Peter Prinsen

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, Rg—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 Rg 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 N0.67±0.01, where N is the number of nucleotides, and that their Rg values vary as 〈MLD〉0.5 in an ideal solvent, and hence as N0.34. An alternative analysis, which explicitly includes all branches, is introduced and shown to yield consistent results.

Multishell structures of virus coat proteins

Under conditions of low ionic strength and a pH ranging between about 3.7 and 5.0, solutions of purified coat proteins of cowpea chlorotic mottle virus (CCMV) form spherical multishell structures in the absence of viral RNA. The outer surfaces of the shells in these structures are negatively charged, whereas the inner surfaces are positively charged due to a disordered cationic N-terminal domain of the capsid protein, the arginine-rich RNA-binding motif that protrudes into the interior. We show that the main forces stabilizing these multishells are counterion release combined with a lower charge density in the RNA-binding motif region of the outer shells due to their larger radii of curvature, arguing that these compensate for the outer shells not being able to adopt the smaller, optimal, radius of curvature of the inner shell. This explains why the structures are only stable at low ionic strengths at pHs for which the outer surface is negatively charged and why the larger outer shells are not observed separately in solution. We show how to calculate the free energy of shells of nonoptimal radius of curvature from the elastic properties of the native shell. The spacing between shells is determined mainly by the entropic elasticity of the RNA-binding motifs. Although we focus on CCMV multishells, we also predict the solution conditions under which multishells formed by CCMV coat protein mutants with a lower RNA-binding motif charge are stable, and we examine other viruses as well. We conclude that at a given surface charge density, the boundaries separating regions of stable multishells with different numbers of shells shift to lower ionic strengths upon either increasing the length of the RNA-binding motif, increasing the stiffness of the shells, or decreasing the charge per RNA-binding motif.

The ends of a large RNA molecule are necessarily close

We show on general theoretical grounds that the two ends of single-stranded (ss) RNA molecules (consisting of roughly equal proportions of A, C, G and U) are necessarily close together, largely independent of their length and sequence. This is demonstrated to be a direct consequence of two generic properties of the equilibrium secondary structures, namely that the average proportion of bases in pairs is ∼60% and that the average duplex length is ∼4. Based on mfold and Vienna computations on large numbers of ssRNAs of various lengths (1000–10 000 nt) and sequences (both random and biological), we find that the 5′–3′ distance—defined as the sum of H-bond and covalent (ss) links separating the ends of the RNA chain—is small, averaging 15–20 for each set of viral sequences tested. For random sequences this distance is ∼12, consistent with the theory. We discuss the relevance of these results to evolved sequence complementarity and specific protein binding effects that are known to be important for keeping the two ends of viral and messenger RNAs in close proximity. Finally we speculate on how our conclusions imply indistinguishability in size and shape of equilibrated forms of linear and covalently circularized ssRNA molecules.

Optimized Baxter model of protein solutions: Electrostatics versus adhesion

A theory is set up of spherical proteins interacting by screened electrostatics and constant adhesion, in which the effective adhesion parameter is optimized by a variational principle for the free energy. An analytical approach to the second virial coefficient is first outlined by balancing the repulsive electrostatics against part of the bare adhesion. A theory similar in spirit is developed at nonzero concentrations by assuming an appropriate Baxter model as the reference state. The first-order term in a functional expansion of the free energy is set equal to zero which determines the effective adhesion as a function of salt and protein concentrations. The resulting theory is shown to have fairly good predictive power for the ionic-strength dependence of both the second virial coefficient and the osmotic pressure or compressibility of lysozyme up to about 0.2 volume fraction.

Parity breaking in nematic tactoids

We theoretically investigate under what conditions the director field in a spindle-shaped nematic droplet or tactoid obtains a twisted, parity-broken structure. By minimizing the sum of the bulk elastic and surface energies, we show that a twisted director field is stable if the twist and bend elastic constants are small enough compared to the splay elastic constant, but only if the droplet volume is larger than some minimum value. We furthermore show that the transition from an untwisted to a twisted director-field structure is a sharp function of the various control parameters. We predict that suspensions of rigid, rod-like particles cannot support droplets with a parity broken structure, whereas they could possibly occur in those of semi-flexible, worm-like particles.

Fluid-crystal coexistence for proteins and inorganic nanocolloids: Dependence on ionic strength.

We investigate theoretically the fluid-crystal coexistence of solutions of globular charged nanoparticles such as proteins and inorganic colloids. The thermodynamic properties of the fluid phase are computed via the optimized Baxter model P. Prinsen and T. Odijk [J. Chem. Phys. 121, 6525 (2004)]. This is done specifically for lysozyme and silicotungstates for which the bare adhesion parameters are evaluated via the experimental second virial coefficients. The electrostatic free energy of the crystal is approximated by supposing the cavities in the interstitial phase between the particles are spherical in form. In the salt-free case a Poisson-Boltzmann equation is solved to calculate the effective charge on a particle and a Donnan approximation is used to derive the chemical potential and osmotic pressure in the presence of salt. The coexistence data of lysozyme and silicotungstates are analyzed within this scheme, especially with regard to the ionic-strength dependence of the chemical potentials. The latter agree within the two phases provided some upward adjustment of the effective charge is allowed for.

Collective diffusion coefficient of proteins with hydrodynamic, electrostatic, and adhesive interactions

A theory is presented for lambdaC, the coefficient of the first-order correction in the density of the collective diffusion coefficient, for protein spheres interacting by electrostatic and adhesive forces. An extensive numerical analysis of the Stokesian hydrodynamics of two moving spheres is given so as to gauge the precise impact of lubrication forces. An effective stickiness is introduced and a simple formula for lambdaC in terms of this variable is put forward. A precise though more elaborate approximation for lambdaC is also developed. These and numerically exact expressions for lambdaC are compared with experimental data on lysozyme at pH 4.5 and a range of ionic strengths between 0.05M and 2M.