Similarities in Self-Assembly of Proteins and Surfactants: An Attempt to Bridge the Gap - Food Colloids: Self-Assembly and Material Science - page 61

Now that we know how the chemical potential depends on aggregate size, we
may answer how the size depends on the value of a . Using Equations (7)-(9) we
may rewrite Equation (2) as 2

X N ¼ N{X 1 exp a } N exp( a )

(10)

for rod-like structures, while for planar and spherical structures we have

X N ¼ N{X 1 exp a } N exp( a N p ), (11)

with p equal to 1/2 or 2/3 for discs and spheres, respectively. We can see from
Equation (11) that, once a is of the order of unity, X N becomes negligible for
larger N. Indeed, a separate phase is then formed consisting of infinitely large
aggregates. This is the case for a constant a , and where m 0 N is a constantly
decreasing function of N, i.e., the chemical potential is smaller the larger the
aggregate, so favouring infinite aggregates. It is only when m 0 N reaches a
minimum value for finite N that we end up with finite aggregates - and more
of them as the concentration of surfactants increases. This minimum in the
chemical potential can be addressed by Equation (5), from which can be
deduced an optimum head-group area of the surfactant and consequently an
optimal number N per aggregate.
Once the concentration becomes high enough to induce interaction between
aggregates, and as such to influence the chemical potential of the surfactants
again, shape transitions may take place. This occurs, e.g., in the case of SDS
exhibiting a spherical to rod transition at a certain concentration. The success
of the above model can be attributed to an accurate enough description of
the shape of the surfactant and the contributions to the chemical potential
of the surfactant in various circumstances. Once a is large, but with p ¼ 1
[i.e., referring to Equation (10)], X N does not decrease rapidly to zero, and large
aggregates of finite size may occur provided that the structure is rod-like.

4.3 Spherical Protein Aggregates
A class of spherical protein aggregates that are in equilibrium with their
monomeric constituents, and so resemble micelles, are empty virus core shells
(also called as capsids). 4-6 Rather detailed experiments have been undertaken
by Ceres and Zlotnick, 7 and these have been analysed in terms of the theory of
micellization by Kegel and van der Schoot. 4 This analysis successfully explains
the influence of salt and temperature on the micellization of the capsids.
The expression for the chemical potential is based on the consideration that
the hydrophobic patches account for a negative contribution to the chemical
potential, while the electrostatic charge distribution accounts for a positive
contribution. The number of units building up the spherical assembly is set as a
constant. Kegel and van der Schoot 4 arrive at an energy of binding for the
assembly which we rewrite as
N m 0 N ¼ N g a h + Na c kT s 2 l B k 1 + constant

(12)