH64, GH67, GH78, GH79, GH81, GH92, GH93, and GH95) and the AA5 family (Table 2). The predominant hemicellulose-degrading enzyme found in the secretome of E. perseae was endo-β-1,3-glucanase (15), which selectively breaks down chains of glucans. This enzyme is found in GH16 (6), GH17 (4), GH128 (3), GH64 (1), and GH81 (1). Interestingly, the GH81 enzyme was found only in E. perseae, suggesting a promising candidate for future research to understand the pathogenicity of E. perseae on avocado. It was also found that the secretome has 10 endo-α-1,5-L-arabinanase enzymes belonging to the CAZyme GH43, 3 α-L-arabinofuranosidase enzymes (GH51 & GH54), 3 β-L-arabinofuranosidase enzymes (GH142 & GH146), and 1 exo-α-1,5-L-arabinofuranosidase enzyme (GH93). These enzymes play a role in breaking down polysaccharides composed of arabinan molecules [65]. In addition, the secretome of E. perseae was revealed to have 27 glycosidic hydrolases that specifically break down the polysaccharide composed of xylan (7), galactose (8), mannose (8), and uronic acids (4) (Table 2) [66][69]. Furthermore, the secretome also contained three esterases that degrade hemicellulose, as indicated in Table 2.

    Pectin is a significant constituent of the primary cell walls found in all terrestrial plants. It consists of several polysaccharides that are rich in galacturonic acid [70]. All primary cell walls are believed to include three primary pectic polysaccharides: homogalacturonan, rhamnogalacturonan-I, and rhamnogalacturonan-II [71],[72]. The secretome of E. perseae contains 40 pectin-degrading enzymes, which are distributed among 11 CAZymes (Table 2). These include 10 endo-β-1,6-galactanase (GH5), 7 endo-polygalacturonase (GH28), 4 pectin methylesterase (CE8), 11 pectate lyase (PL1 and 3), 4 rhamnogalacturonan lyase (PL4), and 4 β-galactosidase (GH2 and 35) (Table 2). Furthermore, individuals belonging to auxiliary activity (AA) families were discovered to possess the capability to break down lignin (Table 2). This category consists of four AA1 multicopper oxidases, one AA2 peroxidase, and eight AA2 aryl alcohol oxidases (Table 2). Furthermore, the refined secretome includes 13 cutinases that break down cutin as well as three amylases that break down starch (Table 2). The role of cutinases in pathogenicity has been demonstrated for many phytopathogenic fungi [73].

    Multiple studies have demonstrated that plant pathogenic fungi secrete proteases that break down plant antimicrobial proteins, as well as protease inhibitors (PIs) to enhance their ability to cause disease [74]. The BlastP search conducted on the MEROPS database yielded the discovery of 49 potential proteases from the 654 refined secretome (Figure 5). These proteases were categorized into various categories based on their catalytic residues, as shown in Table S4. Serine proteases were the most prevalent among the proteases, with a total of 26. They were followed by metalloproteases (17), aspartic proteases (5), and carboxy proteases (1) (Table S4). The serine proteases encompassed families S8, S9, S10, S28, S41, and S51. Among them, the S8 family was identified as the most prevalent (Table S4). The categorization of metalloprotease members was based on their resemblance to the established members of the M6, M20, M28, and M36 families (Table S4). The number of proteases varies among closely related species, ranging from 31 in E. batatas to 56 in E. fawcettii and E. necatrix (Figure 5; Table S5). Comparative analysis showed that serine protease was predominant in E. perseae, E. necatrix, E. arachidis, and E. fawcettii, while metalloprotease was predominant in the remaining species (Figure 5).

    Figure 5.  Secretory proteases in Elsinoe species. The numbers above the bar indicate the number of proteins in each group.

    The combination of EffectorP and manual examination was used to select probable effector proteins with the following features: a signal peptide for secretion, absence of trans-membrane domains, relatively modest size, and high cysteine content [6],[75]. The study led to the discovery of 155 proteins as potential "effector" candidates (Figure 6A, Table S6). Effectors ranged in length from 88 to 395 aa (Table S6). Among these candidates, 43 (27.7%) were between 100 and 200 amino acids long, 56 (36.1%) were between 200 and 300 amino acids long, and 52 (33.5%) were between 300 and 400 amino acids long (Table S6). Four candidates (Ep.00g073550.m01, Ep.00g033060.m01, Ep.00g017120.m01, and Ep.00g059420.m01) were identified as having a length of less than 100 aa (Table S6). The identified effectors exhibited a range of cysteine residues, varying from 2 to 23. Among these putative effectors, 74.8% (116) had more than four cysteine residues, as shown in Table S6. Out of the total 155 potential effectors, EffectorP identified 55 as apoplastic effectors and 20 as cytoplasmic effectors (Figure 6A, Table S6). Of the 155 potential effectors, 80 proteins were selected manually as “putative effectors” based on their small size and number of cysteine residues (Figure 6A, Table S6). Of the 155 putative effector proteins, functional domains were found in 60 proteins (Figure 6A, Table S6). Of these, six apoplastic effectors (Ep.00g005690.m01, Ep.00g005690.m01, Ep.00g005750.m01, Ep.00g005790.m01, Ep.00g033060.m01, and Ep.00g085650.m01) possessed the fungal hydrophobin domain (PF01185) (Table S6). Four potential apoplastic effectors (Ep.00g010150.m01, Ep.00g027650.m01, Ep.00g072050.m01, and Ep.00g064760.m01) were found to have the LysM domain (PF01476) (Table S6). The pathogen effector domain known as putative necrosis-inducing factor (PF14856) was detected in three apoplastic putative effectors (Ep.00g021750.m01, Ep.00g067220.m01, and Ep.00g091030.m01). Additionally, the necrosis-inducing protein (NPP1) domain (PF05630) was observed in one protein (Ep.00g018600.m01) (Table S6). Orthologue analysis revealed that E. perseae had 118 effector clusters, of which 116 were shared among closely related species and 2 were in paralogs (Figure 6B). The inparalogs (Ep.00g005790.m01, Ep.00g005750.m01, Ep.00g018330.m01, and Ep.g018320.m01) exhibited over 90% sequence similarity, suggesting a recent duplication event. Remarkably, a total of 40 effectors were identified to be specific to E. perseae and do not share any similarity with proteins reported in other related species (Table S6). An examination of the functionality of these effectors may provide further understanding of the pathogenic nature of E. perseae on avocado. Notably, the research revealed that E. australis and E. batatas had 40 and 20 distinct protein clusters, respectively (not shown in this study). Examining these proteins could uncover the underlying biological mechanism responsible for scab disease in their specific host plants.

    Figure 6.  Effectors in E. perseae. A. Details of effectors in E. perseae B. Orthologue analysis of putative effectors in seven species of E. perseae. Clusters of orthologous genes were detected utilizing Orthovenn3 and displayed utilizing UpsetR. For the analysis, the e-value cutoff of 1 × 10−10 was utilized. Blue bars represent the number of clusters shared by Elsinoe species, represented by numbers shown above the bar. Line and dot connections between species that share orthologs.

    In order to find the homologs of the E. perseae secretory proteins that are linked to pathogenicity in other phytopathogens, we examined all secretory proteins in E. perseae, including all the potential effectors, against the PHI (Pathogen Host Interactions) database [49]. The protein sequences in the PHI database are categorized based on the outcomes of mutation experiments. These categories include, for instance, loss of pathogenicity, unaffected pathogenicity, reduced virulence, increased virulence, and effector (plant avirulence determinant). For instance, the "loss of pathogenicity" group comprises proteins that, when present in mutant strains, result in failure to induce disease in the host as compared to the natural type. According to the PHI annotation, out of the 654 secretome, 104 had PHI homologues, which included 61 CAZymes, 25 proteases, and 18 putative effectors (Table S7). Out of the 61 CAZymes, 26 were classified as "reduced virulence". These included 8 enzymes that break down cellulose (5 β-glucosidase, 1 cellobiose dehydrogenase, 1 lytic cellulose monooxygenase, and 1 α-glucosidase), 9 enzymes that break down hemicellulose (4 endo-β-1,4-xylanases, 2 endo-β-1,3-glucanases, 1 endo-β-1,4-glucanase, and 2 exo-α-1,6-mannosidases), 8 enzymes that break down pectin (3 endo-β-1,6-galactanases and 5 pectate lyases), and 1 enzyme that breaks down lignin, specifically encoding an aryl alcohol oxidase (Table S7). The counterparts of these enzymes were documented to contribute to the virulence of many phytopathogenic fungi. For example, endo-β-1,4-xylanase has been demonstrated to play a role in the pathogenicity of certain fungal diseases, such as Verticillium dahlia, Ustilago maydis, and Valsa mali [76],[77].

    Furthermore, according to the PHI database, two CAZymes were identified as "effector_(plant_avirulence_determinant)", specifically two lytic cellulose monooxygenases. Chen et al. demonstrated that the homolog of lytic cellulose monooxygenase in Magnaporthe oryzae (MoCDIP) triggered cellular apoptosis upon expression in rice plant cells [78]. Furthermore, a Podosphaera xanthii gene called PHEC27213, which encodes a lytic cellulose monooxygenase, was found to inhibit the immune response triggered by chitin in the cucurbit host [79]. Out of the 61 CAZymes, four were categorized as "increased virulence (hypervirulence)", and 29 were categorized as "unaffected pathogenicity" (Table S7). Out of the 25 proteases with PHI partners, 1 cysteine protease was designated as "loss of pathogenicity", 1 was designated as "effector_(plant_avirulence_determinant)", and 11 were designated as "reduced virulence" (Table S7). Furthermore, two proteases were designated as "unaffected pathogenicity" according to the PHI annotation (Table S7). Among the putative effectors, we discovered PHI partners for 18 proteins, 5 of which were designated as "effector_(plant_avirulence_determinant)". Among them, two proteins (Ep.00g072050.m01, Ep.00g010150.m01) include the LysM domain, whereas one protein (Ep.00g027520.m01) contains the cerato-platanin domain (Table S7). Multiple studies have demonstrated that proteins containing the LysM domain operate as virulence factors in many phytopathogenic fungi by inhibiting the immune response triggered by chitin in host plants [80],[81]. Cerato-platanins are a collection of small proteins that are rich in cysteine and are released by certain plant pathogenic fungi to facilitate virulence on the host plant [82]. Three potential effectors were classified as "reduced virulence" (Table S7). Among these, two genes encode a fungal hydrophobin and a peroxidase. These proteins' homologs have been demonstrated to be essential for the pathogenicity of numerous fungal infections [83].

    The present study employed a bioinformatics pipeline to thoroughly describe the putative secretory proteins of E. perseae. A total of 190 carbohydrate-active enzymes (CAZymes), 90 proteases, and 155 potential effector proteins were detected in the secretome. The investigation showed that E. perseae had multiple families of enzymes capable of breaking down cellulose, hemicellulose, pectin, and lignin, as well as numerous proteases to defeat the initial defense mechanisms of plants. In addition, the comparative study showed that E. perseae had 41 putative effectors that were specific to its species, as well as many putative virulence factors. The present work will serve as a potential resource for research focused on comprehending the pathogenicity mechanism in the interaction between E. perseae and avocado.

    The datasets analyzed in this study can be found in online repositories. The names of the repositories and accession numbers can be found in the article (Table 1). Additional data generated in this study can be found in the supplementary material.

    The author extends his appreciation to the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia, for funding this research work (Project number KFU242183).

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.



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