Similarities in Self-Assembly of Proteins and Surfactants: An Attempt to Bridge the Gap - Food Colloids: Self-Assembly and Material Science - page 58

branches, it also has practical relevance, e.g., because the fibrils act as a vehicle
to form extremely low-weight fraction gels. Another example of a specific type
of protein aggregate is a hollow capsule (serving as a protection device of many
viruses).
In this article, the mesoscopic parameters relevant for describing the aggre-
gate morphology, phase behaviour and topology of surfactant systems will first
be addressed. Then, a similar scheme will be presented for the use of relevant
mesoscopic parameters in describing the morphology and topology of protein
aggregates, for fractal dimensions from 1 to 3. The influences of pH, salt
concentration, salt type and temperature will be considered. Non-equilibrium
effects regarding the particular assembly of proteins into fibrils will also be
addressed. Finally, the similarities and differences regarding surfactant versus
protein assembly will be discussed.

4.2 Surfactant Assembly
Assembly models of surfactants in solution that correctly describe existing
experimental observations are based on a thermodynamic equilibrium descrip-
tion. The fluctuations in sizes of aggregates, the exchange rate of surfactants
between molecules in solution and in the aggregates, as well as the rate
of aggregate formation, are all assumed to be fast in comparison with
the experimental time-scales. We follow here the scheme set out by Israel-
achvili. 1,2
The first step in the scheme of any model of self-assembly is the distinction in
solution between monomers, dimers, trimers, etc., which are all in equilibrium
with one another. Here we denote an aggregate containing a number of n
monomers as an aggregate of size n - or, in short, an n-mer. The second step is
to assume chemical equilibrium and hence to equate the chemical potential of
the surfactant as monomer with the chemical potential of the surfactant within
an n-mer. This leads to the following expression for all aggregates of size N or
M (> 1): 1

kT ln

1
M

X M
M

m 1 ¼ m 1 þ kT ln X 1 ¼ m M ¼ m 0 M þ

kT ln : ð 1 Þ

1
N

X N
N

¼ m N ¼ m 0 N þ

In Equation (1), m 1 denotes the chemical potential of the surfactants as
monomer (i.e., the surfactant in solution), M and N denote different sizes of
aggregate, X N is the mole fraction of surfactants that are present in aggregates
of size n ¼ N, m N the chemical potential of a surfactant within an aggregate of
size N,and m 0 N the self-free energy of the surfactant within an aggregate of size
N. As we assume a dilute solution here, the use of the term kT ln X N is justified,
and the factorization of 1/N ensures that the contribution of each surfactant
within an aggregate of size N is only counted once. We note that mass balance