Conformational Entropy of a Branched Polymer

Long ssRNA undergoes continuous compaction in the presence of polyvalent cations

In the presence of polyvalent cations, long double-stranded DNA (dsDNA) in dilute solution undergoes a single-molecule, first-order, phase transition ("condensation"), a phenomenon that has been documented and analyzed by many years of experimental and theoretical studies. There has been no systematic effort, however, to determine whether long single-stranded RNA (ssRNA) shows an analogous behavior. In this study, using dynamic light scattering, analytical ultracentrifugation, and gel electrophoresis, we examine the effects of increasing polyvalent cation concentrations on the effective size of long ssRNAs ranging from 3000 to 12,000 nucleotides. Our results indicate that ssRNA does not undergo a discontinuous condensation as does dsDNA but rather a "continuous" decrease in size with increasing polyvalent cation concentration. And, instead of the 10-fold decrease in size shown by long dsDNA, we document a 50% decrease, as demonstrated for a range of lengths and sequences of ssRNA.

Generalized Flory Theory for Rotational Symmetry Breaking of Complex Macromolecules

We report on spontaneous rotational symmetry breaking in a minimal model of complex macromolecules with branches and cycles. The transition takes place as the strength of the self-repulsion is increased. At the transition point, the density distribution transforms from isotropic to anisotropic. We analyze this transition using a variational mean-field theory that combines the Gibbs-Bogolyubov-Feynman inequality with the concept of the Laplacian matrix. The density distribution of the broken symmetry state is shown to be determined by the eigenvalues and eigenvectors of this Laplacian matrix. Physically, this reflects the increasing role of the underlying topological structure in determining the density of the macromolecule when repulsive interactions generate internal tension. Eventually, the variational free energy landscape develops a complex structure with multiple competing minima.

Confining annealed branched polymers inside spherical capsids

The Lifshitz equation for the confinement of a linear polymer in a spherical cavity of radius R has the form of the Schrödinger equation for a quantum particle trapped in a potential well with flat bottom and infinite walls at radius R. We show that the Lifshitz equation of a confined annealed branched polymer has the form of the Schrödinger equation for a quantum harmonic oscillator. The resulting confinement energy has a 1/R4 dependence on the confinement radius R, in contrast to the case of confined linear polymers, which have a 1/R2 dependence. We discuss the application of this result to the problem of the confinement of single-stranded RNA molecules inside spherical capsids.

Adsorption of annealed branched polymers on curved surfaces

The behavior of annealed branched polymers near adsorbing surfaces plays a fundamental role in many biological and industrial processes. Most importantly single stranded RNA in solution tends to fold up and self-bind to form a highly branched structure. Using a mean field theory, we both perturbatively and numerically examine the adsorption of branched polymers on surfaces of several different geometries in a good solvent. Independent of the geometry of the wall, we observe that as branching density increases, surface tension decreases. However, we find a coupling between the branching density and curvature in that a further lowering of surface tension occurs when the wall curves towards the polymer, but the amount of lowering of surface tension decreases when the wall curves away from the polymer. We find that for branched polymers confined into spherical cavities, most of branch-points are located in the vicinity of the interior wall and the surface tension is minimized for a critical cavity radius. For branch polymers next to sinusoidal surfaces, we find that branch-points accumulate at the valleys while end-points on the peaks.

Impact of the topology of viral RNAs on their encapsulation by virus coat proteins

Single-stranded RNAs of simple viruses seem to be topologically more compact than other types of single-stranded RNA. It has been suggested that this has an evolutionary purpose: more compact structures are more easily encapsulated in the limited space that the cavity of the virus capsid offers. We employ a simple Flory theory to calculate the optimal amount of polymers confined in a viral shell. We find that the free energy gain or more specifically the efficiency of RNA encapsidation increases substantially with topological compactness. We also find that the optimal length of RNA encapsidated in a capsid increases with the degree of branching of the genome even though this effect is very weak. Further, we show that if the structure of the branching of the polymer is allowed to anneal, the optimal loading increases substantially.

Flory theory of the folding of designed RNA molecules

We construct a Flory theory for the folding of RNA molecules that have a designed, linear ground state.

Size regulation of ss-RNA viruses

While a monodisperse size distribution is common within one kind of spherical virus, the size of viral shells varies from one type of virus to another. In this article, we investigate the physical mechanisms underlying the size selection among spherical viruses. In particular, we study the effect of genome length and genome and protein concentrations on the size of spherical viral capsids in the absence of spontaneous curvature and bending energy. We find that the coat proteins could well adjust the size of the shell to the size of their genome, which in turn depends on the number of charges on it. Furthermore, we find that different stoichiometric mixtures of proteins and genome can produce virus particles of various sizes, consistent with in vitro experiments.

Statistics of ideal randomly branched polymers in a semi-space

We investigate the statistical properties of a randomly branched 3-functional N-link polymer chain without excluded volume, whose one point is fixed at the distance d from the impenetrable surface in a 3-dimensional space. Exactly solving the Dyson-type equation for the partition function Z(N, d )=N(-theta)e(gamma N) in 3D, we find the "surface" critical exponent theta=[Formula: see text], as well as the density profiles of 3-functional units and of dead ends. Our approach enables to compute also the pairwise correlation function of a randomly branched polymer in a 3D semi-space.

Semianalytical calculation of the Rouse dynamics of randomly branched polymers

We present a semianalytical approach to the determination of the dynamic properties of randomly branched polymers under the Rouse approximation. The principal procedure is based on examining an eigenvalue spectrum which represents the average dynamic behavior of various structures. The calculated spectra show that the eigenvalue distribution is random even within a single structure, which in turn produces a continuous spectrum of values for the entire class. The autocorrelation function for the square of the radius of gyration was calculated based on these spectra, which confirms that the dynamics are nonexponential as earlier reported. A universal stretched exponent is also found in this study.