Spec. Matrices 2019; 7:1–19 Research Article Open Access Kazumasa Nomura* and Paul Terwilliger Green Process Synth 2019; 8: 416–420 Self-dual Leonard pairs Yingjie Bu, Bassam Alkotaini, Bipinchandra K. Salunke, Aarti R. Deshmukh, Pathikrit Saha and Beom Soo Kim* https://doi.org/10.1515/spma-2019-0001 Received May 8, 2018; accepted September 22, 2018 Direct ethanol productionAbstract: Let F denote from a eld and cellulose let V denote a vector by space over F with nite positive dimension. Consider a pair A, A∗ of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an consortium of Trichodermairreducible tridiagonal fashion. reesei Such a and pair is called Candida a Leonard pair. We consider the self-dual case in which molischiana there exists an automorphism of the endomorphism algebra of V that swaps A and A∗. Such an automorphism is unique, and called the duality A A∗. In the present paper we give a comprehensive description of this ↔ duality. In particular, we display an invertible F-linear map T on V such that the map X TXT− is the duality https://doi.org/10.1515/gps-2019-0009 waste are attractive alternatives to fossil fuels [1]. New → A A∗. We express T as a polynomial in A and A∗. We describe how T acts on ags, decompositions, Received August 13, 2018; accepted October 09,↔ 2018. technologies to convert plant biomass into alternative and 24 bases for V. biofuels, such as bioethanol, are being developed [2]. Abstract: Industrial cellulosic ethanol production is a Fermentation-derived ethanol can be produced from challenge due to the high cost of cellulasesKeywords: for Leonardhydroly- pair, tridiagonal matrix, self-dual sugar, starch, or lignocellulosic biomass. Sugar and sis when lignocellulosic materials are used as feedstock. starch-based feedstocks are currently predominant at the In this study, direct ethanol productionClassication: from cellulose17B37,15A21 industrial level and are economically advantageous [3]. was performed by consortium of Trichoderma reesei Lignocellulosic materials are important for the production and Candida molischiana. Cellulose was hydrolyzed by of bioethanol due to their abundance [4]. Industrial a fully enzymatic saccharification process using Tricho- cellulosic ethanol production is still a challenge because derma reesei cellulases. The produced Introductionreducing sugar of the high processing cost, for example, the high cost was further utilized by Candida molischiana for ethanol of cellulase for hydrolysis after using lignocellulosic production. Because the optimal temperatureLet F denote for the a eldcel- and let V denote a vector space over F with nite positive dimension. We consider a materials as feedstock [5-7]. During enzymatic hydrolysis, lulase system is approximately 50°C,pair the Aeffect, A∗ of tempe diagonalizable- F-linear maps on V, each of which acts on an eigenbasis for the other one in an cellulase converts cellulose into soluble sugars. rature rise from 30°C to 50°C on celluloseirreducible hydrolysis tridiagonal was fashion. Such a pair is called a Leonard pair (see [13, Denition 1.1]). The Leonard pair Cellulose is a linear polymer of glucose units, which investigated. The results showed thatA the, A temperature∗ is said to berise self-dual whenever there exists an automorphism of the endomorphism algebra of V that are hydrolyzed by endoglucanase, cellobiohydrolase, and from 30°C to 50°C after 36 h of cultivationswaps wasA andthe bestA∗. for In this case such an automorphism is unique, and called the duality A A∗. β-glucosidases [8,9]. Some fungal strains produce large ↔ reducing sugar and glucose production. Under these con- The literature containsamounts many of cellulase. examples For of more self-dual than five Leonard decades, pairs. different For instance (i) the Leonard pair associ- ditions, the maximum concentrations of reducing sugar ated with an irreducibleTrichoderma module for reesei the Terwilliger strains have algebra been screened of the hypercube for their (see [4, Corollaries 6.8, 8.5]); (ii) a and glucose produced by T. reesei were 8.0 g/L and 4.6 g/L Leonard pair of Krawtchoukpotential type production (see [10, of Denition cellulases 6.1]); [9]. T. (iii) reesei the Leonardcellulases pair associated with an irreducible at 60 h, respectively. The maximum production of ethanol module for the Terwilligerare widely algebra used of a distance-regularto hydrolyze lignocellulosic graph that hasbiomass a spin model in the Bose-Mesner alge- by C. molischiana was 3.0 g/L after 120 h. bra (see [1, Theorem],to [3, fermentable Theorems 4.1,sugars. 5.5]); These (iv) an enzymes appropriately synergistically normalized totally bipartite Leonard pair convert cellulose because the action of one enzyme is Keywords: bioethanol; Candida molischiana(see [11, Lemma; cellulose 14.8]); (v) the Leonard pair consisting of any two of a modular Leonard triple A, B, C (see [2, utilized as a substrate by another enzyme, which leads to hydrolysis; Trichoderma reesei Denition 1.4]); (vi) the Leonard pair consisting of a pair of opposite generators for the q-tetrahedron alge- bra, acting on an evaluationthe production module of (see glucose [5, Proposition [10]. 9.2]). The example (i) is a special case of (ii), and the The existing ethanol production process includes a examples (iii), (iv) are special cases of (v). pretreatment step in which the sugar in the raw material Let A, A∗ denote a Leonard pair on V. We can determine whether A, A∗ is self-dual in the following way. is separated and fed to the fermenter as a substrate. 1 Introduction d By [13, Lemma 1.3] eachCandida eigenspace molischiana of A, A is∗ hasonedimension of the few one. yeast Let speciesθi i= denote an ordering of the eigen- { } d The ongoing global dependence valueson fossil of Afuels. For has ≤ icapable≤ d let ofv degradingi denote acellobioseθi-eigenvector into glucose for A [11].. The Because ordering θi i= is said to be standard d { } d produced serious energy crises wheneverand environmentalA∗ acts on theC. molischiana basis vi i= isin highly an irreducible resistant to tridiagonal ethanol and fashion. heat, If the ordering θi i= is standard d { } { } problems. Renewable energy sourcesthen such the orderingas organicθ d−thisi i= yeastis also is likely standard, to be andmore no effective further when ordering introduced is standard. Similar comments apply to d { } d A∗. Let θ denotein a standardindustrial ordering cellulose of ethanol the eigenvalues plants [12]. of AOne. Then mutantA, A ∗ is self-dual if and only if θ { i}i= { i}i= is a standard orderingstrain of the of eigenvalues C. molischiana of Acould∗ (see tolerate [7, Proposition up to 4% 8.7]).ethanol * Corresponding author: Beom Soo Kim, Department of Chemical in the medium when grown at 45°C for 48 h [12]. Recently, Engineering, Chungbuk National University, Cheongju, Chungbuk the highly efficient biodegradation of cellulosic biomass 28644, Republic of Korea, e-mail: [email protected]; through microbial co-cultures or complex communities Tel.: +82 43 261 2372; Fax: +82 43 269 2370 has been proposed [4]. In this study, T. reesei and C. Yingjie Bu, Bassam Alkotaini, Bipinchandra*Corresponding K. Salunke, Author: Kazumasa Nomura: Tokyo Medical and Dental University, Ichikawa, 272-0827,Japan, Aarti R. Deshmukh and Pathikrit Saha, DepartmentE-mail: of [email protected] Chemical molischiana were sequentially cultured to produce Engineering, Chungbuk National University,Paul Cheongju, Terwilliger: ChungbukDepartment ethanol of Mathematics, from cellulose University without of acidic, Wisconsin, ionic, Madison, or chemical WI53706, USA, E-mail: 28644, Republic of Korea [email protected] pretreatments. First, the effect of temperature changes on Open Access. © 2019 Bu et al., published byOpen De Gruyter. Access. ©This2019 work Kazumasa is licensed Nomura under and the Paul Creative Terwilliger, Commons published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Attribution alone 4.0 License. Y. Bu et al.: Direct ethanol production from cellulose 417 cellulosic hydrolysis by T. reesei from 30°C to 50°C were to increasing to 50°C for reducing sugar and glucose investigated. The reducing sugars and glucose obtained production. The initial pH was adjusted to 7 using 3 N were further used for ethanol production by C. molischiana NaOH. T. reesei was grown in a 250 mL flask with 100 mL to develop a fully enzymatic hydrolysis process from of chemically defined medium containing cellulose. cellulose. 2.4 Analytical methods 2 Materials and methods Reducing sugar was measured by the dinitrosalicylic acid (DNS) method as described previously [13]. Glucose 2.1 Microorganisms concentration was measured using an Asan set glucose kit (Asan Pharm, Co. Ltd., Seoul, Korea). Ethanol The fungus T. reesei RUT-C30 (KCTC 6968) and yeast C. concentration was measured by HPLC system (YL 9100, molischiana (ATCC 2516) were purchased from Korean Young-Lin, Inc., Anyang, Korea) using a Biorad Aminex Collection for Type Cultures (Daejeon, Korea) and hpx-87h column (Hercules, CA, USA) with a refractive American Type Culture Collection (Manassas, VA, USA), index detector. respectively. 2.2 Media and cultivation 3 Results and discussion The T. reesei stock culture was aseptically grown on potato Cellulases from T. reesei fungi have been used in the dextrose agar medium consisting of 24 g/L potato dextrose biofuels industry to degrade vegetable feedstocks into broth and 20 g/L agar and incubated at 30°C for 7 days substrates that can be used for other processes. Another until sporulation was sufficient. T. reesei was cultivated in promising microorganism for bioethanol production is chemically defined medium consisting of (per L): glucose C. molischiana, which produces ethanol from substrates 10 g, (NH4)2SO4 1.4 g, KH2PO4 2 g, CaCl2 0.3 g, MgSO4·7H2O containing glucose, fructose, and sucrose [11].