In the chapter, we considered the equilibrium constant for the assembly of a viral capsid, and its salt dependence. In experiments, this equilibrium constant is usually determined by measuring the relative amounts of free capsomers and completed capsids using size exclusion chromatography, for example. Furthermore, in the data analysis leading up to the equilibrium constant, one typically assumes that partially formed capsids are present in negligible amounts in solution, and can be ignored. Here we investigate this assumption in the context of a simple version of an equilibrium model for the assembly of icosahedral viral capsids described by Zlotnick (1994). Our model for a capsid is a dodecahedron that assembles from 12 identical pentagonal subunits. When subunits associate, they make favorable contacts along edges with an energy $\Delta \varepsilon<0$ per edge, and pay a translational entropy penalty due to the loss of translational degrees of freedom once the subunits are part of the larger assembly. We assume that assembly occurs through binding of subunits in such a way that the only allowed species are those consistent with all or part of the final dodecahedron product. We seek to evaluate the equilibrium amount of free capsomers, complete dodecahedrons, and partially assembled structures. The energy for a structure of size $n$ is given by $$\varepsilon_{n}=\sum_{m=1}^{n} f_{m} \Delta \varepsilon$$ where $f_{n}$ is the number of additional contacts created when a capsomer binds to a structure of size $n-1$ to form a structure of size $n$. To simplify matters further, we consider the following form for the number of contacts $$f_{n}=\left\{\begin{array}{ll}
1 & (n=2) \\
2 & (3 \leq n \leq 7) \\
3 & (8 \leq n \leq 10) \\
4 & (n=11) \\
5 & (n=12)
\end{array}\right.$$ Note that within this model, $f_{1}=0,$ meaning that individual capsomers set the zero of energy.
To describe the state of the solution containing $N_{\mathrm{tot}}$ capsomers, we make use of the volume fractions $\phi_{n}$ $n=1,2,3, \ldots, 12,$ which are defined as $\phi_{n}=N_{n} v_{n} / V,$ where $N_{n}$ is the number of partially formed capsids made of $n$ capsomers, $v_{n}$ is the volume of each of these structures, while $V$ is the volume of the solution. The goal of the problem is to compute $\phi_{n}$ for different values of $n,$ as a function of $\Delta \varepsilon$ and $\phi_{\mathrm{T}}=N_{\mathrm{T}} v_{1} / V,$ the total volume fraction for all the capsomers in solution.
(a) Using a lattice model for solution as we have done throughout the book, show that the total free energy of the capsomer solution is
\[
G_{\mathrm{T}}=\sum_{n=1}^{12}\left\{N_{n} \varepsilon_{n}+\frac{V}{v_{n}} k_{\mathrm{B}} T\left[\phi_{n} \ln \phi_{n}+\left(1-\phi_{n}\right) \ln \left(1-\phi_{n}\right)\right]\right\}
\]
Now show that by minimizing this free energy with respect to $N_{n},$ with the constraint that the total number of capsomers is constant (use a Lagrange multiplier to enforce this constraint) and equal to $N_{\mathrm{T}},$ the volume fraction of intermediates of size $n$ is given by
\[
\phi_{n}=\left(\phi_{1}\right)^{n} \mathrm{e}^{-\varepsilon_{n} / k_{\mathrm{B}} T}
\]
To get this result, you will need to use the fact that $\varepsilon_{1}=0$ Finally, show that the constraint on the state variable $N_{n}$ can be rewritten in terms of the volume fractions as
\[
\phi_{\mathrm{T}}=\sum_{n=1}^{12} \phi_{n}
\]
(b) Assume that the only species with significant volume fractions are those of sizes $n=1$ and $n=12 .$ Show that in this case the critical value, $\phi_{C},$ of $\phi_{T}$ for which half of all capsomers are in complete capsids is given by
\[
\ln \left(\phi_{\mathrm{C}} / 2\right)=\frac{\varepsilon_{12}}{11 k_{\mathrm{B}} T}
\]
(c) Carry out a numerical solution for $\phi_{n}, n=1,2, \ldots, 12,$ as a function of $\phi_{\mathrm{T}}$ and $\Delta \varepsilon$. Plot $\phi_{n}$ as a function of $n$ for $\phi_{\mathrm{T}}=\phi_{\mathrm{C}}$ and $\Delta \varepsilon=-1,-5,$ and $-10 k_{\mathrm{B}} T .$ How are the
capsomers distributed among the 12 different structures in each of these cases? What happens to the fraction of capsomers in complete capsids as the total volume fraction is varied from below to above $\phi_{\mathrm{C}},$ in the case $\Delta \varepsilon=-5 k_{\mathrm{B}} T ?$ (Problem courtesy of Mike Hagan.)