Polar Form of Complex Numbers

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Crystal structures of the apo form of b-fructofuranosidase from Bifidobacterium longum and its complex with fructose Anna Bujacz, Marzena Jedrzejczak-Krzepkowska, Stanislaw Bielecki, Izabela Redzynia and Grzegorz Bujacz Institute of Technical Biochemistry, Faculty of Biotechnology and Food Sciences, Technical University of Lodz, Poland Introduction Bifidobacteria are found in human and animal gastro- intestinal tracts, as well as in the oral cavity and the vagina [1]. They are among the first bacteria that colo- nize the sterile digestive system of newborns and they become predominant micro-organisms ($ 95% of the colonic flora) in breast-fed infants [2]. In infants, the most frequently detected bifidobacteria species are Bifidobacterium breve, Bifidobacterium infan- tis, Bifidobacterium bifidum and Bifidobacterium longum. The latter one also inhabits the intestines of adults, despite the fact that the composition of bifidobacterial species changes and the amount of bifidobacteria decreases with age [3–6]. They are Gram-positive, nons- porulating and nonmotile rods, classified as lactic acid bacteria, due to their ability to anaerobically ferment carbohydrates [7,8]. These bacteria are known as micro- Keywords b-fructofuranosidase; Bifidobacterium longum; crystal structure; glycoside hydrolase family GH32; lactic acid bacteria Correspondence A. Bujacz, Institute of Technical Biochemistry, Faculty of Biotechnology and Food Sciences, Technical University of Lodz, Stefanowskiego 4 ⁄ 10, 90-924 Lodz, Poland Fax: 48 42 6366618 Tel: 48 42 6313494 E-mail: anna.bujacz@p.lodz.pl (Received 13 January 2011, revised 25 February 2011, accepted 15 March 2011) doi:10.1111/j.1742-4658.2011.08098.x We solved the 1.8 A ˚ crystal structure of b-fructofuranosidase from Bifido- bacterium longum KN29.1 – a unique enzyme that allows these probiotic bacteria to function in the human digestive system. The sequence of b-fruc- tofuranosidase classifies it as belonging to the glycoside hydrolase family 32 (GH32). GH32 enzymes show a wide range of substrate specificity and different functions in various organisms. All enzymes from this family share a similar fold, containing two domains: an N-terminal five-bladed b-propeller and a C-terminal b-sandwich module. The active site is located in the centre of the b-propeller domain, in the bottom of a ‘funnel’. The binding site, )1, responsible for tight fructose binding, is highly conserved among the GH32 enzymes. Bifidobacterium longum KN29.1 b-fructofura- nosidase has a 35-residue elongation of the N-terminus containing a five- turn a-helix, which distinguishes it from the other known members of the GH32 family. This new structural element could be one of the functional modifications of the enzyme that allows the bacteria to act in a human digestive system. We also solved the 1.8 A ˚ crystal structure of the b-fruc- tofuranosidase complex with b- D-fructose, a hydrolysis product obtained by soaking apo crystal in raffinose. Database Coordinates and structure factors have been deposited in the Protein Data Bank under acces- sion codes: 3PIG and 3PIJ Structured digital abstract l b-fructofuranosidase binds to b-fructofuranosidase by x-ray crystallography (View interaction) Abbreviation GH32, glycoside hydrolase family 32. 1728 FEBS Journal 278 (2011) 1728–1744 ª 2011 The Authors Journal compilation ª 2011 FEBS organisms that Polar Form of Complex Numbers Polar Form of Complex Numbers By: OpenStaxCollege “God made the integers; all else is the work of man.” This rather famous quote by nineteenth-century German mathematician Leopold Kronecker sets the stage for this section on the polar form of a complex number Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras, Descartes, De Moivre, Euler, Gauss, and others Complex numbers answered questions that for centuries had puzzled the greatest minds in science We first encountered complex numbers in Complex Numbers In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre’s Theorem Plotting Complex Numbers in the Complex Plane Plotting a complex number a + bi is similar to plotting a real number, except that the horizontal axis represents the real part of the number, a, and the vertical axis represents the imaginary part of the number, bi How To Given a complex number a + bi, plot it in the complex plane Label the horizontal axis as the real axis and the vertical axis as the imaginary axis Plot the point in the complex plane by moving a units in the horizontal direction and b units in the vertical direction Plotting a Complex Number in the Complex Plane Plot the complex number − 3i in the complex plane 1/25 Polar Form of Complex Numbers From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction See [link] Try It Plot the point + 5i in the complex plane 2/25 Polar Form of Complex Numbers Finding the Absolute Value of a Complex Number The first step toward working with a complex number in polar form is to find the absolute value The absolute value of a complex number is the same as its magnitude, or |z| It measures the distance from the origin to a point in the plane For example, the graph of z = + 4i, in [link], shows |z| A General Note Absolute Value of a Complex Number Given z = x + yi, a complex number, the absolute value of z is defined as |z| = √ x + y It is the distance from the origin to the point (x, y) Notice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin, (0, 0) Finding the Absolute Value of a Complex Number with a Radical Find the absolute value of z = √5 − i Using the formula, we have 3/25 Polar Form of Complex Numbers |z| = √ x + y 2 |z| = √(√5) + ( − 1) |z| = √ + |z| = √ See [link] Try It Find the absolute value of the complex number z = 12 − 5i 13 Finding the Absolute Value of a Complex Number Given z = − 4i, find |z| Using the formula, we have 4/25 Polar Form of Complex Numbers |z| = √ x + y 2 |z| = √(3) + ( − 4) |z| = √9 + 16 |z| = √25 |z| = The absolute value z is See [link] Try It Given z = − 7i, find |z| |z| = √50 = 5√2 Writing Complex Numbers in Polar Form The polar form of a complex number expresses a number in terms of an angle θ and its distance from the origin r Given a complex number in rectangular form expressed as z = x + yi, we use the same conversion formulas as we to write the number in trigonometric form: 5/25 Polar Form of Complex Numbers x = rcos θ y = rsin θ r = √x2 + y2 We review these relationships in [link] We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point (x, y) The modulus, then, is the same as r, the radius in polar form We use θ to indicate the angle of direction (just as with polar coordinates) Substituting, we have z = x + yi z = rcos θ + (rsin θ)i z = r(cos θ + isin θ) a general note label Polar Form of a Complex Number Writing a complex number in polar form involves the following conversion formulas: x = rcos θ y = rsin θ r = √x2 + y2 6/25 Polar Form of Complex Numbers Making a direct substitution, we have z = x + yi z = (rcos θ) + i(rsin θ) z = r(cos θ + isin θ) where r is the modulus and θ is the argument We often use the abbreviation rcis θ to represent r(cos θ + isin θ) Expressing a Complex Number Using Polar Coordinates Express the complex number 4i using polar coordinates On the complex plane, the number z = 4i is the same as z = + 4i Writing it in polar form, we have to calculate r first r = √x2 + y2 r = √02 + 42 r = √16 r=4 π Next, we look at x If x = rcos θ, and x = 0, then θ = In polar coordinates, the ( ( ) + isin( )) or 4cis( ) See complex number z = + 4i can be written as z = cos π π π [link] 7/25 Polar Form of Complex Numbers Try It Express z = 3i as r cis θ in polar form ( ( ) + isin( )) z = cos π π Finding the Polar Form of a Complex Number Find the polar form of − + 4i First, find the value of r r = √x2 + y2 r = ( − 4) + (42) √ r = √32 r = 4√2 Find the angle θ using the ...Logical Form of Complex Sentences in Task-Oriented Dialogues* Cecile T. Balkanski Harvard University, Aiken Computation Lab Cambridge, MA 02138 Introduction Although most NLP researchers agree that a level of "logical form" is a necessary step toward the goal of rep- resenting the meaning of a sentence, few people agree on the content and form of this level of representation. An even smaller number of people have considered the com- plex action sentences that are often expressed in task- oriented dialogues. Most existing logical form represen- tations have been developed for single-clause sentences that express assertions about properties or actual actions and in which time is not a main concern. In contrast, utterances in task-oriented dialogues often express unre- alized actions, e.g., (la), multiple actions and relations between them, e.g., (lb), and temporal information, e.g., (lc): (1) a. What about rereading the Operations manual? b. By getting the key and unlocking the gate, you get ten points. c. When the red fight goes off, push the handle. In the following sections, I discuss the issues that arise in defining the logical form of these three types of sen- tences. The Davidsonian treatment of action sentences is the most appropriate for my purposes because it treats actions as individuals [7]. For example, the logical form of "Jones buttered the toast" is a three place predicate, including an argument position for the action being de- scribed, i.e., 3x butter(jones, toast, x). The presence of the action variable makes it possible to represent op- tional modifiers as predications of actions and to refer to actions in subsequent discourse. Furthermore, and more importantly for the present purpose, it facilitates the representation of sentences about multiple actions and relations between them. Unrealized-action sentences A Davidsonian logical form of sentence (la), namely 3x reread(us, manual, x), makes the claim that there exists a particular action x. But this is not the intended meaning of the sentence. Instead, this sentence con- cerns a hypothetical action. The same problem arises with sentences (lb) and (lc) which state how typical actions are related or when to perform a future action. Apparently, Davidson did not have these types of action in mind when suggesting his theory of logical form. In fact, a closer look at the literature shows that the problem of representing action sentences that do *This research has been supported by U S West Advanced Technologies, by the Air Force Office of Scientific Research under Contract No.AFOSR-89-0273, and by an IBM Grad- uate Fellowship. 331 not make claims about actions that have or are oc- curring (i.e., actual actions) has been virtually ignored. Hobbs, who also adopts a Davidsonian treatment of ac- tion sentences, is one notable exception [11]. His "Pla- tonic universe" contains everything that can be spoken of and the predicate Exist is used to make statements about the existence in the actual universe of individu- als in the Platonic universe. For example, the formula Exists(x) Arun'(x, john) says that the action of John's running exists NANO EXPRESS Temperature Sensitive Nanocapsule of Complex Structural Form for Methane Storage E. I. Volkova • M. V. Suyetin • A. V. Vakhrushev Received: 11 September 2009 / Accepted: 5 October 2009 / Published online: 16 October 2009 Ó to the authors 2009 Abstract The processes of methane adsorption, storage and desorption by the nanocapsule are investigated with molecular-dynamic modeling method. The specific nano- capsule shape defines its functioning uniqueness: methane is adsorbed under 40 MPa and at normal temperature with further blocking of methane molecules the K@C60 1? en- dohedral complex in the nanocapsule by external electric field, the storage is performed under normal external con- ditions, and methane desorption is performed at 350 K. The methane content in the nanocapsule during storage reaches 11.09 mass%. The nanocapsule consists of tree parts: storage chamber, junction and blocking chamber. The storage chamber comprises the nanotube (20,20). The blocking chamber is a short nanotube (20,20) with three holes. The junction consists of the nanotube (10,10) and nanotube (8,8); moreover, the nanotube (8,8) is connected with the storage chamber and nanotube (10,10) with the blocking chamber. The blocking chamber is opened and closed by the transfer of the K@C 60 1? endohedral complex under electrostatic field action. Keywords Methane storage Á Nanocapsule Á Molecular dynamics Introduction The bucky shuttle [1, 2] being the combination of nanosize carbon structures—fullerene [3] and nanotube [4], has many possible applications: nanoscale storage cells [5], devices for directed medicine transfer [6] and containers for effective and safe gas storage [7–13]. Nanosize con- tainers and capsules of various shapes that allow reaching a higher safety level and mass content of gas stored have been investigated for a number of years [11–13]. The engineering of nanostructured carbon opens the ways for the production of nanocapsules of complex structural shapes [14–16]. In this work, the processes of methane molecule adsorption, storage and desorption by the nanocapsule are investigated with molecular-dynamic modeling method. The nanocapsule-specific structure defines its adsorption qualities: at the storage stage under normal conditions, the nanocapsule contains the amount of methane that was adsorbed at normal temperature and under 40 MPa. Methane is stored in the nanocapsule under normal external conditions. The nanocapsule desorption takes place at the temperature elevation up to 350 K. There is no need to apply electric field during storage and desorption. Computational Model and Details Methane adsorption, storage and desorption processes were modeled with the method of molecular dynamics. The calculations were made with the program NAMD [17]in force field CHARMM27. The calculation results obtained were visualized with the program VMD [18]. The values of hydrogen and carbon atom charges in methane molecule were obtained [11] using the combination of Hartree–Fock E. I. Volkova Izhevsk State Technical University, Studencheskaya str., 7, 426069 Izhevsk, Russia M. V. Suyetin (&) Á A. V. Vakhrushev Institute of Applied Mechanics Ub RAS, T.Baramzinoy str., 34, 426067 Izhevsk, Russia e-mail: msuyetin@gmail.com 123 Nanoscale Res Lett (2010) 5:205–210 DOI 10.1007/s11671-009-9466-8 and Becke exchange with Lee–Yang–Parr correlation potential: B3LYP/6-31G(d) [19, 20]. The calculations were made with the program Gaussian [21]. The following atom charge values in methane molecule were obtained: carbon atom -0.628204 Mulliken and Minimum rank of matrices described by a graph or pattern over the rational, real and complex numbers ∗ Avi Berman Faculty of Mathematics Technion Haifa 32000, Israel berman@techunix.technion.ac.il Shmuel Friedland Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL 60607-7045, USA friedlan@uic.edu Leslie Hogben Department of Mathematics Iowa State University Ames, IA 50011, USA lhogben@iastate.edu Uriel G. Rothblum Faculty of Industrial Engineering and Management Technion Haifa 32000, Israel rothblum@ie.technion.ac.il Bryan Shader Department of Mathematics University of Wyoming Laramie, WY 82071, USA bshader@uwyo.edu Submitted: Apr 18, 2007; Accepted: Dec 22, 2007; Published: Feb 4, 2008 Mathematics Subject Classification: 05C50 Abstract We use a technique based on matroids to construct two nonzero patterns Z 1 and Z 2 such that the minimum rank of matrices described by Z 1 is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by Z 2 is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture in [AHKLR] about rational realization of minimum rank of sign patterns. Using Z 1 and Z 2 , we construct symmetric patterns, equivalent to graphs G 1 and G 2 , with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank. Keywords: minimum rank, graph, pattern, zero-nonzero pattern, field, matroid, symmetric matrix, matrix, real, rational, complex. ∗ This research began at the American Institute of Mathematics workshop,“Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns,” and the authors thank AIM and NSF for their support. the electronic journal of combinatorics 15 (2008), #R25 1 1 Introduction The (real symmetric) minimum rank problem (for a graph) is to determine the minimum rank among real symmetric matrices whose zero-nonzero pattern of off-diagonal entries is described by a given (simple) graph G. The zero-nonzero pattern described by the graph has tremendous influence on minimum rank. For example, a matrix associated with a path on n vertices (P n ) is a symmetric tridiagonal matrix with nonzero sub- and super- diagonal, and thus has minimum rank n − 1, whereas the complete graph on n vertices (K n ) has minimum rank 1. For a discussion of the background of the minimum rank problem (and an extensive bibliography), see [FH]. Much of the work on the minimum rank problem has focused on real symmetric ma- trices, but symmetric matrices over other fields have also been studied (see [BHL]). While examples of differences in minimum rank over different fields are known, these examples involve fields of different characteristic or size. We use a technique based on matroids to construct two zero-nonzero patterns C S 1 and C S 2 such that the minimum rank of matrices described by C S 1 is less over the complex numbers than over the real numbers 1 , and the minimum rank of matrices described by C S 2 is less over the real numbers than over the rational numbers. The pattern C S 2 immediately provides a counterexample to a conjec- ture in [AHKLR] about rational realization of minimum rank of sign patterns. We then use C S 1 and C S 2 to construct symmetric patterns, equivalent to graphs G 1 and G 2 , with the analogous minimum rank properties. All graphs discussed in this paper are simple, meaning no loops or multiple edges. The order of a graph G, denoted |G|, is the number of vertices of G. For a symmetric n × n matrix A over a field F , the graph of A, denoted G(A), is the graph with vertices {1, . . . , n} and edges {{i, j}| a ij = 0 and i = j}. Note that the diagonal of A is ignored in determining G(A). The set of symmetric matrices of the graph G over the field F is S F G = {A ∈ F n×n : A T = A and G(A) = G}. Since we will need to consider non-symmetric matrices, as well as matrices over the BioMed Central Page 1 of 12 (page number not for citation purposes) Retrovirology Open Access Research Inhibition of HIV-1 replication by P-TEFb inhibitors DRB, seliciclib and flavopiridol correlates with release of free P-TEFb from the large, inactive form of the complex Sebastian Biglione †1,5 , Sarah A Byers †1,6 , Jason P Price 2 , Van Trung Nguyen 4 , Olivier Bensaude 4 , David H Price 1,3 and Wendy Maury* 1,2 Address: 1 Interdisciplinary Molecular and Cellular Biology Program, University of Iowa, Iowa City, IA, USA, 2 Department of Microbiology, University of Iowa, Iowa City, IA, USA, 3 Department of Biochemistry, University of Iowa, Iowa City, IA, USA, 4 Laboratoire de Regulation de l'Expression Genetique, Ecole Normale Superieure, Paris, France, 5 CBR Institute for Biomedical Research, Harvard Medical School, Boston, MA, 02115, USA and 6 Oregon Health & Science University, Department of Molecular and Medical Genetics, Portland, OR 97239, USA Email: Sebastian Biglione - biglione@cbrinstitute.org; Sarah A Byers - byerssa@ohsu.edu; Jason P Price - jason-price@uiowa.edu; Van Trung Nguyen - vtnguyen@biologie.ens.fr; Olivier Bensaude - bensaude@wotan.ens.fr; David H Price - david-price@uiowa.edu; Wendy Maury* - wendy-maury@uiowa.edu * Corresponding author †Equal contributors Abstract Background: The positive transcription elongation factor, P-TEFb, comprised of cyclin dependent kinase 9 (Cdk9) and cyclin T1, T2 or K regulates the productive elongation phase of RNA polymerase II (Pol II) dependent transcription of cellular and integrated viral genes. P-TEFb containing cyclin T1 is recruited to the HIV long terminal repeat (LTR) by binding to HIV Tat which in turn binds to the nascent HIV transcript. Within the cell, P-TEFb exists as a kinase-active, free form and a larger, kinase-inactive form that is believed to serve as a reservoir for the smaller form. Results: We developed a method to rapidly quantitate the relative amounts of the two forms based on differential nuclear extraction. Using this technique, we found that titration of the P-TEFb inhibitors flavopiridol, DRB and seliciclib onto HeLa cells that support HIV replication led to a dose dependent loss of the large form of P-TEFb. Importantly, the reduction in the large form correlated with a reduction in HIV-1 replication such that when 50% of the large form was gone, HIV-1 replication was reduced by 50%. Some of the compounds were able to effectively block HIV replication without having a significant impact on cell viability. The most effective P-TEFb inhibitor flavopiridol was evaluated against HIV-1 in the physiologically relevant cell types, peripheral blood lymphocytes (PBLs) and monocyte derived macrophages (MDMs). Flavopiridol was found to have a smaller therapeutic index (LD 50 /IC 50 ) in long term HIV-1 infectivity studies in primary cells due to greater cytotoxicity and reduced efficacy at blocking HIV-1 replication. Conclusion: Initial short term studies with P-TEFb inhibitors demonstrated a dose dependent loss of the large form of P-TEFb within the cell and a concomitant reduction in HIV-1 infectivity without significant cytotoxicity. These findings suggested that inhibitors of P-TEFb may serve as effective anti-HIV-1 therapies. However, longer term HIV-1 replication studies indicated that these inhibitors were more cytotoxic and less efficacious against HIV-1 in the primary cell cultures. Published: 11 July 2007 Retrovirology 2007, 4:47 doi:10.1186/1742-4690-4-47 Received: 23 April 2007 Accepted: 11 July 2007 This article is available from: http://www.retrovirology.com/content/4/1/47 © 2007 Biglione et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Retrovirology 2007, 4:47 ... the complex plane + 4i − − 3i − 4i − − 5i 20/25 Polar Form of Complex Numbers + 2i 2i 21/25 Polar Form of Complex Numbers −4 − 2i 22/25 Polar Form of Complex Numbers −2+i − 4i 23/25 Polar Form of. .. represent a formula for finding nth roots of complex numbers in polar form A General Note label The nth Root Theorem 13/25 Polar Form of Complex Numbers To find the nth root of a complex number in polar. .. Quotients of Complex Numbers in Polar Form The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments A General Note Quotients of Complex