Question

A simple model for viral capsid assembly In the chapter we considered the equilibrium constant for the assembly of a viral capsid, and its salt depen dence. In experiments this equilibrium constant is usualiy 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 assem. bles from 12 identical pentagonal subunits. When sub. units associate they make favorable contacts along edges with an energy \(\Delta e<0\) 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 assembied structures. The energy for a structure of size \(n\) is given by $$f_{n}=\sum_{m=1}^{m} f_{m} \Delta t_{*}$$ 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 Niot 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 As 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 $$G_{\mathrm{T}}=\sum_{n=1}^{12}\left(N_{n} \varepsilon_{n}+\frac{V}{v_{n}} k_{B} T\left(\phi_{n} \ln \phi_{n}+\left(1-\phi_{n}\right) \ln \left(1-\phi_{n}\right)\right)\right)$$ Now show that by minimizing this free energy with respect to \(N_{n}\), with the constraint that the total number of capsomers is constant fuse a Lagrange multiplier to enforce this constraint) and equal to \(N_{\text {tot. }}\) the volume fraction of intermediates of size \(n\) is given by \\[\phi_{m}=\left(\phi_{1}\right)^{n} \exp \left(-r_{n} / k_{B} T\right)\\] To get this result you will need to use the fact that \(z_{1}=0\) Finally, show that the constraint on the state variable \(N_{n}\) can be rewritten in terms of the volume fractions as \\[\phi_{T}=\sum_{n=1}^{12} \phi_{m}\\] (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 \(\phi\) for which half of all capsomers are in complete capsids is given by \\[\ln (\phi c / 2)=\frac{\varepsilon_{12}}{11 k_{B} T}\\] (c) Carry out a numerical solution for \(\phi_{n}, n=1,2, \ldots, 12\) as a function of \(\phi_{\mathrm{T}}\) and \(\Delta e\). Plot \(\phi_{n}\) as a function of \(n\) for \(\phi_{T}=\phi_{C}\) and \(\Delta x=-1,-5,\) and \(-10 k_{B} T .\) How are the capsomers distributed arnong the 12 different structures the 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 \(\Delta e=-5 k_{g} T ?\) (Problem courtesy of Mike Hagan.)

Short answer

This problem solution involves a number of steps which include understanding the given equations and the concept of volume fractions, deriving the total free energy equation, finding an equation for the volume fraction of intermediates of size \(n\), rewriting the constraint on the state variable \(N_{n}\) in terms of the volume fractions, and deriving an equation for the critical volume fraction. This is followed by a numerical solution and finally interpretation of the results through graph plotting.

Step-by-step solution

Understanding Equations

Firstly, it is important to understand the given equations and the terms used in them. The exercise provides a formula \(f_{n}=\sum_{m=1}^{m} f_{m} \Delta t_{*}\) for the energy (\(f_n\)) of a structure of size \(n\). In this equation, \(f_{m}\) is the number of additional contacts created when a capsomer binds to a structure of size \(n-1\) to form a structure of size \(n\) and \(\Delta t_{*}\) is the energy per edge. A different model is considered for \(f_{n}\) based on the number of connections.

Understanding Volume Fractions

Next it is important to understand the concept of volume fractions (\(\phi_{n}\)). They 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, and \(V\) is the volume of the solution. The total volume fraction of all the capsomers is defined as \(\phi_{T}=N_{T} v_{1} / V\). The problem goal is to compute \(\phi_{n}\) for different values of \(n\) as a function of \(\Delta e\) and \(\phi_{T}\).

Determining the Free Energy

Now, deriving the total free energy of the capsomer solution can be achieved by using the formula \(G_{T}=\sum_{n=1}^{12}(N_{n} \varepsilon_{n}+\frac{V}{v_{n}} k_{B} T(\phi_{n} \ln \phi_{n}+(1-\phi_{n}) \ln (1-\phi_{n}))\). Here, \(k_{B}\) is the Boltzmann constant, \(T\) is the temperature and \(\varepsilon_{n}\) is the energy level. This can be simplified further by minimizing the function with respect to \(N_{n}\), under the constraint that the total number of capsomers is constant and equal to \(N_{T}\), which results in solving differenctial eauations using Lagrange multipliers.

Numerous Partial Fractions

Solving the above minimization problem using the Lagrange multipliers gives the equation for the volume fraction of intermediates of size \(n\), as \(\phi_{m}=(\phi_{1})^{n} \exp (-r_{n} / k_{B} T)\). It is noted that the additional assumption \(z_{1}=0\), a zero offset, is required to solve this.

Expressing Total Volume Fractions

Further, the constraint on the state variable \(N_{n}\) can be rewritten in terms of the volume fractions as \(\phi_{T}=\sum_{n=1}^{12} \phi_{m}\). This gives a method to express the total volume fractions in terms of the number of units in each state.

Critical Value Calculation

Assuming that the only species with significant volume fractions are those of sizes \(n=1\) and \(n=12\), an expression can be derived for the critical volume fraction \(\phi c\) at which half of all capsomers are in complete capsids. Using natural logarithms, this expression is given as \(\ln (\phi c / 2)=\frac{\varepsilon_{12}}{11 k_{B} T}\). It needs to be noted how the total energy and constants appears in the equation.

Numerical Solution

Now the main task is to solve for \(\phi_{n}, n=1,2, \ldots, 12\) as a function of \(\phi_{T}\) and \(\Delta e\). This involves setting up a system of equations according to the derived formulas and solving it numerically.

Interpretation of Results

Once the numerical solution is obtained, the values of \(\phi_{n}\) are plotted as a function of \(n\) for \(\phi_{T}=\phi_{C}\) and \(\Delta e=-1,-5,\) and \(-10 k_{B} T\). From the plots, the distribution of the capsomers among the 12 different structure sizes and the change in the fraction of capsomers in complete capsids with changes in the volume fraction can be analysed.

Key concepts

Equilibrium Constant

The concept of an equilibrium constant is central to understanding many processes in chemistry and biology, including the assembly of viral capsids.

When a virus assembles, it reaches a point where the rate at which it builds up is equal to the rate at which it disassembles. This is akin to a chemical reaction reaching equilibrium. An equilibrium constant, often denoted as K_eq, quantifies the ratio of product to reactant concentrations at this point. For the assembly of viral capsids, capsomers combine to form a capsid, and the equilibrium constant would express the relative amounts of complete capsids to free capsomers in solution.

In biological systems, this equilibrium is affected by various factors, such as temperature and salt concentration – both of which impact the stability and assembly efficiency of capsids. To aid in understanding, thermodynamic models help explain the behavior of these biological systems within the constraints of temperature and other environmental factors.

Thermodynamic Models in Biology

Thermodynamic models are invaluable tools in biology for predicting the behavior of complex systems like the assembly of viral capsids.

These models incorporate the laws of thermodynamics to describe how biological molecules interact under varying conditions. They often utilize parameters such as Gibbs free energy, entropy, and enthalpy to predict how likely a biological process, such as capsid assembly, will occur.

For example, in the viral capsid assembly model from the exercise, the total free energy G_T of the capsomer solution is used to determine the most stable arrangement of capsomers at any given set of conditions. Managing parameters like the free energy helps researchers and students understand why a virus might assemble more efficiently under one set of conditions versus another, aligning with the notion that biological systems strive towards lower energy states.

Viral Capsid Thermodynamics

Viral capsid thermodynamics specifically refers to the energy changes associated with the assembly of a virus's protein shell, or capsid.

The assembly process is driven by forces such as hydrophobic interactions and is counterbalanced by factors like entropy loss when capsomers become ordered in the capsid structure. In the viral capsid assembly model discussed in the exercise, the energies are derived based on the favorable contacts (negative Delta e per edge) between the subunits and the entropy penalty from the loss of translational freedom upon assembly.

The assumptions in these models can be quite sophisticated as they may ignore partially formed intermediates or consider them in detail depending on their concentrations. Overall, the goal is to predict the distribution of free capsomers, partially assembled capsids, and complete structures in the solution, which is essential for understanding viral life cycles and the development of antiviral therapeutics.