Determination of the molecular mass and dimensions of membrane proteins
by size exclusion chromatography
Edmund R.S. Kunjia,*, Marilyn Hardinga, P. Jonathan G. Butlerb, Pearl Akaminea
aThe Medical Research Council, Dunn Human Nutrition Unit, Hills Road, Cambridge CB2 0XY, UK
bThe Medical Research Council, Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, UK
a r t i c l e i n f o
Accepted 16 October 2008
Available online 24 October 2008
Size exclusion chromatography
a b s t r a c t
Size exclusion chromatography is an established technique for the determination of hydrodynamic vol-
umes of proteins or protein complexes. When applied to membrane proteins, the contribution of the
detergent micelle, which is required to keep the protein soluble in the aqueous phase, needs to be deter-
mined to obtain accurate measurements for the protein. In a detergent series, in which the detergents
differ only by the length of the alkyl chain, the contribution of the detergent micelle to the hydrodynamic
volume is variable, whereas the contribution of the protein is constant. By using this approach, several
parameters of membrane proteins can be estimated by extrapolation, such as the radius at the midpoint
of the membrane, the average radius, the Stokes radius, and the excluded volume. The molecular mass of
the protein can be determined by two independent measurements that arise from the behaviour of the
free detergent micelle and protein–detergent micelle during size exclusion chromatography and the
determination of the detergent–protein ratio. Determining the dimensions of protein–detergent micelles
may facilitate membrane protein purification and crystallization by defining the accessibility of the pro-
? 2008 Elsevier Inc. All rights reserved.
Size exclusion chromatography or gel filtration separates mac-
romolecules according to their hydrodynamic volume, which is de-
fined by the Stokes radius. Size exclusion columns consist of
porous polymer beads designed to have pores of different sizes.
When a mobile phase is passed through the column, particles with
small hydrodynamic volumes have a longer path length, as they
equilibrate into the pores of the beads more often than those with
large hydrodynamic volumes, which will result in their separation.
Size exclusion chromatography was applied first by Lathe and
Ruthven, who used starch as a matrix to separate sugars, amino
acids and proteins [1,2]. Later, dextran was used as the stationary
phase for separation .
There are many applications of size exclusion chromatography.
In group separations, high or low molecular weight components
are removed from the sample, for example in desalting or buffer
exchange. Another application is the separation of proteins in puri-
fication according to differences in hydrodynamic volume. The
method is considered to be a low-resolution technique, as it does
not discern species with similar hydrodynamic volumes very well,
and thus it is often used as a final purification step. Size exclusion
chromatography is used also as a means to assess whether proteins
are mono-disperse prior to crystallization, because aggregated
material is easily resolved from well-folded soluble protein. Final-
ly, the technique is used to determine the oligomeric state of com-
plexes, as they can be purified under native conditions in which
macromolecular interactions are preserved. The migration in the
size exclusion columns is a function of the hydrodynamic volume,
but macromolecules with the same physical volume but with a dif-
ferent shape, will behave differently. When the technique is ap-
contribution of the bound detergent needs to be accounted for.
For membrane proteins in detergents, sedimentation equilibrium
analytical ultracentrifugation  and static light scattering [5,6]
are used preferably, as they can determine the molecular mass of
the protein without interference of shape and bound detergent,
but these techniques require specialised equipment. We have
shown previously that accurate estimates of the dimensions and
molecular masses of mitochondrial transport proteins can be ob-
tained with size exclusion chromatography by determining the
hydrodynamic parameters in a series of detergents that differ in
the lengths of the alkyl chains . Published work had suggested
that mitochondrial transport proteins are dimeric in detergent
solutions by using small-angle neutron scattering , differential
tagging , size exclusion chromatography [10–12], analytical
ultracentrifugation [11,12], native gel electrophoresis [10,13,14],
and cross-linking experiments [15–18]. Yet, the structure of the
mitochondrial ADP/ATP carrier, a member of the membrane pro-
1046-2023/$ - see front matter ? 2008 Elsevier Inc. All rights reserved.
* Corresponding author. Fax: +44 1223 252875.
E-mail address: firstname.lastname@example.org (E.R.S. Kunji).
Methods 46 (2008) 62–72
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/ymeth
tein family, was monomeric in the membrane  and in deter-
gent . This apparent contradiction prompted us to re-evaluate
the oligomeric state in detergent by size exclusion chromatogra-
phy , sedimentation equilibrium analytical ultracentrifugation
, and differential affinity purification , and in the membrane
by functional studies . All of these studies have shown consis-
tently that yeast ADP/ATP carriers are monomeric in structure and
function. Because the main focus was on determining the oligo-
meric state, we did not expand fully on the central ideas behind
the use of a detergent series in size exclusion chromatography.
Here, the simple mathematical concepts behind relationships are
explained and considerations of shape and size are included. We
have expanded the tested detergents with the CYMAL detergent
series and included quantification of the bound lipid, as a minor
but relevant factor.
2. Considerations for the size exclusion chromatography of
There are very good handbooks available that deal with the the-
oretical and practical aspects of size exclusion chromatography
[23,24], but we will discuss briefly a few practical considerations
when this technique is applied to membrane proteins. Matrices,
such as Sephadex, Superose, Superdex, or Sephacryl, are suitable
for detergent-solubilised membrane proteins, as they are designed
to have minimal interactions. However, aggregated membrane
proteins may bind tightly to the columns. To select a column with
a suitable separation range, it is important to consider the calcu-
lated molecular mass of the membrane protein, its potential oligo-
meric state, and the mass contribution of the bound detergent. As a
guide, typical detergent–protein weight ratios are in the range of
1–5, depending on the volume ratio of the hydrophobic and hydro-
philic domains of the membrane proteins. It is necessary to equil-
ibrate the column with at least two column volumes of buffer
containing detergent at preferably two to three times the critical
micellar concentration (CMC), which is the concentration of deter-
gent above which micelles are formed spontaneously. Although it
is possible to carry out the purification close to the CMC, in practice
many proteins will become unstable as they are starved of deter-
gent, especially when the protein concentration is high and the
free-detergent concentration low, e.g. in the case of low CMC
detergents (tridecyl-maltoside (13M) 0.033 mM). Another possible
complication with low CMC detergents, such as 13M and tetra-
decyl-maltoside, is their poor solubility at low temperatures.
The protein needs to be applied in a small volume (1–2% of col-
umn volume) to obtain sharp peaks, as migration causes peak
broadening. As membrane proteins are difficult to purify at high
concentration, this requirement often means that a concentration
step is introduced prior to loading. However, high concentrations
may promote non-specific interactions, leading to partial aggrega-
tion of the protein. Another consideration is that size exclusion
chromatography can be a delipidating step, as the free detergent
micelles extract lipid from the protein–detergent micelles, unless
lipids are added to the detergent-containing buffers [25,26].
3. Size exclusion chromatography of the ADP/ATP carrier in
alkyl-maltoside and CYMAL detergents
As an example of the procedures, we will discuss the purifica-
tion of the carboxy-atractyloside-inhibited ADP/ATP carrier Aac3p1
by size exclusion chromatography in alkyl-maltoside  and CYM-
AL detergents (Fig. 1A). The elution volumes of the Aac3p-deter-
gent micelles decreased with increasing chain length of the
alkyl-maltoside and CYMAL detergents (Fig. 1B and C), whereas
those of the soluble protein standards were unaffected by the
detergents (data not shown). The dependence of the elution vol-
ume on alkyl chain length of the detergents has been observed
also for the Escherichia coli metal transporter YiiP  and the
osmoregulatory ATP-binding cassette transporter OpuA from Lacto-
coccus lactis (Bert Poolman, personal communication). Free deter-
micelles on the size exclusion column (Fig. 2) and they were pres-
ent at a higher concentration in the sample than in the buffer, be-
cause they were concentrated together with the protein in the
The theoretical relationship between elution volume and
hydrodynamic volume was derived by Killander and Laurent based
on the principle of volume exclusion . The partition coefficient
Kav, which is the fractional volume of the column available to the
protein, is defined by
Kav¼ ðVe? VoÞ=ðVt? VoÞ
where Ve the elution volume of the protein, Vt the total volume
of the packed bed, and Vo the void volume, which is the elution
volume of molecules that pass through the column without retar-
dation, as they are larger than the largest pores of the beads. The
column is calibrated by using standard proteins of known molec-
ular mass and Stokes radius (Fig. 1B and C) and by using blue
dextran to determine the void volume Vo. Calibration curves for
the molecular mass M and the Stokes radius R of the particles
are obtained by plotting for all protein standards logM against
Kav and R against
way, the Stokes radii and molecular masses of the protein–deter-
gent and detergent micelles were determined for all alkyl-malto-
side and CYMAL detergents (Table 1).
In addition, the detergent–protein weight ratios were deter-
mined in decyl-maltoside (10M), undecyl-maltoside (11M), dode-
cyl-maltoside (12M), 13M, CYMAL-6 (C6) and CYMAL-7 (C7) by
measuring the detergent and Aac3p concentrations of the peak
fractions separately (Fig. 2 and Table 1).
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p
[28–30], respectively. In this
4. Why is it preferable to use a detergent series in size exclusion
As shown for the ADP/ATP carrier, the migration of proteins
with relatively small extra-membranous hydrophilic domains is
dependent on the contribution of the detergent micelle (Fig. 1B
and C, and Fig. 3A). If the hydrodynamic volume of the bound
detergent is relatively small compared to that of the protein or pro-
tein micelle, then the migration may be virtually independent of
the bound detergent, as exemplified by ATP-synthase (Fig. 3B).
Thus, for a protein with unknown hydrodynamic volume, a single
observation does not reveal whether the observed Stokes radius
or molecular mass represents mainly the protein or the protein–
detergent micelle. The fact that the elution volume increases with
decreasing alkyl chain lengths, shows that in the series the protein
is the constant contribution to the hydrodynamic volume, whereas
the bound detergent micelles are the variable contributions. There-
fore, it is possible to obtain parameters for the protein by extrapo-
lation. Continuous linear relationships between parameters in the
detergent series indicate that the membrane protein has the same
oligomeric state in all detergents.
Another important consideration is shape. The average rota-
tional behaviour of ellipsoids of revolution, due to Brownian mo-
tion, has been studied in some detail with respect to its effect on
1Abbreviations used: CATR, carboxy-atractyloside; 8M, octyl-b-D-maltoside; 9M,
nonyl-b-D-maltoside; 10M, decyl-b-D-maltoside; 11M, undecyl-b-D-maltoside; 12M,
dodecyl-b-D-maltoside; 13M, tridecyl-b-D-maltoside; C4, CYMAL-4; C5, CYMAL-5; C6,
CYMAL-6; C7, CYMAL-7; AAC1, ADP/ATP carrier 1 from Bos Taurus; Aac3p, ADP/ATP
carrier 3 from Saccharomyces cerevisiae.
E.R.S. Kunji et al./Methods 46 (2008) 62–72
diffusion coefficients, particularly as this affects the sedimentation
coefficient of a macromolecule [31,32]. This relationship is usually
described as the frictional ratio (f/f0), comparing the frictional coef-
ficient of the ellipsoid of revolution with that of an equivalent
sphere. The original calculation of f/f0was by Perrin , who de-
rived the following equations, where qð¼ b=aÞ is the axial ratio for
the ellipsoid of revolution, with b as the equatorial radius and a as
the half axis of revolution:
;whereq<1;i:e: for aprolate ellipsoid
Þ;whereq>1;i:e: for anoblate ellipsoid: ð3Þ
However, according to Stokes 
f ¼ 6pgr
hence it follows that r=r0¼ f=f0and Perrin’s formulae can be used
to calculate the rotationally averaged radial ratios as a function of
axial ratio for oblate and prolate spheroids (Fig. 4). Most membrane
proteins of known structure are prolate spheroids with an axial ra-
tio of less than two, and thus the overestimation of the Stokes ra-
dius due to shape will be less than 5%. In detergent solution, the
detergent band that surrounds the protein will make prolate spher-
oids more spherical, and thus the effect will be even smaller. Thus,
the effects of shape on the determination of the Stokes radius are
negligible for most membrane proteins. In the following, we will
show that several parameters of the protein can be determined
accurately by considering the hydrodynamic behaviour of the pro-
tein in a series of detergents.
5. The Stokes radius of the ADP/ATP carrier
The Stokes radius of the protein–detergent micelle in octyl-
maltoside (8M) is 34.0 Å. Thus, Aac3p is expected to be smaller
than 34.0 Å, as approximately 70 8M molecules are bound, pro-
viding a maximal estimate of the Stokes radius of the protein
(Table 1). The plot of the logarithm of the Stokes radius against
the number of detergent molecules is linear and shows that the
protein is the constant and the detergent micelle the variable
componentoftheStokes radiusin the detergent series
Fig. 1. Elution profiles of ADP/ATP carrier Aac3p in the alkyl-maltoside and CYMAL detergents in size exclusion chromatography. (A) Structures of the detergents from the
alkyl-maltoside and CYMAL series. The carbon atoms of 8M–13M are coloured in red, orange, yellow, green, blue and purple, respectively, whereas those of C4–C7 are shown
in chartreuse, dark blue, dark red and cyan, respectively. (B and C) Size exclusion chromatography traces for Aac3p in alkyl-maltoside and CYMAL detergents, respectively.
Colour coding as in (A). The elution volumes of standard proteins are indicated by triangles in the following order, ferritin (molecular mass 440 kDa, Stokes radius 61.0 Å),
catalase (232 kDa, 52.2 Å), aldolase (158 kDa, 48.1 Å), albumin (67 kDa, 35.5 Å), ovalbumin (43 kDa, 30.5 Å), and chymotrypsinogen A (25 kDa, 20.9 Å).
E.R.S. Kunji et al./Methods 46 (2008) 62–72
(Fig. 5A). The intercept, where no detergent molecules are bound
to the protein, is at ?30 Å, which represents the smallest possi-
ble estimate. This estimate is not for the protein alone, but for a
hypothetical protein–detergent micelle with pentyl-maltoside
(5M) (see below).
Mitochondrial carriers have an axis of three-fold pseudo-sym-
metry, which allows a radial distribution of all atoms to be calcu-
lated (Fig. 5B). In agreement, the atoms of the bovine AAC1 fall
within the minimal and maximal size estimates of the protein–
detergent micelles, showing that Aac3p is monomeric in all of
the detergents in the series.
6. The radius of the protein at the midpoint of the membrane
The Stokes radii of the detergent micelles and protein–deter-
gent micelles increase linearly in the maltosides detergent series,
but their difference values are in the range of 7.4-12.7 Å (Table
1). Given a semi-circular cross-section of the bound detergent
, the longest axis of the protein–detergent micelle is most
likely at the middle of the hydrophobic band on the surface of
the protein, which interacts with the hydrophobic core of the lipid
bilayer. The equatorial axis of the protein–detergent micelle is the
biggest contribution to the hydrodynamic volume. Thus, Rpdmis a
reasonable estimate of the equatorial axis of the protein–detergent
micelle, but an underestimation, as the polar axis is shorter and
thus theequatorial axislonger than
Rpdm? Rdm is most accurate for the smaller detergents, such as
12.6 Å for 8M, 12.7 Å for 9M and 10.3 Å for C4, because the error
is smaller for smaller Stokes radii (Table 1). The difference value
Rpdm? Rdm is broadly independent of the type of detergent and
was determined to be 13.3, 12.3, 12.5 and 10.0 Å for the yeast
ADP/ATP carrier Aac2p purified in 3-laurylamido-N,N0-dimethyl-
propylaminoxide, digitonin, octa-ethyleneglycol mono-n-dodecyl
ether and Triton X-100, respectively . The difference value was
?12 Å for the bovine AAC1 in Triton X-100 . As said above,
Rpdmis an underestimate of the equatorial axis and thus the differ-
ence Rpdm? Rdmis expected to be an underestimation of the equa-
torial radius of the protein. An estimate of the systematic error can
be obtained by representing the protein–detergent micelle by a
spheroid with equatorial axes Req and a polar axis Rdm with the
same volume as a sphere with radius Rpdm. Calculated in this way
Rpdmis an underestimation of the equatorial axis Req by ?17% on
average. The average equatorial radius of the protein at the middle
Fig. 2. Elution peaks of the protein–detergent micelle and the free-detergent
micelle in size exclusion chromatography. Protein determinations and detergent
measurements  for Aac3p purified in 12M are shown in closed and open circles,
respectively. The elution volumes of standard proteins are indicated by triangles in
the following order, aldolase (158 kDa, 48.1 Å), albumin (67 kDa, 35.5 Å), and
ovalbumin (43 kDa, 30.5 Å).
Parameters for the detergent and protein–detergent micelles for the size exclusion chromatography of Aac3p in alkyl-maltosides and CYMAL detergents.
aDetergents. The data for alkyl-maltosides are published  and the data for the CYMAL detergents have been obtained similar methods.
bVdtcalculated van der Waals volume of the hydrophobic tail (Å3), according to . The maltoside head group has a van der Waals volume of ?281 Å3.
cMdmolecular mass of the detergent molecule (Da).
dMdmmass of the free detergent micelle (kDa).
eMpdmmass of the protein–detergent micelles (kDa).
fWdpdetergent to protein weight ratio, based on measurement of the concentrations of detergent and protein in the protein–detergent micelle peak fractions.
gNdmnumber of detergent molecules in the free detergent micelle, according to Mdm/Md.
hNpdmnumber of detergent molecules in the protein–detergent micelle based on Wdp, according to WdpMp=Md, where Mpis the molecular mass of Aac3p.
iRdmStokes radius (Å) of the free detergent micelle.
jRpdmStokes radius (Å) of the protein–detergent micelles.
kVeexcluded volume (kÅ3), based on Eq. (19) and on Ndmand Npdmderived from the molecular masses.
lRav average radius of Aac3p calculated from the excluded volume (Å).
mND not determined, because of low protein yields combined with high detergent concentrations in the buffer arising from high critical micelle concentrations for these
nBased on ðMpdm? MpÞ=Md, when Wdpcould not be determined.
oBased on the weight ratio Wdp.
E.R.S. Kunji et al./Methods 46 (2008) 62–72
of the membrane adjusted in this way is ?15 ± 3 Å. Based on the
coordinates of AAC1 , the average radius on the outside of
the protein at the middle of the membrane is ?14.6 Å (average ra-
dius 13 Å plus van der Waals radius 1.6 Å) (Fig. 5B).
Another estimate of equatorial radius of the protein can be ob-
tained from the relationship between the Stokes radii of the deter-
gent and the protein–detergent micelles (Fig. 6), which gives a
value of ?18 Å at the intercept for both the alkyl-maltoside and
the CYMAL detergent series. This method is not so accurate be-
cause of the long-range extrapolation and will provide an overesti-
mation of the equatorial radius of the protein. However, both
determinations show unequivocally that the yeast and bovine
ADP/ATP carrier are monomeric in the protein–detergent micelles
of the series.
7. Determination of the molecular mass of the membrane
When the molecular mass of the protein–detergent micelle is
plotted against the weight ratio, a linear relationship is observed
(Fig. 7A) that can be explained in the following way. The molecular
mass of protein–detergent micelle Mpdmis the total of the contrib-
Mpdm¼ Mpþ Mblþ Mbd;
where, Mp, Mpland Mbdare the masses of the protein, bound lipid
and bound detergent, respectively.
By definition the detergent–protein weight ratio Wdpis
Combining Eqs. (5) and (6) gives
Mpdm¼ Mpþ Mblþ MpWdp;
indicating that the plot of Mpdmagainst Wdpshould give a straight
line with Mpas the slope and Mpþ Mblas the intercept. Two esti-
mates of Mp can be obtained when the mass of the bound lipid
Mbl is determined. It is important to note that Mpdm and Wdp are
two independently determined parameters, i.e. from the elution
volume by size exclusion chromatography and from the weight ra-
tio by protein and detergent quantifications, respectively. The inter-
cept Mpþ Mblis 39 kDa (Fig. 7A). The molecular mass of the bound
lipid was determined to be 7.6 kDa, consisting of 2.8 cardiolipin and
4.8 phosphatidyl-choline/ethanolamine lipids, which correspond
Fig. 3. Influence of the detergent micelle on the migration of membrane proteins in size exclusion chromatography. Hypothetical elution profiles of (A) the ADP/ATP carrier
and (B) ATP-synthase, as an example of a large micelle, which have small and large volumes compared to the associated detergent micelles, respectively.
Fig. 4. The effect of shape on the determination of the Stokes radius for spheroid
objects with the same volume. (A) Oblate and prolate spheroids with the same
volume and three different axial ratios. (B) The rotationally averaged radial ratios R/
R0as a function of the axis ratio for prolate and oblate spheroids, which are shown
in blue and red lines, respectively.
E.R.S. Kunji et al./Methods 46 (2008) 62–72
well with the numbers observed for the bovine AAC1 [20,36]. Thus,
the estimate of Mpfrom the intercept is 38.7?7.6 = 31.1 kDa, which
corresponds well with the value obtained from the slope, 31.5 kDa.
The molecular mass of modified Aac3p is 35 kDa, and thus the esti-
mates are accurate within ?10%.
The plot of the molecular mass of the detergent micelle against
that of the protein–detergent micelle is also linear, which can be
explained in the following way (Fig. 7B). The molecular mass
Mdmis determined by the number of detergent molecules Ndmin
the micelle and the molecular mass of the detergent molecule Md.
Eq. (8) rearranged gives
Similarly, the molecular mass of the bound detergent Mbd is the
determined by the number of molecules Npdmand Md
Combining Eqs. (5) and (10) gives
Mpdm¼ Mpþ Mblþ NpdmMd:
Combining Eqs. (9) and (11) gives
Mpdm¼ Mpþ Mblþ ðNpdm=NdmÞMdm:
Thus, the plot of Mpdmagainst Mdmshould give a straight line with
Npdm=Ndm as the slope and Mpþ Mbl as the intercept, but only if
Npdm=Ndm is constant for different detergents. Notably, Mpdm and
Mdmare two independent parameters, as they are determined by
elution volume of the protein–detergent micelle and free detergent
micelle in size exclusion chromatography, respectively. Within a
detergent series, the plot of Npdm against Ndm is a straight line
through the origin by approximation (Fig. 8C), and thus empirically
the linear relationship between Mpdmand Mdmholds for the deter-
gent series. The intercept is 36.2 kDa (Fig. 7B), meaning that Mp is
36.2?7.6 = 28.6 kDa, which is not as good an estimate as obtained
from the relationship between Mpdmagainst Wdpfor the reasons sta-
ted above (Fig. 7A). Both independent methods show that Aac3p
was monomeric in all protein–detergent micelles of the series.
Fig. 5. Relationship between Stokes radii and the number of detergent molecules bound to Aac3p. (A) For the alkyl-maltoside (closed symbols) and CYMAL detergents (open
symbols) the Stokes radius of the protein–detergent micelle was plotted against the number of detergent molecules bound to Aac3p, determined from the weight ratio Wdp
(circles) or from ðMpdm? MpÞ=Md, when Wdpcould not be determined (squares) (Table 1). The maximal possible Stokes radius is the Stokes radius of the AAC3-8M micelle,
whereas the minimal possible estimate is the intercept, where the number of bound detergent molecules is zero. (B) Comparison of the Stokes radii of Aac3p-detergent
micelles with the structure of the bovine ADP/ATP carrier AAC1 (PDB entry 1OKC) . The distances to the pseudo three-fold axis of symmetry of atoms are plotted against
the vertical height of the atom. Atoms coordinates of hydrophobic and hydrophilic residues on the surface of AAC1 are shown in orange and blue circles, respectively, whereas
all others are shown as white circles. The average contour of the surface atoms is shown as a black line. The grey area indicates the void of the cavity. The middle of the
membrane is shown as a dashed grey line. The minimal and maximal Stokes radius estimates are shown as dashed and continuous red lines, respectively. The average radius
at the midpoint of the membrane is shown as a green line. The contours of the 8M–13M detergent micelles are coloured in red, orange, yellow, green, blue and purple,
respectively. The detergent micelles are represented by rounded discs with an equatorial radius Rpdmand height 2Rdm.
Fig. 6. Determination of the radius of the protein at the midpoint of the membrane.
The Stokes radii of the free-detergent micelle and protein–detergent micelles are
plotted. The values for the weight ratio for 8M–13M (closed circles), C4–C7 (open
circles) were taken from Table 1. The intercept is where the contribution of the
detergent to the protein–detergent micelle is zero, assuming that the two micelles
have a similar cross-sectional shape and surface area.
E.R.S. Kunji et al./Methods 46 (2008) 62–72
We have shown that it is possible to obtain fairly accurate
molecular mass estimates for Mp, but there are two issues that de-
serve attention. First, Kavis inversely related to the log Mp, mean-
ing that the method is much less accurate for large proteins or
complexes and for large detergent or protein–detergent micelles.
Second, size exclusion chromatography separates particles by
hydrodynamic volume, and proteins, lipids and detergents have
different partial specific volumes v (ml mg?1). For example, the
partial specific volumes are 0.744 , 0.775, 0.820  and 0.981
 for Aac3p, 10M, 12M and phosphatidyl-choline lipids, respec-
tively. Thus, the molecular masses of detergent micelles and pro-
tein–detergent micelles obtained by comparison to protein
Fig. 7. Determination of the molecular mass of the protein. (A) Relationship between the detergent–protein weight ratio and the molecular mass of the protein–detergent
micelle. The values for the weight ratio for 10M, 11M, 12M, 13M (closed circles), C6 and C7 (open circles) were taken from Table 1. When the weight ratio could not be
determined experimentally, the weight ratio was calculated from the mass of the protein–detergent micelle (dashed circles). According to Eq. (7) the slope (black box) is an
estimate for molecular mass of Aac3p and the intercept (white box), where no detergent is bound, is an estimate of the molecular mass of Aac3p plus bound lipid Mpþ Mbl. (B)
Relationship between the molecular mass of the free-detergent micelle and protein–detergent micelles. According to Eq. (12) the slopes (black boxes) are estimates for
Npdm=Ndmand the intercept (white box) for the molecular mass of Aac3p plus bound lipid Mpþ Mbl.
Fig. 8. Properties of detergents in the free-detergent micelles and protein–detergent micelles. (A) Relationship between the van der Waals volume of the hydrophobic tail of
the detergent molecule and the number of detergent molecules in the detergent micelle. (B) Relationship between the head group-tail van der Waals ratio and the curvature,
which is the inverse of the Stokes radius. (C) Relationship between the number of detergent molecules in the free-detergent micelle and protein–detergent micelles. The
values for 8M–13M (closed circles) and C4–C7 (open circles) were taken from Table 1.
E.R.S. Kunji et al./Methods 46 (2008) 62–72
standards are an overestimation. It is possible to get an estimate of
the systematic error by calculating correction factors on the basis
of the partial specific volumes. On this basis, the molecular masses
of the 12M and 10M detergent micelles are overestimated by
approximately 12% and 6%, respectively. The weight contributions
of the three components to the protein–detergent micelles need to
be accounted for to obtain a value for the overall partial specific
vpdm¼ ðvpþ Wdpvdþ WlpvlÞ=ð1 þ Wdpþ WlpÞ;
where, vp, vdand vlare the partial specific volumes of the protein,
detergent and lipid, respectively, and Wdpand Wlpare the deter-
gent–protein and lipid–protein weight ratios, respectively. For the
10M-Aac3p and 12M-Aac3p micelles, the molecular masses are
overestimated by 10% and 6%, respectively. Thus, the molecular
masses for the detergent and protein–detergent micelles by com-
parison to protein standards will be an overestimation of ?10% at
the most. These considerations show another advantage to using
a detergent series rather than a single detergent, as the extrapola-
tion reduces the systematic error induced by differences in the par-
tial specific volumes, because they decrease with decreasing alkyl
chain length (Fig. 7A). When the molecular masses of the free deter-
gent micelle and protein–detergent micelles are compared (Fig. 7B),
both values are overestimated by a similar amount and thus the val-
ues of the slope and intercept are fairly accurate.
8. What have we learned about detergent micelles?
Detergent micelles, whether free or associated with proteins,
are well-defined entities with small variations in the number of
aggregated detergent molecules and in the dimensions, as they
are strictly determined by the physical properties of the detergent
molecules. The peak width of standard proteins is very similar to
that of the detergent and protein–detergent micelles, confirming
this notion. Several parameters influence the formation of micelles.
First, the hydrophobic alkyl chains of the detergents will aggregate
to shield from the water phase. The length and volume of the alkyl
chain will determine the dimensions and the number of detergent
molecules through hydrophobic interactions. The average increase
in the Stokes radius of the detergent micelle Rdmwas 3.1 Å for the
alkyl-maltosides and 3.4 Å for the CYMAL detergents per CH2(Ta-
ble 1). The increase in Rdmis not related directly to an increase in
covalent bond lengths through the addition of CH2, which is about
1.3 Å. The alkyl chains are not in the extended conformation (as of-
ten seen in diagrams of a detergent micelle), but can assume a
range of conformations because of the rotational freedom of the
carbon–carbon bonds. Larger contributing factors to the increase
in Rdmare the increase in the number of detergent molecules in
the micelle (Table 1), and concomitantly, the increase in hydropho-
bic interactions between the alkyl chain lengths, which occur at
larger distances than covalent bonds (?3.5–4.0 Å) and the increase
in hydration, because of a larger surface area. Empirically, the cor-
relation between the calculated van der Waals volume and the
number of detergent molecules in the free detergent micelle is lin-
ear for both types of detergents, showing that the hydrophobic ef-
fect is an important underlying principle for the formation of
micelles (Fig. 8A).
Another important principle for the formation of micelles is the
curvature, which may be determined by the mismatch in volume
between the detergent head group and the hydrophobic tail. The
correlation between the calculated head group-tail volume ratio
and the curvature, which is the inverse of the Stokes radius, is
empirically linear (Fig. 8B). This relationship will hold only for mal-
toside-containing detergents, as the chemical interactions and
hydration of the head group are the same for each detergent in
the series. The pressure of the surrounding water is another factor
that will induce spherical shapes for micelles.
The hydrophobic effect and the volume ratio may explain the
physical limits of micelle formation. Micelles of the detergent ser-
ies have strictly defined sizes, as they may be limited by the curva-
ture, which is a function of the volume ratio. Excluding the
potential contribution of water, the calculated volume ratios of
heptyl-maltoside (7M), hexyl-maltoside (6M) and pentyl-malto-
side (5M) are 2.189, 2.530 and 2.996, respectively. The relation-
ships in Figs. 5A, 7A and 8A and C all suggest that 7M, 6M and
CYMAL-1 may form micelles, but 5M may not. Where the number
of detergent molecules in the free detergent micelle is zero, the
volume of the hydrophobic tail is ?100 Å3(Fig. 8A). The calculated
volumes of the hydrophobic tails of CYMAL-1, 6M and 5M are 116,
111 and 93 Å3, respectively. On this basis 5M, which has a volume
ratio of ?3, may not form a micelle.
The calculated volume ratios for 13M and C7 are 1.210 and
1.278, respectively. It is possible that detergents with volume ra-
tios close to 1 could form disc-like micelles, which may have the
Do the CYMAL detergents differ from the alkyl-maltosides in
their physical properties? For the same number of carbon atoms
in the hydrophobic tail (10M–C4, 11M–C5, 12M–C6 and 13M–
C7) the free detergent micelles of the CYMAL series have smaller
Stokes radii (Table 1). The reason is that the terminal hexyl group
has a smaller van der Waals volume than the equivalent alkyl
group of six carbon atoms (Fig. 8A). The volume ratio is smaller
for alkyl-maltosides than for CYMAL detergents with the same
number of carbon atoms, meaning that the curvature is smaller
and the Stokes radius larger (Fig. 8B). The relationships between
the volume of the hydrophobic tail versus the number of detergent
molecules in the free detergent micelle, and the curvature versus
the volume ratio are very similar for both types of detergents,
showing that the underlying physical properties do not differ.
Where the two types of the detergents may differ in a more subtle
way is in packing arrangements, as the hexyl ring of CYMAL deter-
gents is more rigid than the equivalent alkyl group, which may af-
How are the free detergent micelles and the protein–detergent
micelles related? The ratio of the number of detergent molecules in
the protein–detergent micelles and in the free detergent micelle is
?1.3 for the alkyl-maltosides and ?1.1 for the CYMAL detergents
(Table 1 and Fig. 8C). The relationship between the number of
detergent molecules in the free detergent micelle and the pro-
tein–detergent micelle passes through the origin by approximation
(Fig. 8C). Taken together, these observations indicate that the
detergent molecules in the protein–detergent micelle have similar
properties as those in the free detergent micelle. In other words,
the protein does not affect the micelle very much other than by
increasing the number of bound molecules by a factor of ?1.2.
Thus, the increase in the number of detergent molecules is a func-
tion of the radius and the hydrophobic surface area of the protein,
as argued previously . But rather than a monolayer model ,
the shape of a protein–detergent micelle is likely to be an annulus
with a half-circular cross-section, as shown by coarse-grain model-
ling [40,41] and by neutron diffraction . Mathematically, the
two types of micelles differ only in the radius of the annulus, which
is zero in the free-detergent micelle and the radius of the protein in
the protein–detergent micelle. These models explain the difference
in the number of detergent molecules between the two types of
micelles and the effect on volume well  (see also below). The
relationships show that the interactions between the detergent
molecules are the dominating factors in assembly rather than the
interactions between the detergent molecules and the protein.
The curvature is fixed by the difference in volume between the
E.R.S. Kunji et al./Methods 46 (2008) 62–72
head group and hydrophobic tail (Fig. 8B), and thus it is likely that
the cross-sections are semi-circular in the protein–detergent mi-
celle also. The inability to adapt to the protein may explain differ-
ences in the stability of Aac3p in the different detergents, which
can be deduced from the yields in purification starting from the
same amount of material. The purification yield decreased in deter-
gents in the order 10M, 11M, 12M, 13M, 9M and 8M (Fig. 1). In the
case of 8M, the head group is forced to interact with the hydropho-
bic surface of the protein, whereas in the case of 13M the alkyl
chain is interacting with the hydrophilic parts of the protein to
some extent (Fig. 9). Another reason is that the 8M micelle is not
able to cover fully the hydrophobic surface of the protein. The opti-
mal arrangement can be observed for 10M (Fig. 9).
9. The volume excluded by the protein in the protein–detergent
Using the mathematical descriptions of the two types of mi-
celles and the difference in number of detergent molecules as a ba-
sis, it is possible to obtain an estimate of the excluded volume of
the protein Ve, which is the volume of the protein plus any other
volume excluded by the protein, such as an aqueous cavity. The
volume of the protein–detergent micelle Vpdmconsists of the sum
of volumes taken up by the bound detergent and the protein. The
contribution of the bound detergent to the volume can by ex-
pressed as the number of bound detergent molecules Npdmtimes
the volume Vdthat a single detergent molecule takes up.
Vpdm¼ Veþ NpdmVd;
Vdof a single detergent molecule is also given by
As the number of associated detergent molecules is known for
both types of micelles (Table 1), the contribution of the detergent
to the volume of the protein–detergent micelle can be expressed
in terms of the volume of the detergent micelle  by rearranging
Eqs. (14) and (15),
By approximation, the detergent micelle is a sphere and the pro-
tein–detergent micelle a rounded disc [35,40,41], assuming that
the protein is circular in cross-section. The volume of the free-
detergent micelle is
The volume of the protein–detergent micelle is :
3p ?p2þ 2p
From Eqs. (16)–(18), it follows
dmþ ðp2? 4pÞR2
3p ?p2þ 2p
dmþ ðp2? 4pÞR2
The volume excluded by the protein Vecan be represented by a
cylinder with an average radius Ravand height 2Rdm.
Fig. 9. Hydrophobic mismatch of the detergents in the micelle with the protein. The surface representation of AAC1  is shown with hydrophobic amino acid residues
(leucine, valine, isoleucine, tryptophan, methionine, cysteine, phenylalanine and alanine) in orange. The 8M, 10M and 13M detergent molecules are shown in sphere
representation for size comparisons with the associated micelles and the protein.
Fig. 10. Dimensions of the detergent micelles and the Aac3p-detergent micelles for
alkyl-maltoside and CYMAL detergents. (A and B) Detergent micelles and (C and D)
Aac3p-detergent micelles approximated by rounded discs. (A and C) The boundaries
of the micelles for 8M–13M are in red, orange, yellow, green, blue and purple,
respectively, and (B and D) for the C4–C7 in chartreuse, dark blue, dark red and
cyan. The aggregation numbers for the two types of micelles are based on the
molecular masses. For size comparisons the surface representation of AAC1  is
shown. Hydrophobic amino acid residues (leucine, valine, isoleucine, tryptophan,
methionine, cysteine, phenylalanine and alanine) are orange. For 8M (C) and C4 (D),
the excluded volume is represented by a cylinder with radius Rav, shown in
projection (Table 1 and Eq. (20)). The difference in excluded volume between
successive micelles is represented by two cylinders at the interface above and
below, each with height DRdm. The volumes that are not excluded by the protein in
the protein–detergent micelle are coloured blue, most likely representing bound
lipid and water cavities.
E.R.S. Kunji et al./Methods 46 (2008) 62–72
For the alkyl-maltosides 8M–13M and CYMAL detergents C4–
C7, the excluded volumes increased with increasing chain length
from 67 to 90 kÅ3and 84 to 93 kÅ3, respectively (Table 1). The
average radius of the cylinder, approximating the protein, was in
the range 19.7–22.6 Å or 20.3–22.8 Å, respectively. The average
outside radius of AAC1, based on the structure, is approximately
18 Å (Fig. 5B), which is in fairly good agreement with these esti-
mates, which were based on four independent parameters (Eq.
(19)). These approximations do not take into account the volume
of bound lipid or the underestimation of the equatorial radius by
using Rpdm. The former will reduce the estimated volume of the
protein and the latter will increase it.
The differences in excluded volumes for successive detergent
micelles in the alkyl-maltoside and CYMAL detergent series can
be used to calculate a symmetrised representation of the protein
in the form of a set of cylinders, which follows the contours of
the protein quite well (Fig. 10). All the remaining volumes not ex-
cluded by the protein in the larger protein–detergent micelles
most likely represent bound lipid or water cavities, since the head
groups cannot reach around the protein to close the micelle fully in
the centre (‘donut’ shaped micelles) (blue areas in Fig. 10). Of
course, the determination of the shape of the protein in this way
works well for mitochondrial carriers, because they are highly
symmetrical barrels .
The volume and dimensions of the protein–detergent micelles
predict that in some detergents the protein is not exposed
(Fig. 10). In the development of the first purification procedures,
it was observed that the bovine AAC1 in Triton X-100 does not bind
to hydroxylapatite columns, which are used in ion exchange chro-
matography . The dimensions of the Triton and AAC1-Triton
micelles indicate that the protein is not exposed to column mate-
rial, explaining why it does not bind. To test this hypothesis, the
same amount of AAC1 was solubilised and loaded onto a cation ex-
change column in 9M, 10M, 11M and 12M, which are predicted to
expose the protein consecutively less (Fig. 11A). The chromato-
graphic profiles show that the flow-through fraction in 11M and
12M contain more mitochondrial protein than in 9M and 10M
(Fig. 11B), whereas the converse is true for the elution fractions.
Western blots show that AAC1 is bound to the column in 9M and
to a lesser extent in 10M, whereas it does not bind to the column
in 11M and 12M (Fig. 11C), in agreement with the predictions
(Fig. 11A). These results show that the dimensions of the micelles
based on the determined Stokes radii are fairly good estimates and
that the micelles are not highly dynamic entities, exposing the sur-
face intermittently. There are only small variations in the dimen-
sions of the protein–detergent micelles, as only 5–10% of the
population shows the opposite of the expected behaviour.
Fig. 11. The dimensions of the bovine AAC1-detergent micelles in 9M–12M as probed by cation exchange chromatography. (A) The bovine AAC1-detergent micelles are
represented by rounded discs with an equatorial radius Rpdmand height 2Rdm. A model of the AAC1 with electrostatic charges is shown in surface representation. (B) The UV
profiles for cation chromatography of AAC1 in 9M, 10M, 11M and 12M (colour of the lines as in the legend of Fig. 1). Bovine mitochondria membranes (3 mg) in 20 mM HEPES
pH 7.5, 1 mM EDTA were incubated with 20 lM CATR for 10 min. Detergent was added (4% 9M, 4% 10M, 2% 11M or 2% 12M) and the sample was incubated for 10 min at 4 ?C,
after which the insoluble fraction was removed by centrifugation (MLA-130) at 20,000g for 10 min. Approximately 1.4 mg of solubilisate was loaded onto a 1-mL HiTrap
Mono SP column, equilibrated with 20 mM HEPES pH 7.5, 1 mM EDTA, containing either 0.7% 9M, 0.2% 10M, 0.05% 11M or 0.03% 12M. After collection of the flow-through (F),
the bound protein was washed with 10 ml of buffer (W) and eluted with 10 ml of buffer with 500 mM NaCl (E), all at 0.5 ml min?1. (C) Western blots of the solubilisate S,
flow-through F and elution E fractions in 9M–12M. The protein amounts in the fractions were related to those of the solubilisate (?20 lg) by correcting for differences in
volume. Proteins were separated on a 12% sodium-dodecyl sulfate polyacrylamide gel and transferred onto polyvinylidene difluoride membranes (Millipore) by using a
BioRad Semi-dry blotter at 240 mA for 90 min. The membranes were blocked with milk protein and incubated with anti-AAC chicken antibody (1:25,000) and rabbit a-chick
IgY peroxidase conjugate (Sigma) (1:25,000). The blot was developed by using chemiluminescence (ECL) (G.E. Healthcare).
E.R.S. Kunji et al./Methods 46 (2008) 62–72
10. Detailed protocols
10.1. Purification of yeast Aac3p
Isolated yeast mitochondria (600 mg of protein), expressing
Aac3p with nine histidines,  were incubated for 20 min at 4 ?C
with the inhibitor carboxy-atractyloside (CATR) (Calbiochem) at a
concentration of 2 nmol/mg of protein. Membrane proteins were
solubilized in 100 ml of 1% 12M (Anatrace) dissolved in a buffer
consisting of 20 mM imidazole, 150 mM NaCl, 10 mM Tris, pH
7.4, and two tablets of complete protease inhibitor minus EDTA
(Roche Diagnostics Ltd.) by mixing at 4 ?C for 45 min. Particulate
material was removed by ultracentrifugation at 140,000g at 4 ?C
for 35 min. Chromatography was performed with an ÄKTAprime
(GE Healthcare), but most high performance liquid chromatogra-
phy systems would be suitable. The soluble fraction was loaded
onto a Ni-Sepharose (High performance) column (Amersham Bio-
sciences) at 1 ml min?1. The column was washed with 50 column
volumes of buffer containing 40 mM and 50 column volumes of
buffer containing 60 mM imidazole and 0.1% 12M, 150 mM NaCl,
10 mM Tris pH 7.4 and 1 lM CATR. Bound protein was eluted with
300 mM imidazole, 0.1% 12M, 150 mM NaCl, 10 mM Tris pH 7.4,
1 lM CATR, at a flow rate of 0.3 ml min?1. 12M was exchanged
for other detergents on the Ni-Sepharose column in the same buf-
fers but with detergents at the following concentrations; 13M 0.1%,
11M 0.1%, 10M 0.2%, 9M 0.7%, 8M 1.8%, C4 0.9%, C5 0.3%, C6 0.1% or
C7 0.05% (Anatrace). The protein was concentrated to 1–2 ml with
a Centricon YM30 (Amicon) at 4800g and 4 ?C and the concen-
trated sample was loaded onto a Superdex 200 XK16/60 column
(Amersham Biosciences) (flow rate 0.5 ml min?1) pre-equilibrated
with gel filtration buffer (0.1% 12M or other alkyl-maltosides or
CYMAL detergents with concentrations as above, 10 mM Tris, pH
7.4, 150 mM NaCl and 1 lM CATR). The Superdex 200 column
was calibrated with aldolase, albumin, ovalbumin and chymotryp-
sinogen A (Amersham Pharmacia Biotech).
10.2. Determination of protein, detergent and lipid concentration
Protein concentrations were determined by BCA assay (Pierce)
and by amino acid analysis (Department of Biochemistry, Cam-
bridge University). The detergent concentrations were determined
by a colorimetric assay that detects the sugar component of the
detergent . Lipid analysis was carried out by thin layer chroma-
tography. Purified protein samples (100 lg) were extracted with
20 volumes of CHCl3/MeOH (2:1) plus 0.25% HCl. The sample was
vortexed and left to stand for 15 min prior to the addition of 4 vol-
umes of 0.05 M CaCl2. Following centrifugation, the lower layer
was removed and the upper layer re-extracted. The lower layers
were combined and then washed with 50 mM Tris–HCl pH 8.25,
0.1 M NaCl, 0.1 mM EGTA. The sample was evaporated under nitro-
gen and dissolved in CHCl3. Lipid extracts (10 ll) were spotted
onto a TLC plate (silica gel 60 F254, Merck) along with standards.
Separation wasobtained in
(160:20:4:1.5, by volume). Detection was carried out using a char-
ring solution (7.5%w/v copper acetate, 2.5% w/v copper sulphate,
8% v/v phosphoric acid) and incubating at 160 ?C for 10–20 min
until the appearance of spots. The spots were quantified by densi-
tometry with background correction.
We thank the European consortium on Membrane Proteins
(EMeP) and the Medical Research Council UK for funding of this
 G.H. Lathe, C.R. Ruthven, Biochem. J. 60 (1955) xxxiv.
 G.H. Lathe, C.R. Ruthven, Biochem. J. 62 (1956) 665–674.
 J. Porath, P. Flodin, Nature 183 (1959) 1657–1659.
 P.J.G. Butler, C.G. Tate, in: D.J. Scott, S.E. Harding, A.J. Rowe (Eds.), Analytical
Ultracentrifugation: Techniques and Methods, Royal Society of Chemistry
(Great Britain), Cambridge, 2005, pp. 133–149.
 Y. Hayashi, H. Matsui, T. Takagi, Methods Enzymol. 172 (1989) 514–528.
 D.J. Slotboom, R.H. Duurkens, K. Olieman, G.B. Erkens, Methods 46 (2008) 73–
 L. Bamber, M. Harding, P.J. Butler, E.R.S. Kunji, Proc. Natl. Acad. Sci. USA 103
 M.R. Block, G. Zaccai, G.J. Lauquin, P.V. Vignais, Biochem. Biophys. Res.
Commun. 109 (1982) 471–477.
 A. Schroers, A. Burkovski, H. Wohlrab, R. Kramer, J. Biol. Chem. 273 (1998)
 R. Kotaria, J.A. Mayor, D.E. Walters, R.S. Kaplan, J. Bioenerg. Biomembr. 31
 H. Hackenberg, M. Klingenberg, Biochemistry 19 (1980) 548–555.
 C.S. Lin, H. Hackenberg, E.M. Klingenberg, FEBS Lett. 113 (1980) 304–306.
 S.D. Dyall, S.C. Agius, C. De Marcos Lousa, V. Trezeguet, K. Tokatlidis, J. Biol.
Chem. 278 (2003) 26757–26764.
 L. Capobianco, A. Ferramosca, V. Zara, J. Protein Chem. 21 (2002) 515–521.
 M. Klingenberg, M. Appel, Eur. J. Biochem. 180 (1989) 123–131.
 F. Bisaccia, V. Zara, L. Capobianco, V. Iacobazzi, M. Mazzeo, F. Palmieri,
Biochim. Biophys. Acta 1292 (1996) 281–288.
 M. Hashimoto, E. Majima, S. Goto, Y. Shinohara, H. Terada, Biochemistry 38
 C.S. Lin, M. Klingenberg, Biochemistry 21 (1982) 2950–2956.
 E.R.S. Kunji, M. Harding, J. Biol. Chem. 278 (2003) 36985–36988.
 E. Pebay-Peyroula, C. Dahout-Gonzalez, R. Kahn, V. Trezeguet, G.J. Lauquin, G.
Brandolin, Nature 426 (2003) 39–44.
 L. Bamber, D.J. Slotboom, E.R.S. Kunji, J. Mol. Biol. 371 (2007) 388–395.
 L. Bamber, M. Harding, M. Monne, D.J. Slotboom, E.R.S. Kunji, Proc. Natl. Acad.
Sci. USA 104 (2007) 10830–10834.
 S. Mori, H.G. Barth, Size Exclusion Chromatography, Springer Verlag,
 G. Healthcare, Gel Filtration Principles and Methods, Uppsala, 2002.
 L. Guan, O. Mirza, G. Verner, S. Iwata, H.R. Kaback, Proc. Natl. Acad. Sci. USA
104 (2007) 15294–15298.
 S.B. Long, E.B. Campbell, R. Mackinnon, Science 309 (2005) 897–903.
 Y. Wei, H. Li, D. Fu, J. Biol. Chem. 279 (2004) 39251–39259.
 T.C. Laurent, J. Killander, J. Chromatogr. 14 (1964) 317–330.
 L.M. Siegel, K.J. Monty, Biochim. Biophys. Acta 112 (1966) 346–362.
 D. Rodbard, A. Chrambach, Proc. Natl. Acad. Sci. USA 65 (1970) 970–
 T. Svedberg, K.O. Pedersen, The Ultracentrifuge, Oxford University Press,
London and New York, 1940.
 C. Tanford, Physical Chemistry of Macromolecules, John Wiley & Sons Inc., New
 F. Perrin, J. Phys. Et Radium 7 (1936) 1–11.
 G.G. Stokes, Trans. Cambridge Philos. Soc. 9 (1856) 8–166.
 E. Pebay-Peyroula, R.M. Garavito, J.P. Rosenbusch, M. Zulauf, P.A. Timmins,
Structure 3 (1995) 1051–1059.
 M. Schlame, K. Beyer, M. Hayer-Hartl, M. Klingenberg, Eur. J. Biochem. 199
 S. Lu, P. Somasundaran, Langmuir 23 (2007) 9188–9194.
 C. Huang, J.P. Charlton, J. Biol. Chem. 246 (1971) 2555–2560.
 J.V. Moller, M. le Maire, J. Biol. Chem. 268 (1993) 18659–18672.
 P.J. Bond, M.S. Sansom, J. Mol. Biol. 329 (2003) 1035–1053.
 P.J. Bond, J. Cuthbertson, M.S. Sansom, Biochem. Soc. Trans. 33 (2005) 910–
 P. Riccio, H. Aquila, M. Klingenberg, FEBS Lett. 56 (1975) 133–138.
 A. Urbani, T. Warne, Anal. Biochem. 336 (2005) 117–124.
 Y.H. Zhao, M.H. Abraham, A.M. Zissimos, J. Org. Chem. 68 (2003) 7368–
E.R.S. Kunji et al./Methods 46 (2008) 62–72