(a) Time-resolved light scatter for capsid assembly on nanoparticles with a functionalized surface density of acid groups/. (b) Packaging efficiency or the fraction of nanoparticles incorporated into capsids (measured from TEM micrographs) varies with . The capsid protein-nanoparticle stoichiometric ratio is , the core diameter is 12 nm (which promotes capsid formation), the capsid protein concentration is 0.4 mg/ml ( dimer subunit), and there are 100 mM 1:1 salt and 5 mM 2:1 salt. Packaging efficiencies were measured at . The surface density of acid groups is estimated from the fraction of carboxylated TEG molecules, assuming a total surface density of 3 TEG molecules/ on the nanoparticle surface. Data thanks to Dragnea (Ref. 57).
The model system. Surface functionalization molecules (TEG) are end grafted to an impenetrable gold sphere. The cationic capsid protein N-terminal tails are modeled as polyelectrolytes grafted to the inner surface of a sphere (the capsid), which is impermeable to TEG and polyelectrolyte but permeable to solvent and ions.
The volume fractions of (a) polyelectrolyte segments and (b) carboxylate groups in the grafted TEG layer are shown as a function of layer above the nanoparticle surface for several surface acid group densities with the nanoparticle radius and the layer dimension .
The free energy per area at the capsid surface for the adsorption of a concave polymer brush onto a spherical brush with opposite charge. For varying surface densities of polyelectrolyte , the free energy (relative to ) is shown for nanoparticle functionalization with weak acid groups at acid groups/ (◼), (▲), and strong acid groups at , (○). There are 100 mM 1:1 salt and 5 mM 2:1 salt and .
The equilibrium surface density of adsorbed polyelectrolytes normalized by the surface density of polyelectrolyte on a core with a complete capsid. Predictions are shown for surface functionalization with weak acid (◼) and strong acid groups. The bulk subunit concentration is with other parameters as in Fig. 4.
The electrostatic potential at the nanoparticle surface for varying surface functionalization with strong and weak (◼) acid groups. There are 100 mM 1:1 salt and 5 mM 2:1 salt.
The free energy per area at the capsid surface of an empty capsid due to electrostatic repulsions and ion and solvent entropy is shown as a function of the charge density on capsid subunits, as predicted by SF calculations (solid lines with symbols) and the linearized PB equation (dashed line). The calculations use 100 mM 1:1 salt and 5 mM 2:1 salt.
Packaging efficiencies, or the fraction of nanoparticles incorporated in well-formed capsids, at varying surface charge densities of weak acid groups predicted by (a) the equilibrium theory and (b) the kinetic theory at . Parameters are as listed in Fig. 1 with , and subunit binding free energies are indicated on the plots.
The predicted time dependence of light scattering for functionalization with strong acid groups. [(a) and (b)] Indicated functionalization densities with rate constants (a) and (b) . (c) Indicated surface assembly rate constants with . Other parameters for [(a)–(c)] are , , , and .
The predicted time dependence of light scattering for surface charge groups with and for (a) indicated functionalization densities , (b) indicated functionalization densities with , and (c) indicated surface assembly rate constants with . Other parameters are the same as in Fig. 9.
Estimated light scatter from simulations in which subunits assemble around model nanoparticle with a size mismatch. The preferred empty-capsid morphology is a capsid but the lowest free energy configuration is a capsid assembled around the size nanoparticle. Estimated light scatter is shown as a function of time for different values of the parameter (in units of ), which gives the free energy cost for adopting a subunit conformation consistent with a morphology. As described in Ref. 44, the larger values of lead to the formation of frustrated nanoparticle-associated partial capsids with a morphology, which cannot close around the nanoparticle and block assembly into the lowest free energy configuration. The data are replotted from simulations reported in Fig. 6 of Ref. 44 with the nanoparticle-subunit interaction energy and the simulation time unit is defined in Appendix A.
Packaging efficiencies depend on the capsid protein/core stoichiometric ratio. The predictions of the kinetic theory for the variation in packaging efficiencies with the capsid protein/core stoichiometric ratio at a fixed subunit concentration of are shown for functionalization with weak acid groups with other parameters as in Fig. 8.
Packaging efficiencies depend only weakly on the initial subunit concentration for fixed . The predictions of the kinetic theory for the variation in packaging efficiencies with subunit concentration are shown for functionalization with weak acid groups with other parameters as in Fig. 12.
The kinetic theory predictions (smooth lines) and simulation results (noisy lines) for the time dependence of adsorbed and assembled subunits are shown for the neutral subunit model [Eq. (A1)]. The simulations consider capsids and commensurate cores so a complete capsid is comprised of dimer subunits. Curves at increasing height correspond to reduced subunit densities of , 4.07, 8.14, 20.4, and 40.7 with a surface free energy of and subunit-subunit binding energies of (a) and (b) .
Parameter values used for calculations in this work. The parameters used for capsid assembly are as follows. The number of subunit-subunit contacts are for , for , , and . This choice obviates the need for different nucleation and elongation rate constants (Ref. 31). The nucleus size is . The forward and reverse reaction degeneracies are , for (the average value calculated from simulations in Ref. 45), and , . The binding degeneracy entropy [Eq. (9)] is .
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