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Modeling Virus Capsids using Tiling Theory

Aziza Jefferson Department of Mathematics Rutgers University Advisor: Professor Stanley Dunn. Modeling Virus Capsids using Tiling Theory. How do viruses effect us?. Several viruses that effect humans are Rhinoviruses (common cold) Orthomyxoviridae (Influenza) Rhabdoviridae (Rabies)

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Modeling Virus Capsids using Tiling Theory

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  1. Aziza Jefferson Department of Mathematics Rutgers University Advisor: Professor Stanley Dunn Modeling Virus Capsids using Tiling Theory

  2. How do viruses effect us? • Several viruses that effect humans are • Rhinoviruses (common cold) • Orthomyxoviridae (Influenza) • Rhabdoviridae (Rabies) • Hepadnaviridae (Hepatitis B) • Flaviviridae (Yellow Fever)

  3. Virus Structure • A simple virus contains nucleic acid and a capsid • The nucleic acid is normally RNA or DNA • The capsid is made up of proteins. http://www.pinkmonkey.com/studyguides/subjects/biology-edited/chap14/b1400001.asp

  4. Virus Structure Other viruses such as the HIV virus have a more complex structure and may include • Virus membrane • Shell membrane • Reverse transcriptase http://www.schoolscience.co.uk/content/4/biology/abpi/immune/immune10.htm

  5. Importance of the Capsid • The virus is fragile inside of the capsid • The capsid introduces the virus to its host cell • Once the structure is known anti-viral medications to penetrate the capsid can be developed. http://www.microbiology.wustl.edu/sindbis/sin_genes

  6. Capsid structure • In a simple virus the capsid is created from a tiling of one type of protein • Because of the size of a virus, the nucleic acid can only code for several proteins maximum • The capsid can contain one or two layers of proteins http://pathmicro.med.sc.edu/int6.jpg

  7. Theoretical Problem • We need to know all possible configurations of the capsid in order to design effective anti-viral therapy. • Experimental evidence alone can not effectively give us every possible capsid structure • We need a mathematical theory that will allow us to predict the number and types of capsids for each virus

  8. Background Several people have examined the virus capsid and developed a mathematical theory from experimental findings • Caspar, D.L.D., and A Klug. "Physical Principles in the Construction of Regular Viruses." Cold Spring Harbor Symposia on Quantitative Biology 27 (1962): 1-24 • Twarock, R. "A tiling approach to virus capsid assembly explaining a structural puzzle in virology." Journal of Theoretical Biology226 (2004): 477-482. • Twarock, R. "Mathematical models for tubular structures in the family of Papovaviridae." Bulletin of Mathematical Biology (2004): 1-15

  9. Caspar-Klug Theory • Studied simple viruses with Icosahedral shaped capsids • Used triangulation to predict the shape and position of proteins in the capsid • T=Pf 2 • P=h2+hk+k2 http://www.tulane.edu/~dmsander/WWW/335/335Structure.html

  10. Twarock 2004 • Relaxed the assumption of triangular shaped subunits of proteins • Re-evaluated the family of Icosahedral shaped capsids • Uses tiling theory to determine the structure of the capsid

  11. Twarock 2004 • Looked at tubular shaped capsids the family of Papovaviridae • Compared predicted results with experimental results • Predicted locations and orientations of the pentamers

  12. Topological disk- bounded, connected and simply connected set Patch- finite number of tiles of the tiling such that their union is a topological disk Incident- the relation of a tile to each of its edges or vertices and also of an edge to each of its endpoints Tiling Theory • Definitions • Tilings- tessellations in terms of a set of basic building blocks • Decorations- Location of protein subunits on tiles • Plane tiling(T)- countable family of closed sets which cover the plane without gaps or overlaps • Simply connected- tile does not enclose any holes

  13. Tiling Theory • Well-behaved tiles, tilings- each tile is a closed topological disk • Monohedral tilings- every tile in tiling T is congruent to one fixed set T • Prototile of T- the set T

  14. Motivation • The number of viruses being discovered is increasing at a faster rate then our ability to develop anti-viral therapies. What we do not know is if these theories of virus capsid structure apply to or are made up of newer viruses such as the emerging viruses: • E. coli O157:H7 disease • Cryptosporidiosis • Human Immunodeficiency Virus • Ebola

  15. This Summer... • Identify a emerging or re-merging virus which crystallographic information about its structure exists • Gain a better understanding of the underlying assumptions of the virus capsid tile theories and see if the current theories apply to this target virus • Generalize the theory as appropriate

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