Modelling biological macromoleculesin
solution: 1. The ellipsoidof revolution
StephenE. Harding
DepartmentoJ Biochemisty, Uniuersity of Bristol, Bristol BS8 lTD, U K
and Arthur J. Rowe
Departmentof Biochemistry,Uniuersity.of Leicester,Leicester LEI 7RH, U K
( R e c e i v e dl 4 A u g u s t l 9 8 l ; r e v i s e dl 8 N o v e m b e r l 9 8 l )
in sohttion employitrgthe ellipsoid oJ
The problemsoJdeterminingthe axial ratio of biological mac'romolecules
reoolutionas u model are discussedand unalysedin termsoJ the sensitiuitiesol the uariousuolume-irulependent
Junctionsarailable. It is shown that ouer the whole range oJ'ctxiulratio only the R function (Rowe, 1977) is
applicable,but the newly deriuetlil and AJilnt'tions may hare upplication to macromoleculesttf uxial ratio > 3.
The widely employed[]Jinction is shown to be entirell,urutsablein terms oJ the tleJinedcriteria.
Keywords: Mathematicalmodels;ellipsoidmodel;volumeindependent
function;sensitivities
Introduction
Theory
X-ray crystallography,whereapplicable,is by far the most
accurate method for determining the conformation of
Unfortunately,this techniqueis also the
macromolecules.
most laborious, and the calculated structures are of a
static form of the macromolecule,which may be only an
approximationto the dynamic structurein solution.The
study of the transport propertiesof macromoleculesin
solution (hydrodynamics) yields information directly
r e l e v a n tt o t h e a c t u a l s t r u c t u r e i n s o l u t i o n , b u t t h e
interpretation of evidencefrom such techniquesposes
certain problems which severaldecadesof investigation
have not yet overcome.
There are two basic approachesfor determining the
grossconformationusinghydrodynamictechniques.One
rnethodis to assumea structureand then to calculateits
hydrodynamic properties, for example the intrinsic
viscosity, sedimentation coefficient or translational
diliusion coefficient,and then to see how much these
predicted properties differ from the experimentally
determined properties for the unknown structure. The
rnodel is then 'successivelychanged (refined) until the
predicted propertiesconvergeto agree with the actual
properties. This method has been developed by
Bloomiield,Garcia de la Torre and coworkerst 8. There
rs.however,a seriousdrawback in that the final calculated
structure may not be the only one that gives these
propertles.
The alternative approach is to calculate a structure
directly from the known hydrodynamicproperties.Some
generalmodel must,of course,be assumed,but, although
thc modelsavailablefrom this approach are lessprecise,
the method doesnot sufferfrom the uniquenessproblem.
The most general,and almost universallyapplied,model
is the ellipsoidof revolution,i.e.an ellipsoidwith two equal
axes' 'o. In this paperwe considerthe optimal procedures
for a structurecalculationbasedupon this model, and in
show how the restriction
the secondpaperofthis seriesas
to two equal axesmay, in appropriate cases,be relaxed.
Vo Iume- independent shapeI unctions
There are several shape functions arising from the
various hydrodynamic properties of a macromolecular
s o l u t i o n .T h e m o s t s a l i e n ta r e :
(l ) the viscosityincrement:
0l4l 8t 10/82103016M5$01.00
Ltd
@ 1982Butterworth& Co. (Publishers)
160 Int. J. Biol. Macromol..1982.Vol
I' April
y:L
tnl-__L,tlM,
r-
(l)
Ne4
w h e r e[ 4 ] i s t h e i n t r i n s i cv i s c o s i t y( m l g - r ) , r ' .t h e s w o l l e n
specificvolume (ml g- r), M, the molecularweight, I{ the
v o l u m e o f a m a c r o m o l e c u l e{ m l ) a n d N o A v o g a d r o ' s
numbcr.
{ l } t h e t r a n s l a t i o n a fl r i c t i o n a lr a t i o :
'
! = MNl t - L . p o ) r * ) '
fo-
,r,
o 6 n 4 n t\ 3 4 . /
where4o is the solventviscosity,po the solventdensityand
.sis the sedimentationcoefficient(extrapolatedto infinite
d i l u t i o n ) . F o l l o w i n g t h e c o n v e n t i o n o f S c h e r a g aa n d
Mandelkernr8,Tolana 0o, to below) refersto a hydrated
sphereof the same volume.
t3) the reduced molecularcovolume:
r t
u
,ea: y
U
(3)
oy"
w h e r e U i s t h e m o l e c u l a rc o v o l u m e( m l m o l
(4) the rotational diffusion ratios:
0i
64oV.
il::;t"-o'
li:a' b'c)
-t).
(4)
where a, b, c' refer to the semiaxesof the ellipsoid (for a
p r o l a t ee l l i p s o i da > b : c a n d f o r a n o b l a t e a < b : c ) .
Modelling mat'romoleculesin solution: StephenE. Harding and Arthur J. Rowe
1
Explicit relationsfor vrr'r2, J'lfot', {J,^,o, 0J0ott'ro,
r i l r o t 5 ' ' 6 ( a n d h e n c er n l r o ) t ' i n t e r m s o f a x i a l r a t i o i o r
ellipsoidsof revolution have been given.
The determination of all these functions. however.
requires a knowledge of the molecular volume
V.:(M,u,lN ). V" can be eliminated by various
combinationsof equations(1H7) to yield shapefunctions
which are volume-independent:
r,'r
Nos[n]ttqo
Nlr
,,
_:
-, ) : (-f . -Dpo)100t,
6200n2;'tJ'UM
' o ,- 2 ' 3 U
(8)
(Refl8)
5
t
lo
Prolore
Axiol rotio
,t': :;h(i)'=ff$;
n:?=
lr)+,
(Rer,4)
(e)
(Rers
to,4e)
(lo)
:(;)"'(*)=(#)"'i*;#(|
*
.
(Refs17,19)
n : f " ) , = 3 ' . 1 1 ! ! ' ( R e2io )1 r r )
NAA'7rrr,
\t,/
^:;['.(*)']=#
( R e i2 l )
t
6
i
Obloie
I
5
(l-.])
where k, is the sedimentationconcentration regression
coefficient given by:
prolole
Axial rotio
Figure I Rateofchange(lst derivative)ofthe relativeerror in
various volume-independent
hydrodynamic functions(defined
in the text) with respectto the relativeimprecisionin the axial
ratio of the assumed ellipsoid of revolution. Numerical
differentiationhas been performed on values of the various
functionscomputedat intervalsof I in axial ratio, by taking rhe
analyticalderivativeof the local coefficientsol a sliding strip
quadratic fit. The horizontal broken line in each
least-squares
case indicatesthe minimum value which this derivative mr,rst
haveif an axial ratio preciseto +20'd is to be retrievedfrom the
measuredfunction, the latter being assumedto be preciseto
*3_".,,.
Ratesfor the ry'and Y functionsare not plotted, in the
interestsofclarity.They arecloseto the baseline(i.e.the function
is very insensitive)for all valuesof axial ratio
. \ . : . s ( 1 - k , c ) : . s ( l+ k . r ' ) - '
(t+)
and where .s.and .sare the sedimentationcoefficientsat
concentrationc' and infinite dilution, respectively,
0i
611o0,fu1f
M,
,-r , : % r = : , v . o t i
(15)
( R e i s1 8 ,1 9 ,l 0 )
/
.n
,
/ \'
t':U,/
,, =s+7f,tr
* : r t - r ="1 , n )
.tryJ?,
(16r
( R e il 9 )
(5) the dielectric dispersion relaxation time rarios:
ri
kT
i:34ov"'
Results and discussion
{i:u' b' t')
(5)
where,for ellipsoids
of revolution
I
tu:
*
/rt
Th:T":
0"+0o
(6)
(6) the harmonicmean rotational relaxationtime ratio
Th
to
-
KT
(rolt,)+(2rolro) 3 r l o V " ' o
(tl
M a n y o f t h e v o l u m e - i n d e p e n d e nf tu n c t i o n s g i v e n i r r
e q u a t i o n s( 8 F ( l 6 ) a b o v ea r e e x t r e m e l yi n s e n s i t i vteo a x i a l
ratio and sensitiveto experimentalerror. Nevertheless.
many workers haveappliedthem, notably the p function,
without adequateregard for the inaccuraciesinvolved in
their use.It is thereforeboth important and interestingto
c o m p a r e q u a n t i t a t i v e l yt h e i r s e n s i t i v i t i e tso a x i a l r a t i o
and to experimentalerror.
Serr.siriultl,
ln Figure / we plot the fractionalchangein the function
(p, R . . ) arisingfrom a givenfractionalchangein the axial
Int. J. Biol. Macromol.,1982,Vol 4, April
16l
Modelling macromolecules in solution: Stephen E. Harding and Arthur J. Rowe
Table l
model
Useof the R fu nction to predictthe conformationof variousmacromoleculesin solution in termsof an ellipsoidof revolution
Protein
^
h,*'',k,*-',
,t*]r-',
Apoferritin2a
BSA2*
t2
5.5
Fibrinogenr5
Ovalbumina('
C-protein:t
M y o s i n28 ' :o
SyntheticA-filamentsro
Collagensonicatesir
M.: -152000
M,:330000
M,:273000
M,:227ffiO
7.7
t4
5.45.
6.6
II
t5.4
85
92
160.8 366
308
29t
241
193
880
756
564
428
5.16 t.55
Axial
ratio
1.45"
Modeldependent
1I./r,-)
2.6' '"
2.75 2.0
7.8 0.9
3.49 1.56
12.6 0.87
234
0.18
176
0.9
1252
1078
639
400
A.246
0.270
0.377
0.483
Modelindepen
dent
(t ,lD)
Conclusion
1.5
1.4
6.1,
t . 5 "b
26.0b,6.65"
10"
19.5'
80"
64
30'
18"
I.l'
2.0
r.5".,
1.2
1.4
l.l
2.3
oqh ) t),
4.3'
t6
2.28
2.85'
6.12'
9.ll'
2.85
2.60
2.34
2.22
Approx. spherical:agree:
with X-ray crystallography and electron
microscopy3e
Nor a hydrodynamic
etlipsoid(c.t.f <2.1)a6
Prolate eltipsoid -6:1.
Agrees with electron
microscopy26
Approx. sphericat
, O b l a t e e l l i p s o i d- 2 5 : I
f Not hydrodynamic
[ ellipsoidsof revolution
Prolate - 80:l
Prolare -65:l
/Not hydrodynamic
[ ellipsoidsof revolution
' Prolate ellipsoid; D
oblate ellipsoid;. corrected to solution densitv2r
ratio. A large change in the function denotes high
sensitivity,and it is clear that the various functionsdiffir
widely, both as a function of axial ratio and amons
themselves.
To evaluatethe practicaluseof the functioni
we definethe following criteria:
(1) Estimates of axial ratio of worse than +20%
precisionare of little or no interest.
(2) Functions can be calculated from experimental
data to a precision of better than * 3r/,, 6ut seldom to
muchbetterprecision(seebelow).In termsof thesecriteria
- admittedly slightly subjectiveones - we
see from
F igure 1 that many of the definedfunctionsare unusable,
especiallyat low axial ratio. The R and li. functionsare the
most sensitivefunctionsfor the whole rangebut the newly
deiined lI and A functions may have application for
particlesof small (but not very small)asymmetry,and the
rotational functions(dr, 7" and yr) for prolate ellipsoids.
In addition to its purely mathematicalsensitivity to
shape variation, a shape function must be judged with
respectto its insensitivity to experimental error. It is
readilyseenfrom equarions(8),(9),(l l) and (16)that the Il,
,lt,Y, ^i, and 7, functions require a relatively large number
of measurements
to be made,and many terms appearing
in theseequationsare either squared or cubed. On the
other hand, the A function [equation (12)] requires
knowledgeof the harmonic mean rotational relaxation
tlme, r,,, the measurementof which suffersfrom problems
of internal rotation of the chromophore and segmental
rotation of the macromolecule,a good example being
fibrinogen2o'22.In order to determine 6u or 7", a
knowledge of the rotational diffusion coefficient g, (or,
alternatively,?o,seeequation(6))is required.Accordingto
Benoita2,for ellipsoidsof revolution there will be one
electricbirefringencerelaxationtime rn relatedto 6)6byre
t,:ll60a). Using this technique,0,has beenmeasuredro
a precision of +1.5'/, for haemoglobin and Squirere,
determining the correspondingvalue of 7,, has found that
162 Int. J. Biol. Macromol.,1982,Vol 4, April
the axial ratio of haemoglobinlies within the range 1.2
(prolate)to 2.6(oblate),correspondingto an error in of
;,,
-
+Sol
! " / o -
The evaluation of do and 7, is more complicated since
they require 0" and rr, respectively,and hence,from equation (6), the resolution of two dielectric relaxation times.
Resolutionis almost impossiblewithout constraintsin the
analysisof the dispersioncurve,but Moser et al.a3.aa
have
developeda techniquewhereby 0, (or equivalentlyr,) is
first obtained by electric birefringencedecay as above and
then used as a constraint in the analysisof the dielectric
dispersion curve to obtain ru (and hence 0"). In their
programs,Moser et al.have added the further constraint
that the correspondingy, and 7, functionsgive the same
axial ratio, and for bovine serum albumin they obtain a
prolate ellipsoid of axial ratio I 0, compared with
'unconstrained'
valuesfrom 7oand 7r, respectively,of 3.5
and 5.1.The ratio of the two dielectricrelaxationtimesas
is also a volume-independentfunction of axial ratio; an
axial ratio of 3.0 is again obtained for BSAaa. There
remainssomedoubt, however,from other measuremenrs
as to whetherBSA is ellipsoidalar all(seeTable I and Ref
46\.
Although these algorithms apparently produce
resolutionof the two-term dielectricdispersioncurve for
an axial ratio -3, it is not known whetheradequate
resolution is possiblelor more symmetric particlei, ie.
where the relaxation times will be closer. Another
difficLrltyteis that the merhodis limited to solutionsof low
conductivity so that macromoleculesin physiological
conditions cannot be examined, measurementsbeins
restrictedto thosepH valuesand ionic strengthsat whicf,
proteinshave minimum solubility. Finally, the relaxation
times are concentration dependent and have to be
extrapolatedto infinite dilutionaT.
The degreeof experimentaluncertaintyassociatedwith
a calculationof valuesfor all the variousfunctionscan be
Modellingl macromoleculesin solution: StephenE. Harding und Arthur J. Rowe
Table 2
proteins
Crystallographic dimensions of some globular
Protein
Dimensions
(A)
Carboxypeptidase
Myoglobin
Cytochrome c
Lysozyme
Ribonuclease
Pre-albumin
Haemoglobin
50x42x38
43x35x23
25x25x 35
45x30x30
38x28x22
70x55x50
64x55x50
Reference
)z
J-t
34
35
36
J I
38
estimated from values assigned to the errors in t h e
molecular parametersfrom which these functions are
calculated.We assigntheseas follows:
s,o.*O.2l/o
o"2Y"
M,1.0%
0b t.5%
r " 1.5/"
UtlQ+0.5*\%
'J
+. b
\o/
u 0.5*7"
L.J /o
rn2.O\
k, (1.0+0.5*)7/"
c 0.5\
U(1.5+0.5*)\
The asterisked quantities reflect the contribution of
uncertainty in the concentration measurementwhich
beinga systematicerror will cancelin certaincases.From
theseassignederrors,usingnormal statisticalprocedures
we may estimatethe resultinguncertaintyin the derived
functionsas:
p 1.7% 6,2.7?;
(' 4.4% 6b2.3%
n 2.1% y"2.6%
Y 2.0% % 3.3y"
L 2.7%
R t.4%
The error in the final estimatedparametersis -2)" in
most cases,although the R function is a little better than
this and the ry'function significantlyworse. The purely
mathematical sensitivity of the various functions
discussed
aboveand illustratedin Figure 1 is thereforethe
dominating factor in decidingwhich function is the most
readily usable.
The R function requires knowledge only of the
sedimentationregressioncoefficientk, and the intrinsic
viscosity[4] which can both be determinedaccuratelvbv
fitting data to a new universalequationfor transportit ail
solute concentrations up to the critical packing
fraction2r'ro. This insensitivity to experimental error
strengthensthe conclusion from consideration of the
sensitivityto axial ratio that the R function is by far the
most applicableto protein systems.Also, any systematic
errors in soluteconcentrationcancel in the ratio k,l\tl.
Criterion Jbr the cloodness
of fit oJ a chosenmotlel
Although, using procedures defined above, a
h y d r o d y n a m i c a l ley q u i v a l e n tp r o l a t e o r o b l a t e e l l i p s o i d
of revolutioncan be fitted with reasonableprecisionto a
protelnstructurein solution it could still be differentfrom
eitherellipsoid,i.e.the choiceof an ellipsoidof revolution
in the first instancemay be a poor approximation to the
true structure.
A u s e f u l c r i t e r i o n f o r d e t e r m i n i n gw h e t h e r t h e t r u e
structure resemblesan ellipsoid of revolution
or any
model for which data are available-- is a comparisonof
the model-dependent with the model-independent
estimate of the swollen molecular volume, /", swollen
specific volume, r.., or the 'swelling' ratio, r../r-,for the
protein.
The model-dependentestimatecan be found by back_
substitution of the axial ratio for the ellipsoid of
r e v o l u t i o n ,f o u n d f r o m t h e R f u n c t i o n ,i n t o e q u a t i o n( l )
for the viscosityincrement,from which r./l can be found.
An estimate that is intlepentlentof any assumedmodel can
be found from the ratio o[ the viscosity resression
coefficient, k,r, to the sedimentation reiression
coefficient2l.i.e.:
r.,- kr
r k ,
Sucha comparisonis givenfor severalproteinsin Tuhte t.
Coiiclusions
It is evident {rom Tuhle 1 that the application of an
ellipsoid of revolution model to many protein systemsin
solution is not valid. One exception is fibrinogen, for
which the result predicted by the hydrodynamic data
appearsto agreequite accuratelywith that from electron
mrcroscopy.
The most likely protein systemto which an ellipsoidof
revolution would be a valid model is that of globular
proteins, whose shapes, as their name implies, are
reasonably regular. A perusal of the shapes and
dimensions of globular proteins predicted by X-ray
crystallographyillustratesthat in somecasesan ellipsoid
of revolution model could well be valid, e.g. lvsozvme
and cytochrome c (Titble2: see also Table I oi Squir.
and Himmelal). In many cases, however, such as
carboxypeptidaseand myoglobin, the distinction as ro
whether the protein is bettermodelledeither by a prolate
or oblate eltipsoid may be arbitrary and. indeed.
i r n p o s s i b l ei n s o m ec a s e s .
It would be a significantstep forward, therefore,if the
restriction of two equal axes on the ellipsoid were
removed to allow use of the more general .triaxial'
ellipsoid.However,due either to the lack of the necessary
theoreticalrelationshipslinking the axial dimensionsof
the ellipsoidwith experimentalparameters,
or, evenif they
are available, due to the lack of the necessary
experimental precision,numerical inversion procedures
or data analysistechniques,this model has not yet been
available.A very recentstudy has shown, however.that
this restriction can now be removedas.
References
I
I
-l
4
Bloomfield, V. A., Dalton, W. O. and Van Holde, K. E.
Biopolymers 1967,5. 135
C a r c i a d e l a T o r r e ,J . a n d B l o o m f i e l d ,V . A . B i o p o l r m e r s1 9 7 7 , 1 6 .
1147
y . A. Biopolr.nter..r
Garciade la Torre,J. and Bloomfield,
1977.t6.
1765
Garciade la Torre,J. and Bloomfield.y . A. Biopoly.nrers
1977
, 16,
1779
Int. J. Biol. Macromol.,1982,Vol 4, April 163
Modelling macromoleculesin solution: StephenE. Harding and Arthur J. Rowe
5
6
7
8
9
l0
ll
12
ll
14
I5
16
l7
18
19
20
2l
22
23
24
25
26
27
28
29
l0
G a r c i a , d e l a T o r r e , J . a n d B l o o m f i e l d , Y . A . B i o p o l y m e r s l 9 T 81,7 ,
1605
Wilson, R. W. and Bloomfield, Y. A. Biopolymer.s1979,18, 1205
Wilson, R. W. and Bloomfield, Y. A. Biopolymers 1979, 18, l54l
Garcia Bernal,J. M. and Garcia de la Torre, J. Biopolymers 1980,
19,751
'Physical
Tanford, C.
Chemistry of Macromolecules' J. Wiley,
New York, 196l
Harding, S. E. Pr?D lfta.sis Leicester University (1980)
S i m h a , R . J . P h y . r .C h e m . 1 9 4 0 , 4 4 , 2 5
S a i t o , N . J . P h y s . S o c .( J p n ) 1 9 5 1 , 6 , 2 9 7
P e r r i n ,F . J . P h y s . R a d i u m 1 9 3 6 , 7 , l
J e f f r e y ,P . D . , N i c h o l , L . W . , T u r n e r , D . R . a n d W i n z o r , D . J . J .
P h y s .C h e m . 1 9 7 7 , 8 1 ,7 7 6
Perrin, F. J . Phys. Radium 1934,5, 497
K o e n i g , S . H . B i o p o l y m e r s1 9 7 5 ,1 4 , 2 4 2 1
Squire, P. G. Biochim. Biophys. Acta 197O,221, 425
Scheraga,H. A. and Mandelkern, L. J. Am. Chem, Soc. 1953,79,
179
Squire, P. G. Electro-Opt-Ser. 1978,2, 565
Harding, S. E. Biochem.J. 1980, 189, 359
R o w e , A . J . B i o p o l y m e r s1 9 7 7 , 1 6 , 2 5 9 5
Johnson, P. and Mihalyi, E. Biochim.Biophys.Acta 1965,102,476
Rowe, A. J. manuscript in preparation
Rowe, A. J. and Pancholi, A. unpublished work
Rowe, A. J. and Mihalyi, E. unpublished work
Fowler,A. E. and Erickson, H. P. J. Mol. Biol. 1979, 134.241
Offer, G., Moos, C. and Starr, R. J. MoL Biol. 1973,74,653
Emes, C. H. PhD Thesis Leicester University (1977)
Emes,C. H.and Rowe,A. J. Biochim.Biophys.Acta1978,537,ll0
E m e s , C . H . a n d R o w e , A .J . B i o c h i m . B i o p h y s . A c t a 1 9 7 8 , 5 3 7 , 1 2 5
164 Int. J. Biol. Macromol.,1982,Vol 4, April
3l
32
l3
34
15
36
37
18
39
40
4l
42
43
44
45
46
47
48
49
from Nishihara,T. and Doty, P. Proc. Natl. Acad. Sci. USA
1 9 5 8 . 4 44
.l I
Lipscomb, W. Proc. Robert A. Welch Found. Conl. Chem. Res.
1971, 15, 134
Kendrew, J. C., Bodo, G., Dintzis, H. M., Parrish, W. G., Wycoff,
H . a n d P h i l l i p s , D . C . N a t u r e 1 9 5 8 ,l E l , 6 6 2
'The
Dickerson, R. and Geiss. I.
Structure and Action of
P r o t e i n s ' ,B e n j a m i n ,C a l i f o r n i a , 1 9 6 9
Blake, C. C. F., Loenig, D. F., Mair, G. A., North, A. C. T.,
P h i l l i p s , D , C . a n d S a r m a ,V . R . N a t u r e 1 9 6 5 , 2 M , 7 5 7
K a r t h a , G . , B e l l o w ,J . a n d H a r k e r , D . N u t u r e 1 9 6 7 , 2 1 3 , 8 6 2
B l a k e , C . C . F . , G e i s o w , M . J . a n d O a t l e y , SJ . J . M o l . B i o l . 1 9 7 8 ,
t2l, 339
P e r u t z ,M . F . , R o s s m a n n M
, . C . , C u l l i s ,A . F . , M u i r h e a d ,H . , W i l l ,
C. and North, A. C. T. N ature 1960, 185, 4 I 6
H a r r i s o n , P . M . J . M o l . B i o l . 1 9 5 9 ,l , 6 9
from Holt, J. C. PhD. fiesi.s London University (1970);seealso
Harding, S. E. Inr. J. Biol. Macromol. 1981, 3, 398
Squire, P. G. and Himmel Arch. Biochem. Biophys. 1979, 196,
165
Benoit, H. Ann. Physik 1951,6, 561
M o s e r , P . , S q u i r e , P . G . a n d O ' K o n s k i , C . T . J . P h y s . C h e m1. 9 6 6 ,
70, 744
Squire, P. C., Moser, P. and O'Konski, C. T. Biothemistry 1968,7,
4261
Wyman, J. and Ingalls, E. N. J. Biol. Chem. 1943, 147,297
McCammon, J. A., Deutch, J. M. and Bloomfield, V. A.
Biopolymers 1975, 14,2479
Riddiford, C. L. and Jennings. B. Biopolymers 1967,5,757
Harding, S. E. and Rowe, A. J. Biopolymersin press
. 981,3,340
F l a r d i n g ,S . E . I a t . J . B i o l . M e t c r o m o l 1