Supporting Information
Kinetics versus Thermodynamics in Virus Capsid Polymorphism
1*
Pepijn Moerman , Paul van der Schoot,
1
2,3
and Willem Kegel
1*
Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Utrecht University, The Netherlands
2
3
Institute for Theoretical Physics, Utrecht University, The Netherlands
Department of Applied Physics, Eindhoven University of Technology, The Netherlands
Comparison between Steady-State and Time Dependent Rate of Capsid Assembly.
We model the increase of the capsid density during an assembly reaction as following a steady-state assembly
rate until the free assembly monomer is depleted. After monomer depletion, capsids cannot assemble or
disassemble anymore. The steady-state assembly rate is given by equation 6 according to classical nucleation
theory of virus capsids and depends on the density of free monomers. As the assembly reaction proceeds, the
monomer concentration decreases, which causes the rate of assembly to decrease. The number densities of
capsids at any point in time can be found numerically by solving for the monomer concentration and the
concomitant steady-state assembly rate in each time increment. This implies that we presume the reaction
cascade to respond swiftly to the variation of the pseudo-instantaneous steady state nucleation rates. The fact
that the lag time is small on the time scale of that required for significant increases in complete product,
supports this assumption.
One can consider a simplification to this model, in which the assembly rate only depends on the initial protein
concentration and does not gradually decrease with monomer concentration but stays constant until the
monomer is depleted and the assembly stops. Then the capsid concentrations after assembly are simply
proportional to the initial steady-state assembly rate and can hence be calculated analytically. Figure 5A
depicts the difference between the model where the steady-state assembly rate adapts to the monomer
concentration in each time increment (continuous lines) and the model where it stays constant during the
S1
entire assembly process (dashed lines). Notice that the assembly rates start identical as they should, that is the
slopes in the graph are the same, but the assembly rate in the time incremented calculation slows down.
Figure 5. A) Increase of capsid density with time, calculated using a constant steady-state assembly rate
(dashed lines) and an assembly rate that adapts to the decreasing monomer concentration (continuous line).
Parameters used: εn = -18.5 kBT, n0 = 120, κ = 340 kBT, ν = 10
-2
-1
s
and the initial assembly monomer
5
concentration is 300 µM. The numerical calculations constist of 10 time steps and are stopped when the
monomer concentration reaches 1.3 times the critical monomer concentration. B) Fraction T=4 capsids as
function of initial monomer concentration calculated using a constant steady-state assembly rate (dashed lines)
and an assembly rate that adapts to the decreasing monomer concentration (continuous line). Parameters are
the same as for Figure 5A, except the initial monomer concentration was varied from 1.5 times the critical
assembly concentration to 300 µM. The black line is the same as the red line in Figure 4.
This calculation was repeated for various initial monomer concentrations to obtain the dependence of the
fraction T=4 capsids (NT=4/Ntot) on capsid protein concentration, as shown in Figure 5B. The trends obtained
with the single steady-state assembly rate and the time incremented one are identical and there is only a
marginal quantitative difference, confirming that the use of the constant assembly rate is justified.
S2
The interaction free energy between protein building blocks in hexagonal and pentagonal symmetry.
We consider the assembly process of viral protein building blocks into icosahedral capsids. Initially all building
blocks are identical, but as the icosahedral capsid forms, some proteins are built into hexagons and others into
pentagons. The interaction between proteins in these geometries varies slightly and so does their binding free
energy. Because the pentagon to hexagon ratio varies with capsid geometry, the free energies of assembly of
these capsids must also change. This is probably the difference in free energy that determines in which ratio
polymorphs are formed. Now the question rises whether an average, size dependent free energy of association
can be used to describe the competition, or that a distinction needs to be made between building blocks that
end up in the two different geometries.
In order to address this issue, we first realize that, even though an icosahedral structure contains pentagons
and hexagons, the proteins are positioned on a triangular lattice such that they all make four contacts (the
quasi-equivalence principle). This is illustrated in Figure 6, where geometrical models of different T-number
structures are depicted. The model is based on the Hepatitis B Virus, but also applies to some other small
icosahedral viruses like the Norwalk Virus. The spheres correspond to symmetry elements. This is where all
protein-protein contacts are located. The rods correspond to the assembly monomers, which in the case of the
Hepatitis B Virus are protein dimers. The red rods make two contacts in a five-fold symmetry and two contacts
in a six-fold symmetry. The yellow rods make all four contacts in a six-fold symmetry.
Figure 6 Geometrical models of icosahedral virus capsids based on the Hepatitis B Virus made using Geomag.
The spheres correspond to symmetry elements and the rods represent the (dimeric) protein building blocks.
S3
Yellow rods have all contacts in hexagons whereas two contacts of the red rods are located in a pentagon and
two in a hexagon. The yellow and green rods in the T=7 capsid have the same geometry and have a different
color just for clarity.
Next we note that, since all building blocks have four contacts with their neighbors, the only variation in the
free energy between contacts in pentagons and hexagons can arise from the angle of the interaction. That
variation with angle maybe due to a difference in hydrophobic contact area or it could originate from the free
energies of the different conformations the proteins adopt in order to facilitate the different contact angles. It
is however not clear if the preferred angle is the one in five-fold arrangements, the angle in six-fold
arrangements or an angle in between. Moreover, actual proteins, unlike the building blocks of this geometrical
model, are not rigid sticks, so the proteins themselves could bend to minimize the difference in interaction
angles. This means that the difference in free energy between various capsid sizes is not necessarily located in
the interactions in pentagons, but might be spread out over the entire structure. Therefore it makes sense to
consider an average free energy which relates to the curvature of the entire structure, rather than looking at
the specific angle dependence of the binding free energy.
S4