Chapter7
ProteinHydrodynamics
S T E P H EEN.H A R D I N C
Abstract
Introduction
Hydrodynamic Techniques
Molar Mass (Molecular Weight) and Quaternary Structure
Gel Filtration and Size Exclusion Chromatography
Dynamic Light Scattering (DLS)
SedimentationVelocity in the Analytical Ultracentrifuge
SedimentationEquilibrium
Shape Measurement
Modelling Strategies:Spheres,Ellipsoids, Beads, and Bends
Intrinsic Viscosity
SedimentationVelocity and Dynamic Light Scattering
Use of Concentration DependenceParameters,Combined Shape
Functions, and the Radius of Gyration Rs
Measurementand Use of Rotational Hydrodynamic ShapeFunctions
FluorescenceDepolarization Decay
Some Computer Programs for Conformational Analysis
Protein: A Comprehensive
Tleatise
Volume2,pages27l-NS
Copyright @1999by JAI PressInc.
AU rights of reproduction in any form reserved.
ISBN: 1-55938-672-X
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STEPHEN
E. HARDING
ABSTRACT
This article provides,a pointer for the non-specialist to the various
hydrodynamic
methodologies available for the characterisationof the size, conformaiion
in dilute
solution and interaction properties of proteins. The virtue of
combining data from
different techniquesis stressed,particularly in connectionwith conformation
analysis
and its associateduniquenessand hydration problems.
INTRODUCTION
Hydrodynamics provides the protein scientistwith a powerful array
of methodolo_
gies for investigatingthe mass,conformation, and interactionproperties
of proteins
in solution conditions-conditions in which they largely function
in vivo. These
methodscan alsoprovidean importantsupportingrole to the so-called.,high-resolution" structuralprobesofX-ray crystallographyand nuclear
magneticresonance.
In the caseof nuclearmagneticresonance,bicauseof the high
Joncentrationsof
mass/volumerequired to give satisfactory spectra,simple seJimentation
velocity
or equilibriumrunsin the analyticalultracentrifugecanprovidevital
checksagainst
any self-associationbehavior that can give rise to misinterpretation
of the chemical
shift or relatedspectra.Hydrodynamic methodsaregenerallyrapid
and nondestructive: these particular features have not escaped the notice
of peopte such as
molecularbiologists,who oftenhaveonly very smallamountsof
materialavailable,
They can provide early "low-resolution" information on a macromolecular
struc_
ture prior to detailed crystallographic or high-resolution nuclear
magnetic reso_
nance analysis' or conversely,they can provide the finishing
touchel refining a
crystallographic model to account for dilute solution behaviorl especially
in terms
of intermolecular interaction phenomena(Schachman,l9g9).
The delicateintra_
molecular relationships between subunits of a multienzyme complex
can also be
explored.In work now almostconsideredclassical,H. K. Schachman
and coworkers (Schachmanet al, 1984) showed, using a combination
of high-precision
analytical ultracentrifuge measurementswith the tools of molecular
biology (production of point mutants)how such interactionsin aspartate
transcarbamoylase
produce powerful allostery.
There have been many classic reviews on the application
of hydrodynamic
protes' Despiteits age,c. Tanford'sbook (196r) is still regarded
by many as the
authority on the subject,although the subjecthas advancedJonsiaerauty
since that
time particularlyin terms of molecularweight and molecular
weight iirt ibution
analysis,analysisof interactionparameters,and hydrodynamic
conflrmation modelling (tri-axialellipsoids,beadmodels,flexibleparticleanalysis,
etc.).The purpose
of this articleis thusto attemptto indicaterorn" Lf th" ,.latei990's"
siate-of-the-art
of hydrodynamic methodology for the investigationof macromolecular
conformation in dilute solution.This article,with the gineral protein
scientistin mind, will
Protein Hydrodynamics
273
not give a comprehensivereview of the theory,experimentation,and applications
of all hydrodynamic methodologies,but aims to provide a pointer to the various
methodologies,and for each of the two classesof hydrodynamic measurementmass and shapeanalysis-it will focus on certain techniquesin more detail than
others(this is merely a result of the particular expertiseof the author).For example,
for mass analysis,we focus on gel filtration and size exclusionchromatography
(including on-line coupling with multianglelaserlight scattering),dynamic light
scattering,sedimentationvelocity, and sedimentationequilibrium in the analytical
ultracentrifuge. For shapemeasurement,we focus on sedimentationvelocity and
dynamic light scatteringagain,togetherwith intrinsic viscosity,steady-state
fluorescence depolarization, and the use of concentration dependenceparameters,
combinedshapefunctions,and the radiusof gyration.The treatmentgiven hereis
by no meanscomprehensive,but certain key follow-up referenceswill be given.
HYDRODYNAMTC
TECHNTQUES
By "hydrodynamic" (Greek for "water-movement") techniques, we mean any
technique involving motion of a macromoleculewith, or relative to, the aqueous
solventin which it is dissolvedor suspended.
This thereforeincludesnot only gel
filtration and size-exclusionchromatography,
viscometry,sedimentation(velocity
and equilibrium), and rotational diffusion probes (fluorescenceanisotropy depolarization and electric-opticalmethods)but also "classical" and "dynamic" light
scattering that both (even "classical") derive from the relative motions of the
(macromolecular)solutein relationto the solvent.Although this definition technically also includes electrophoreticmethods,thesewill not be consideredhere.Let
it suffice to say here however that electrophoreticmethods,besidesbeing powerful
tools for separation, purification, and identification of proteins, can also, with
"SDS"
methodology, be used to provide an estimate of polypeptide molecular
weight.Carefuluseofcross-linkingagentscanalsogive an indicationofquaternary
structure,although correct application of other hydrodynamic methodsgive a more
precise picture.
This article therefore considers the hydrodynamic determination of molecular
weight, or "molar mass", and quaternary structure (subunit composition and
arrangement,self-associationphenomena,and polydispersity). We will also consider the measurementof protein conformation in dilute solution.
MOLARMASS(MOLECULAR
WEICHT)AND QUATERNARY
STRUCTURE
For an unglycosylated polypeptide, a value to +l g/mol can be obtained from
sequenceinformation or from mass spectrometry . A similar precision cannot be
obtained for glycosylated proteins becauseof polydispersity deriving from the
variability of a cell's glycosylation process.Many proteins-and glycoproteins-
274
STEPHEN
E.HARDING
contain more than one noncovalentlylinked protein chain, particularly at higher
concentrations,and important roles of hydrodynamic methodsfor massanalysisin
protein chemistry areto give the molar massof the "intact" or "quaternary" structure
and to provide an idea of the strengthof binding of these noncovalententities
throughmeasurementof associationconstants.
Cel Filtrationand Size ExclusionChromatography
The simplest hydrodynamic method for measuring molar mass is gel filtration
(Ackers, 1975),commonly referredto as "gel permeationchromatography" or now
"size
exclusionchromatography"since the chemicalintertnessof the separation
medium is assumed.This was originally conceivedas a method for the separation
and purification of macromoleculesbut has developedover the years in its .,calibrated" form as a very popular method for measuringprotein molar massesboth in
nativeand dissociativeconditions.
The separationmediumof this methodis a crosslinkedgel. Traditionally,this has
been made by using cross-linked polysaccharide or polyacrylamide beads and
allowing them to swell in water; this is then packed into a glass or metal walled
column, which is then equilibrated with the buffer in which the macromoleculesto
be separatedare dissolved. Control of the degree of crosslinking will dictate the
separation range of the gel: looser gels will separatelarger molecules. proper
packing of columns requires some skill, and the user manuals as supplied by the
commercial manufacturers are usually very comprehensive.The availability of
prepacked,metal-walled columns for usein so-called"high-pressure"or "high-performance liquid chromatography"(HPLC) with positive pressureapplied upstream
of the column to acceleratethe separationprocessmakesthe measurementparticularly attractivefor protein chemists.
Gel frltration or size exclusion chromatographydepends on the principle that
some of the space inside the gel particle is available to smaller molecules but
unavailableto larger moleculesthat are excluded.Thus, when a solution is applied
to the top of a properly packedgel column, only the deadspacebetweengel particles
is available to the excluded molecules, which therefore come off first when
"elution"
is commenced(addition of the buffer at a continuousrate, or equivalently
with HPLC, injection of the solution into an already continuously running buffer
system).The excludedmolecules-the largermolecules-will thushavea smaller
elution volume, V", and will elute first from the column. Smaller macromolecules.
having progressively more and more space available to them as molar mass
"Biggest
decreases,are accordingly eluted only at higher valuesof
come off the
\.
column first" is the rule of thumb for sizeexclusionchromatography.The separation
is sometimesgiven in terms of the partition coefficient, K"u as defined by
V"=Vo+Kn"(V,-Vo)
(l)
where vo and v, are the void volume and total volume of the column, respectively,
determinedfrom separateelutions using solutespecieshaving partition coeffrcients
Protein Hydrodynamics
275
of 0 (totally excluded) and l (non-excluded),respectively.
Elution of proteins as
they emergefrom the column is monitored by the use of a
spectropholmeter set
for either 280 nm in the uv (trp and tyr residue absorption) or
thetore sensitive
far-uv (210-230 nm-peptide bond), provided the buffer is reasonably
transparent
in the selectedregion. Reagentslike ATB azide, and so forth
in buffers can cause
seriousproblems for detection by causingan effective uv blackout.
In thesecases,
use of a differential refractometerinstead of a spectrophotometer
is appropriate.
Highly sensitive differential refractometers are now available,
which are now
arguably more preferable generally as the detectionmethod
of choice. The broad_
nessof a peak eluting from a column doesnot necessarilymean
the componentis
polydisperse:it more probablyis a likely result of diffusion
effects.
Empirically, the volume at which a protein erutesv" and its
molar mass M are
related by the logarithmic expression(Ackers, lglis)
% = A-g
log,oM
e)
where A and B are properties of the column. This equation
is valid over the
fractionation rangeof the gel and forms the basisof caribiated gel
chromatography
(Ackers, 1975): To obtain the molar mass of a protein
molecule or mixture of
(a)
tto
Sucro:t
230
(u
Cyloch.ona-
t90
Myoglobin
?.5
Chtmlrypdrcgrtr
(9
t70
a
o r50
o
t30
a
ilo
Ovolbumin
dahydtoganota
Orcmucolt*o
\
/'_Molola
pttot*trlota
Bovi^a 3.rm olbmh
E,6t
otycarold.hyd.5.
Trontftrr
_
pho3ghotrdchydrog.n6r.
pho39holr
dchydrooandra
- - . Loclotrrorid6r-/\.
/./
o
L*o h drtttd'os'mt'
gi,,',lft-.
::::s-J,g!'Yaolt
olcohol dahyd.oe.m3a
trctDn<
?.o
E
y-Globulinz-/
p
o
-+Farritin
Fibfitg.n
+.
Uraorc
o -Cryrlollh
tot
Molar
mass, M
(g/mol)
Slrr drrlron
tos
Gontinuedl
Figure 1. size exclusionchromatography.
(a) sEC logarithmiccaribrationprot for
proteinselutingfrom a sephadexc-200 corumn.Repr6duced
with permission
from
Andrews(1965).(b)SECelutionvolume/molecularweightrelation
obtaineddirectly
(pigsastric
=Booh
mucin,
(adapted
stycosylated)
f:
:=ffllft
rromJumer
et ar',l.jj-F,y:oprotein
r 996).(c)Morarmass
distribution
corresponding
to (b).
STEPHEN
E. HARDING
(b)
F
\
d0
v
F
a
u)
q
F
F
li
a
ts
a
r.o
t.t
10.o
to.s
Elution volume, Ve (ml)
(c)
o
I
cg
L
5i
a0
e)
:>
1, O€+7
Molar mass, M (g/mol)
Figurel. Continued
molecules,the column is first calibratedby the useof standardproteinsor .,markers"
of known molar mass.Linear regressionanalysisis then usedto evaluateA and B,
and hencefrom the measuredvalue ofv" ofthe unknown protein, M can be found.
The calibration can only be applied within the gel's fractionation range,which will
dependon the pore size (Figure 1a).Fractionation ability is normally enhancedby
277
Protein Hydrodynamics
running differing gel columns in series,a practice comlnon with HPLC systems
becauseof the much shorter elution times.
Equation 1 assumesthe fractionation is based on the size-exclusion principle
alone. Separationmechanismsnot governedby the size of the molecules will tend
to decouple the molecular size-migration velocity relation and the experimental
elution profile will not reflect differencesin size (Barth, 1980). Equation 1, which
fails also outside the fractionation range of the gel, works only for molecules of
similar shapeand conformation. Thus calibration using globular protein standards
would be inappropriate for fibrinogen and muscle proteins like myosin and titin
(asymmetric) and also heavily glycosylated glycoproteins.
The theory behind equation 1 is not rigorous, but for globular proteins at least it
seems to represent the data very well. For linear macromolecules of limited
stiffness, there appearsto be growing acceptancethat the separation is more a
logarithmic function of the hydrodynamic volume of a macromolecule (=M'[n]
"effecwhere [r1] is the intrinsic viscosity) and its conespondinghydrodynamic or
"universal
(Dubin
and
calibration"
tive" radius r", culminating in a proposal for a
Principi, 1989). This may be more appropriatefor proteins in denaturing solvents
such as proteins in the presenceof mercaptoethanol(disulfide bond breaker) and
6M GuHCI; for these substances,wider pore gels (e.g., sepharose)are a more
appropriateseparationmedium.
These calibration problems can be avoided completely by coupling an absolute
molar mass detector (a light scatteringphotometer)downstreamfrom the column
(Wyatt, 1992). This coupling, called "SEC/IvIALLS;' is particularly valuable for
the characterizationof polydisperseheavily glycosylated protein systemssuch as
mucus glycoproteins since it provides the elution volume to weight-average molar
mass relationship without recourse to calibration standards(Figure lb) and also
provides the molar mass,or for a heterogeneoussystem,the molar massdistribution
(Figure lc) and its associatedmolar mass averages(number average,Mn, weight
average,M*, and z-average,Mr). The coupled light scatteringand refractive index
detectorsare so sensitivethat only low concentrationsare required and problems
through thermodynamic nonideality are usually negligible.
Dynamic Light Scattering(DLS)
Although the light scatteringphotometerdescribedin the SEC/IvIALLS applica"static" light scattering)is not thought of asa classical
tion above(often describedas
"hydrodynamic" probe (although technically it is derived from motions of macromoleculesrelativeto solvent),the techniqueof dynamiclight scatteringhaswithout
doubt a firm hydrodynamic basis and now appearsto be the method of choice for
the measurementof translationaldiffusion coefficients.In addition, via an approximation or by combination with sedimentation measurements,(see below) this
method also provides an estimate for the molar mass.The appearanceof simpleto-use, fixed- angle (90') dynamic light scatteringphotometershas made dynamic
278
E. HARDING
STEPHEN
light scatteringan increasinglypopular tool amongstprotein chemists.After certain
assumptionsand approximations, largely involving an assumedspherical shape,
surprisingly reliable estimatesfor the molar mass of globular proteins have been
obtained (Claes et al., 1992). When used in isolation, this method for molar mass
measurementis, like gel filtration, a relative one, requiring a calibration using
standardproteins of known molar mass.For asymmetric proteins like fibrinogen
and myosin, the single-angleapproximation fails, but extraction of molar massand
related parameters is still possible if multiangle instruments are used and the
primary parameter,which comesfrom dynamic light scatteringmeasurements,the
translationaldiffusion coefhcient D (cm2 s-l;, is combined with results from
sedimentationanalysisin the analytical ultracentrifuge (seebelow).
For a recent comprehensivetreatmentof the technique,the reader is referred to
Brown's book (1995), and for a more introductorytext, Schmitz (1990) and an
articleby Johnson(1984).Althoughthetheoryis complex,theprincipleof dynarnic
light scatteringexperimentsis simple and is basedon the high intensity,monochromaticity, collimation, and coherenceof laser light. Laser light is directed onto a
protein solution in a controlled temperaturebath, and the intensity at either a single
or multiple angles recorded using a photomultiplier/photodetector system. The
(a)
(a)Experimental
set-up.(b) NormalizedautocorFigure2. Dynamiclight scattering.
relajiondecayplot for the proteinassemblypynein(in 40 mM NaCl)D!0,*= 1.1 x
t a9l 9. ,0 )
1 O - ' c m 's - ' ; M ( f r o m e q u a t i o n l=02) . 5 x 1 0 " 8 / m o(la d a p t e d f r o m W e l l s e 1
(2)
(c)"MHKS"double-logarithmic
calibrationplot of rn versusM : (1)thyroglobulin;
(6)
(5)
(4)
(3)
hexokinase;
amylogluyeastalcoholdehydrogenase;
apoferritin; lgC;
(9)bovineserumalbumin;
(8)transferrin;
(7)horsealcoholdehydrogenase;
cosidase;
('l'l) hexokinase
(10)hemoglobin;
subuniU(12)ovalbumin;('l3) carbonicanhydrase;
(15)myoglobin;(16)lysozyme;(17) ribonuclease
(14)chymotrypsinogen;
A. ReprofromClaeset al. (1992).
ducedwith permission
279
Protein Hydrodynamics
intensitiesrecordedwill fluctuate with time becauseof Brownian diffusive motions
"Doppler" type of wavelength
of the macromolecules; this movement causesa
broadeningof the otherwisemonochromaticlight incident on the protein molecules
and the beating between waves of different but similar wavelength causes the
intensity fluctuation. How rapid the intensity fluctuates (ns-ps time intervals)
@.0
(b)
-0.5
-t.0
I
i
-o
-J
-2.@
-2.5
-3.0
6
a4
t
o
2
0
3
0
4
A
5
0
6
@
ChonneL number
(c)
L
0.?
o0
0.5
'i.h
r.io
r.io
rio
r.io
t.bo
5io
5.io
log16M
Figure2. Continued
3.t0
ilo
1.00
280
STEPHEN
E. HARDINC
dependson the mobility or diffusivity of the protein molecules. A purpose-built
computer called an autocorrelator,as indicated by its name, "correlates" or interprets thesefluctuations. It doesthis by evaluatinga "normalized intensity autocorrelation function,"^g(2),as a function of "delay time", ..T (ms-ps)". The decay of
the correlation, g(2\r), as a function of T , averagedover longer time intervals
(usually = minutes) can then be used, by an interfaced pc (or the equivalent) to
obtain D. (Larger and/or asymmetricparticles that move more sluggishly will have
slower intensity fluctuations, slower decay of g(2)(t) with t, and hence smaller D
values compared to smaller and./ormore globular particles). The delay time r is
itself the product of the "channel number" b (taking on all integral valuesbetween
I and 64 or up to 128 or 256 dependingon how expensivethe correlator is) with a
user-set"sample time", r,, (typically - 100 ns for a rapidly diffusing low molar mass
[M - 20000 g/mol] enzyme, and increasing up to around milliseconds for microbes).In the past,T. was selectedby trial and error, but now modern data-capture
software usually does this automatically.
For spherical particles, a single term exponential describesthe decay of f with
t.
,tz)g)-l =s-Dk2r
(3)
where k is the Bragg wave vectorwhosemagnitudeis definedby
I = {4nnl}.} sin (0/2)
(4)
and where n is the refractive index of the medium, 0 is the scatteringangle, and .1,
(cm) is the wavelengthof the incidentlight. Equation3 can be reasonablyapplied
to quasisphericalparticles like globular proteins or spheroidalprotein assemblies
(Figure 2b).
Fixed-Angle (90') DLS Photometer
For globular proteins and spheroidalassemblies,application of equation3 at only
a single fixed angle is usually sufficient. Low angles are usually avoided because
they magnify problems due to any contaminationwith dust or other supramolecular
particles and thus an angle of 90' is normally used. For a given laser power at a
given protein concentration, the smaller the protein the lower the intensity of
scatteredlight and hencethe longer the averagingrequiredto give a sufficient signal.
A commercial instrument is available based on this single fixed angle principle
(claes et al., 1992).To obtain molar massinformation from D, a calibration curve
of log D versuslog M is produced(this is known asan "MHKS"
{Mark-HouwinkKuhn-Sakurada)relation; for example, see Harding, 1995) based on globular
protein standards,and the approximation is made that this relation holds for the
protein whosemolar massis being sought.Figure 2c showssucha calibrationplot
(The D values have been convertedto hydrodynamic radius values,seebelow).
281
Protein Hydrodynamics
Other approximations and practical requirementswith the operation of this type
of fixed-angle instrument have to be made:
1. Solutions have to be as free as possible from dust and supramolecular
aggegates. This requirement is met by injection of the sample into the (scrupulously
clean)scatteringcell via a millipore filter(s) of appropriatesize(0.1-0.45 pm).
2. The diffrrsion coefficient is a sensitive function of temperature and the
viscosity ofthe solvent (also sensitiveto temperature)and the log D versuslog M
relationship must correspondto the sametemperature.
3. The diffusion coefficient measuredat a single concentrationis an apparent
one, D".^, becauseof nonideality effects (finite volume and charge).These effects
becomJ'vanishingly small as the concentrationc-+0' The approximation which is
usually reasonablefor proteins, is made that Duoo= D or that nonideality effects are
the same as for the calibration standards.
Despite theseapproximations,diffusion coefftcientsand molar massesobtained
in this way with thesefixed-angle instrumentshave been remarkably reliable. For
nonglobular proteins, however both the log D versus log M calibration becomes
invaiid and also equation 3 no longer applies: an instrument with a multiangle
facility must then be resortedto.
Mu ltiangl e Instru ments
Measurementsusing multiangle equipment are more time-consuming and the
instrumentationlarger and more expensive.Data analysisis alsomore complicated'
Equation 3 no longer applieslargely becauseof the addedcomplication ofrotational
-+ 0' It is
diffusion effects.These effects vanish however as the scatteringangle 0
coefficient
diffusion
an
apparent
therefore possible to use equation 3 in terms of
D^-- with contributions from both concentrationand totational diffusion affects'
O"ll ir measuredat severalanglesand extrapolatedback to zero angle to give D if
coilentration effects are negligible. However, ifconcentration dependenceaffects
aresuspected,then a double extrapolationcan be performedon the sameplot (called
a"DynarnicZimm plot") of Duoo(or the equivalentautocorrelationfunction) to zero
angle and zero concentration(burchard, 1992).The common intercept gives the
"iJeaf ' (in a thermodynamic sense)diffusion coefficient, D0. Becausethis quantity
is not only an intrinsic property ofthe protein but also ofthe viscosity q(poise) and
temperature T (K) of the buffer, it has to be corrected to standard conditions
"C,
(viscosity of pure water at2o
4zo,*)either before or after the extrapolation (van
Holde, 1985) as shown in the following.
= Do'{n/nr,*}'{T1293'l5l
D3o.*
(s)
The size of a protein, as representedby its equivalenthydrodynamic radius r", is
related to D!0,* by the Stokesequation according to the relation
rn = knT/(6nr1zo,*D%,*
)
(6)
282
E. HARDING
STEPHEN
where k" is Boltzmann's constant.To obtain an absolutemeasureof molar mass,
M, of aprotein from Doro,*without assumptionsconcerningthe shapeof the protein
requires combination with the sedimentation coefficient from the analytical ultracentrifuge, as describedbelow. Some modern software attempts to evaluateM
directly from the diffusion coefficient; this should be treated with some caution.
For multiangle measurements,preferencesvary in terms of the type of cuvets
used.Squarecuvetsareoptically more reliable,but cell cornersobviouslyprohibit
somescatteringangles.Cylindrical cuvets,if used,shouldbe of the wide diameter
type (>2 cm) to avoid internal and stray reflections. Scrupulousattention to sample
and cuvet clarity is mandatory, particularly for macromolecules of M<100000
g/mol, which give low scatteringsignals,and also if low anglesareemployed where
the effects of supramolecularcontaminants are at their maximum: special cuvet
hlling anangements are used for clarification purposes (Sanders and Cannell,
1980). The angular extrapolationof Duoocan provide an estimatefor the rotational
diffusion coeffrcient, albeit to a lower precision than conventional methods (fluorescencedepolarization, electric birefringence). If the protein is polydisperse or
self-associating,the logarithmic plot of the type shown in Figure 2b will tend to be
curved, and the corresponding diffusion coefficient will be a z-average(Pusey,
1974)The spreadof diffusion coefficientsis indicatedby a parameterknown asthe
"Polydispersity Factor" (Pusey, 1974), which most software packages evaluate.
Various computer packagesare available from the commercial manufacturer for
datacaptureand evaluation.In our laboratory,we prefer to capturethe datain ASCII
format using the datacapturesoftware of the commercial manufacturerand then to
"PROTEPS" (Harding et al., 1997) or the evaluation
use our own in-house routine
of diffusion coefficients and polydispersity factors. More advancedroutines are
"CONTIN", which was designedfor the study of heterogeneavailable,including
ous systemsby going beyond the use of polydispersity factors and inverting the
autocorrelation data directly to give distributions of particle size. These methods
havebeenrecentlyreviewed(Johnsenand Brown, 1992;Step6nek,1993).
Dynamic light scatteringis particularly valuablefor the investigation of changes
in macromolecular systemsas long as the timescale of changesis of the order of
minutes or hours, and not secondsor lower (Harding, 1986).Finally, it is worth
pointing out that dynamic light scatteringalso provides a useful tool for monitoring
electrophoretic mobilities (Langley, 1992) and commercial instrumentation is
availablefor this purpose.
SedimentationVelocity in the Analytical Ultracentrifuge
Combination of the sedimentationcoefftcient, s, from sedimentation velocity
with the diffusion coefficient, D, from dynamic light scattering gives an absolute
value for the molar massof a protein without assumptionsover conformation. This
method for molar massmeasurementwas given by T. Svedberg(seeSvedbergand
Pedersen,1940).
Protein Hydrodynamics
283
The basic principle of the technique is as follows: a solution of the protein is
placed in a specially designed sector-shapedcell with transparentend windows.
This in turn is placed in an appropriatelybalancedrotor and run in high vacuum at
the appropriatespeed(typically = Jgg6g-60000rev/min for aprotein of molarmass
A light sourcepositioned
10000-100000g/mol, lower speedsfor largermolecules).
below the rotor transmits light via a monochromatoror filter through the solution
and a variety of optical components.The moving boundary can then be recorded
at appropriate time intervals on photographic frlm, on chart paper, or as digital
output fed directly into a PC. Measurementof the rate of the movement of the
boundary (per unit centrifugal field) enables evaluation of the sedimentation
coefficient. (For an introduction,see van Holde, 1985; for more detail, see two
recent books: Harding et al., 1992a; Schusterand Laue, 1994). There are three
principal opticalsystemswhich canbe employed:(i) absorptionoptics(in therange
"Schlieren"refractiveindex gradientoptics, and (iii) Rayleigh
200-7OOnm), (ii)
interferenceoptics. The simplest system is the absorptionsystem and the only
commercially available analytical ultracentrifuge currently available is based
around this (we will describe the operation of this here). Use of the other optical
systems requires more specializedknowledge and the interestedprotein chemist
needsreally to consult an expert.
LJseof an Analytical Ultracentrifuge with a Scanning Absorption Optics
Detection Systemand On-line Data Capture to a PC
Double sectorcells are used with the solution (0.2-0.4 ml) in one sector and the
referencebuffer or solvent in the other, the latter filled to a slightly higher level to
avoid complications causedby the signal coming from the solvent meniscus.The
scanningsystem subtractsthe absorptionof the referencebuffer from the solution.
Electronic multiplexing allows multiple hole rotors to be used so that samplescan
be run severalat a time.
In Figure 3a, examples of sedimenting boundaries recorded using absorption
optics are shown. Fig 3a (top) is for a highly purified preparation of an enzyme
(methylmalonyl mutase). Fig. 3a (lower) is for a heterogeneouspreparation of a
DNA-binding protein (Pfl) with a macromolecularcomponent and a fast moving
aggregate; the virtue of the technique for assaying the purity of a preparation
(number and asymmetry of boundary/ boundariesfor a given scan) can be directly
seen.Although commercial software is availablefor identifying the center of the
"second moment" of the boundary is more
sedimenting boundary (strictly the
appropriate;practically there is no real difference), in practice the simplest way is
(i) to plot out the boundaries(recordedat appropriatetime intervals) using a high
resolution printer or plotter and to graphically draw a line through the user-identified boundary centers and then (ii) use a graphics tablet to recapture the central
boundary positions as a function of radial position. Computer routines such as
XLA-VEL (Crilfen and Harding, unpublished)yield the sedimentationcoefficient
and a correction to the loading concentrationfor averageradial dilution during the
STEPHEN
E. HARDING
run (causedby the sector shapeofthe cell channels).Other routines are available
basedon the total concentrationdistribution suchasSVEDBERG (Philo, 1994)and
measurementof the apparentdistribution of sedimentationcoeffrcients,g(s) such
as DCDT (Stafford, 1992).The sedimentationcoeffrcient, s, equalsrate of movement of boundary/ unit centrifugal field, that is
s = (dr/dt)/ro2r
(7)
where r is the radial position of the boundary at time t and or is the angular velocity
in rads/sec(= rpm x 2ttl60). For a small globular protein of sedimentationcoeffia rotor speedof50000 rpm
cient of about 2 Svedbergs(S, where I S = 10-13sec),
will give a measurableset of optical records after some hours. For larger protein
systems(e.g. l25 globulinsor 30Sribosomes)speedsof <30000rpm areappropriate.The standardtemperatureat which sedimentationcoefficientsarequotedis now
20.0 "C (sometimes 25.0 "C). If the protein is thermally unstable, temperatures
down to around4oCcanbe usedwithout difficulty.The concentrationuseddepends
on the extinction coefficient of the protein. The lower the protein concentrationthe
better, since it minimizes problems of nonideality. For proteins of averageextincaslow as0.2 mg/ml arepossible
tion at 280 nm (=500 ml g-r cm-r), concentrations
with the standard 12 mm optical path length cells. This limit can be pushed even
lower if the peptide bond wavelength is used (210-230 nm) and the buffer is
transparent.For absorbancesgreater than 3, shorter path length cells need to be
employed instead (minimum = 3 mm: below this, cell window problems become
"off-maximum" wavelengthsused(with caution), or more desirably,
significant), or
a different optical system used (interferenceor Schlieren).For each concentration
used, the sedimentationcoefficient, s, is correctedto standardconditions of bufferlsolventdensityand viscosity(waterat 20.0'C):
sr,* = s.{q/qzo,*}'{(1-upzo,*)/(1-vp)}
(8)
where p is the density of the solvent. Knowledge of a parameter known as the
"partial specific volume", V (essentially the reciprocal of the anhydrous macromolecular density), is needed;this can usually be obtainedfor proteins from amino
acid composition data (Perkins, 1986) or measuredwith a precision density meter
(Iftatky et al., 1973).Typically, t = 0.73rnllg for proteins.
Extrapolation to Zero Concentration
As with D2o.*,S20.*
is plotted versusc (the latter correctedfor radial dilution) and
extrapolated(usually linearly) to zeroconcentration(Figure 3b) to give aparameter,
s!0,*, which can be directly relatedto the frictional propertiesof the macromolecule
(the so-called"frictional ratio") and from which size and shapeinformation can be
inferred. (If the protein is very asymmetric or solvated,plotting l/sro,* versus c
generally gives a more useful extrapolation).The downward slope of a plot of sro.*
Protein Hydrodynamics
285
(a)
8.5
(b)
0o 8.0
o)
6
I
R
7.5
x
=
7.O
6.s6
c (mg/ml)
Figure3. sedimentation
velocityin the analyticalultracentrifuge
usingscanning
absorption optics. (a) Sedimentation,,diagrams,,,
Methylmaloiyl mut"ase,c=0.2
mgiml.Monochromator
= 29snm;scaninterval9 min,rotorspeed44000
wavelength
= 2o.o oC, measuredsrn = (7.14+0.04)s.
rev/min,temperature
(b) sedimentation
diagrams,cene 5 DNA bindingprotein,c= oi"milml. Monochromator
wavelengh
=n278nm; scan interval8 min, rotor speed40000 rev/min,temperature= 2O.O"C,
tiq,* = (35-5tl .4)S(faster
(slowerboundary).
boundary)
and (2.6+0.1)S
(c)Sedimensro,*
versus
concentration
plot for an antibody (rat lgE).,30,* =
j1,1l: ^.":-fl.ient
(7.9210.06)S
STEPHEN
E.HARDING
versusconcentrationis a result of nonideality behavior and is characterizedby the
"Gralen" parameterk. in the equation
(9)
Szo.*= S3o.*(l - krc)
k., which dependson nonideality effects of the system, will depend on the size,
shape,and chargeon the protein. If the solvent usedis of a sufficient ionic strength,
I, then thesechargeeffects can be suppressed.
The molar mass,M, can then be found by combination of s!0,* with D!0,* using
the Svedbergequation (Svedbergand Pedersen,1940):
M = { r30,./D80,*
}.{RT(l-vpzo,*)}
(10)
An accurateestimate for V as describedabove is normally required, because,for
proteins, errors are triplified; for example, an error of +lVo in V results in an error
of +3Voin M. This meansthat care has to be made if the protein is glycosylated
sincethe V of carbohydrateis typically = 0.6 mVg.
For a heterogeneoussystem, s!0,* will be a weight averageand D!0,* will be a
z-average;the M calculated will also be a weight average (Pusey, 1974) thus
distinguishingit from molar mass obtainedby osmometry(seeTombs and Peacocke,1974), which yields a number average.
A further estimate can be obtained by combining s!0,* simply with k, (Rowe,
1977)'
- (vr/v)l}05
M = (6nqro.*r30.*)t''
{(3v)/an).tG.lzv)
(11)
where v, is a specific volume allowing for hydration of the protein, and since the
ratio (vs/V) is usually small in comparison with (k./2V), an approximate estimate
normally suffrces. This method has given reliable values for standard protein
molecules of known molar mass(Rowe, 197-l).k" itself is a valuable parameterfor
shapemeasurementas is discussedbelow. The form of the concentrationdependence can also be used as an assayfor self-associatingsystems(Rowe, 1977),
although sedimentationequilibrium methods (seebelow) are usually superior.
SedimentationEquil ibrium
The "sedimentation-diffusion" method (Equation 10) for giving molar mass,
althoughabsolute,is ratherinconvenientin requiringtwo setsof measurements.
A
simpler method is to use the analytical ultracentrifuge by itself with the technique
known as sedimentationequilibrium, and it is probably the method of choice for
molar mass determination of intact protein assembliesand particularly for the
investigationof interacting systemsof proteins (Schachman,1989). The same
instrumentand optical system(s)for sedimentationvelocity are used,the principal
differences being (i) the much lower rotor speedsemployed, (ii) the longer run
times, and (iii) the shorter solution (and buffer) columns in the ultracentrifuge
cell-hence the smaller amount of material required.
Protein Hydrodynamics
287
Sedimentation equilibrium, unlike sedimentation velocity, gel filtration, and
dynamic light scattering,is not a transportmethod. In a sedimentationequilibrium
experiment, the rotor speed is chosen to be low enough so that the forces of
sedimentation and diffusion on the macromolecular solute become comparable
allowing an equilibrium distributionof soluteto be attained.This equilibrium can
be establishedafter a periodof 2 to 96 hoursdependingon the macromolecule,the
solvent, and the run conditions. Since there is no net transport of solute at
equilibrium,the recordingand analysisof the final equilibriumdistribution(Figure
4) will give an absoluteestimatefor the molar massand associatedparameterssince
frictional (i.e., shape)effectsare not involved.
In this description,we again,for simplicity,refer only to the absorptionsystem,
becauseof its simplicity and availability,for recordingthe distributionof solutein
the ultracentrifuge cell-this time an equilibrium distribution. The most accurate
method is in fact the interference system, but this requires considerable more
expertiseto operatecorrectly (the readeris referred to referencesVan Holde, 1985;
Harding et al., 1992a;Schusterand Laue, 1994.)The concentrationand volume
requirementsfor the macromolecular solute depend more critically, compared to
sedimentationvelocity,on the extinctioncoeffrcientof the protein.Like sedimentation velocity and dynamic light scattering,the lower the protein concentrationthe
better, since it minimizes problems of thermodynamic nonideality. At higher
concentrations(necessary
phenomenaarebeing investigated
if possibleassociative
- such as at the concentrationsused for NMR measurements).the limitation is the
Lambert Beer law. The proportionality c * absorbance(A) fails aboveabsorbances
of about 1.4 to 1.5.For concentrationsof 1 mg/ml and above,shorterpath length
cells need to be employed or an ultracentrifuge with Schlieren optics employed.
Volume requirementsare lower than for sedimentationvelocity: generally0.1 to
0.2 ml. The longer the column, the greaterthe precisionand the more information
that can be extracted. The shorter the column, the quicker equilibrium can be
reached.Experimentaltimes can be long. For moleculesof M<10000, <24 h arc
required;large,slowerdiffusing moleculestake48 to72h, althoughfor the latter,
time to equilibrium can be decreasedby initial "overspeeding",that is, running at
higher speedfor a few hours before setting to the final equilibrium speed.It may,
in some applications,be desirableto use shortercolumns (as low as 0.5 mm);
although the accuracyof the molar masseswill be lower, this "short column"
method offers the advantageof fast equilibrium (few hours) (Correia and Yphantis,
1992), which may be important if many samples need to be run and/or the
macromoleculeis relatively stable.As with sedimentationvelocity, a temperature
of 4 oC can be used without difficulty.
If scanningabsorptionoptics are used,equilibrium patternssuch as in Figure 4
can be readdirectly into an attachedPC. As with sedimentationvelocity,cells can
be run multiply in multihole rotors and electronicallymultiplexed.In addition,
specialmultichannelcells containingthreesolution/solventpairs can be used,and
this is illustrated in Figure 4. So for a four-hole ultracentrifuge rotor (with t hole
288
STEPHEN
E.HARDING
neededfor the counteqpoisewith referenceslits for calibrating radial positions in
the cell), nine solutions can be run simultaneously.
Before interpretation in terms of molar mass, a baseline is normally required.
After the final equilibrium pattern has been recorded (equilibrium checked by
comparing scansseparatedby a few hours), the rotor is run for a short time at a
higher speed(up to 60000 rev/min or the upper limit for a particular centerpiece)
to deplete the solution-or at least the meniscus region-of solute: the residual
absorbancegives the baselinecorrection (absorbanceof nonmacromolecularspecies).This is not so easywith small proteinswhoseequilibrium speedwill be quite
1.5
tr
€
F r.o
0.5
6.0
6.2
6.4
6.6
6.8
7.0
7.2
Radius, r (cm)
Figure4. Sedimentation
equilibriumprofiles
for p-lactoglobulin
B.Absorption
optics,
wavelength= 280 nm. Rotorspeed= 15000 rev/min,temperature
= 20.0.C. A
multichannel
cell (l2 mm opticalpathlength)was usedallowingthreesolution/solv e n tp a i r sw i t h = 0 . 1 2m l i n s o l v e nct h a n n e l s=,0 . 1 0m l i n s o l u t i o nc h a n n e l sI .n n e r
profile:loadingconcentration
c = 0.1 m/ml; middle:O.2
mg/ml;outer= 0.3 mg/ml.
Becauseof restrictions
from the Lambert-Beer
law, with the outer channel,only
absorbances
<1.5 could be used.This difficultycould be offsetby usinga higher
wavelength.
With the innerchannel,the signalcould be increased
by usingfar-uv
o p t i c s( 2 1 0 - 2 3 0n m ) .
Protein Hydrodynamics
289
high anyway: careful dialysis of solution versusthe referencesolvent before the
run (and use of the dialysate as reference)may be necessary.
The averageslope of a plot of ln A versusr2, the squareof the radial distance
from the center of the rotor, will yield the molar mass:
- up) co2
M = (dlnA/drz1xZFttt1t
(t2)
At finite concentrations,
this will be an apparentmolar mass(becauseof the effects
of thermodynamic nonideality; seebelow), but for macromolecularsystemsof M<
100,000g/mol in aqueoussolventsof reasonable
ionic strength(0.05M and above),
these effects are small at loading concentrationsof 0.5 mg/ml and less: in these
cases,it is reasonableto assumeM = M*.uoo.
If the protein solutionis heterogeneous
(Containinginteractingor noninteracting
species of different molar mass), then the plot of ln A versus r2 will be curved
upwards.This situationoccurswith self-associating
systemsand heavily glycosylated protein systems such as mucus glycoproteins. In this case, the data can be
treatedin one of two ways: (i) an averageslope is obtained.This yields, as with
equation12, the weight averagemolar mass,M*. For stronglycurving systemsor
for systems where the cell base is not clearly defined, a procedure that uses a
function known as M' (creeth and Harding, 1982;Harding et at., lgg2b)is useful
for this purpose;(ii) local slopesusinga sliding stripprocedure(Teller,1973)along
the ln A versusr2 curve can be obtainedto give what is called apparent.,point"
weight averagemolar masses,M*,uoo(r),asfunction of eitherradialposition (or the
equivalentlocal concentrationor absorbance).
This procedureis pariicularlyuseful
for theinvestigationofself-association
phenomenaandothertypesofheterogeneity
and also providesa methodfor extractingthe z-averagemolar mass:
= { M*Gu).A(rJ- M*(r").A(r,)}/[A(ro)-A(r")
M,,uoo
J
(l 3 )
where (ru, 16) are the radial positions of the solution meniscus and cell base
respectively,und Mr,uoo-) M, as the concentration(in absorbanceunits, A) -+
0.
The ratio MA4* can be usedas an index of the heterogeneityof the sample,and,
,
for noninteracting systems,is a measureof the inherentpolydispersity of a system;
this is particularly relevant to the study of heavily glycosylated systems, for
example.
If the systemis self-associating
or involved in "heterologous"association(i.e.,
complex formation phenomena),then either the A(r) versusr plot (Figure 5a), the
M*,"pp(r) versus A(r) plot, or a plot of M*uoo versus c for different loading
concentrations,c, can be used to assayfor the itoichiometry and strength of an
interaction. There are severalcommercial softwarepackagesavailable: seeccilfen
et al' (1997).Assaysarealsoavailablefordistinguishingbetweena self-association
from noninteractingmixtures (Roark and yphantis, 1969).
S T E P H EE
N.H A R D I N G
(a)
g
002
G
'E
-oo,
tr
0.4
o
o
t!
.cl
o
,4 0.2
7.0
7.1
Radius(cm)
0.2m
(b)
0.195
L
A
0.190
q
' 0.r85
13
a
0.180
\o
0.r75
0.170
t.0
1.5
c(r) mg/ml
Figure5. Analysisof sedimentation
equilibriumdata self-association
analysis.(a)
Self-association:
AbsorbanceA(r) versusradial displacement(r) data for protein
(PDl).Rotorspeed= 12000rpm,temperature
disulphideisomerase
= 4 oC,loading
concentration,
c = 0.4 mglml. Line fitted is for a reversibleideal dimerization,
dissociationconstant,Ka = 180 pM (from Darby et al., 1997).(b) Thermodynamic
nonideality:
Plotof the reciprocal
point(apparen0
average
molarmassM*,.00.(r)
asa
functionof radialposition,r,versusconcentration,
c(r),forturnip-yellowmbsaicvirus
(TYMV). M* (fromextrapolation
= (5.g10.2) x 106g/mol.
to zero concentration)
(19S5).
Adaptedfrom Hardingand.lohnson
291
Protein Hydrodynamics
(M > 100000)suchasproteinassembliesand heavily
For largermacromolecules
solutions,nonideality(through
glycosylatedsystemsand/orfor more concentrated
macromolecular exclusion and any unsuppresedcharge effects) may become significant,and this will tendto causedownwardcurvaturein the ln A versusr2 plots:
this can often obscure heterogeneityphenomenaand the two effects (nonideality
and heterogeneitycan occasionallycancelto give a linearplot that can be misleading, a problem that can be avoidedby running at more than one loadingconcentrathen a simpleextrapolation
tion). If the solutionis not significantlyheterogeneous,
from a single experiment of point (apparent) molar mass to zero concentration
"ideal" value (in
(absorbance)can be made in order to give the infinite dilution
general, reciprocals are usually plotted; see Figure 5b). Alternatively, several
sedimentationequilibrium experimentsperformed at different loading concentra"whole cell" molar massesM*,uooto zero concentrations, c, and extrapolationof
tion are necessary.
Insofar as modern computing packagesare concerned,software currently available from the commercial manufacturertends to require an assumedmodel prior
etc.).We
nonidealself-association,
to the analysis(idealmonomer,self-association,
find a generalpackage,of usethatdoesnot requireassumedmodels.This is MSTAR
(Harding et al., 1992b), now available for PC (Cdlfen and Harding, 1997). This
programevaluatesM*uoo(usingthe M* function),M*,upp(r)versusr or A-,and also
M,,upp(r),if the data is 6i suffrcientlyhigh quality. After thesemodel independent
analyies have been performed, resort can then be made to the more specialized
polydispersity,etc.)'
packages(self-association,
SHAPEMEASUREMENT
Hydrodynamic methods provide a relatively quick method to acquire average
or "gross" conformation information aboutproteinsand protein assemblies,and
in some casesto give rather detailedrepresentations,as for example for T-even
bacteriophagesand antibodies.Limited flexibility information is also possible.
"low-resolution" compared to the
Although such information may seem to be
information possible from the powerful structural probes of x-ray crystallography and high-resolution NMR, it should be borne in mind that the latter
are sometimesnot applicable for the following reasons:(i) high enough aggregation-free concentrations necessaryfor high-resolution NMR may not be
attainable for a given protein system or assembly; (ii) the protein or protein
assemblymay not be crystallizable,or molecularflexibility effectsmay obscure
attemptsto interpret electron density maps: the latter is the reasonwhy crystallographershave had considerabledifficulty in evaluatingthe structureof intact,
immunologically active antibody molecules'
In both these cases, hydrodynamic methods are particularly valuable (i) to
monitor possibleassociativebehavior at higher concentration(using any of the
techniquesabove, particularly sedimentationvelocity and equilibrium) and (ii) to
292
STEPHEN
E. HARDING
provide conformation information of the protein or protein assembly in in-vivo
solution conditions;this can be either in terms of an overall shapeor in terms of
refinement of a crystal structureof a protein or electron microscopic structureof a
protein assembly (arrangementof subunits). A good example is the case of antibodieswith a usefulearly attemptmadeby Gregoryand colleagues(1987).
M o d e l l i n g S t r a t e g i e sS:p h e r e sE
, l l i p s o i d sB, e a d s ,a n d B e n d s
Hydrodynamicrepresentation
of proteinshapeis in termsof modelsthatprogress
in sophisticationfrom a sphereto bead models (Figure 6). The simplest is the
equivalenthydrodynamic(or "Stokes") sphere,of radius rs (cf. equation6). The
next steptowardbetterrepresentation
is theellipsoidof revolution,anellipsoidwith
two equal axesof which there are two: the prolate ellipsoid (cigar shape)with two
equalminor axesandthe oblateellipsoid(discoid)with two equalmajor axes,both
characterizedby the axial ratio a./bwith the semiaxesaZb.In the limit of u>b, the
prolatebecomesa rod and the oblatea disc. The next step in sophisticationis the
generaltriaxial ellipsoid of semiaxesa > b > c and axial ratios { a/b, b/c }, which in
the limits go to spheres{alb = l; blc = | }, oblateellipsoids lalb=l }, and prolate
ellipsoids {b/c=l}, the latter two going to discs and rods respectively.Another
extreme of the general ellipsoid is the tape (a >> b >> c). The final degree of
sophistication is the bead model: many macromoleculessuch as antibodies and
multisubunit proteins are difficult to representby symmetric shapeslike ellipsoids.
(a)
ftC.|. txiproit
ObLl. gliptdt
Figure 6, Hydrodynamicmodels for conformation.(a) Ellipsoidsof revolution
(adapted
(c)Beadmodels.(r)T-even
fromTanford,
1961).(b)Generaltriaxial
ellipsoids.
bacteriophage
in slow(s)and fast(f)forms(s!0.*= 710Sand 10205,respectively).
Modelledon sedimentation
and diffusioncoefficient
data.FromCarciade la Torre
(l989). (ir)C1 complexfrom the complementsystern.
Modelledon sedimentation
coefficientand R*data.FromPerkins(1989).(iir)CyclicAMP receptorassociated
with
BO-bpDNA. Only maximumbendingof the DNA reproduces
the measured
rotational
diffusiondecay constant(from electric dichroismdecay).From Prirschkeand Antosiewicz(1989).
293
Protein Hydrodynamics
(b)
x
l']o-.
(c)
(r)
8?
(rr)
(rr,)
Figure6. Continued
Bead modelling (arraysof touching or overlappingspheres)allows very sophistiA successfulvariantofthis is bead-shellmodelling,
catedshapesto be represented.
where the surfaceof the macromoleculeis representedby beads.Filling strategies,
however, such as those based on crystallographic coordinates, have sadly been
shown (Carrasco,1998)to be unreliable'
As the degreeof sophisticationincreases,the uniquenessproblem also increases.
What this means is that a model may be consistent with a particular measured
hydrodynamic parameter such as a sedimentationcoefficient s!0,* or a radius of
gyrationR, (from solutionx-ray scatteringor light scattering;see,for example,Van
Holde, tlSl5; Uutso may othermodels.For example,a valuefor the sedimentation
coefftcientcan corespond to one equivalentsphere,fwo ellipsoidsof revolution,a
line solution of triaxial ellipsoids, and almost an infinity of bead models. There is
294
STEPHEN
E. HARDING
a further problem from ellipsoids upward: hydration or the degreeof buffer/solvent
associatedwith (chemicallyboundor physicallyentrapped)by the protein - which
alsocontributesto s!0,*amongother things andhasto be "ith". -"uru.ed separately,
assumed,or eliminatedby combinationof measurements.
As the degreeof sophistication in the model increases,there is a greater need for independentmeasure_
ments (two for ellipsoids of revolution, three for triaxial ellipsoids to give
a
"unique"
answer).Beadmodelling^,
normally performedwith at leasttwo hydrodynamic measurements
(popularlyj8o,* o, D!0,* and R, , althoughrotationalprobes
havebeenused;Antosiewiczandpcirschk.,tisg; p6ti"hke andlntosiewicz,
l9g9)
is bestusedto refine a structurefrom crystallographyor to selectbetween
certaln
plausiblestructures.A further refinementto beadmodelling is in the modelling
of
molecular flexibility, the bending or "segmentalflexibility" in the molecules.
Detailsof this and its applicationto flexibility phenomenairmyosin can be
found
in Garciade la Torre (1989),Garciade la Tone and Bloomfieti
ltolls; Garciade
la Torre (1992), and Garcia de la Torre (rgg4). Finally, bead ani bead-shell
strategieshave been developedbased on shapealone, without the ambisuities
causedby size (Garciade la Torreet al.. 1997).
IntrinsicViscosity
The simplesthydrodynamicconformationmeasurement
is the intrinsicviscosity.
The classicalreferenceon the theoryandpracticeofprotein viscometryis an
article
by J.T. Yang (1961). A more recenteffort has beenwritten by the piesent
author
(Harding, 1997). The viscosity of an aqueoussolvent will be
increasedby the
additionof a macromolecularsoluteto an extentdependingon (i) the concentration,
(ii) the size (including the degreeof hydration), ana (iii) the
shape.Increased
concentration,size,and shapeall increasethe viscosityof a solution.
viscosity measurements
on proteinsin dilute solutionarenormally performedin
a capillary (or "ostwald") viscometerwith the flow time unde. g.auity (between
two referencepoints)of thesolution(t) comparedto thatof thesotvent
1t"y,although
differential microviscometersbased on a pressureimbalance princifte
appear
highly promising (Haney,1985).
with conventionalcapilraryviscometers,pumping of liquid and timing
is now
usually done automatically,employing photodetectors(using, for
example, a
Schcitt-Gerdte
(Hofheim, Germany)system)and becauseviscJsity is a sensitive
function of temperature,a water bath is requiredwith the temperature
controlled
and measuredto within t 0.005 oc. From the flow times (averagedover
consistent
measurements),
the relativeviscosityis 11,obtainedfrom
rl,= (t/t).(p/po)
(r4)
with (p/pJ the ratio of the solution to solvent density.This can
be measured
separatelyfor each concentration using a precision density meter (Kratky
et al.,
1973;Rowe,1978),butmoreconvenientlythiscanbeavoidedifweuseakinematic
295
Protein Hydrodynamics
relative viscosity Tl', = (yto) and use a correctionfactor in the data analysis(see
below) (Thnford,1955).
A (kinematic) reducedspecific viscosity is then defined
tl'r.o- (t'l'r- 1)/c
( 1s)
so if c is in g/rnl, I'."0 is in mVg. To eliminatenonidealityeffects,l'l'r"6is measured
at a seriesof concentrationsand extrapolatedto zero concentrationto yield the
(kinematic)intrinsic viscosityI11'] which can then be correctedfor densityto give
the ("dynamic") intrinsic viscosity [ 4 ].
[n]= {(1-Vp")/p"}+ [ n']
(16)
"viscosity increment" v (see for example,
The shapeparameter,known as the
Tanford, 1961;Harding, 1995)is obtainedfrom
y=[q] /v,
(r7)
"swollen specific volume", is the volume of the "hydrated"
where vr(ml/g), the
protein per unit mass of dry protein and is related to the partial specific volume v
"w") is known as the "hydration",
by v, = v + (5/p"), where6 (sometimessymbol
the number of grams of solvent bound per gram of dry protein. Or, in terms of
protein volume, V, v, = VNo/JvIwhereV (ml) is the (hydrated)volume of the protein
and No is Avogadro'snumber.Sincev is the Einstein (1906, l9ll) 2.5 value for
spheresand since y = (4/3)nr3,the hydrodynamic radius can be found thus
providing an alternativeprocedureto dynamic light scatteringfor its measurement.
v has also been evaluatedfor prolate and oblate ellipsoid models.Although the
direct formulae are complicated(Harding, 1995),simple polynomial approximations that are accurateto +l%oareavailable(HardingandCcilfen,1995)and hence,
provideda valuefor v. (or 6) is known or assumed,the axial ratio a/b can be found.
The value typically taken for 6 for proteinsis about 0.35 (: v,=l), although for
unconjugatedproteinsit can vary by about !l00%o, and for heavily glycosylated
proteinssuchasthosefrom mucussecretions,
6 canbe ashigh asabout70 (Harding
when
(Caution
assigninga conformationfrom
has
to
be
expressed
et al., 1983).
evaluation
of v merely specifiesa llne
For
triaxial
ellipsoids,
viscositydataalone.)
(a/b,
the
extremes
of prolateellipsoid
possible
values
b/c)
between
of
solution of
(b/c = l) and oblateellipsoid (a/b=l) (Figure7). Besidesan assumptionover 6, a
to providea graphical
is necessary
furtherindependenthydrodynamicmeasurement
(a/b,b/c)
directly.
intersectionwith the v-line to specify
For thecaseof beadmodelllng,computerprogramsareavailablesuchasHYDRO
SOLPRO
(Garcia de la Torre et al., 1994) or the more recent size-independent
algorithm (Garcia de la Torre et al., 1991) for predicting u (or [q]) for a given
specified set of coordinatesfor the beads;this procedurecan thus be used for
selectingwhich model givesthe desired[q] (afterassuminga valuefor 6). Because
of the uniquenessproblems referred to above,for bead modelling the [q] data
E. HARDING
STEPHEN
296
Truc (a./b,b/c) = (2.0,2.O)
L)
. o 3
P
1.5
2.s 2.5
1.0
4.0
alb
"line solutions")
for v and P as a functionof
Figure7, Plotsof constantvalues(i.e.,
moleculeof
for
a hypothetical
data,
Simulated
ratios.
thl two triaxialellipsoidaxial
"real" la/b,1o1gy
=(2.0,2.0).AdaptedfromHardingandRowe(1983).Theintersection
to give a uniquevaluefor (alb;blc),althoughthis particularchoiceof
is supposed
an intersection.
shapefunctionsgivestoo-shallow
cannot be used in isolation but has to be combined with other hydrodynamic
(e.g.,sedimentation,
diffusion,x-ray scattering,rotationaldiffusion,
measurements
etc.).
SedimentationVelocity and Dynamic Light Scattering
The principal conformation parameterto come out of both thesemeasurements
is known as the frictional ratio (f/f"). This is the ratio of the frictional coefhcient of
the protein to the frictional coeffrcientof a rigid sphericalparticle^of the same
anhydrousmassand volume. This can be related to either tlo,* o. Dlo.* by
(f/f) = (v(l -npo)/Nn.6nqos!0,*
) (4nN^l3iLDtt3
(l 8 )
rur (+nro\t'3
(le)
(I/to)=*rrlTil)
I
N
(seefor example,Tanford,l96l; Harding, 1995)where,qo is the viscosityof water
at2O.O"C. In order to get shapeinformationfrom equations18 or 19, hrst of all a
297
Protein Hydrodynamics
function P (named in recognition of F. Perrin, who worked out the theory for the
frictional coefficientsfor ellipsoids)is defined:
(20)
p= (f/fo) . [(6/tp2o,*) + 1] - r/3
and then,similarly to viscometry,if 6 is known or assumed,P can be obtained.The
"Perrin function" P is analogousto the viscosity incrementv, and the axial ratio
(a/b) for an ellipsoid of revolution can be found either by a rather complicated
expressioninvolving an elliptic integral or by simple polynomial expansions
availablefor both prolate and oblate ellipsoids(Harding and Cijlfen, 1995).For
generaltriaxial ellipsoids,as with v, there is a line solutionof possiblevaluesfor
P (Figure 7). In principle, {a/b, b/c} can be found from the graphicalintersection
but as is clearfrom Figure 7, this is too shallowto cope with any dataerror.Other
combinationsinvolving theseor other shapefunctionsneedto be employed.
Use of ConcentrationDependenceParameters,Combined Shape
Functions,and the Radiusof Gyration Rg
A simple way in principle to solving the hydrationproblem is to combine two
shapefunctions together in such a way that the experimentalrequirement for 5 or
"hydration-independent"
shapefunction. The
v. is eliminatedto give a combined
p-function
and comes from combination of
simplest of these is known as the
equations17 with 18 or 19 (see,for example,Tanford, 1961;Van Holde, 1985;
Harding, 1995).This function is unfortunatelyhighly insensitiveto shapeand of
very limited use for conformation analysis; in fact it has found more use as a
quasi-constantparameterfor enablingM to be calculatedfrom [n] and s!0,* or
D!0,* (Yang, 196l). A more useful combinationis [q] with k,, the concentration
dependenceregressionparameterfrom sedimentationvelocity measurements(cf'
equation 9), provided the sedimentationmeasurementshave been made in a buffer
of suffrcient ionic strength, I, to suppresscharge effects. To an approximation,
(Rowe, 1992;Rowe, 1977)the ratio
R = {k,/[n]]=2(l +PJ)lv
(2r)
Another is a combinationof the secondthermodynamicvirial coefficient,B (from
the concentration dependenceof the apparent molar mass measurementsusing
shape,
sedimenrationequilibrium), with [r1] to define the hydration-independent
(Harding,
1995).
1981;
fI
function
rI = {2BM/[q]] - f(z,D/{tnlM}
(22)
where the 2nd term on the RHS [a function of molecular chargeor valency (Z) and
ionic strength (I)l goesto zero if the ionic strengthis sufficient (normally > 0.3M).
As with v and P above,both R and lI are availableas simple polynomial expansions
in terms of axial ratio a./bfor ellipsoids of revolution (Harding and Ciilfen, 1995).
They are also availableas line solutions for {a/b, b/c} for triaxial ellipsoids and of
S T E P H E NE . H A R D I N G
298
coursehavethe advantageover t/and P of not requiringan assumptionconcerning
Unfortunately,plotting R with II givesan equally
hydrationfor their measurement.
poor intersectionas that shown in Figure 7. A bettercombinationis lI with the
radiusof gyration shapefunction G definedby (Harding, 1987)
6 = {(4nN6)/(3vM }2/3.R1
(23)
and
Rnderivesfrom a light scattering(or x-ray or neutronscattering)measurement,
indistinguishif the surface(aq.)solventon the proteinis to a good approximation
able from surroundingsolvent,and if the protein is not internally swollenthrough
hydration,the specificvolume term in equation23 refersto the anhydrousprotein
(v = V) and no assumedvaluefor the hydrationis required.G alsohasa line solution
for triaxial ellipsoids,but graphicalcombinationof G with fI doesgive a reasonable
intersectionand hasbeenusedto investigatethe overallconformationof myosin in
solution (Harding, 1987).
Insofar as bead modelling is concerned,Rn (i.e., G) from x-ray and neutron
scatteringand s!0,* (or P) have beenusedwitiin the limitations referredto above
with the earliermodelling programTRV (Garciade la Tone, 1989)to distinguish
plausibleconformationsfor antibodymodels(Gregoryet al., 1987) and has been
usedto show thesemoleculesareclearly not coplanaras sometimesrather misleadingly depicted in textbooks.R, with s!0,* has been used to select appropriate
modelsfor the complementsystem(Perkins,1989)(seeFigure 6) and a combination of D!0,* and s!0,* used to model the self-assemblyof T:even bacteriophages
(Garciade la Torre, 1989;Garciade la Torre and Bloomfield, 1977).R, combined
with [r1] and electroopticdatahas beenusedto model the flexibility of regionsof
myosin betweenthe 52 head and low meromyosin(LMM) in terms of bending
energies(Iniestaet al., 1989;Garciade la Tone, 1989).
Measurementand Use of RotationalHydrodynamic Shape Functions:
FluorescenceDepolarizationDecay
A protein in solution will be subjectto Brownian rotationalforces.The easeor
rate at which a protein rotateswill dependon its size, shape,and hydration-in
common with the three factors that also determine rate of translational diffusion.
Therefore,if the size and hydration areknown (or can be eliminated by combination
with anothermeasurement),then measurementof the rotational diffusion property
can be usedas anotherprobe to measureshape.Although thesemeasurementstend
to be more diffrcult, the incentive is that the shapefunctions so derived are more
sensitivefunctions of shape.The principal methods have been flourescencedepolarization, electro-optics and, more recently, nuclear magnetic resonance(Garcia
de la Torre et al., 1998).
The most popular rotational diffusion probe is fluorescencedepolarization (Weber, 1952; Van Holde, 1985). With the fluorescencedepolarization method, fluorescent light emanating from a stimulated (by polarized light at the appropriate
Protein Hydrodynamics
299
wavelength)protein with a suitablefluorescentchromophore(either intrinsictryptophan,or syntheticallyattached)will be planepolarized.As theproteinsrotate
under rotational Brownian forces, the degree of polarization will decay at a rate
dependenton the speedof rotation of the molecules.Detectorsfitted with polarizers
are used to measurethe intensity of light parallel (Ir) and perpendicular(I,, ) to the
incidentpulse and the anisotropymeasured
A=(1, -Ir)l(1, +2Ir)
(24)
In the "steady-statemethod",the protein solutionis continuouslyirradiatedand
by making measurementsof A in solutions at a variety of temperaturesand
viscosities(usually with the addition of glycerol) and with knowledge of the
fluorescentlifetime of the chromophore,the harmonic mean relaxation time tn
(units: sec.)can be measuredfrom extrapolatinga plot of 1/A versusT/qo to T/qo
= 0 (Van Holde, 1985;Weber, 1952)11"is the solventviscosity at temperatureT.
As with other hydrodynamic parameters,in principle, rn needsto be extrapolated
to zero concentrationto eliminate any possiblecontributionsfrom nonideality
effects.
To obtain shapeinformation from tn, a ratio 1tn/r"l is defined (by analogy with
the Perrin P function) where
{'cr,!r
ol = lkTrn)/(noV)
(2s)
and where the volume of the protein V = v.M/}.{a.
To remove the requirementof knowledge of v, (i.e., hydration), {tnlro} is
combined with [q] to produce the hydration-independentparameterA (Harding,
1980, 1995;Harding and Rowe, 1982a).
A = v/{tr,/%}= (q"[r1JM)
(NAkTrh)
(26)
As with the other shapefunctionsrefened to above,simplepolynomial equations
are availablethat relate A to the axial ratio of ellipsoids of revolution, and an
example of its applicationto the globular protein neurophysincan be found in
Rholam and Nicholas (1981). It is also availablefor triaxial ellipsoids, and a
graphical combination of A with R can be used to obtain {a/b, b/c} uniquely
(Harding and Rowe, 1982b).Indeed,this methodhas beenusedto confirm measurementspreviously made using the ellipsoid of revolution model (Rholam and
Nicolas, 1981) that the dimerization of neurophysinclearly occurs through a
side-by-sideas opposedto an end-to-endprocess(Figure 8). Theselatter references
also illustrate respectivelythe extraction of [q], k,, and tn (and henceR and A) for
a dimerizing system.
some words of caution:althoughfluorescencedepolarization,along with other
rotational diffusion techniques,are particularly sensitiveprobes for conformation,
it should be stressedthat particularly for synthetically attachedfluorescent chromophores, it must be establishedthat there is no free rotation of the fluorescent
chromophore with respectto the rest of the molecule; also for proteins containing
more than one domain, segmentalflexibility can obscurethe shapemeasurement
Protein Hydrodynamics
301
However, a significant recent advancehas been the design of an instrument with
adequateshielding againstsuch affects(pdrschkeand obst, I 99 I to permit the
use
)
of solventsat physiological ionic strengths.The applicationof electric birefringencemethodsto triaxial ellipsoid modelling can be found in Harding and
Rowe
(1983) and to beadmodellingin pcirschkeand Antosiewicz(19g9).
rinatty, nmr as
a route for obtaining time-resolved rotational relaxation time appears highly
promising (Garciade la Torreet al., l99g).
Some Computer programsfor ConformationalAnalysis
For ellipsoid modelling,the ELLIpS seriesof programfor rhe pc (BASIC and
FORTRAN)havebeendeveloped(Hardingandccilfen,
rgg5;Hardingetal.,1997).
ELLIPSI evaluates the axial ratio a,/b for prolate and oblate elfpsoids
for a
user-specifiedvalue for a hydrodynamicparameterand is basedon polynomial
approximationsto thefull hydrodynamicequations:accuracyof this appioximation
is normally well within the precisionof the measurement.
ELLIps2 usesthe full
hydrodynamicequationsfor generaltriaxial ellipsoidsto specifythe set ofhydrodynamicparametersfor any given valueof the axial ratios
{a/b, ulc1.eLLms: ana
ELLIPS4 do the reverseprocedure using a variety of graphical combinations
of
hydration-independent
triaxial shapefunctions(cf. Figures1 and4). Elsewhere,the
routine soLPRo (Garciade la Torre et al., l99i,l99g) is particularlyuseful for
the applicationof beadmodels.
R EFERENCES
Ackers, G. (1975). Molecular sieve methods of analysis. rn: The proteins.
Third ed. (Neurath, H. and
Hill, R.L., Eds.),p.1. Academicpress,New york.
Andrews, P. (1965). Estimation of molecular weights of proteins by Sephadex
gel filtration. Biochem.
J. 91,22.
Antonsiewicz, J. and P<irschke,D. (1989). An unusualelectrooptical effect observed
for DNA fragments
and its apparent relation to a Permanentelectric moment associatedwith bent
DNA. Biophys.
C h e m . 3 3 ,1 9 .
Amer, E'C. and Kirkland, J.J. (1992).ln: Analytical lJltracentifugation in
Biochemistry and polymer
science.(Harding, s E.' Rowe, A.J., and Horton, J.c., Eds.),p. 209. Royal
society of chemistry,
Cambridge, England.
Barth' H'G' (1980).A practicalapproachto stericexclusionchromatographyofwater-soluble
polymers.
J. Chromatog. Sci. l 8, 409.
Brown , w. (ed) (1993). Dynamic Light scattering. The Method and
some Apprications. oxford
University Press,Oxford.
Burchard, w. (1992). Static and dynamic light scanering approachesto
structure determination of
biopolymers. rn: Laser Light scatteing in Biochemistry (Harding, S.8.,
Sattelre, D.B., and
Bloomfield, V.A., Eds.),p.3-22. Royal Societyof Chemistry,Cambridge,
England.
claes, P., Dunford, M.' Kennedy, A., and vardy, p. (rgg2). An online
aynu,nr" light-scattenng
instrument for macromolecular characterization. In: Laser Light Scatteing
in-Biochemistry.
(Harding,s'E', satrelle,D.B., and Broomfierd,V.A., Eds.),p.66-26.
Royal so"cietyof chemisrry,
Cambridge, England.
3O2
STEPHENE. HARDINC
cctlfen,H. andHarding,s.E. (r997).
MSTARA andMSTARI:Interactivepc
algonrhmsfor simpre,
model-independent
evaruation
of sedimentation
"q"'iui",n i"i". il;;;;.
J. 25, 333_346.
"t*:;,f;;l;1if;jf
D.J.
(teelj.Lowtemperature
:wt"t, s.r, ."*,i";, ii, .* winzor,
byanaryticar
a,tuay
""^*""T;'l#:r"lfi:T:.,iffi:::|;:;r"ft"".*"'r";ng-niuoprot"in,
Correia' J'J' and Yphantis'D'A (1992).
rq"tiir""r
sedimentation in short sorution
corumns. In:
Anarvticar urtracentrifugation in
Bioch)mistry and porvmer sci"r;;.i;;;
,.8., Rowe, A.J.,
and Horton, J.C., Eds.), p.231_252.noyuf
S*i"ty of Chemistry Cambridge,_England.
Creeth,J.M. and Harding, S.E. (19g2).
Sor" oUr"*uti ons on a new
type of point averagemolecular
weight. J. Biocheml Biophys.
Meth. 7,25-34.
Darby, N., Harding,S.E.,and
Creighton,T.E. 11997).(n eress.;
Dubin' p'L' and principi, J M. (19'9)
N" p,""io"rrf zuggesteddimensional
paramerer contrors peak
insize
frflll]:
exclusion.tro'nutogrupr,y.
oi".o"rrr. ct".., a,n.Ci",n..io".rr"p.int.,:0,
Einstein,
A. (1906).Ann.physik.19,289_305;and
corrigenda
( lglt) 34,591_592.
Garciade ta Torre,t ,tnlll..,llg"dynamic
l,r-j**,
:!..cromotecular u.r"_bti"r. rn:Dynamic
" -on"*",e-r.ia,i,pp:-:r,n"y"r
^ :::'","t[!^.:;';*:f;;*r!:;^:X;e"'.
Garciadela Torre'J' (1992)'Sedimentation-coefficients
ofcomplexbiologicalpartic les.In:
ultracentrifusation
Analytical
in Biochemistry
""a rrly..)"ili1...1H*aing,
I.e., R";;.r.,
J.C.,Eds.),p. 333_345.
"nd Horron,
RoyalSociey"iCi"rfr"y, Cambridge,
London.
o^'"';:::;l:r:;.1.
(tee4). Hvdrodynamics
orr";";;ry
n"xibremacromorecures.
Err Biophys.l.
*'1"tll,f;,11;.i?1"t'"1rliil or|:hrffi;r"vdrodvnamics
ormacromerecurar
comprexes
3.
Garciade ra Torre'J" canasco,8.,
and Harding,s.E. (lgg7). SoLpRo:
Theory and compurerprogram
j:;n?fl;T"
ofsoLution
PRop."i.i"i'igiir"cromolecules
anaui"p*,i"r"r.err.Biophys.
*'':i;1},ff;';,i,T,lli?.i,1;llJ$"fJjl";?;!11e8)'carcurarionorNMRreraxation,covorum
v'vl'rlres ur oeao models using the soLPRo
computer program. Eur.
Biophys. l. r;i pr"*r
Garcia de la Tone, J., Navarro,
S., Lopez Martinez,M.C., Diaz,
F.G., and Lopez Cascales,J.J. (1994)
iJr"rl?'r1;::5i!;;;**'tt"h"p;i;1";;;hvdrodvnamicp.";;l;,;;;;romorecures.
Gregory L., Davis, K.G., Sheth,
B., Boyd,
J., Jefferis, R., Nave, C. and Burron,
D.R. (1987) The solution
::i*fi i'ff ll:#ffffi'"::lUru::,:#r'o''"oi'"ntuti;;;;;;"i,"",,eX-rav
Han' M'K'' Knutson, J'R', and
Brand, L. (1989). Fluorescence
studies of protein-subunit interactions.
tn: Dynamic ptoperties of Biomotecutar'A;;;;;;;;.(Harding,
S.E. and Rowe, A.J., Eds.), p.
I l5--134.
RoyalSocieryof Chemistry,
C.riiae","l;"a"".
Haney'M'A' (1985)'A differentiarviscomerer.
AmerilanLaboratory17, 4r-56.
Harding's'E' (1980)'The combination
of the viscosiflnlrement with the
harmonicmeanrotarional
;:ff:;:r:rffi,?li"r,;i'",
tr,""o"r".,"iiinir uiorogi"ur
,"",;;;;;;
insorution.
Harding's'E' (1981)'A comnoyd
lv{rodynamicshapefunctionderivedfrom viscosityandmolecurar
covolume
measurements.
tnt.l. giot. Uac-r"ii:,:oO_:+1.
"*oT3;:51t986)'
Appriqxl;ensof light ,"",onrg t"
-,"robiologv. Biorech.Appl. Biochem.
8,
Harding,s.E. ( l ggT).A generalmethod
for moderingmacromolecurar
shapein sorution_a graphical
(11-C)intersectionprocedurefortriaxiat"rifprifirlglpnys.J.5l,6T3_680.
Protein Hydrodynamics
303
Harding' s'E' (1995)' on the hydrodynamic
analysisof macromolecularconformation.
Biophys. chem.
55, 69_93.
Harding' s E' ( 1997) The intrinsic viscosity
of biological macromolecules.progress
in measurement,
t;#:;:rr.rrt",
and application ro structure iniilute solution. prog.
Bifhys. Mol. Biot.68,
Harding' s'E and ccilfen. H' (1995). Inversion
formulae for ellipsoid of revolution macromorecular
shapefunctions.Analyt. Biochem.22g, l3l_142.
Harding, s.E. and Johnson,p. (19g5). physicochemical
studies on tumip-yelrow-mosaic vrus. Homo_
geneity' relative molecular masses,hydrodynamic
radii and concentration-dependenceofparameters in nondissociating solvents. Biochem.
J. 231, 549_555.
Harding' s'E and Rowe,AJ (1982a).Modeling
biotogicalmacromolecules
in solution.l. The ellipsoid
of revolution.Int. J. Biol. Macromol. q, ilO_ia.Int.
J. Biol. Macromol. q, rcL,_.rcq.
Harding' s'E' and Rowe, A.J. (1982b).
Modeling biological macromoleculesin sotution.
3. The
lambda-Rintersectionmethodfor triaxial elrif,soids.
Int. J. Biol. Macromor.q, lil _lat.
Harding' S'E' and Rowe, A'J'.(1983).Modeling
biologicalmacromoleculesin solution.2.
The general
triaxial ellipsoid. Biopolymers 22, 1gl3_1829.
Harding' S'E' and Rowe, A J.(1984). Modeling
biological macromoleculesin solution.2.
The general
triaxial ellipsoid.Biopolymers23, g43.
Harding, S.E. and Rowe, A.J., and creeth,
J.M. (r9s3). Further evidencefor a flexible
and highly
expandedspheroidalmodelformucusglycoproteinsinsorution.Biochem.
J.2w,g93_g96.
Harding, s.E', Rowe, A.J., and Horton, J.c. (Bos.j(tsqza).
Anarytical urtracentrifugation in biochemrstry and polymer science.Royal Society of
Chemistry, Cambridge, England.
Harding, S E', Horton, J.c.' and Morgan, p.J. (1gg2b).
MSTAR: A FORTRAN program tbr rhe
model-independentmolecular weight analysis
of macromolecuresusing low speedof high
speed
sedimentation analysis.In: Ana lytical IJttiacentifugation
in Biochemiitry or) rory^"r science.
(Harding, S.8., Rowe, A.J., and Ho(on,
J.C., Ejr.l] pp 275_2g4.RoyalSociety of
Chemistry,
Cambridge, England.
Harding, S'E', Horton, J.c., and ctilfen (1997).
The ELLIpS suite of macromolecular conformation
algorithms. Eur. Biophys. J. 25, 347_35g.
Harding,S.E.,Horton, J.C., and Johnson,p. (t992).
(To be published.)
Iniesta, A', Diaz, F.G., and-Garcia de ra
Torre, J. (19g9). Transport properties of
rigid bent-rod
macromorecules and of semi-flexible broken
rods in the ;giJ-uoay treatment: Anatysis
of the
fl exibility of of myosin rod. Biophys, J. 54,
_27
269
6.
Johnsen,R.M. and Brown, w. (1992). An overview
of current methodsof anarysing
eLS data. In:Laser
Light scatteing in Biochemistry. (Harding, s.8.,
Satrete, D.B., and si"".;n"td, v.e., eor.;, p.
77-91. Royal Society ofChemistry, Cambridge,
England.
Johnson' P' (1984)' Light-scattering and correlation-measurement.
Biochem. soc. Trans. 12,623-62s.
Johnson,P' and Mihalyi, E. (1965).Physicochemical
studiesofbovine fibrinogen.2.Depolarization
of
fluorescencestudies. Biochim. Biophys. Acta.
102, 476_4g6.
Jumel, K', Fiebrig, I' and Harding, s.E. (1996).
Rapid size.distributionand purity analysrsofgasrric
mucus glycoproteins by size exclusion chromatography/multiangle
laser light scattering. Int. J.
B i o l . M a c r o m o l .1 8 , 1 3 3 _ 1 3 9 .
Kratky, o., lropold, H., and stabinger, H. (1923).
The determination of the partial specific vorume
of
proteinsby mechanicaloscillatortechnique.
Meth. Enzymol. 27D,gg_110.
Langley' K'H' (1992)' Developments in electrophoretic
laser right scattering and some brochemical
applications' rn: rttser Light scatteinS
in Biochemistry (Harding, s.E., Sattelle,
D.B., and
Bloomfield, V.A., Eds.),p.151_160.Royal
Societyof Chemistry CimOriage,e"gi""a.
Livesey, A K and Brochon' J (lggg] Maximum
entropy data analysis ofdynamic parameters
from
pulsed-fluorescent decays.^rn:_Dynamicprcpexies'of
Biomorecurar Assembries.(Harding, S.E.
and Rowe, A.J., Eds.), p.135. Royal Society
of Chemistry, Cambridge, England.
304
STEPHEN
E. HARDING
Perkins, S.J.(1986). Protein volumes and hydration effects: The
calculationsofpartial specific volumes,
neuron-scattenng match points and 280-nm absorptioncoefficients
forproteins and glycoproteins
from amino-acidsequences.
Eur. J. Biochem. 157, 169_1g0.
Perkins, s'J' (1989). Hydrodynamic modelling of complement.
In: Dynamic properties of Biomolecular
Assemblies (Harding, S.E. and Rowe, A.J., Eds.), p. 226-245.
Royal sociery of chemisrry,
Cambridge, England.
Perkins, s.J. (1994). rn: Microscopy, opticar spectroscopy
and Macmscopic Techniques.(Jones, c.,
Mulloy, B., and Thomas,A.H., Eds.),yo|.22,p.39. Humanapress,
NJ.
Philo'J' (1994). Measuring sedimentation,diffusion, and molecular
weights of small moleculesby direct
fitting of sedimentation velocity profiles. rn: Modem Analytica[ (Jltracentifugation.
(Schuster,
T.M., and Laue, T.M., Eds.),p. 156_120.Birkhiiuser.Bosron.
Ptirschke, D. and Antonsiewicz, J. 09g9). Analysis of macromolecular
structures in solution by
electrooptical procedures.ln: Dynamic Properties ofBiomolecular
Assemblies.(Harding, S.E. and
Rowe, A.J., Eds.),p. 103-114. Royal Socieryof Chemistry,Cambridge,
England.
Prirschke, D. and obst, A. (1991). An erectric fierd jump
apparatus with ns time resorution for
electrooptical measurementsat physiological salt concentrations.
Rev. sci. Instrum. 62, g I g-g20.
Pusey,P.N. (1974). Macromolecular diffusio n. In: Photon coffelation
and Light Beating spectroscopy.
(Cummins,H.Z. and pike, 8.R., Eds.).p.3g7_428,plenum,
New york.
Rholam, M' and Nicolas,P' (1981).Side-by-sidedimerization
ofneurophysin:Sedimentatron-velocity,
viscometry, and fluorescencepolarization studies.Biochemistry
iO, SafZ_Sa+f.
Ridgeway, D. (1966). Transient electric birefringence of suspensions
of asymmetric ellipsoids. J. Am.
Chem. Soc. 88, I 104-l 112.
Roark, D' and Yphantis,D.A. (1969). Studiesofself-associating
systemsby equilibrium ultracentrifugation.Ann. N.Y. Acad. Sci. 164,245_2j9.
Rowe, A'J (1977). concentration-dependenceof transport processes-general
description applicable
to sedimentation, translational diffusion, and viscosity coefficients
of macromolecular solutes.
Biopolymers 16, 2595-2611.
*or": o;1. (1978).Techniquesfor determiningmolecurarweight.
Techn.Life sci.: Biochem. Bl05a,
l-Jl.
Rowe, A J. (1992). The concentration-dependence
of sedimentation.ln: Analytical IJltrctcentifugation
in Biochemistry and polymer Science. (Harding, S.8., Rowe,
n.i., ana Horton, J.C., Eds.), p.
394-4M. Royal Society of Chemistry, Cambridge, England.
Sanders,A'H' and Cannell, D.s. (1980). ln: Light scatteing in Liquids
and Macrcmolecular solutions.
(Degiorgio, V, Corti, M., and Giglio, M., Eds.), p. 173, plenum,
New york.
schachman, H'K. (1989). Analytical ultracentrifugation rebom.
Nature 34r, 25g-260.
Schachman,H.K., Pauza,C.D., Navre, M., Karela, M.J., Wu,
L., and yang, y.R. (19g4). Location of
amino-acid alterations in mutants of aspartatetranscarbamylase-structural
aspectsof interallelic
complementation.Proc. Natl. Acad. Sci. USA, gl, I l5_119.
Schmitz, K's' (1990). An Introduction to Dynamic Light Scatteing
by Macromolecules. Academic
Press,New York.
Schuster, T.M. and Laue, T.M. (Eds.)(199a). Modem Anaryticar
urtracentifugalron, Birkhiiuser,
Boston.
Stafford, w'F (1992). Methods for obtaining sedimentation
coefficient distributions. rn: Analytical
ultracentrifugation in Biochemistry and porymer science. (Harding,
S.8., Rowe, A.J., and Horton,
J.C., Eds.) p. 359-393. Royal SocieryofChemistry, Cambridge,
England.
Stepdnek,P.(1993). Data analysisin dynamic lights curelng.ln:
Dynamic Light scattering. TheMethod
and SomeApplicarions. (Brown, W., Ed.) Oxford University press,
Oxford, Englarid.
svedberg, T. and Pedersen, K.o. (1940). The IJltracentifuge.
oxford universit-y press, oxford,
England.
Tanford, c. ( I 955). Intrinsic viscosity and kinematic viscosity. J. phys.
chem. 59, 7gg-7gg.
Tanford, C. (1961). Physical Chemistry of Macromolecules.!.Wiley
&Sons, New york.
Protein Hydrodynamics
305
Teller' D'c' (1973) characterization ofproteins
by sedimentationequilibrium in the ultracentrifuge.
Meth. Enzymol. 2'tD, 346_441.
Tombs, M P and Peacocke,A.R. (1974). The
osmotic pressure of Biological Macromolecules,
oxford
University press, Oxford.
van Holde' K'E (1971)' Physical Biochernistry.
First ed. Prentice Hall, Englewood cliffs,
New Jersey.
Van Holde, KE' (1985). Physical Biochemistry.
second ed. prentice Hall, Englewood cliffs,
New
Jersey.
weber, G. (1952). polaization of the fluorescence
of macromorecules.I. Theory and experimental
m e t h o d .B i o c h e m .J . 5 1 , 1 4 5 _ t 5 5 .
weber' G' (1952) Polarization of the fluorescence
of macromolecules. II. Fluorescent conJugates
of
ovalbuminand bovine serumalbumin.Biochem.
J. 51, 155_162.
wells' C'' Molina-Garcia,A'D., Harding, s.E.
and nowe, e.t. (r990). Self-interactionof dynein
from
Tetrahymenacilia. J. Mus. Res. Cell Motil.
11, 344_350.
wl"t111 t12lz).rn: LaserLight scatterinS
in Biori"mirtrv.(Harding, S.8., Sanelre,D.B.,
Boomfield,
V.A., Fns.), p. 35-58, Royal Society of Chemistry,
Cambridge, England.
Yang'J'T (1961) viscosity of macromolecules
in relationto molecularconformation.Adv. prot.
chem.
16,323_400.
S T E P H EE
N.H A R D I N G
300
(b)
(a)
alb
monofor (a)neurophysin
evaluations
B. Triaxialellipsoidgrossconformation
Figure
"line solutions"
R
and
for
A
values
constant
of
Plots
d'r"mers'
mersand(b)neurophysin
knowledgeof threehydrodynamic
in the {a/b,b/c} plane.To performtheseanalyses'
t6 (from
(for monomerand dimer)is required:[n] (intrinsicviscosity),
parameters
velocity)'Monoandk, (fromsedimentation
depolarization),
fluorescence
steady-state
adaptedfrom
and
()'8'
Redrawn
=
2'l:D'
t>lcl
=
la/b,
mers:{a/b, blcl @,1);olt""'
H a r d i n ga n dR o w e( 1 9 8 2 b ) .
(JohnsonandMihalyi,lg65);finally,inthesteady-statemethoddescribedabove'
theuseofsolventsofdifferingTandrlmustcausenosignificantconformation
change.
rotationalrelaxationmodesof
The harmonicmeanitself is a meanover different
theprotein,eachcontainingpotentialshapeinformation.Toresolvetheserequires
uput,"alightsource,time-resolvedmeasurements,andmathematicalalgorithms
function and resolution of
f- ua"quui" deconvolution of the light source decay
(see'
for example' Han et al"
multiexponential terms, a by no means simple task
1989;LiveseyandBrochon,lg8g)'Electricbirefringence(ordichroism)decayis,
fluorescence anisotropy
however, another attractive alternative to time-resolved
decaymeasurementssince,foragivenisotropicmonodisperseasymmetricscat(Ridgeway'1966)'A seriousrestricterer,therearejusttwo exponentiaito resolve
the restriction to solutions of low
been
tion of electroopticalmethods,however,has
ionicstrengthsbecauseofheatingeffectscausedbythestrongelectricfieldsused.