Mathematical Virology Viruses under the mathematical microscope: An opportunity to demonstrate the impact of in the classroom mathematics in biology in the classroom environment Reidun Twarock Departments of Mathematics and Biology York Cross-disciplinary Centre for Systems Analysis H. C. Ørsted Institute, Copenhagen, November 2018
Central Questions 1. How can mathematics help to make discoveries in virology and find novel anti-viral solutions? 2. Which aspects can be covered in the classroom? -> Suggestions are given in the Teacher’s Packs
The biological challenge Viruses are responsible for a wide spectrum of devastating diseases in humans animals and plants. Examples: •HIV •Hepatitis C •Cancer-causing viruses •Picornaviruses linked with type 1 diabetes •Common Cold • Options for anti-viral interventions are limited. • Therapy resistant mutant strains provide a challenge for therapy
Protein Containers Viral capsids are like Trojan horses, hiding the genome from the defense mechanisms of their host. = Challenges: • What are the mathematical rules underpinning their structure? • Can this insight be used to combat viruses by preventing their formation?
Viruses and Geometry An understanding of Viral Geometry enables discovery in virology and creates new opportunities in bionanotechnology and anti-viral therapy Symmetry in Virology
What is icosahedral symmetry? The icosahedron has •6 axes of 5-fold symmetry •10 axes of 3-fold symmetry •15 axes of 2-fold symmetry
Part I: What are the mathematical rules?
The architecture of larger viruses Caspar and Klug’s Theory of Quasiequivalence (1962) : ``The local environments of all capsid proteins look similar. ’’ ’’ Dots mark the positions of capsid proteins
Which triangulations are the right ones to use?
Surface of an icosahedron http://agrega.educacion.es/repositorio/24052014/07/es_2014052412_9134736/poliedros_regulares.html
Virus architecture according to Caspar and Klug icosahedron planar representation superposition
Surface lattices predicting virus architecture
Application of Pythagoras’ Theorem Question: In how many different ways can this be done? Caspar and Klug (1962) predict virus architecture based only on geometrical considerations T=S 2 =(H+K/2) 2 +3/4K 2 = H 2 +HK+K 2
Meaning of the T-number S Area = T=4 : 80 small triangles; 60T=240 proteins 4 T T counts the number of small triangles per triangular face of the icosahedron The T-number can be used to enumerate different virus structures
Examples • Find the icosahedral triangle. • What is the T-number of this virus? Chikungunya virus Herpes Simplex virus Rotavirus (T=4; H=2, K=0) (T=16; H=4, K=0) (T=13; H=3, K=1)
Viral designs are picked up in architecture Large viruses look like Buckminster Fuller’s Domes
Why is new mathematics needed? 2. It does not provide 1. Caspar-Klug Theory information at different is too restrictive to radial levels capture all virus architectures The cancer-causing papilloma virus falls Pariacoto virus out of this remit
The mathematical problem You cannot tile your bathroom with pentagons without gaps and overlaps Viral Tiling Theory There are no lattices with 5-fold symmetry! The solution: Sir Roger Penrose
Quasi-lattices via projection 6D - minimal embedding dimension for icosahedral symmetry 6D Lattice 5D Lattice 3D “ Control Space ” 3D “ Control Space ” 3D Quasilattice 2D Quasilattice depends on lattice type
A New Group Theoretical Approach Construct point arrays from orbits in the hyperoctahedral group B 6 via projection Emilio Zappa Classification of subgroups of B 6 containing the icosahedral group as a subgroup. R. Twarock, M. Valiunas & E. Zappa (2015) Acta Cryst. A71, 569-582.
Virus structure at different radial levels Develop new (affine extended) group structures and 3D tilings Pariacoto virus 2D 3D
Applications Vaccine Design: Predict the Fullerenes: Model the structures of self-assembling structures of carbon onions protein nanoparticles Sir Harald Kroto Adapted from Nobel Prize in Chemistryworld (June 2014) Chemistry 1996 C 60 C 240 With Peter Burkhard C 540 CEO AOPeptides (vaccine design)
Why do viruses use symmetry? Crick and Watson, 1956: The principle of genetic economy Viruses code for a small number of building blocks that are repeatedly used to form containers with symmetry. Containers with icosahedral symmetry are largest given fixed protein size, thus viruses minimise the length of the genome required to code for a protein container of sufficient volume to fit the genome. If the position of one red disk is known, then the positions of all others are implied by symmetry. F. Crick and J.D. Watson, Structure of Small Viruses, Nature 177 (1956), 473-475.
Part II Can we understand the mechanisms by which viruses form, and then use this to inhibit it or repurpose them for therapy?
A simple model of virus assembly Assemble an icosahedron from 20 triangles: 20 x
Enumerate assembly pathways 2 2 1 3 1 3 5 5 4 4 Characterise each assembly pathway by a Hamiltonian paths (a connected path visiting every vertex precisely once) on the inscribed polyhedron: 2 2 1 1 3 3 5 4 5 4
Viruses play the Icosian Game A board game designed by Hamilton in 1857 based on the concept of Hamiltonian circuit (cycle)
Opportunity for the classroom • Find connected paths on the Schlegel diagram of the dodecahedron that visit every vertex precisely once ( Hamiltonian paths ) • Find circular paths of this type.
The Hamiltonian Paths Approach Viral capsid + Viral genome
Viral geometry and code breaking Hamiltonian Paths Analysis enabled a discovery There is an “assembly code” hidden in the viral genome (i.e. in the code for the protein components) Prevelige (2015) Follow the Yellow Brick Road: A Paradigm Shift in Virus Assembly. JMB Note: This is challenging via bioinformatics alone due to the sequence/structure variation of the capsid Viral Enigma Machine protein recognition motif.
A paradigm shift in our understanding of virus assembly Viral genomes play vital roles in the formation of viral capsids
The mechanism: Viruses behave like “self-packing suitcases” = Article by Prof Peter Stockley, Leeds – Huffington Post
Anti-viral strategies Can we break the mechanism?
Opportunities for translation 1: drug treatment Small molecular weight compounds inhibiting assembly: Eric Dykeman Richard Bingham Example: Hepatitis B virus PS-binding drugs are distinguished by: •the speed of viral clearance Ligands binding HBV •large numbers of misencapsidated cellular PS1 inhibit virus RNAs formation •a high barrier to drug resistance R. Bingham, E.C. Dykeman & R. Twarock (2017) RNA virus evolution via a quasispecies-based model reveals a drug target with high barrier to resistance. Viruses 9, 347.
Drug delivery Can we customise the mechanism to fulfill a specific purpose?
Opportunities for translation 2: VLP production Combine geometry with Cooperativity can be optimised: biophysical modelling: Cooperative action of packaging signals enables selective and efficient genome packaging Example: STNV RNAs with optimised initiation cassette outcompete viral particles in a ratio 2:1 Develop stable particles as vaccines and for drug/gene delivery: • Lentiviral vectors (with Greg Towers) • Picornaviruses (with Peter Stockley) • E.C. Dykeman, P.G. Stockley, R Twarock, PNAS 2014 • N. Patel, E. Wroblewski, G. Leonov, S.E.V. Phillips, R. Tuma, R. Twarock and P.G. Stockley, PNAS 2017.
New opportunities for therapy Viruses covered by our patents (with experimental collaborators at the Universities of Leeds and Helsinki) include: Opportunities: • Hepatitis C • Hepatitis B •New drugs • HIV •Virus-like particles for vaccine • Human Parechovirus design & drug delivery • A number of plant and bacterial viruses With Prof Peter Stockley Astbury Centre for Structural Molecular Biology University of Leeds
A webpage for teachers: Mathematical Virology in the classroom More material is available for download from our Teacher’s Resource Pack website: www-users.york.ac.uk/~rt507/teaching_resources.html We would like to hear from you! We would be very grateful for any comments and suggestions, as this will enable us to improve our content and apply for more funding to keep this initiative going!
Summary Our interdisciplinary approach (iterative theory-experiment cycles) has uncovered a new virus assembly paradigm. •It occurs across different viral families •It is highly conserved ⇒ New applications: •Drug design – inhibit virus assembly •Nanotechnology – VLP production
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