Modeling Virus Capsids using Tiling Theory Aziza Jefferson Department 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. ects/biology-edited/chap14/b asp

Virus Structure Other viruses such as the HIV virus have a more complex structure and may include ● Virus membrane ● Shell membrane ● Reverse transcriptase y/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. _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

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 Biology 226 (2004): ● 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=h 2 +hk+k 2 WW/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 T ● Plane tiling( T )- 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