Modeling Virus Capsids using Tiling Theory

Aziza JeffersonDepartment of Mathematics

Rutgers University

Advisor:Professor Stanley Dunn

How do viruses effect us?

Several viruses that effect humans are

● Rhinoviruses (common cold)

● Orthomyxoviridae (Influenza)

● Rhabdoviridae (Rabies)

● Hepadnaviridae (Hepatitis B)

● Flaviviridae (Yellow Fever)

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

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

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

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

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

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

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

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

Twarock 2004

● Looked at tubular shaped capsids the family of Papovaviridae

● Compared predicted results with experimental results

● Predicted locations and orientations of the pentamers

Tiling Theory

● Tilings- tessellations in terms of a set of basic building blocks

● Decorations- Location of protein subunits on tiles

● Plane tiling(TT)- countable family of closed sets which cover the plane without gaps or overlaps

● Simply connected- tile does not enclose any holes

● 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

Definitions

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

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

Motivation

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