Cantù, Laura

Corti, Mario

Del Favero, Elena

Elsevier - Publisher Connector 

Consistency of Microstructural Modeling of Micelles: Letter Concerning “Thermotropic Behavior and Stability of Monosialoganglioside Micelles in Aqueous Solution” 

Letter to the Editor
Consistency of Microstructural Modeling of Micelles: Letter Concerning
“Thermotropic Behavior and Stability of Monosialoganglioside Micelles in
Aqueous Solution”
Electronic densities and radii of self-aggregating objects,
like micelles and vesicles, constituted by molecules of
known volume, are not at all independent variables. In
interpreting scattering results on binary or pseudobinary
surfactant systems, this property should always be used.
Improper account of the packing of molecules can lead to
unphysical results, and wild use of the inverse Fourier
transform method, independent of physical constraints, may
produce meaningless results, as we show on the specific
example of a published paper regarding gangliosides:
“Thermotropic behavior and stability of monosialoganglio-
side micelles in aqueous solution,” which appeared in Bio-
physical Journal (1996) 70:1761–1768.
Micelles are made of self-assembled amphiphilic mole-
cules. The most accepted model for their microstructure
(Tanford, 1980) consists of an apolar core containing the
hydrophobic chains surrounded by a polar shell including
hydrophilic headgroups and some solvent molecules. No
solvent penetration of the apolar core is allowed, except for
core surface roughness. The composition of the solvent
outside the polar shell is the bulk composition.
Let us call Va and Vp the apolar and polar volumes of a
micelle of any shape, and a and p the apolar and polar
scattering length densities, equivalent to electronic densities
in the case of X-rays (Stuhrmann, 1978). In addition, let us
call vtail and vhead the tail and headgroup volumes of a single
amphiphile molecule and vsolv the volume of a water mol-
ecule. For any possible aggregation number N of the mi-
celle, the molecular packing for given molecular volumes
requires
Va Nvtail and Vp Nvhead hvsolv
where h is the number of water molecules per amphiphile
molecule included in the outer shell, not necessarily bound
energetically.
These statements concerning the conservation of vol-
umes, although rather obvious, have nevertheless been ex-
plicitly pointed out in the literature by Hayter and Penfold
(1983). Furthermore, even if vtail and vhead cannot be mea-
sured separately, they cannot be arbitrarily chosen, because
they have to be consistent with the specific volume of the
whole amphiphilic molecule, which can be assessed pre-
cisely by density measurements on the micellar solution. In
addition, some reasonable guess about the molecular apolar
volume can often be made, for example after Tanford
(1980) or Small (1986).
The above statements should be taken carefully into
consideration, whatever the chosen procedure in the inter-
pretation of data regarding micellar solutions. Two ap-
proaches are mainly used in the literature. On one hand, a
model of the micelle is made and the theoretical scattering
spectrum is calculated to be compared with the measured
one. In this case it is rather straightforward to account for
molecular constraints, as the monomer can be explicitly
taken as the building unit of a micelle. On the other hand,
the smoothed experimental scattering spectrum is mathe-
matically inverted to give a distance distribution function, a
scattering length density profile inside the micelle, and then
a geometrical representation of it. This last procedure does
not account for molecular constraints, and the results have
then to be checked a posteriori for molecular consistency. A
detailed comparison between the results obtained according
to the two guidelines for the interpretation of scattering data,
taking into account the molecular constraints, has recently
been made on small micelles of a double-chain surfactant
(Arleth, 1997).
The above considerations can be put in other words by
saying that, in any modeling of micellar shape and mass, the
choice of a set of independently adjustable parameters has
to be made in such a way that they are truly independent
from each other, and not connected via molecular con-
straints. For example, aggregation number N and included
water molecules h are suitable to be chosen, whereas radii
and densities are not, as they cannot be assumed as inde-
pendent parameters, once the reasonable and widely agreed-
upon guidelines for the autoaggregation of amphiphiles,
described at the beginning, have been accepted.
We show by the following example that absurd structures
can be proposed if an unsuitable choice of adjustable pa-
rameters is made, without considering molecular con-
straints.
In the paper by Hirai et al. (1996), a GM1 micelle is
proposed to be reproduced by a double-shell prolate (rod-
like) ellipsoid of revolution. Different sets of physicochem-
ical parameters are given at different temperatures and
conditions. As an example, let us consider the set at 6°C:
1. hydrophobic core minor semiaxis ac  26.7 Å; whole
micelle minor semiaxis at  47.5 Å;
2. hydrophobic core axial ratio ARc  1.63; whole micelle
axial ratio ARt  1.53;
3. average scattering density relative to the solvent of the
hydrophobic core  0.573;
Received for publication 1 May 1997 and in final form 30 September 1997.
© 1998 by the Biophysical Society
0006-3495/98/03/1600/04 $2.00
1600 Biophysical Journal Volume 74 March 1998 1600–1603
4. average scattering density relative to the solvent of the
outer shell  1.56.
In addition, the following values are used:
5. average scattering density of the hydrophilic head of the
GM1 molecule  12.3  1010 cm2, equivalent to head
 0.435 electrons/Å3;
6. average scattering density of the hydrophobic tail, the
ceramide, of the GM1 molecule  8.7  1010 cm2,
equivalent to tail  0.308 electrons/Å3;
7. average scattering density of the water solvent  9.4 
1010 cm2, equivalent to solv  0.333 electrons/Å3.
Some values are known from the chemistry of the GM1
molecule:
8. number of electrons of the hydrophobic part, the ceram-
ide, netail  317;
9. number of electrons of the hydrophilic headgroup, nehead
 528.
Some values used by the authors, not explicitly mentioned,
can then be inferred after the ones they quote:
10. volume of the hydrophobic moiety of GM1, from 6 and
8, vtail  317/0.308  1029 Å3;
11. volume of the hydrophilic moiety of GM1, from 5 and
9, vhead  528/0.435  1213 Å3.
Without going into the details of the choices of the
authors, we wish to show the internal inconsistency of their
results, points 1–4, starting from their own assumptions.
First of all, the average scattering densities relative to the
solvent recalled in points 3 and 4 can be expressed in terms
of electron densities to give a  0.573  0.333  0.191
electrons/Å3 for the hydrophobic core of the micelle and
p 1.56 0.333 0.519 electrons/Å3 for the hydrophilic
shell.
At a glance, these values should give a warning about
consistency, as the following observations can readily be
made:
• A core of such very very low electron density (and
density, of course) should be built up by ceramides,
which have the much higher electron density of 0.308
electrons/Å3 (see 6).
• Nothing but highly compressed sugar headgroups should
be allowed to participate to the hydrophilic shell of the
micelle, not even a few water molecules, to have an
electron density that is already significatively higher than
the one quoted in 5 for the headgroup of the individual
GM1 molecule (0.435 electrons/Å3).
In any event, let us assume that the quoted densities are right
and try to draw the microstructure of the GM1 micelle. As
already noticed, in this case no water is allowed into the
hydrophilic shell, which is usually determined by solving
the following equation:
p nehead hnesolv/vhead hvsolv
which expresses that the polar shell volume is made up of
water and headgroups and which reconstructs the average
electronic density of the polar shell p starting from the
numbers of electrons and volumes of the polar headgroup
and solvent molecules. In the present case h comes out to be
negative, as, astonishingly, the determined p is higher than
the one of the headgroup of the individual GM1 molecule.
The quoted geometrical dimensions of the GM1 micelles
recalled in 1 and 2 allow us to calculate the volumes of the
core Va and the shell Vp according to the prolate ellipsoidal
shape:
Va 4/3ac3ARc 129,960 Å3
and
Vp 4/3a t3ARt ac3ARc 556,888 Å3
The total number of electrons in the core is necore  Va 
a  24,692, corresponding to 78 ceramides of 317 elec-
trons each, whereas the total number of electrons in the shell
is neshell  Vp  p  289,025, corresponding to 547
headgroups of 528 electrons each. If the calculation is
carried out on the whole micelle, disregarding the attribu-
tion of volumes and electrons to the hydrophobic or hydro-
philic parts of the GM1 molecules, one finds that the total
volume of the micelle, V Vp Va 686,848 Å3, contains
371 whole molecules with 528 317 845 electrons each.
It is hard to imagine how 78 ceramides can associate with
547 headgroups to make a micelle of 371 whole GM1
molecules. One could argue that not all 371 ceramides
resulting from the global evaluation of the electrons content
of the micelle are wholly embedded in the core; then a
volume corresponding to (371  78)  293 ceramides, that
is, (129,960/78)  293  488,183 Å3, has to be included in
the outer shell, leaving for the sugar headgroups a volume of
only (556,888  488,183)  68,705 Å3, a very shallow
place to host 371 headgroups. In addition, the presence of
tails in the outer shell would impose a dramatic reduction of
its scattering length density, which is against the initial
assumptions of the correctness of its value (0.519 electrons/
Å3, which is already too high).
The same kind of internal inconsistencies are found even
if the purely geometrical features of the proposed micellar
model are considered. In fact, if one compares the molecular
volumes recalled in points 10 and 11 with the already
calculated volumes of the core and shell of the proposed
micelle, one finds that (129,960/1029)  126 ceramides
included in the core combine with (556,888/1213)  459
headgroups included in the outer shell to form a micelle of
((129,960  556,888)/(1029  1213))  306 whole GM1
molecules, revealing a problem of chemical and physical
balance as before, although with different numbers.
No water has been included in this last evaluation, which
is, of course, unphysical, but the paper we are dealing with
gives no explicit value for the water content of the hydro-
philic region of the micelle; as pointed out before, one could
at most deduce from the already criticized proposed electron
Cantu` et al. Consistency of Microstructural Modeling of Micelles 1601
densities that absolutely no water is assumed to be allowed
into the outer shell. Inclusion of water would reduce both
the 459 headgroups and the 306 whole molecules, leaving
their values unmatched.
One could argue that only 126 GM1 molecules form the
micelle, according to the 126 ceramides of the core, so that
only 126 headgroups participate in the outer shell, occupy-
ing a volume of (126 1213) 152,838 Å3, the remaining
(556,888  152,838)  404,050 Å3 being occupied by
water molecules, each one taking the well-known volume of
30 Å3. The number of water molecules comes out to be
(404,050/30)  13,468, that is, 107 for a single GM1
headgroup, which is a notably large amount of water. Of
course, the inclusion of water in the outer shell reduces its
scattering density to a value significantly lower than the one
quoted by the authors. The only way to come out of this
trouble would be to assume for only those water molecules
which are embedded in the hydrophilic shell an electron
density70% higher than usual, corresponding to a specific
gravity of 1.7 g/cm3 and to a molecular volume of less
than 18 Å3, and then to fill up the shell with 180 densest
H2O molecules for each ganglioside headgroup. After all,
the authors of the paper we are dealing with should know
from the literature that the aggregation number of GM1
micelles is not 126 but more than twice as large, as deter-
mined by means of experimental techniques, such as laser
light scattering, which are more straightforward than X-ray
scattering for micellar mass assessment.
As a result, an unphysical micelle is drawn starting from
the physical parameters proposed. Exactly the same incon-
sistencies are present in the rest of the data sets proposed for
ganglioside micelles at different temperatures and for dif-
ferent histories.
Overlooking molecular constraints in the modeling of
micelles is not at all new in the literature, giving rise to
similar problems of inconsistency (for example, in the mod-
eling of the CTAB micelle proposed by Tabony, 1984),
which are avoided by taking into proper account the mo-
lecular volumes, as shown by Cabane and Zemb (1985) and
by Zemb and Charpin (1985).
Unfortunately, as already said, the inverse Fourier trans-
form method (Glatter, 1982), which is additionally used by
the authors of the paper we are dealing with, also ignores
molecules as building units of micelles, so that it is not a
way out of the inconsistency loop. This last method for the
mathematical treatment of the scattering results in the ab-
sence of interactions carries to the determination of an
explicit distance distribution function, and may give directly
the value of the maximum chord inside the scattering object.
Then, in a second step, this distance distribution function is
reproduced, in the case of micelles, with a core-shell model
with no molecular constraints. It is not surprising then that,
also in this case, users are not prevented from getting
inconsistent and even unrealistic values for the micellar
physical parameters, such as densities and dimensions,
which have at least to be checked a posteriori.
Finally, we should say that extending the significant data
to q-values that are as large as possible is always useful, as
it enhances resolution. In any event, most small-angle scat-
tering measurements, both x-ray and neutron, (q 	 0.4
Å1), do not have better resolution than a CH2 group size,
so that the limit between polar and apolar parts of the
micelle is a matter of definition. In addition, it has to be
pointed out that geometrically distinct models predict sim-
ilar shapes for the form factor up to the second and third
oscillations. Therefore, the check with molecular packing
constraints, when feasible as in the case of micellar systems,
is important (Cabane et al., 1985).
Two warnings are then to be given regarding the paper.
One is general, and refers to the correct accounting of
consistency constraints, which have to stand together with
any method for data interpretation. Such a warning is gen-
eral; that is, it applies to both small-angle x-ray and neutron-
scattering data interpretation.
The second warning is specific and refers to the topic of
the paper itself. The evident and nonnegligible problems of
internal consistency, which have been explicitly and exten-
sively shown to affect the data interpretation, clearly pre-
vent the reader from attributing any reliability to the results
and to the picture that has been drawn regarding the ther-
motropic behavior of ganglioside micelles. In addition, be-
sides some details like the questionable choice of the GM1
hydrophilic volume, it should be underlined that ganglio-
sides have been quite well assessed to form oblate (disklike)
rather than prolate (rodlike) micelles, which is obviously
effective in the modeling procedure.
Indeed, ganglioside micelles do exhibit a peculiar and
very interesting thermotropic behavior, including thermal
hysteresis and bistability, which was first accidentally en-
countered (Cantu` et al., 1986; Corti, 1994) and then widely
and deeply investigated with both light and x-ray scattering
techniques. Quite a detailed landscape of results, including
the comparison among different gangliosides together with
a quite complete theoretical interpretation of the observed
behavior in terms of a cooperative conformational transition
of the ganglioside headgroups, can be gained by looking at
the papers by Sonnino et al. (1995), Cantu` et al. (1996a, b),
and Corti et al. (1996).
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Laura Cantu`
Mario Corti
Elena Del Favero
Department of Medical Chemistry and Biochemistry
INFM
University of Milan
Milan, Italy
Monique Dubois
Thomas Zemb
CEA
Gif-sur-Yvette
France
Cantu` et al. Consistency of Microstructural Modeling of Micelles 1603


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Consistency of Microstructural Modeling of Micelles: Letter Concerning “Thermotropic Behavior and Stability of Monosialoganglioside Micelles in Aqueous Solution”

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