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ATP Synthase: Structure,Abstract: Functionlet F Denote a Eld and and Let Inhibitionv Denote a Vector Space Over F with Nite Positive Dimension

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ATP Synthase: Structure,Abstract: Functionlet F Denote a Eld and and Let Inhibitionv Denote a Vector Space Over F with Nite Positive Dimension Spec. Matrices 2019; 7:1–19 Research Article Open Access Kazumasa Nomura* and Paul Terwilliger BioMol Concepts 2019; 10: 1–10 Self-dual Leonard pairs Research Article Open Access https://doi.org/10.1515/spma-2019-0001 Prashant Neupane*, Sudina Bhuju, Nita Thapa,Received Hitesh May 8, 2018; Kumar accepted Bhattarai September 22, 2018 ATP Synthase: Structure,Abstract: FunctionLet F denote a eld and and let InhibitionV denote a vector 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 irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which https://doi.org/10.1515/bmc-2019-0001 there exists an automorphism of the endomorphism algebra of V that swaps A and A∗. Such an automorphism phosphate (Pi), along with considerable release of energy. received September 18, 2018; accepted December 21, 2018. is unique, and called the duality A A∗. In the present paper we give a comprehensive description of this ADP can absorb energy and regain↔ the group to regenerate duality. In particular, we display an invertible F-linear map T on V such that the map X TXT− is the duality Abstract: Oxidative phosphorylation is carried out by an ATP molecule to maintain constant ATP concentration. → A A∗. We express T as a polynomial in A and A∗. We describe how T acts on ags, decompositions, five complexes, which are the sites for electron transport↔ Other than supporting almost all the cellular and ATP synthesis. Among those, Complex V (also knownand 24 basesfunctions for V that. require energy, ATP also works as a as the F F ATP Synthase or ATPase) is responsible for coenzyme during phosphorylation reactions. Besides, ATP 1 0 Keywords: Leonard pair, tridiagonal matrix, self-dual the generation of ATP through phosphorylation of ADP has crucial role in RNA and DNA synthesis and in amino by using electrochemical energy generated by protonClassication: acid activation17B37,15A21 during protein synthesis. Khakh and gradient across the inner membrane of mitochondria. A Burnstock, established that ATP also serves as a critical multi subunit structure that works like a pump functions signaling molecule that allows inter and intracellular along the proton gradient across the membranes which communications. The ubiquitous distribution of ATP not only results in ATP synthesis and breakdown, but alsoIntroduction allows for signaling functions that have a uniquely broad facilitates electron transport. Since ATP is the major energy influence on physiological functioning [1]. currency in all living cells, its synthesis and functionLet haveF denoteEstimates a eld andshow let thatV denotethe normal a vector body spaceuses 40 over kg ofF with nite positive dimension. We consider a widely been studied over the last few decades uncoveringpair A , AATP∗ of on diagonalizable a daily basis; henceF-linear ATP maps production on V, eachis one of of which the acts on an eigenbasis for the other one in an several aspects of ATP synthase. This review intendsirreducible to most tridiagonal frequent processes fashion. Suchoccurring a pair in is the called body a Leonard[2]. The pair (see [13, Denition 1.1]). The Leonard pair summarize the structure, function and inhibition of the aggregate amount of ATP in the human body is about 0.1 A, A∗ is said to be self-dual whenever there exists an automorphism of the endomorphism algebra of V that ATP synthase. Mole. The energy utilized every day by an adult requires swaps A and A∗. In this case such an automorphism is unique, and called the duality A A∗. the hydrolysis of 100 to 150 Moles of ATP. This implies ↔ Keywords: ATP, ATP Synthase, Electron Transport Chain,The literatureevery ATP molecule contains must many be examples reused 1000 of self-dualor more multiple Leonard pairs. For instance (i) the Leonard pair associ- Proton motive force, Rotational Catalysis, ATP synthaseated withtimes an irreducible in a day [3]. moduleATP cannot for be the stored Terwilliger and so its algebra synthesis of the hypercube (see [4, Corollaries 6.8, 8.5]); (ii) a Inhibition. Leonardis pair closely of Krawtchouk linked to its type consumption. (see [10, Denition ATP is 6.1]);produced (iii) the Leonard pair associated with an irreducible module forunder the aerobic Terwilliger conditions algebra through of a distance-regular glycolysis, citric graph acid that has a spin model in the Bose-Mesner alge- bra (seecycle/oxidative [1, Theorem], [3,phosphorylation Theorems 4.1, 5.5]);and (iv)beta-oxidation an appropriately normalized totally bipartite Leonard pair ATP: The Fuel for Life (see [11,and Lemma under 14.8]); anaerobic (v) the condition Leonard pairthrough consisting fermentation, of any two of a modular Leonard triple A, B, C (see [2, Denitionphotophosphorylation 1.4]); (vi) the Leonard and pair replenishment consisting of areactions pair of opposite generators for the q-tetrahedron alge- Often referred as “molecular currency” for intracellularbra, acting catalyzed on an evaluationby nucleoside module diphosphate (see [5, kinase Proposition [4]. 9.2]). The example (i) is a special case of (ii), and the energy transfer, Adenosine Triphosphate (ATP) functions examples (iii), (iv) are special cases of (v). as a chemical fuel by powering many organic processes Let A, A∗ denote a Leonard pair on V. We can determine whether A, A∗ is self-dual in the following way. of life. ATP generation is the principle energy generating ATP Synthase d procedure found in all forms of life. ATP is the fuel forBy the [13, Lemma 1.3] each eigenspace of A, A∗ has dimension one. Let θi i= denote an ordering of the eigen- { } d operation of almost all metabolic pathways of thevalues cell. ofATPA. Foris synthesized ≤ i ≤ d letfromvi denoteits precursor, a θi-eigenvector ADP, by ATP for A. The ordering θi i= is said to be standard d { } d In an ATP molecule, two high-energy phosphate bonds,whenever synthases.A∗ acts onThese the basisenzymesvi arei= infound an irreduciblein the cristae tridiagonal and fashion. If the ordering θi i= is standard d { } { } called phosphoanhydride bonds, are responsible forthen high the the ordering inner membraneθd−i i= is alsoof mitochondria, standard, and the no furtherthylakoid ordering is standard. Similar comments apply to d { } d energy content. Hydrolysis of the third phosphate Agroup∗. Let membraneθ denote of chloroplasts, a standard orderingand the plasma of the eigenvaluesmembrane of of A. Then A, A∗ is self-dual if and only if θ { i}i= { i}i= produces adenosine diphosphate (ADP) and inorganicis a standard bacteria ordering [5]. Usually, of the there eigenvalues is a general of understandingA∗ (see [7, Proposition that 8.7]). ATP generation occurs in mitochondria. However, in the case of bacteria and archaea that lack mitochondria, ATP *Corresponding author: Prashant Neupane, Kathmandu University, synthase is found in their plasma membrane. Additionally, Dhulikhel, Nepal, E-mail: [email protected]*Corresponding ATP synthases Author: Kazumasaare licensed Nomura: to inhabitTokyo the Medical chloroplast and Dental of University, Ichikawa, 272-0827,Japan, Sudina Bhuju: Department of Biotechnology, Kathmandu UniversityE-mail: [email protected] cells. The structure and procedure of ATP synthesis is Nita Thapa: Department of Biotechnology, Kathmandu UniversityPaul Terwilliger:similar inDepartment all three locations of Mathematics, except that University light energy of Wisconsin, excites Madison, WI53706, USA, E-mail: Hitesh Kumar Bhattarai: Department of Biotechnology, [email protected] electrons enabling transmembrane movement of H+ ions in University Journal xyz 2017; 1 (2): 122–135 Open Access. © 2019 Prashant Neupane 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-NonCommercial NoDerivatives 4.0 License. Attribution alone 4.0 License. The First Decade (1964-1972) Research Article Max Musterman, Paul Placeholder What Is So Different About Neuroenhancement? Was ist so anders am Neuroenhancement? Pharmacological and Mental Self-transformation in Ethic Comparison Pharmakologische und mentale Selbstveränderung im ethischen Vergleich https://doi.org/10.1515/xyz-2017-0010 received February 9, 2013; accepted March 25, 2013; published online July 12, 2014 Abstract: In the concept of the aesthetic formation of knowledge and its as soon as possible and success-oriented application, insights and profits without the reference to the arguments developed around 1900. The main investigation also includes the period between the entry into force and the presentation in its current version. Their function as part of the literary portrayal and narrative technique. Keywords: Function, transmission, investigation, principal, period Dedicated to Paul Placeholder 1 Studies and Investigations The main investigation also includes the period between the entry into force and the presentation in its current version. Their function as part of the literary por- trayal and narrative technique. *Max Musterman: Institute of Marine Biology, National Taiwan Ocean University, 2 Pei-Ning Road Keelung 20224, Taiwan (R.O.C), e-mail: [email protected] Paul Placeholder: Institute of Marine Biology, National Taiwan Ocean University, 2 Pei-Ning Road Keelung 20224, Taiwan (R.O.C), e-mail: [email protected] Open
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