Similarities in Self-Assembly of Proteins and Surfactants: An Attempt to Bridge the Gap - Food Colloids: Self-Assembly and Material Science - page 63

In Equation (14), f is the volume fraction of material within the aggregate, r
the size of the constituents building up the aggregate and R the size of the
aggregate. In the case of surfactants, the dimensionality of a planar aggregate is
2, although there exists in this case a particular arrangement due to the
anisotropic nature of the surfactant. In the case of proteins, there is usually
less structural anisotropy present per molecule, which means that one expects
arrangements that are more isotropic. In principle, D may range from 1 to 3,
with D ¼ 1 resembling rods, and D ¼ 3 resembling a close-packed spherical
assembly. The extent to which an aggregate exhibits linearity - and thus the less
it exhibits branching - can be expressed in terms of the closeness to unity of the
value of D. The number of endpoints, defined as the outermost points of a
fractal aggregate, is

n endpoints ¼ DN (D 1)/D . (15)

So, taking into account also the endpoint contribution, we find

m 0 N ¼ð a þ b Þ kT þ ð a b Þ kTD
N 1 = D

; ð 16 Þ

with a ¼ g a h /kT and b ¼ a c s 2 l B k 1 ; the term ( a b )kT denotes the bonding
energy per particle. Again one finds that, if ( a b ) is of the order of 1 or larger,
there is phase separation for N 4 1. If the bonding energy is higher than kT,
only finite aggregates are formed for D ¼ 1, i.e., rod-like structures.
The question remaining is whether we may expect equilibrium structures with
a fractal dimension larger than 1. The idea behind surfactant assembly is that,
also for D 4 1, there can be finite aggregates once the bonding energy is of the
order kT, since there is usually an optimum in the amount of surface area of the
surfactant exposed to the water phase. This optimum in the case of surfactants
is caused by two surface energy contributions per particle, which together can
be minimized, yielding the optimal surface area per surfactant within the
aggregate. Subsequently, it is possible to deduce minimal packing constraints
for the surfactants, thus defining the preferred shape of the aggregate. We
propose to follow an analogous procedure now for proteins.
We assume a number n p of hydrophobic patches with equal surface area a p ;a
patch may be 1-30 nm 2 in size. 9 We assume a number n b of bonds per protein
within the aggregate, i.e., neglecting the endpoints for the moment. The
exposed area of the protein surface to the surrounding solution is then a exp
¼ a n b a p , where a denotes the total surface area of the protein. There exist
two contributions to the chemical potential arising from the exposed area of the
protein within the aggregate to water. Their sum may be approximated by

Dm g ð n p a p a Þþ g a exp þ 4 p q 2 l B k 1 kT
a exp

ð 17 Þ

with q being the number of unit charges. Minimization of Equation (17) with
respect to a exp leads to a minimal exposed surface area a exp given by
a exp ¼ |q|(4 pl B k 1 kT/ g ) 1/2 (18)