Eur. Phys. J. E 19, 303{310 (2006) THE EUROPEAN DOI 10.1140/epje/i2005-10071-1 PHYSICAL JOURNAL E Physics of RNA and viral assembly R.F. Bruinsmaa Department of Physics and Astronomy, The University of California at Los Angeles, Los Angeles 90049, CA, USA Received 11 August 2005 / Published online: 23 March 2006 { °c EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag 2006 Abstract. The overview discusses the application of physical arguments to structure and function of single- stranded viral RNA genomes. PACS. 87.15.Nn Properties of solutions; aggregation and crystallization of macromolecules { 61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling { 81.16.Dn Self-assembly The other contributions to this Focus Point discuss how physical descriptions of DNA and RNA can illumi- nate our understanding of cell functioning. Here, the fo- cus will be on dysfunctional nucleic acids, speci¯cally the genomes of infectious viruses. A virus is a shadowy citizen of a borderland between living and dead matter; a par- asite that reproduces only in the environment of a host cell, in fact by the mobilization of the macromolecular machinery of the host. A virus also does not carry out any metabolic activity, which means that, unlike living cells, it e®ectively is in a state of thermal equilibrium with its environment; one reason why physics |and statistical Fig. 1. Image reconstruction of the MS2 virus, viewed along physics in particular| can be a useful tool. Another rea- a 5-fold icosahedral symmetry axis. From reference [2]. son is apparent from an image reconstruction of a virus. Many spherical viruses, such as the MS2 virus shown in Figure 1, exhibit icosahedral symmetry. bacteria (E. Coli in this case), and also something of a Note that the image is oriented along one of the twelve venereal-disease carrier since infection of an E. Coli bac- ¯ve-fold icosahedral axes. MS2 belongs to the T = 3 struc- terium can only take place when it exchanges genetic ma- tural class, an extended group of small RNA viruses that terial with another bacterium. Figure 2 shows a schematic also includes the polio and common cold viruses. The im- cross-section of a T = 3 virus like MS2. age only shows the outer protein shell, or capsid, that The outer radius R¤ of the shell is in the 10 nm range encloses the nucleic acid material. For a T = 3 virus this and the inner radius is about 8 nm. The MS2 genome is capsid consists of precisely 180 proteins. The pioneers of composed of a single ss RNA molecule of about 4000 bases structural studies of viruses |some of which had a back- and contains four genes [3]. The \coat" gene is the code ground in physics such as Francis Crick, Donald Caspar, for the synthesis of a protein that is the main component and Aaron Klug| viewed the capsid as a curved, two- of the MS2 capsid. A second gene encodes the \Repli- dimensional crystal closed on itself, and they borrowed case" protein, a molecular copying machine of viral RNA concepts from crystallography to describe the di®erent molecules. Infection starts when a viral RNA molecule icosahedral tiling patterns of viral shells [1]. is released out of the capsid and injected into the cyto- The genome enclosed by the capsid is composed of one plasm of a host cell. Next, a host Ribosome attaches to the or more DNA or RNA molecules that are either single RNA molecule and synthesizes a Replicase protein from stranded (ss) or double stranded (ds). The focus of this the replicase gene template. This Replicase protein can contribution will be on ss RNA genomes enclosed by a make new copies of the viral genome. As this process is T = 3 capsid, and the MS2 virus will serve as our pro- iterated again and again, more and more Replicase pro- totype. It is a \bacteriophage", i.e. a virus that infects teins and viral RNA molecules are synthesized, as well as capsid proteins. Manfred Eigen studied the population dy- a e-mail: [email protected] namics of this process and found a hyperbolic divergence 304 The European Physical Journal E R* R RNA Protein Fig. 2. Schematic cross-section of a T = 3, single-stranded RNA virus. The 180 protein capsid shell has an outer radius ¤ R of about 10 nm and an inner radius R of about 8 nm. It Fig. 3. A cross-section of the electron density map of the encloses a genome of one or more RNA molecules with a total Flock-House Virus. The density shown in red on-line corre- of about 4000 bases. sponds to the protein capsid, while density in green on-line corresponds to the RNA genome or to disordered protein seg- ments. From reference [7]. of both the viral protein and the viral RNA molecules after a certain delay time [3]. Now, in vitro studies of so- lutions of MS2 capsid proteins and RNA molecules show So not only can we view the viral capsid as a two- that |under the right conditions of pH and salinity| dimensional protein crystal but the viral interior should be they can spontaneously self-assemble into infective MS2 compared with a three-dimensional RNA solid. This RNA viruses [4]. A similar self-assembly process in a host cell solid has unusual structural properties. We noted that the would lead to the production of a swarm of new viruses. capsid has icosahedral crystal symmetry. This same icosa- A third gene, \lysis", generates a protein that stimulates hedral symmetry is imposed as well, to some degree, on the dissolution of the host cell. The swarm is released into the outer part of the genome. Figure 3 shows the recon- the environment, ready to infect other bacteria. struction of the 4800{base-pair ss RNA genome inside the T = 3 Flock-House Virus (FHV), as obtained from an We will focus here on two questions: ¯rst, what are the X-ray di®raction study by the group of J. Johnson [7]. structure and physical conditions of an RNA molecule in- Icosahedral symmetry is imposed on the outer part side a virus and, second, what is the physics of the amazing of the genome, but not on the inner part. Raman spec- protein/RNA self-assembly process. To illustrate the fact troscopy studies [8] indicate that the genome of T = 3 ss that physical considerations can be useful in this respect, RNA viruses largely is in double-helical A-form (about let us estimate the maximum number of genes of a T = 3 70%). Condensation apparently favors the paired state, virus like MS2 [5]. Reasonably, the maximum density of presumably because of the reduced conformational en- an enclosed viral RNA genome is that of RNA molecules tropy of unpaired bases in the crowded environment of the condensed into a crystal by condensing agents such as virus interior. Icosahedral genome patterns are only ob- polyvalents ions. Molecular biologists choose to express served for single-stranded genomes. For instance, the lin- packing densities in units of cubic angstroms per dalton ear, double-stranded genome of ds DNA bacteriophages is (really the inverse of a density). Hydrated RNA crystals arranged in a spool-like, toroidal structure with no icosa- have a packing density Vm of about 2.2 cubic angstrom hedral symmetry at all. per dalton. It follows that we can pack a maximum of It is interesting to compare these neatly folded, icosa- (V=Vm) daltons worth of RNA in an enclosing volume V . hedral RNA packages with the structure of the same viral For a T = 3 virus, this amounts to about 103 kilo daltons 4 3 6 RNA molecule outside the virus. Stretched out, an MS2 (kD), since V = 3 ¼R is about 2 £ 10 cubic angstroms. genome would have a total length of about 1000 nm. The The molecular mass of one RNA nucleotide happens to actual size of the genome molecule in solution is smaller be about 320 daltons, while the RNA code of one gene in- than that, not only because of the usual polymer entropic volves about 700{1000 nucleotides. The molecular mass of elasticity, but also because of secondary and tertiary struc- one gene is then about 200{300 kD so the molecular mass ture formation. Scattering studies of the MS2 genome in of four genes is about 800{1200 kD. solution under physiological conditions [9] indicate that It follows that four to ¯ve genes is about the maxi- it has the shape of a cylindrical particle with a length of mum amount of genetic information that can be packed about 300 nm and a lateral diameter of about 10 nm. The on the genome of a T = 3 virus, an economy of design size of the particle is rather sensitive to the ambient Mg++ that compares rather favorably with the large amount of ion concentration, which is in fact an RNA condensing non-coding DNA we carry around. The key point here is agent. The nucleotide density of viral RNA in solution is that a purely physical constraint appears to be limiting thus about ten times less than that of the interior of the the maximum size of the genetic program that a T = 3 MS2 virus, so a substantial level of RNA compaction is virus can carry out [6]. required during assembly. R.F. Bruinsma: Physics of RNA and viral assembly 305 Fig. 5. Solid line: Hamiltonian cycle on a dodecahedral cage. Bubbles or branch points must be placed at the vertices (ar- rows) in order to accommodate sharp turns.