A Prüfer-Sequence Based Algorithm for Calculating the Size of Ideal Randomly Branched Polymers.

Branched polymers can be represented as tree graphs. A one-to-one correspondence exists between a tree graph comprised of N labeled vertices and a sequence of N - 2 integers, known as the Prüfer sequence. Permutations of this sequence yield sequences corresponding to tree graphs with the same vertex-degree distribution but (generally) different branching patterns. Repeatedly shuffling the Prüfer sequence we have generated large ensembles of random tree graphs, all with the same degree distributions. We also present and apply an efficient algorithm to determine graph distances directly from their Prüfer sequences. From the (Prüfer sequence derived) graph distances, 3D size metrics, e.g., the polymer's radius of gyration, Rg, and average end-to-end distance, were then calculated using several different theoretical approaches. Applying our method to ideal randomly branched polymers of different vertex-degree distributions, all their 3D size measures are found to obey the usual N(1/4) scaling law. Among the branched polymers analyzed are RNA molecules comprised of equal proportions of the four-randomly distributed-nucleotides. Prior to Prüfer shuffling, the vertices of their representative tree graphs, these "random-sequence" RNAs exhibit an Rg ∼ N(1/3) scaling.

Visualizing the global secondary structure of a viral RNA genome with cryo-electron microscopy.

The lifecycle, and therefore the virulence, of single-stranded (ss)-RNA viruses is regulated not only by their particular protein gene products, but also by the secondary and tertiary structure of their genomes. The secondary structure of the entire genomic RNA of satellite tobacco mosaic virus (STMV) was recently determined by selective 2'-hydroxyl acylation analyzed by primer extension (SHAPE). The SHAPE analysis suggested a single highly extended secondary structure with much less branching than occurs in the ensemble of structures predicted by purely thermodynamic algorithms. Here we examine the solution-equilibrated STMV genome by direct visualization with cryo-electron microscopy (cryo-EM), using an RNA of similar length transcribed from the yeast genome as a control. The cryo-EM data reveal an ensemble of branching patterns that are collectively consistent with the SHAPE-derived secondary structure model. Thus, our results both elucidate the statistical nature of the secondary structure of large ss-RNAs and give visual support for modern RNA structure determination methods. Additionally, this work introduces cryo-EM as a means to distinguish between competing secondary structure models if the models differ significantly in terms of the number and/or length of branches. Furthermore, with the latest advances in cryo-EM technology, we suggest the possibility of developing methods that incorporate restraints from cryo-EM into the next generation of algorithms for the determination of RNA secondary and tertiary structures.

Viral RNAs are unusually compact.

A majority of viruses are composed of long single-stranded genomic RNA molecules encapsulated by protein shells with diameters of just a few tens of nanometers. We examine the extent to which these viral RNAs have evolved to be physically compact molecules to facilitate encapsulation. Measurements of equal-length viral, non-viral, coding and non-coding RNAs show viral RNAs to have among the smallest sizes in solution, i.e., the highest gel-electrophoretic mobilities and the smallest hydrodynamic radii. Using graph-theoretical analyses we demonstrate that their sizes correlate with the compactness of branching patterns in predicted secondary structure ensembles. The density of branching is determined by the number and relative positions of 3-helix junctions, and is highly sensitive to the presence of rare higher-order junctions with 4 or more helices. Compact branching arises from a preponderance of base pairing between nucleotides close to each other in the primary sequence. The density of branching represents a degree of freedom optimized by viral RNA genomes in response to the evolutionary pressure to be packaged reliably. Several families of viruses are analyzed to delineate the effects of capsid geometry, size and charge stabilization on the selective pressure for RNA compactness. Compact branching has important implications for RNA folding and viral assembly.

The assembly pathway of an icosahedral single-stranded RNA virus depends on the strength of inter-subunit attractions.

The strength of attraction between capsid proteins (CPs) of cowpea chlorotic mottle virus (CCMV) is controlled by the solution pH. Additionally, the strength of attraction between CP and the single-stranded RNA viral genome is controlled by ionic strength. By exploiting these properties, we are able to control and monitor the in vitro co-assembly of CCMV CP and single-stranded RNA as a function of the strength of CP-CP and CP-RNA attractions. Using the techniques of velocity sedimentation and electron microscopy, we find that the successful assembly of nuclease-resistant virus-like particles (VLPs) depends delicately on the strength of CP-CP attraction relative to CP-RNA attraction. If the attractions are too weak, the capsid cannot form; if they are too strong, the assembly suffers from kinetic traps. Separating the process into two steps-by first turning on CP-RNA attraction and then turning on CP-CP attraction-allows for the assembly of well-formed VLPs under a wide range of attraction strengths. These observations establish a protocol for the efficient in vitro assembly of CCMV VLPs and suggest potential strategies that the virus may employ in vivo.

Visualizing large RNA molecules in solution.

Single-stranded RNAs (ssRNAs) longer than a few hundred nucleotides do not have a unique structure in solution. Their equilibrium properties therefore reflect the average of an ensemble of structures. We use cryo-electron microscopy to image projections of individual long ssRNA molecules and characterize the anisotropy of their ensembles in solution. A flattened prolate volume is found to best represent the shapes of these ensembles. The measured sizes and anisotropies are in good agreement with complementary determinations using small-angle X-ray scattering and coarse-grained molecular dynamics simulations. A long viral ssRNA is compared with shorter noncoding transcripts to demonstrate that prolate geometry and flatness are generic properties independent of sequence length and origin. The anisotropy persists under physiological as well as low-ionic-strength conditions, revealing a direct correlation between secondary structure asymmetry and 3D shape and size. We discuss the physical origin of the generic anisotropy and its biological implications.

Vaults are dynamically unconstrained cytoplasmic nanoparticles capable of half vault exchange.

Vaults are naturally occurring ribonucleoprotein particles with an enormous interior volume, large enough to encapsulate hundreds of proteins. They are highly conserved and are present in nearly all eukaryotic cells ranging from 10(4) to 10(7) particles per cell. Recombinant vaults can be produced in vitro and engineered to allow cell targeting and protein packaging. These nanometer-sized particles have many desirable characteristics that may give them advantages for use as drug delivery vehicles. Using photoactivatable green fluorescent protein (PAGFP) labeled vaults, we demonstrate that the particles rapidly diffuse throughout the cytoplasm following single pixel photoactivation in live cells. Their in vivo movement remained relatively unchanged despite exposure to a variety of cellular stresses, suggesting that vaults are largely unconstrained in the cytoplasm. Fluorescence resonance energy transfer (FRET) was observed from polyethylene glycol (PEG) fused hybrid cells that expressed either CFP or YFP labeled vaults, indicating that vaults can exchange major vault protein (MVP) subunits in vivo. Investigation into the mechanism of this exchange in vitro using recombinant vaults demonstrated that they were capable of rapidly separating at the particle waist and reassembling back into whole vaults, supporting a half vault exchange mechanism. This data suggests a means whereby vaults can functionally interact with their cellular environment and deliver materials packaged within their interior.

Predicting the sizes of large RNA molecules.

We present a theory of the dependence on sequence of the three-dimensional size of large single-stranded (ss) RNA molecules. The work is motivated by the fact that the genomes of many viruses are large ssRNA molecules-often several thousand nucleotides long-and that these RNAs are spontaneously packaged into small rigid protein shells. We argue that there has been evolutionary pressure for the genome to have overall spatial properties-including an appropriate radius of gyration, R(g)-that facilitate this assembly process. For an arbitrary RNA sequence, we introduce the (thermal) average maximum ladder distance (MLD) and use it as a measure of the "extendedness" of the RNA secondary structure. The MLD values of viral ssRNAs that package into capsids of fixed size are shown to be consistently smaller than those for randomly permuted sequences of the same length and base composition, and also smaller than those of natural ssRNAs that are not under evolutionary pressure to have a compact native form. By mapping these secondary structures onto a linear polymer model and by using MLD as a measure of effective contour length, we predict the R(g) values of viral ssRNAs are smaller than those of nonviral sequences. More generally, we predict the average MLD values of large nonviral ssRNAs scale as N(0.67+/-0.01), where N is the number of nucleotides, and that their R(g) values vary as MLD(0.5) in an ideal solvent, and hence as N(0.34). An alternative analysis, which explicitly includes all branches, is introduced and shown to yield consistent results.

Encapsulation of semiconducting polymers in vault protein cages.

We demonstrate that a semiconducting polymer [poly(2-methoxy-5-propyloxy sulfonate phenylene vinylene), MPS-PPV] can be encapsulated inside recombinant, self-assembling protein nanocapsules called "vaults". Polymer incorporation into these nanosized protein cages, found naturally at approximately 10,000 copies per human cell, was confirmed by fluorescence spectroscopy and small-angle X-ray scattering. Although vault cellular functions and gating mechanisms remain unknown, their large internal volume and natural prevalence within the human body suggests they could be used as carriers for therapeutics and medical imaging reagents. This study provides the groundwork for the use of vaults in encapsulation and delivery applications.

Headgroup percolation and collapse of condensed langmuir monolayers.

We present a study of Langmuir isotherms and 2D bulk moduli of binary lipid mixtures, where changes in monolayer collapse pressure (Pic) are followed while varying the relative amounts of the two components. For monolayers containing dipalmitoylphosphocholine (DPPC) with either hexadecanol (HD) or palmitic acid (PA), a distinctly non-monotonic change in Pic is observed with varying composition. At low mole fractions, there is a slight decrease in Pic as films get richer in DPPC, while a sharp increase to pure DPPC-like values is observed when the mole fraction exceeds approximately 0.7. The sudden transition in collapse pressure is explained using the principles of rigidity percolation, and important ramifications of this phenomenon for biological surfactant are discussed.