Past exam of the biological physics course of the University of Oxford / 2021 / 4 / e / Solution

The specified screened-sphere charging model gives $8.00\times 10^5\mathrm{pN}\mathrm{nm}$, about $3.8$ times the supplied experimental magnitude. It is much closer than the unscreened result. The separate ideal counterion-compression estimate gives $1.91\times 10^5\mathrm{pN}\mathrm{nm}$, within about $9\%$ of that magnitude, but this numerical proximity does not validate its rough geometry.

Remaining differences can arise from the unpacked reference self-energy, counterion condensation, ion exchange with the surrounding reservoir, nonuniform charge density, nonlinear electrostatic screening, finite DNA thickness and the actual capsid shape. DNA bending energy, hydration forces and configurational entropy also contribute. The experiment measures a process-dependent work or inferred free energy, whereas the simple screened sphere computes just one model contribution. Neither a zero-net-charge argument nor agreement of one approximate number establishes an exact total energy.