Reidun Twarock

The University of York · York Centre for Complex Systems Analysis

Hepatitis B virus (HBV) is a global health threat, and its elimination by 2030 has been prioritised by the World Health Organisation. Here we present an age-structured model for the immune response to an HBV infection, which takes into account contributions from both cell-mediated and humoral immunity. The model has been validated using published p...

The roles of RNA sequence/structure motifs, Packaging Signals (PSs), for regulating assembly of an HBV genome transcript have been investigated in an efficient in vitro assay containing only core protein (Cp) and RNA. Variants of three conserved PSs, within the genome of a strain not used previously, preventing correct presentation of a Cp-recognit...

Defective interfering particles arise spontaneously during a viral infection as mutants lacking essential parts of the viral genome. Their ability to replicate in the presence of the wild-type (WT) virus (at the expense of viable viral particles) is mimicked and exploited by therapeutic interfering particles. We propose a strategy for the design of...

Within-host models of COVID-19 infection dynamics enable the merits of different forms of antiviral therapy to be assessed in individual patients. A stochastic agent-based model of COVID-19 intracellular dynamics is introduced here, that incorporates essential steps of the viral life cycle targeted by treatment options. Integration of model predict...

Cryo-electron microscopy (EM) is undergoing a revolution, enabling the study of viral pathogens in unprecedented detail. The asymmetric EM reconstruction of bacteriophage MS2 at medium resolution (8.7 Å) by Koning et al.1, and the subsequent reconstruction at even higher resolution (3.6 Å) by Dai et al.2 revealed the structures of both the protein...

Significance Viruses composed of a shell of coat proteins enclosing ssRNA genomes are among the simplest biological entities. Their lifecycles include a range of processes, such as specific genome encapsidation and efficient capsid self-assembly. Until recently, these were not linked, but we have shown that many viruses in this class encode multipl...

Formation of the Hepatitis B (HBV) nucleocapsid (NC) is an essential step in the viral lifecycle but its assembly is not fully understood. We report the discovery of sequencespecific interactions between the viral pre-genome and HBV core protein (Cp) that play roles in defining the NC assembly pathway. Using RNA SELEX and bioinformatics we identifi...

We introduce here a mathematical procedure for the structural classification of a specific class of self-assembling protein nanoparticles (SAPNs) that are used as a platform for repetitive antigen display systems. These SAPNs have distinctive geometries as a consequence of the fact that their peptide building blocks are formed from two linked coile...

Assembly of the major viral pathogens of the Picornaviridae family is poorly understood. Human parechovirus 1 is an example of such viruses that contains sixty short regions of ordered RNA density making identical contacts with the protein shell. We show here via a combination of RNA SELEX, bioinformatics analysis and reverse genetics that these RN...

The specific packaging of the hepatitis C virus (HCV) genome is hypothesised to be driven by Core-RNA interactions. To identify the regions of the viral genome involved in this process, we used SELEX (systematic evolution of ligands by exponential enrichment) to identify RNA aptamers which bind specifically to Core in vitro. Comparison of these apt...

Viruses are remarkable examples of order at the nanoscale, exhibiting protein containers that in the vast majority of cases are organized with icosahedral symmetry. Janner used lattice theory to provide blueprints for the organization of material in viruses. An alternative approach is provided here in terms of icosahedral tilings, motivated by the...

Cryo-electron microscopy permits 3-D structures of viral pathogens to be determined in remarkable detail. In particular, the protein containers encapsulating viral genomes have been determined to high resolution using symmetry averaging techniques that exploit the icosahedral architecture seen in many viruses. By contrast, structure determination o...

Significance Single-stranded RNA viruses self-assemble protective protein containers around their cognate genomes rapidly and efficiently at low concentrations. RNA encapsidation in vivo occurs preferentially with the cognate genome, in contrast to many in vitro reassembly experiments. We describe in molecular detail how this specificity and effici...

The architecture of infinite structures with non-crystallographic symmetries can be modeled via aperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. We present here a group theoretical method for the construction of finite nested point set with...

For many viruses, structural transitions of the viral protein containers, which encapsulate and hence provide protection for the viral genome, form an integral part of their life cycle. We review here two complementary mathematical models for the expansion of an icosahedral viral capsid. The first is based on a geometrical description of the capsid...

Significance One of the important puzzles in virology is how viruses assemble protective protein containers for their genomes rapidly and efficiently during an infection. Recent advances in the field of RNA viruses suggests that multiple specific contacts between the genomic RNA and the proteins in these containers play crucial roles in this proces...

The principle of affine symmetry is applied here to the nested fullerene cages (carbon onions) that arise in the context of carbon chemistry. Previous work on affine extensions of the icosahedral group has revealed a new organizational principle in virus structure and assembly. This group-theoretic framework is adapted here to the physical requirem...

We investigate the subgroup structure of the hyperoctahedral group in six dimensions. In particular, we study the subgroups isomorphic to the icosahedral group. We classify the orthogonal crystallographic representations of the icosahedral group and analyse their intersections and subgroups, using results from graph theory and their spectra.

Key steps in a viral life-cycle, such as self-assembly of a protective protein container or in some cases also subsequent maturation events, are governed by the interplay of physico-chemical mechanisms involving various spatial and temporal scales. These salient aspects of a viral life cycle are hence well described and rationalised from a mesoscop...

For a significant number of viruses a structural transition of the protein container that encapsulates the viral genome forms an important part of the life cycle and is a prerequisite for the particle becoming infectious. Despite many recent efforts the mechanism of this process is still not fully understood, and a complete characterization of the...

The current paradigm for assembly of single-stranded RNA viruses is based on a mechanism involving non-sequence-specific packaging of genomic RNA driven by electrostatic interactions. Recent experiments, however, provide compelling evidence for sequence-specificity in this process both in vitro and in vivo. The existence of multiple RNA packaging s...

Recent work with RNA phages and an ssRNA plant satellite virus challenges the widely held view that the sequences and structures of genomic RNAs are unimportant for virion assembly. In the T=3 phages, RNA-coat protein interactions occur throughout the genome, defining the quasiconformers of their protein shells. In the plant virus, there are multip...

Understanding the fundamental principles of virus architecture is one of the most important challenges in biology and medicine. Crick and Watson were the first to propose that viruses exhibit symmetry in the organization of their protein containers for reasons of genetic economy. Based on this, Caspar and Klug introduced quasi-equivalence theory to...

The formation of a protective protein container is an essential step in the life-cycle of most viruses. In the case of single-stranded (ss)RNA viruses, this step occurs in parallel with genome packaging in a co-assembly process. Previously, it had been thought that this process can be explained entirely by electrostatics. Inspired by recent single-...

Virus capsid assembly has traditionally been considered as a process that can be described primarily via self-assembly of the capsid proteins, neglecting interactions with other viral or cellular components. Our recent work on several ssRNA viruses, a major class of viral pathogens containing important human, animal, and plant viruses, has shown th...

The interaction of p53 with its regulators MDM2 and MDMX plays a major role in regulating the cell cycle. Inhibition of this interaction has become an important therapeutic strategy in oncology. Although MDM2 and MDMX share a very high degree of sequence/structural similarity, the small-molecule inhibitor nutlin appears to be an efficient inhibitor...

We study the structural transformations induced, via the cut-and-project method, in quasicrystals and tilings by lattice transition in higher dimensions, with a focus on transition paths preserving at least some symmetry in intermediate lattices. We discus the effect of such transformations on planar aperiodic Penrose tilings, and on three-dimensio...

The surface lattices of all possible tubular structures in the papilloma- polyoma family of viruses are classied based on tiling theory and com- pared with experimental results. The papilloma-polyoma family of viruses contains tumour causing viruses, and mathematical models for the structure of these viruses and their tubular variants are hence of...

We investigated the potential of small peptide segments to function as broad-spectrum antiviral drug leads. We extracted the α-helical peptide segments that share common secondary-structure environments in the capsid protein-protein interfaces of three unrelated virus classes (PRD1-like, HK97-like, and BTV-like) that encompass different levels of p...

The following talk has been given in a special session dedicated to Professor Heinz-Dietrich Doebner at QTS in Prague in August 2011 on the occasion of his 80th birthday. It documents my journey from being a PhD student in Mathematical Physics at the Arnold Sommerfeld Institute in Clausthal under his supervision, to becoming a Professor of Mathemat...

We have expressed a recombinant form of the coat protein of satellite tobacco necrosis virus (STNV) in E. Coli using a codon-optimized gene, and shown it assembles spontaneously into capsids closely resembling the wild-type virus. The T=1 virus-like particles (VLPs) package the recombinant RNA transcript, and conditions have been established for di...

The majority of viruses have protein containers, called capsids, in which proteins are arranged with icosahedral symmetry. The capsids of these viruses follow structural blueprints that can be modelled as subsets of 3-dimensional aperiodic lattices called quasicrystals. These, in turn, can be obained by projection from 6-dimensional Bravais lattice...

The study of drug-receptor interactions has largely been framed in terms of the equilibrium thermodynamic binding affinity, an in vitro measure of the stability of the drug-receptor complex that is commonly used as a proxy measure of in vivo biological activity. In response to the growing realization of the importance of binding kinetics to in vivo...

Motivated by recent results in mathematical virology, we present novel asymmetric Z[tau]-integer-valued affine extensions of the non-crystallographic Coxeter groups H_2, H_3 and H_4 derived in a Kac-Moody-type formalism. In particular, we show that the affine reflection planes which extend the Coxeter group H_3 generate (twist) translations along 2...

In this paper, we show that affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of affine extended versions of the root systems E_8, D_6 and A_4. We show that the induced affine extensions of the non-crystallographic groups H_4, H_3 and H_2 correspond to a distinguished subset...

Single-stranded RNA (ssRNA) viruses form a major class that includes important human, animal, and plant pathogens. While the principles underlying the structures of their protein capsids are generally well understood, much less is known about the organization of their encapsulated genomic RNAs. Cryo-electron microscopy and x-ray crystallography hav...

Using a recombinant, T=1 Satellite Tobacco Necrosis Virus (STNV)-like particle expressed in Escherichia coli, we have established conditions for in vitro disassembly and reassembly of the viral capsid. In vivo assembly is dependent on the presence of the coat protein (CP) N-terminal region, and in vitro assembly requires RNA. Using immobilised CP m...

Viruses with icosahedral capsids, which form the largest class of all viruses and contain a number of important human pathogens, can be modelled via suitable icosahedrally invariant finite subsets of icosahedral 3D quasicrystals. We combine concepts from the theory of 3D quasicrystals, and from the theory of structural phase transformations in crys...

Viruses are pathogens in every kingdom of life and are major causes of human disease and suffering. They are known to encompass a size range that overlaps with that of the smallest bacterial cells, and the largest viruses now seem to be hosts of their own viral pathogens. Recent genomic sequencing efforts show that many organisms have genes that ar...

Many single-stranded RNA viruses self-assemble their protein containers around their genomes. The roles that the RNA plays in this assembly process have mostly been ignored, resulting in a protein-centric view of assembly that is unable to explain adequately the fidelity and speed of assembly in such viruses. Using bacteriophage MS2, we demonstrate...

A large number of single-stranded RNA viruses, which form a major class of all viruses, co-assemble their protein container and their genomic material. The multiple roles of the viral genome in this process are presently only partly understood. Recent experimental results indicate that RNA, in addition to its function as a repository for genetic in...

Emerging Topics in Physical Virology is a state-of-the-art account of recent advances in the experimental analysis and modeling of structure, function and dynamics of viruses. It is the first interdisciplinary book that integrates a review of relevant experimental techniques, such as cryo-electron microscopy, atomic force microscopy and mass spectr...

Assembly of the T=3 bacteriophage MS2 is initiated by the binding of a 19 nucleotide RNA stem loop from within the phage genome to a symmetric coat protein dimer. This binding event effects a folding of the FG loop in one of the protein subunits of the dimer and results in the formation of an asymmetric dimer. Since both the symmetric and asymmetri...

Previously, an RNA stem-loop (TR) encompassing 19 nt of the genome of bacteriophage MS2 was shown to act as an allosteric effector of conformational switching in the coat protein during in vitro capsid assembly. TR RNA binding to symmetric coat protein dimers results in conformational changes, principally at the FG-loop connecting the F and G beta-...

Blueprints for polyhedral cages with icosahedral symmetry made of circular DNA molecules are provided. The basic rule is that every edge of the cage is met twice in opposite directions by the DNA strand(s), and vertex junctions are realized by a set of admissible junction types. As nanocontainers for cargo storage and delivery, the icosidodecahedra...

Since the seminal work of Caspar and Klug on the structure of the protein containers that encapsulate and hence protect the viral genome, it has been recognised that icosahedral symmetry is crucial for the structural organisation of viruses. In particular, icosahedral symmetry has been invoked in order to predict the surface structures of viral cap...

Since the seminal work of Caspar and Klug on the structure of the protein containers that encapsulate and hence protect the viral genome, it has been recognized that icosahedral symmetry is crucial for the structural organization of viruses. In particular, icosahedral symmetry has been invoked in order to predict the surface structures of viral cap...

Polyomaviridae assemble in vitro into different aggregates depending on experimental conditions. We use an energy landscape approach using empirical energy calculations to quantify how the formation of these different aggregates depends on pH, the presence of bound calcium ions and disulfide linkages. Computations are carried out for SV40, a member...

Cage structures engineered from nucleic acids are of interest in nanotechnology, for example as a means of drug delivery (Destito et al 2007). Until now, most experimentally realized DNA cages have crystallographic symmetry, such as the shape of a cube (Chen and Seeman 1991 Nature 350 631-3), a tetrahedron (Goodman et al 2005 Science 310 1661-5), a...

In a seminal paper, Caspar and Klug [1962. Physical principles in the construction of regular viruses. Cold Spring Harbor Symp. Quant. Biol. 27, 1-24] derived a family of surface lattices as blueprints for the structural organisation of the protein shells, called viral capsids, which encapsulate and hence protect the viral genome. These lattices sc...

The Caspar-Klug classification of viruses whose protein shell, called viral capsid, exhibits icosahedral symmetry, has recently been extended to incorporate viruses whose capsid proteins are exclusively organised in pentamers. The approach, named 'Viral Tiling Theory', is inspired by the theory of quasicrystals, where aperiodic Penrose tilings enjo...

Using cryo-electron microscopy, single particle image processing and three-dimensional reconstruction with icosahedral averaging, we have determined the three-dimensional solution structure of bacteriophage MS2 capsids reassembled from recombinant protein in the presence of short oligonucleotides. We have also significantly extended the resolution...

A construction method for duplex cage structures with icosahedral sym- metry made out of single-stranded DNA molecules is presented and applied to an icosidodecahedral cage. It is shown via a mixture of analytic and computer techniques that there exist realisations of this graph in terms of two circular DNA molecules. These blueprints for the organ...

Understanding the structure and life cycle of viruses is a fascinating challenge with a crucial impact on the public health sector. In the early 1960s, Caspar & Klug (Caspar & Klug 1962 Cold Spring Harbor Symp. Quant. Biol. 27, 1-24) established a theory for the prediction of the surface structures of the protein shells, called viral capsids, which...

The structural organisation of the viral genome within its protein container, called the viral capsid, is an important aspect of virus architecture. Many single-stranded (ss) RNA viruses organise a significant part of their genome in a dodecahedral cage as a RNA duplex structure that mirrors the symmetry of the capsid. Bruinsma and Rudnick have sug...

The distribution of inequivalent geometries occurring during self-assembly of the major capsid protein in thermodynamic equilibrium is determined based on a master equation approach. These results are implemented to characterize the assembly of SV40 virus and to obtain information on the putative pathways controlling the progressive build-up of the...

A vital part of a virus is its protein shell, called the viral capsid, that encapsulates and hence protects the viral genome. It has been shown in Twarock [2004. A tiling approach to vius capsids assembly explaining a structural puzzle in virology. J. Theor. Biol. 226, 477-482] that the surface structures of viruses with icosahedrally symmetric cap...

In a seminal paper Caspar and Klug established a theory that provides a family of polyhedra as blueprints for the structural organisation of viral capsids. In particular, they encode the locations of the proteins in the shells that encapsulate, and hence provide protection for, the viral genome. Despite of its huge success and numerous applications...

A vital constituent of a virus is its protein shell, called the viral capsid, that encapsulates and hence provides protection for the viral genome. Viral capsids are usually spherical, and for a significant number of viruses exhibit overall icosahedral symmetry. The corresponding surface lattices, that encode the locations of the capsid proteins an...

A vital constituent of a virus is its protein shell, called the viral capsid, that encapsulates and hence provides protection for the viral genome. Assembly models are developed for viral capsids built from protein building blocks that can assume different local bonding structures in the capsid. This situation occurs, for example, for viruses in th...

An important part of a virus is its protein shell, called the viral capsid, that protects the viral genome. While the viral capsids of viruses in the family of Papovaviridae are usually spherical, their protein building blocks are known to assemble also as tubular structures [Kiselev, N.A., Klug, A., 1969. J. Mol. Biol. 40, 155]. In Twarock [2004....

A formalism is developed which allows to determine the locations of all local symmetry axes of three-dimensional particles with overall icosahedral symmetry. It relies on the fact that the root system of the non-crystallographic Coxeter group H_3 encodes the locations of the planes of reflection that generate the discrete rotational symmetries of t...

The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, is shown to be multiplicative for appropriate choices of acceptance windows. This leads to the definition of an associative additive graded composition law and permits the introduction of Lie algebras over such aperiodic point sets. These infinite...

A vital constituent of a virus is its protein shell, called the viral capsid, that encapsulates and hence protects the viral genome. The surface structures of a large number of icosahedral viruses can be modelled via Caspar-Klug Theory, which has hence become one of the fundamental concepts in virology. However, growing experimental evidence have s...

A formulation of quantum mechanics on S1 or its N-point discretisation based on different types of q-deformations of subalgebras of the kinematical algebra of the system was discussed ( [1] ) in the framework of Borel quantisation. We review this method and introduce new Hopf q-deformations of the full kinematical algebra, i.e. q-deformations of bo...

A method for the construction of infinite-dimensional Lie algebras of Virasoro-type is discussed, which uses aperiodic point sets as basic building blocks. The corresponding algebras have generators in a one-to-one correspondence with aperiodic point sets that are obtained via a projection formalism from higher dimensional lattices. They share stru...

A novel approach for the description of the protein stoichiometry of viral capsids, that is the protein shells protecting the viral genome, is introduced based on tiling theory. This approach generalizes Caspar-Klug theory of quasi-equivalence to account also for non-quasi-equivalent subunit arrangements in icosahedral virus capsids that have been...

A family of infinite-dimensional Lie algebras with generators in a one-to-one correspondence with the points of a Penrose tiling is introduced. Central extensions, leading to Virasoro-type algebras, are constructed, and highest weight representations for these algebras are considered. Furthermore, extensions to a super-symmetric setting and thus ap...

The basic features of traffic flow problems can be modelled via the asymmetric simple exclusion process. It is explained how quadratic algebras can be used to obtain the stationary state of the model in a matrix-product form, from which the quantities of interest can be calculated. In particular, we focus on the role of diffusion algebras in this c...

We determine completely all tiles of one-dimensional cut-and-project quasicrystals which have a cubic irrationality as self-similarity factor and are given in terms of connected rectangular acceptance windows. It is shown that there are at least five and at most nine tiles, and they are indicated explicitly depending on the lengths of the two accep...

A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding quantisation method. After a short discussion this method is translated step-by-step to a framework based on difference...

We present results underlining the conjecture that affine extensions for non-crystallographic Coxeter groups are suitable mathematical objects for the description of the symmetries of Carbon onions and Carbon nanotubes. It is the hope that these considerations will shed new light on open questions concerning structure, stability and formation of th...

A. P. Isaev, P. N. Pyatov and V. Rittenberg [J. Phys. A, Math. Gen. 34, No. 29, 5815-5834 (2001; Zbl 1053.16019)] introduced diffusion algebras in the context of one-dimensional stochastic processes with exclusion in statistical mechanics. While that paper is based on the needs of the physicist reader and thus states results without proofs and focu...

Unique affine extensions Haff2, Haff3 and Haff4 are determined for the noncrystallographic Coxeter groups H2, H3 and H4. They are used for the construction of new mathematical models for quasicrystal fragments with tenfold symmetry. The case of Haff2 corresponding to planar point sets is discussed in detail. In contrast to the cut-and-project schem...

Similarly as in the theory of Kac-Moody algebras, affine extensions of the non-crystallographic Coxeter groupsH k, (k=2, …, 4) can be derived via an appropriate extension of the Cartan matrix. These groups lead to novel applications in the theory of quasicrystals and integrable models. In the former case, a new model for quasicrystals with five-fol...

A new type of infinite-dimensional Lie algebra, called aperiodic Virasoro algebra is presented which is by construction compatible with the structure of cut-and-project quasicrystals. The properties of the algebra depend on the acceptance window and thus on the structure of the cut-and-project quasicrystal, and its generators are in a one-to-one co...

We derive a new model of Calogero-Sutherland type based on the aperiodic Virasoro algebra, which is an aperiodic analog of the Virasoro algebra with generators in a one-to-one correspondence with an aperiodic point set. It is shown that the Hamiltonian obtained in this setting contains an additional term with respect to the corresponding model base...

An exactly solvable model of Calogero type is constructed for the finite subgroups of the affinisation H3aff of the noncrystallographic Coxeter group H3 based on their invariants. This demonstrates that the construction scheme based on group invariants can be successfully applied also to groups with non-integrally laced graphs.

An aperiodic analog to the Virasoro algebra is introduced and its representation theory is investigated. In particular, highest and lowest weight representations are constructed. An analog to the Kac determinant formula is derived and the implications for unitarity are discussed. © 2000 American Institute of Physics.

We determine all (statistical) rotation-dilation symmetries of a planar quasiperiodic tiling with fivefold symmetry, with a two-dimensional continuous wavelet transform, using a modified Cauchy wavelet and the scale-angle measure. The tiling is constructed via an affine extension of the Coxeter group H2 and its statistical symmetries were unknown.

In an earlier paper a q-Schrödinger equation was obtained based on a particular quantization procedure, called Borel quantization, and a related q-deformation of the Witt algebra. This q-deformation is a deformation in the category of Lie algebras and hence the corresponding q-Witt algebra has a trivial Hopf algebra structure. In this paper, we ext...

We review and extend the results about quasicrystal Lie algebras of Patera et al. [Phys. Lett. A 246 (1998) 209], which is a new family of infinite dimensional Lie algebras over the real and complex number fields, whose generators are in a one-to-one correspondence with the points of a one-dimensional quasicrystal. Some new properties of quasicryst...

In the method of q-Borel quantization, quantum observables - that is, self-adjoint operators in a suitably chosen Hilbert space - can be assigned a particular set of q-deformed position and momentum observables. In this pattern, the momentum operator appears to be a q-difference operator - and not a differential operator - so that the result is use...

A family of new infinite-dimensional Lie algebras over the real and complex number fields, closely related to the Witt (or Virasoro) algebra, is introduced and some unusual properties are pointed out. Generators of the Lie algebras are in a one-to-one correspondence with the points of a one-dimensional quasicrystal.