Similarities in Self-Assembly of Proteins and Surfactants: An Attempt to Bridge the Gap - Food Colloids: Self-Assembly and Material Science - page 62

Here g is the surface tension between the hydrophobic site of area a h which is
involved in self-assembly, a c the area of interaction per protein involved in the
Coulombic interaction, s denotes the charge per surface area, l B the Bjerrum
length and k 1 the electrostatic Debye screening length. In fact, the electrostatic
contribution, the second term in Equation (12), is based on similar consider-
ations that led to Equation (5). The end effect that is taken into account in
Equation (5) on the basis of the surface area of hydrophobic molecular parts
that are exposed to water is omitted in Equation (12). It is assumed that the
main contribution to the chemical potential due to hydrophobicity arises from
the efficient shielding of hydrophobic sites, giving rise to a negative term.
The expression above can also explain effects of pH as observed with a
closely related protein. The expression for s originates from the Henderson-
Hasselbalch equation 8 according to

s b
1 þ 10 pH pK b

s a
1 þ 10 pK a pH

s ¼ ð 13 Þ

where pK b denotes an effective value for the basic groups of the protein
sequence all taken together, and analogously for pK a .

4.4 Protein Assemblies of Arbitrary Morphology: A
First Attempt
From the surfactant point of view, we may determine a packing criterion
originating from minimizing the sum of the two surface contributions to the
chemical potential. The surfactant can be divided into hydrophobic and
hydrophilic parts, which are clearly separated. The bulk contribution to the
chemical potential due to the replacement of water molecules around hydro-
carbon chains by its own hydrocarbon chains is the driving force for assembly.
The packing criterion and the consequent optimal number of surfactants in a
micelle is determined by the minimization of the surface contribution at finite
N. The first step in testing the model of surfactant assembly is to test the size of
spherical micelles as a function of experimental parameters like salt concen-
tration and temperature.
By analogy with the surfactant approach, the assembly of proteins into
empty virus core shells has been modelled as a case of spherical assembly, as
described in Section 3. We propose, as a first ansatz, a simple extension to
Equation (12) in order to account for non-spherical and/or branched struc-
tures, and at the same time to take into account endpoint defects (i.e., the
endpoints do not fully contribute to the chemical potential, compared to when
an aggregate would be infinite, in the same manner as done in Equation (7) for
aggregates of dimension 1). Taking into account an arbitrary shape, and/or a
degree of branching, we may denote the dimensionality by its fractal dimension
D, defined by

f ¼ (R/r) D . (14)