Chemistry Reference
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or
a
exp
/a
¼
|
s
|(4
pl
B
k
1
kT/
g
)
1/2
,
(19)
where we have introduced the surface charge density
s
. In terms of the
minimum number of neighbours, n
b
, we find
h
i
:
n
b
¼
a
a
exp
1
=
2
¼
a
a
p
1
j
s
j
4
pl
B
k
1
kT
=
g
ð
20
Þ
a
p
We assume
l
B
¼
0.7 nm, values of
k
1
in the range 0.25-1.75 nm
(the experimentally existing window for a fibrillar b-lactoglobulin assembly
at pH
¼
2), kT
¼
4
10
21
J,
g
¼
50 mN m
1
and
s
¼
4.2
10
17
(using 21 unit
charges at pH
¼
2, and a radius of 4 nm of the protein). We then arrive at a
value for n
b
ranging between 0.4 a/a
p
and 0.8 a/a
p
for
k
1
ranging between 0.25
and 1.75 nm. Experimentally it is known that fibrils are being formed in the
above-mentioned regime with almost no branches, i.e., n
b
¼
2, implying that
a/a
p
ranges between 2.5 and 5, i.e., 20-40% hydrophobic surface area per
protein, a seemingly reasonable value. From Equation (13) it is clear that the
charge density is also a strong function of pH, which in turn sharply affects the
value of n
b
, and thus the experimentally accessible n
b
as is observed experi-
mentally.
10-14
It is clear that the above equation needs further refinement; it will be deferred
to a future publication. The above represents a first crude estimate to predict
equilibrium aggregate forms of proteins in solution. It needs to be emphasized
that the description applies to an equilibrium aggregate morphology. This
approach may also have merit for describing the behaviour within the so-called
crystallization slot,
15
and beyond, as well as in describing the role of different
salts via the effect on
g
(the Hofmeister series).
One may take Equation (20) further by realizing that n
b
should not decrease
below 1. If this should nonetheless happen, one may use Equation (20) as a
criterion for aggregate formation where the single protein molecule is not the
building block, but instead dimers or n-mers form the building blocks of the
aggregate, since the presence of n-mers effectively decreases the exposed surface
area. Thus, Equation (20) may also be used to identify transitions from single-
stranded fibrils to fibrils having more than one strand wound around each
another, or to fibrils with two strands having a twist with respect to one another
(tubules or other such topologies).
4.5
b
-Lactoglobulin Fibrils: Equilibrium Assembly or
Not?
An example of fibrillar protein assembly is b-lactoglobulin assembling into
fibrils from strands of one or two monomers thick, on heating above 801Cat
pH
¼
2 and low ionic strength.
10-14
The length distribution of the mature fibrils
is a polydisperse single-peaked distribution with a smooth linear decrease to
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