Cyclic GMP–AMP Signalling Protects Bacteria Against Viral Infection
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Engineering Curves – I
Engineering Curves – I 1. Classification 2. Conic sections - explanation 3. Common Definition 4. Ellipse – ( six methods of construction) 5. Parabola – ( Three methods of construction) 6. Hyperbola – ( Three methods of construction ) 7. Methods of drawing Tangents & Normals ( four cases) Engineering Curves – II 1. Classification 2. Definitions 3. Involutes - (five cases) 4. Cycloid 5. Trochoids – (Superior and Inferior) 6. Epic cycloid and Hypo - cycloid 7. Spiral (Two cases) 8. Helix – on cylinder & on cone 9. Methods of drawing Tangents and Normals (Three cases) ENGINEERING CURVES Part- I {Conic Sections} ELLIPSE PARABOLA HYPERBOLA 1.Concentric Circle Method 1.Rectangle Method 1.Rectangular Hyperbola (coordinates given) 2.Rectangle Method 2 Method of Tangents ( Triangle Method) 2 Rectangular Hyperbola 3.Oblong Method (P-V diagram - Equation given) 3.Basic Locus Method 4.Arcs of Circle Method (Directrix – focus) 3.Basic Locus Method (Directrix – focus) 5.Rhombus Metho 6.Basic Locus Method Methods of Drawing (Directrix – focus) Tangents & Normals To These Curves. CONIC SECTIONS ELLIPSE, PARABOLA AND HYPERBOLA ARE CALLED CONIC SECTIONS BECAUSE THESE CURVES APPEAR ON THE SURFACE OF A CONE WHEN IT IS CUT BY SOME TYPICAL CUTTING PLANES. OBSERVE ILLUSTRATIONS GIVEN BELOW.. Ellipse Section Plane Section Plane Hyperbola Through Generators Parallel to Axis. Section Plane Parallel to end generator. COMMON DEFINATION OF ELLIPSE, PARABOLA & HYPERBOLA: These are the loci of points moving in a plane such that the ratio of it’s distances from a fixed point And a fixed line always remains constant. The Ratio is called ECCENTRICITY. (E) A) For Ellipse E<1 B) For Parabola E=1 C) For Hyperbola E>1 Refer Problem nos. -
A Characterization of Helical Polynomial Curves of Any Degree
A CHARACTERIZATION OF HELICAL POLYNOMIAL CURVES OF ANY DEGREE J. MONTERDE Abstract. We give a full characterization of helical polynomial curves of any degree and a simple way to construct them. Existing results about Hermite interpolation are revisited. A simple method to select the best quintic interpolant among all possible solutions is suggested. 1. Introduction The notion of helical polynomial curves, i.e, polynomial curves which made a constant angle with a ¯xed line in space, have been studied by di®erent authors. Let us cite the papers [5, 6, 7] where the main results about the cubical and quintic cases are stablished. In [5] the authors gives a necessary condition a polynomial curve must satisfy in order to be a helix. The condition is expressed in terms of its hodograph, i.e., its derivative. If a polynomial curve, ®, is a helix then its hodograph, ®0, must be Pythagorean, i.e., jj®0jj2 is a perfect square of a polynomial. Moreover, this condition is su±cient in the cubical case: all PH cubical curves are helices. In the same paper it is also stated that not only ®0 must be Pythagorean, but also ®0^®00. Following the same ideas, in [1] it is proved that both conditions are su±cient in the quintic case. Unfortunately, this characterization is no longer true for higher degrees. It is possible to construct examples of polynomial curves of degree 7 verifying both conditions but being not a helix. The aim of this paper is just to show how a simple geometric trick can clarify the proofs of some previous results and to simplify the needed compu- tations to solve some related problems as for instance, the Hermite problem using helical polynomial curves. -
Linguistic Presentation of Objects
Zygmunt Ryznar dr emeritus Cracow Poland [email protected] Linguistic presentation of objects (types,structures,relations) Abstract Paper presents phrases for object specification in terms of structure, relations and dynamics. An applied notation provides a very concise (less words more contents) description of object. Keywords object relations, role, object type, object dynamics, geometric presentation Introduction This paper is based on OSL (Object Specification Language) [3] dedicated to present the various structures of objects, their relations and dynamics (events, actions and processes). Jay W.Forrester author of fundamental work "Industrial Dynamics”[4] considers the importance of relations: “the structure of interconnections and the interactions are often far more important than the parts of system”. Notation <!...> comment < > container of (phrase,name...) ≡> link to something external (outside of definition)) <def > </def> start-end of definition <spec> </spec> start-end of specification <beg> <end> start-end of section [..] or {..} 1 list of assigned keywords :[ or :{ structure ^<name> optional item (..) list of items xxxx(..) name of list = value assignment @ mark of special attribute,feature,property @dark unknown, to obtain, to discover :: belongs to : equivalent name (e.g.shortname) # number of |name| executive/operational object ppppXxxx name of item Xxxx with prefix ‘pppp ‘ XXXX basic object UUUU.xxxx xxxx object belonged to UUUU object class & / conjunctions ‘and’ ‘or’ 1 If using Latex editor we suggest [..] brackets -
Helical Curves and Double Pythagorean Hodographs
Helical polynomial curves and “double” Pythagorean-hodograph curves Rida T. Farouki Department of Mechanical & Aeronautical Engineering, University of California, Davis — synopsis — • introduction: properties of Pythagorean-hodograph curves • computing rotation-minimizing frames on spatial PH curves • helical polynomial space curves — are always PH curves • standard quaternion representation for spatial PH curves • “double” Pythagorean hodograph structure — requires both |r0(t)| and |r0(t) × r00(t)| to be polynomials in t • Hermite interpolation problem: selection of free parameters Pythagorean-hodograph (PH) curves r(ξ) = PH curve ⇐⇒ coordinate components of r0(ξ) comprise a “Pythagorean n-tuple of polynomials” in Rn PH curves incorporate special algebraic structures in their hodographs (complex number & quaternion models for planar & spatial PH curves) • rational offset curves rd(ξ) = r(ξ) + d n(ξ) Z ξ • polynomial arc-length function s(ξ) = |r0(ξ)| dξ 0 Z 1 • closed-form evaluation of energy integral E = κ2 ds 0 • real-time CNC interpolators, rotation-minimizing frames, etc. helical polynomial space curves several equivalent characterizations of helical curves • tangent t maintains constant inclination ψ with fixed vector a • a · t = cos ψ, where ψ = pitch angle and a = axis vector of helix • fixed curvature/torsion ratio, κ/τ = tan ψ (Theorem of Lancret) • curve has a circular tangent indicatrix on the unit sphere (small circle for space curve, great circle for planar curve) • (r(2) × r(3)) · r(4) ≡ 0 — where r(k) = kth arc–length derivative -
Chapter 2. Parameterized Curves in R3
Chapter 2. Parameterized Curves in R3 Def. A smooth curve in R3 is a smooth map σ :(a, b) → R3. For each t ∈ (a, b), σ(t) ∈ R3. As t increases from a to b, σ(t) traces out a curve in R3. In terms of components, σ(t) = (x(t), y(t), z(t)) , (1) or x = x(t) σ : y = y(t) a < t < b , z = z(t) dσ velocity at time t: (t) = σ0(t) = (x0(t), y0(t), z0(t)) . dt dσ speed at time t: (t) = |σ0(t)| dt Ex. σ : R → R3, σ(t) = (r cos t, r sin t, 0) - the standard parameterization of the unit circle, x = r cos t σ : y = r sin t z = 0 σ0(t) = (−r sin t, r cos t, 0) |σ0(t)| = r (constant speed) 1 Ex. σ : R → R3, σ(t) = (r cos t, r sin t, ht), r, h > 0 constants (helix). σ0(t) = (−r sin t, r cos t, h) √ |σ0(t)| = r2 + h2 (constant) Def A regular curve in R3 is a smooth curve σ :(a, b) → R3 such that σ0(t) 6= 0 for all t ∈ (a, b). That is, a regular curve is a smooth curve with everywhere nonzero velocity. Ex. Examples above are regular. Ex. σ : R → R3, σ(t) = (t3, t2, 0). σ is smooth, but not regular: σ0(t) = (3t2, 2t, 0) , σ0(0) = (0, 0, 0) Graph: x = t3 y = t2 = (x1/3)2 σ : y = t2 ⇒ y = x2/3 z = 0 There is a cusp, not because the curve isn’t smooth, but because the velocity = 0 at the origin. -
Eukaryotic Genome Annotation
Comparative Features of Multicellular Eukaryotic Genomes (2017) (First three statistics from www.ensembl.org; other from original papers) C. elegans A. thaliana D. melanogaster M. musculus H. sapiens Species name Nematode Thale Cress Fruit Fly Mouse Human Size (Mb) 103 136 143 3,482 3,555 # Protein-coding genes 20,362 27,655 13,918 22,598 20,338 (25,498 (13,601 original (30,000 (30,000 original est.) original est.) original est.) est.) Transcripts 58,941 55,157 34,749 131,195 200,310 Gene density (#/kb) 1/5 1/4.5 1/8.8 1/83 1/97 LINE/SINE (%) 0.4 0.5 0.7 27.4 33.6 LTR (%) 0.0 4.8 1.5 9.9 8.6 DNA Elements 5.3 5.1 0.7 0.9 3.1 Total repeats 6.5 10.5 3.1 38.6 46.4 Exons % genome size 27 28.8 24.0 per gene 4.0 5.4 4.1 8.4 8.7 average size (bp) 250 506 Introns % genome size 15.6 average size (bp) 168 Arabidopsis Chromosome Structures Sorghum Whole Genome Details Characterizing the Proteome The Protein World • Sequencing has defined o Many, many proteins • How can we use this data to: o Define genes in new genomes o Look for evolutionarily related genes o Follow evolution of genes ▪ Mixing of domains to create new proteins o Uncover important subsets of genes that ▪ That deep phylogenies • Plants vs. animals • Placental vs. non-placental animals • Monocots vs. dicots plants • Common nomenclature needed o Ensure consistency of interpretations InterPro (http://www.ebi.ac.uk/interpro/) Classification of Protein Families • Intergrated documentation resource for protein super families, families, domains and functional sites o Mitchell AL, Attwood TK, Babbitt PC, et al. -
STACK: a Toolkit for Analysing Β-Helix Proteins
STACK: a toolkit for analysing ¯-helix proteins Master of Science Thesis (20 points) Salvatore Cappadona, Lars Diestelhorst Abstract ¯-helix proteins contain a solenoid fold consisting of repeated coils forming parallel ¯-sheets. Our goal is to formalise the intuitive notion of a ¯-helix in an objective algorithm. Our approach is based on first identifying residues stacks — linear spatial arrangements of residues with similar conformations — and then combining these elementary patterns to form ¯-coils and ¯-helices. Our algorithm has been implemented within STACK, a toolkit for analyzing ¯-helix proteins. STACK distinguishes aromatic, aliphatic and amidic stacks such as the asparagine ladder. Geometrical features are computed and stored in a relational database. These features include the axis of the ¯-helix, the stacks, the cross-sectional shape, the area of the coils and related packing information. An interface between STACK and a molecular visualisation program enables structural features to be highlighted automatically. i Contents 1 Introduction 1 2 Biological Background 2 2.1 Basic Concepts of Protein Structure ....................... 2 2.2 Secondary Structure ................................ 2 2.3 The ¯-Helix Fold .................................. 3 3 Parallel ¯-Helices 6 3.1 Introduction ..................................... 6 3.2 Nomenclature .................................... 6 3.2.1 Parallel ¯-Helix and its ¯-Sheets ..................... 6 3.2.2 Stacks ................................... 8 3.2.3 Coils ..................................... 8 3.2.4 The Core Region .............................. 8 3.3 Description of Known Structures ......................... 8 3.3.1 Helix Handedness .............................. 8 3.3.2 Right-Handed Parallel ¯-Helices ..................... 13 3.3.3 Left-Handed Parallel ¯-Helices ...................... 19 3.4 Amyloidosis .................................... 20 4 The STACK Toolkit 24 4.1 Identification of Structural Elements ....................... 24 4.1.1 Stacks ................................... -
On Helices and Bertrand Curves in Euclidean 3-Space
ON HELİCES AND BERTRAND CURVES IN EUCLİDEAN 3-SPACE Murat Babaarslan Department of Mathematics, Faculty of Science and Arts Bozok University, Yozgat, Turkey [email protected] Yusuf Yaylı Department of Mathematics, Faculty of Science Ankara University, Tandoğan, Ankara, Turkey [email protected] Abstract- In this article, we investigate Bertrand curves corresponding to spherical images of the tangent indicatrix, binormal indicatrix, principal normal indicatrix and Darboux indicatrix of a space curve in Euclidean 3-space. As a result, in case of a space curve is general helix, we show that curve corresponding to spherical images of its tangent indicatrix and binormal indicatrix are circular helices and Bertrand curves. Similarly, in case of a space curve is slant helix, we demonstrate that the curve corresponding to spherical image of its principal normal indicatrix is circular helix and Bertrand curve. Key Words- Helix, Bertrand curve, Spherical Images 1. INTRODUCTION In the differential geometry of a regular curve in Euclidean 3-space E3 , it is well known that, one of the important problems is characterization of a regular curve. The curvature and the torsion of a regular curve play an important role to determine the shape and size of the curve 1,2. Natural scientists have long held a fascination, sometimes bordering on mystical obsession for helical structures in nature. Helices arise in nanosprings, carbon nonotubes, helices, DNA double and collagen triple helix, the double helix shape is commonly associated with DNA, since the double helix is structure of DNA 3 . This fact was published for the first time by Watson and Crick in 1953 4. -
Calculating the Structure-Based Phylogenetic Relationship
CALCULATING THE STRUCTURE-BASED PHYLOGENETIC RELATIONSHIP OF DISTANTLY RELATED HOMOLOGOUS PROTEINS UTILIZING MAXIMUM LIKELIHOOD STRUCTURAL ALIGNMENT COMBINATORICS AND A NOVEL STRUCTURAL MOLECULAR CLOCK HYPOTHESIS A DISSERTATION IN Molecular Biology and Biochemistry and Cell Biology and Biophysics Presented to the Faculty of the University of Missouri-Kansas City in partial fulfillment of the requirements for the degree Doctor of Philosophy by SCOTT GARRETT FOY B.S., Southwest Baptist University, 2005 B.A., Truman State University, 2007 M.S., University of Missouri-Kansas City, 2009 Kansas City, Missouri 2013 © 2013 SCOTT GARRETT FOY ALL RIGHTS RESERVED CALCULATING THE STRUCTURE-BASED PHYLOGENETIC RELATIONSHIP OF DISTANTLY RELATED HOMOLOGOUS PROTEINS UTILIZING MAXIMUM LIKELIHOOD STRUCTURAL ALIGNMENT COMBINATORICS AND A NOVEL STRUCTURAL MOLECULAR CLOCK HYPOTHESIS Scott Garrett Foy, Candidate for the Doctor of Philosophy Degree University of Missouri-Kansas City, 2013 ABSTRACT Dendrograms establish the evolutionary relationships and homology of species, proteins, or genes. Homology modeling, ligand binding, and pharmaceutical testing all depend upon the homology ascertained by dendrograms. Regardless of the specific algorithm, all dendrograms that ascertain protein evolutionary homology are generated utilizing polypeptide sequences. However, because protein structures superiorly conserve homology and contain more biochemical information than their associated protein sequences, I hypothesize that utilizing the structure of a protein instead -
And Beta-Helical Protein Motifs
Soft Matter Mechanical Unfolding of Alpha- and Beta-helical Protein Motifs Journal: Soft Matter Manuscript ID SM-ART-10-2018-002046.R1 Article Type: Paper Date Submitted by the 28-Nov-2018 Author: Complete List of Authors: DeBenedictis, Elizabeth; Northwestern University Keten, Sinan; Northwestern University, Mechanical Engineering Page 1 of 10 Please doSoft not Matter adjust margins Soft Matter ARTICLE Mechanical Unfolding of Alpha- and Beta-helical Protein Motifs E. P. DeBenedictis and S. Keten* Received 24th September 2018, Alpha helices and beta sheets are the two most common secondary structure motifs in proteins. Beta-helical structures Accepted 00th January 20xx merge features of the two motifs, containing two or three beta-sheet faces connected by loops or turns in a single protein. Beta-helical structures form the basis of proteins with diverse mechanical functions such as bacterial adhesins, phage cell- DOI: 10.1039/x0xx00000x puncture devices, antifreeze proteins, and extracellular matrices. Alpha helices are commonly found in cellular and extracellular matrix components, whereas beta-helices such as curli fibrils are more common as bacterial and biofilm matrix www.rsc.org/ components. It is currently not known whether it may be advantageous to use one helical motif over the other for different structural and mechanical functions. To better understand the mechanical implications of using different helix motifs in networks, here we use Steered Molecular Dynamics (SMD) simulations to mechanically unfold multiple alpha- and beta- helical proteins at constant velocity at the single molecule scale. We focus on the energy dissipated during unfolding as a means of comparison between proteins and work normalized by protein characteristics (initial and final length, # H-bonds, # residues, etc.). -
Effect of Superhelical Structure on the Secondary Structure of DNA Rings
VOL. 5, PP. 691-696 (1967) Effect of Superhelical Structure on the Secondary Structure of DNA Rings DANIEL GLAUBIGER and JOHN E. HEARST, Departnient of Chemistry, University of California, Berkeley, California Synopsis A quantity, called the linking number, is defined, which specifies the total number of t,wists in a circular helix. The linking number is invariant under continuous deforma- tions of the ring and therefore enables one to calculate the influence of superhelical structures on the secondary helix of a circular molecule. The linking number can be determined by projecting the helix into a plane and counting strand crosses in the projection as described. For example, it has been shown that for each 180" twist in a left-handed superhelix, a right-handed 360" twist is removed from the secondary helix, thus allowing local unwinding. Introduction It is of interest to relate superhelical structure of helical molecules to their secondary structure.' By defining a quantity characteristic of two- stranded circular helical niolecules, related to the helical structure, and in- variant under continuous deformations of the molecule, we are able to evaluate the changes in helical structure brought about by the imposition of superhelical configurations on such molecules. The quantity of interest, called the linking number, is related to the number of turns/cycle for a circular helical molecule.* It will be shown that superhelical structures which are made from helical polymers con- tribute to this number and alter the contribution made by the secondary structure of the molecule to this number. This is made manifest as a change in the number of turns/cycle or as a change in the number of turns/ unit length. -
Introduction to the Local Theory of Curves and Surfaces
Introduction to the Local Theory of Curves and Surfaces Notes of a course for the ICTP-CUI Master of Science in Mathematics Marco Abate, Fabrizio Bianchi (with thanks to Francesca Tovena) (Much more on this subject can be found in [1]) CHAPTER 1 Local theory of curves Elementary geometry gives a fairly accurate and well-established notion of what is a straight line, whereas is somewhat vague about curves in general. Intuitively, the difference between a straight line and a curve is that the former is, well, straight while the latter is curved. But is it possible to measure how curved a curve is, that is, how far it is from being straight? And what, exactly, is a curve? The main goal of this chapter is to answer these questions. After comparing in the first two sections advantages and disadvantages of several ways of giving a formal definition of a curve, in the third section we shall show how Differential Calculus enables us to accurately measure the curvature of a curve. For curves in space, we shall also measure the torsion of a curve, that is, how far a curve is from being contained in a plane, and we shall show how curvature and torsion completely describe a curve in space. 1.1. How to define a curve n What is a curve (in a plane, in space, in R )? Since we are in a mathematical course, rather than in a course about military history of Prussian light cavalry, the only acceptable answer to such a question is a precise definition, identifying exactly the objects that deserve being called curves and those that do not.