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

The named bacteriophage, Bacillus virus phi29, infects Bacillus subtilis; the paper's identification of Escherichia coli as its host is incorrect. This correction does not alter the idealized geometry or numerical inputs used below.^[ref]

The stipulated DNA volume is $V_D=20000\mathrm{n}{\mathrm{m}}^3$ and the spherical capsid volume is $V_c=4\pi (20\mathrm{nm}{)}^3/3=33510\mathrm{n}{\mathrm{m}}^3$. ThusThe packed DNA occupies a substantial fraction of the available space: the remaining nominal fraction is $40.3\%$. Compared with the paper's stated $0.71$ reference, it reaches about $84\%$ of that value. This indicates dense confinement, but does not prove crystalline order or specify interstrand spacing. DNA is a connected elongated polymer, rather than a collection of independent hard spheres.

There is also a numerical qualification: the exact densest equal-sphere packing fraction is $\pi /(3\sqrt{2})\simeq 0.74048$, not $0.71$. Using it changes the comparison to about $81\%$ without changing the calculated $0.597$ volume fraction or its physical interpretation.

The polymer contour length and magnitude of total electric charge areUse a uniform charged-sphere model for the packed DNA, a homogeneous dielectric with $ϵ={ϵ}_r{ϵ}_0$, and initially neglect electrostatic screening. Gauss's law gives $E(r)=Qr/(4\pi ϵR^3)$ inside the sphere and $E(r)=Q/(4\pi ϵr^2)$ outside. Integrating the electrostatic energy both inside and outside,For an illustrative aqueous relative permittivity ${ϵ}_r=80$ and $R=20\mathrm{nm}$,The question does not specify permittivity; the result scales as $80/{ϵ}_r$. Using vacuum instead would increase it eightyfold and would not describe an aqueous interior.

Strictly, the work of packaging is a difference of electrostatic energies, rather than the packed self-energy alone. An extended charged rod also has self-energy. With a microscopic cutoff $a$ of order the DNA radius, the thin-rod approximation isOne derivation integrates the pair Coulomb potential along the rod, replacing separation by $\sqrt{z^2+a^2}$: the exact regularized expression is ${\lambda }^2[L\mathrm{arsinh}(L/a)-\sqrt{L^2+a^2}+a]/(4\pi ϵ)$. The diagram's $1\mathrm{nm}$ transverse label suggests $a\simeq 0.5\mathrm{nm}$, giving $U_{\mathrm{rod}}\simeq 6.25\times 10^6\mathrm{pN}\mathrm{nm}$. Hence this explicit reference-state estimate is $\boxed{\Delta U\simeq 1.32\times 10^8\mathrm{pN}\mathrm{nm}}$. The cutoff and precise initial conformation are additional model assumptions, but their correction is small compared with the unscreened sphere value.

The bare packed-sphere estimate exceeds $2.1\times 10^5\mathrm{pN}\mathrm{nm}$ by approximately $659$; subtracting the estimated extended-rod self-energy still leaves a factor of about $629$. The dominant missing effect is electrostatic screening: mobile positive counterions accompany the negatively charged DNA, greatly reducing long-range repulsion. Counterion condensation can further reduce its effective electric charge. Treating the genome's bare charge as uncompensated throughout and outside a $20\mathrm{nm}$ sphere therefore grossly overestimates its long-range electrostatic energy.

An optical tweezers measurement integrates an imposed force over a displacement. A reversible extraction or packaging comparison concerns a free energy difference relative to the unconfined DNA and ionic reservoirs. It can include entropy, bending and hydration contributions; dissipative pulling can add further work. Extraction can also be assisted by stored internal free energy, so the sign of externally supplied work requires the experimental convention. Here the supplied positive value is compared as an energy magnitude.

Adding $40000$ monovalent counterions makes the combined genome-plus-ion electric charge nominally zero. It does not make the local electric field or free energy vanish: opposite charges have spatial distributions, and confining mobile ions costs entropy.

A definite screened continuation of b replaces the bare Coulomb potential by $e^{-\kappa r}/(4\pi ϵr)$, with $\kappa =1/{\lambda }_D$ and ${\lambda }_D=1\mathrm{nm}$. In the Debye–Hückel approximation, a uniformly distributed fixed charge density $\rho =Q/V_c$ satisfies $({\nabla }^2-{\kappa }^2)\phi =-\rho /ϵ$ inside, with zero fixed charge outside and matching potential and radial field at $R$. Its interior solution isThe screened charging free energy $U_{\kappa }=\frac{1}{2}\int \rho \phi dV$ is thereforeFor $R\gg {\lambda }_D$, the bracket approaches one: only charges within roughly a screening length contribute appreciably, yielding the local estimate $U_{\kappa }\simeq Q^2{\lambda }_D^2/(2ϵV_c)$. With the same ${ϵ}_r=80$ as before, $x=20$ gives bracket $0.92519$ andabout $173$ times smaller than the unscreened packed-sphere self-energy. This is a specified continuum estimate of packed-state charging free energy; a full packaging free energy also subtracts the screened unpacked reference and includes ion redistribution. It should not be presented as a unique exact packaging work fixed by the problem's few inputs.

The supplied counterion count also permits an independent estimate of the entropic contribution. For $N$ ideal confined ions, $F_{\mathrm{ion}}=-N{k}_{B}T\ln V+\text{constant}$, so compression from an accessible reference volume $V_0$ to $V_f$ costsFor example, approximate the original screening atmosphere as an annulus of thickness ${\lambda }_D$ around a rod. With $a=0.5\mathrm{nm}$, $V_0=\pi [(a+{\lambda }_D{)}^2-a^2]L=4.27\times 10^4\mathrm{n}{\mathrm{m}}^3$. Taking $V_f=V_c-V_D=1.35\times 10^4\mathrm{n}{\mathrm{m}}^3$ gives $\Delta F_{\mathrm{ion}}\simeq 1.91\times 10^5\mathrm{pN}\mathrm{nm}$. This useful order-of-magnitude ion-entropy model is distinct from the screened charging model; the two estimates must not simply be added as independent terms without a consistent ionic free energy functional. The rod cutoff, fixed DNA volume and screening-atmosphere geometry are rough geometric assumptions, not a microscopic reconstruction of the source sketch.

There is a further consistency limit. The nominal ion number density is $n=N/V_c=1.19\mathrm{n}{\mathrm{m}}^{-3}$. For a uniform ideal monovalent screening gas, ${\lambda }_D^2=ϵk_{B}T/(n{e}^2)$ would give ${\lambda }_D\simeq 0.309\mathrm{nm}$ at $300\mathrm{K}$, rather than $1\mathrm{nm}$; added salt would shorten it further. Thus the stated ion inventory and screening length cannot both be interpreted as a self-consistent dilute uniform screening gas with this dielectric. Bound counterions, nonuniform distributions and an effective phenomenological screening length can change that interpretation. Linear screening is itself only an estimate near highly charged DNA.

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.

For ideal freely mobile counterions, their gross osmotic pressure isIf the DNA volume is excluded from the ion-accessible volume, use $V_f=V_c-V_D$ instead, giving $\Pi \simeq 1.23\times 10^7\mathrm{Pa}\simeq 123\mathrm{bar}$. These are ideal-ion estimates; bound counterions contribute less like independent mobile particles, and the net outward osmotic pressure also subtracts the external osmotic contribution.

Stored free energy associated with this pressure, together with bending and interstrand repulsion, can favor initial genome release when a delivery pathway becomes available. This explains a possible mechanical benefit of dense packaging at a conceptual level. It does not imply that internal pressure necessarily supplies all the work of complete genome delivery into a host, whose own environment and interactions also matter.