an equilibrium model for linear and closed-loop amyloid fibril formation

doi:10.1016/j.jmb.2012.02.026 J. Mol. Biol. (2012) 421, 364–377

Contents lists available at www.sciencedirect.com

Journal of Molecular Biologyj ourna l homepage: ht tp : / /ees .e lsev ie r.com. jmb

An Equilibrium Model for Linear and Closed-LoopAmyloid Fibril Formation

Shuo Yang 1, Michael D. W. Griffin 1, Katrina J. Binger 1,Peter Schuck 2 and Geoffrey J. Howlett 1⁎1Department of Biochemistry and Molecular Biology, Bio21 Molecular Science and Biotechnology Institute,University of Melbourne, Victoria 3010, Australia2Dynamics of Macromolecular Assembly Section, Laboratory of Cellular Imaging and Macromolecular Biophysics,National Institute of Biomedical Imaging and Bioengineering, National Institutes of Health, Bethesda,MD 20892, USA

Received 29 September 2011;received in revised form8 February 2012;accepted 18 February 2012Available online24 February 2012

Edited by S. Radford

Keywords:fluorescence detection;sedimentation velocity;isodesmic;apolipoprotein C-II;amyloid subunits

*Corresponding author. DepartmenMolecular Biology, The University ofAustralia. E-mail address: ghowlett@Abbreviation used: apo, apolipop

0022-2836/$ - see front matter © 2012 E

Amyloid fibrils and their soluble oligomeric intermediates are implicatedin several age-related diseases including Alzheimer's and Parkinson'sdiseases. The distribution of oligomers and fibrils is related to toxicityand is dependent on the pathways for fibril assembly, generallyconsidered to occur via a slow nucleation step that precedes fibrilelongation. Human apolipoprotein (apo) C-II forms amyloid fibrils via areversible self-assembly process accompanied by closed-loop formationand fibril breaking and joining. Our fluorescence quenching andsedimentation velocity experiments with Alexa488-labeled apoC-II indi-cated a time-dependent subunit interchange for both linear and closed-loop fibrils, while dilution experiments using mature fibrils indicated ashift to smaller size distributions consistent with a reversible assemblypathway. To account for this behavior, we developed an equilibrium self-association model that describes the final size distributions of apoC-IIfibrils formed at different starting concentrations. The model proposes areversible isomerization of apoC-II monomer to form an active conformerthat self-assembles into fibrils via an isodesmic self-association pathwaycoupled to fibril length-dependent closed-loop formation. The modeladequately described fibril size distributions and the proportion of closedloops as a function of total apoC-II concentration over the concentrationrange 0.1–0.5 mg/ml. Extension of the model to include the rates ofisomerization, self-association and fibril breaking and joining providedsatisfactory global fits to kinetic data on fibril formation and changes inaverage fibril size at different apoC-II starting concentrations. The modelprovides a simple thermodynamic description of the processes governingthe size distribution of apoC-II fibrils at equilibrium and the formation ofdiscrete oligomeric intermediates.

© 2012 Elsevier Ltd. All rights reserved.

t of Biochemistry andMelbourne, Vic. 3010,unimelb.edu.au.rotein.

lsevier Ltd. All rights reserve

Introduction

Several common neurological and systemic dis-eases are accompanied by the aggregation ofmisfolded proteins to form insoluble amyloidfibrils.1,2 These fibrils are defined by a characteristic

d.

365Equilibrium Model for Amyloid Fibril Formation

cross-β structure and the ability to interact with thedyes thioflavin T and Congo Red.3 While amyloidfibril deposits have been implicated in the diseaseprocess, it is widely considered that toxicity ismediated by small oligomeric intermediates in theassembly pathway.4,5 The observation of a distinctlag phase in the kinetics of amyloid fibril formationforms the basis for nucleated kinetic models for fibrilassembly that involve the initial slow formation of anucleus that serves as a template for fibrilelongation.6,7 A feature of simple nucleated kineticmodels is the transient nature of the nucleus andlack of stable intermediates. More recent modelshave incorporated fibril breaking, joining and lateralassociation events as additional mechanisms todescribe the kinetics of fibril formation and theheterogeneity of the end product.8–13 A significantfinding is the observation that fibril fragmentsproduced from breaking and joining enhanceamyloid fibril cytotoxicity.14

We have developed amyloid fibril formation byhuman apolipoprotein (apo) C-II as a system toexamine the steps involved in fibril assembly.ApoC-II is a 79-amino-acid cofactor of lipoproteinlipase and an integral component of the lipid-richvery low density lipoproteins and chylomicrons,which transport lipids through the bloodstream.Under lipid-free conditions in vitro, apoC-II sponta-neously aggregates to form homogenous fibrils withall the hallmarks of amyloid.15 Fibrils composed ofapoC-II activate a pro-inflammatory macrophagesignaling cascade mediated by the CD36 scavengerreceptor,16 which is a key feature of vascularinflammation leading to lesion development. Kinet-ic analysis of the rate of formation of apoC-II fibrilsas a function of starting concentration coupled withan analysis of the size distribution of the fibrillarproducts led to the development of a model for fibrilformation that involves reversible nucleated assem-bly from monomers coupled with fibril breakageand joining.17 Direct evidence for fibril breakageand joining was provided by antibody-labelingtransmission electron microscopy studies.17 A lim-itation of the analysis was the observed correlationof some of the fitting parameters and the difficulty ofdefining the size of the nucleus. In addition, theanalysis neglected the formation of closed-loopstructures by apoC-II.18

Structural analyses of apoC-II amyloid fibrilsindicate that the apoC-II monomer within fibrilsadopts a “letter-G-like” conformation that stacks toform an in-register structure composed of twoparallel β-sheets.19 This structural model suggesteda simple mechanism for fibril assembly involving areversible isomerization of apoC-II subunits to forma conformer that self-assembles into linear fibrils, viaan isodesmic self-association process governed by asingle equilibrium constant. Isodesmic, noncooper-ative self-association models have been described for

a number of interacting protein systems.20–23 Anequilibrium isodesmic model for apoC-II fibril for-mation could account for the formation of closed-loopstructures by assuming circularization via a length-dependent probability of loop closure.15,18 Closed-loop fibrils,24 discrete oligomers25 and pore-likeannular protofibrils26 have been observed in severalamyloid forming systems, accounting for much of theheterogeneity inherent in fibril populations. In thepresent study, we have developed a generallyapplicable and comprehensive thermodynamicmodel to describe the formation and equilibriumsize distribution of linear and closed-loop apoC-IIamyloid fibrils.

Results

ApoC-II fibril size distributions

Studies of the concentration dependence of apoC-II fibril formation have shown that the averagesedimentation coefficients of fibrils, formed in theconcentration range 0.1–0.6 mg/ml, reach stabletime-independent values after 6–7 days,17 indicat-ing an equilibrium condition. Sedimentation veloc-ity data for apoC-II fibrils at equilibrium after 7 daysat 0.3 mg/ml (preformed fibrils) are presented inFig. 1. Fitting the data to a continuous sizedistribution indicated two separate populations offibrils, with modal sedimentation coefficients ofapproximately 36 and 88 S. Assuming a worm-likechain model for apoC-II fibrils, calculated for apersistence length of 36 nm and rise per subunit of0.47 nm,18 these sedimentation coefficients indicateweight-average molecular mass values of 3.0×106

and 2.9×107, corresponding to fibrils composed ofapproximately 340 and 3300 subunits, respectively.The slower sedimenting population, which accountsfor approximately 6.6% (on a mass scale) of totalfibrils at 0.3 mg/ml, is ascribed to the formation ofclosed loops.17 Size distribution analysis of fibrilsformed at different total concentrations of apoC-IIindicated a systematic increase in the averagesedimentation coefficient of the larger subpopula-tion of fibrils over the concentration range 0.1–0.6 mg/ml (Fig. 1). As indicated previously, thistrend is not consistent with a simple nucleation–elongation model predicting that low concentrationswill restrict nucleation leading to the formation oflonger fibrils than those formed at higherconcentrations.17 An explanation for the concentra-tion dependence of fibril size shown in Fig. 1,consistent with the structural model for apoC-IIfibrils,19 is that the major population of fibrils isformed by the isodesmic self-association of an activeconformer, yielding an equilibrium mixture ofinteracting fibrils. In contrast, values for the

Sedimentation coefficient (S)0 50 100 150 200 250 300

c(s)

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Fig. 1. Sedimentation velocity analysis of apoC-II fibrilsformed at different apoC-II concentrations. ApoC-IIsamples were prepared by incubating apoC-II in refoldingbuffer for 14 days prior to analytical ultracentrifugation at8000 rpm at 20 °C with radial scans taken at 8-minintervals. (a) Radial scans for apoC-II sample (0.3 mg/ml),monitored at 280 nm (open circles). The fit of the data to ac(s) model45 is also shown (continuous lines). (b)Sedimentation coefficient distributions for fibril samplesformed at 0.1 mg/ml (dotted line), 0.3 mg/ml (continuousline) and 0.5 mg/ml (broken line). (c) Dependence of theweight-average sedimentation coefficient on total apoC-IIconcentration for the faster moving and slower movingpopulation of fibrils (filled circles and filled squares,respectively).

Table 1. Characterization of free pool and closed-loopfraction in apoC-II fibril samples

Total apoC-II(mg/ml)

Non-sedimentingapoC-II (mg/ml)

Closed loops(mg/ml)

0.1 0.0193 0.00990.2 0.0195 0.01570.3 0.0197 0.01970.4 0.0231 0.01980.5 0.0257 0.01930.6 0.0264 —a

ApoC-II samples were prepared by incubating apoC-II inrefolding buffer for 14 days prior to sedimentation velocityanalysis. The concentration of non-sedimenting apoC-II wasdetermined from the optical density in the supernatant at theconclusion of centrifugation.

a A closed-loop population was not observed for fibrils formedat 0.6 mg/ml.

366 Equilibrium Model for Amyloid Fibril Formation

weight-average sedimentation coefficient of theslower moving, “closed-loop” fraction did notshow a consistent trend over the same concentrationrange (Fig. 1 and Table 1). The results in Table 1

show that the free pool of non-sedimenting apoC-II and the concentration of the closed-loop fractionare relatively independent of the total apoC-IIconcentration.

Recovery of equilibrium size distributionsfollowing shearing of apoC-II amyloid fibrils

Preformed apoC-II fibrils were subjected to anumber of shear processes to further establish theequilibrium nature of fibrils formed. The resultsshow that extended vortexing or bath sonication ofpreformed fibrils had little effect on either theweight-average sedimentation coefficient or therelative proportion of the two subpopulations offibrils (Fig. 2). In contrast, freeze–thaw cycling ofpreformed apoC-II fibrils led to an initial decrease inthe average size of the fibrils and an increase in theproportion of the smaller fibril subpopulation. Animportant observation, however, is that incubationof the freeze–thawed sample for 24 h led to arecovery of the size distribution toward the distri-bution observed for a control untreated sample,consistent with the return toward an equilibriumcondition. Similar recoveries of fibril size distribu-tions following probe sonication have been reportedpreviously.17

Purification of apoC-II closed-loop fibrils

The results in Fig. 1 indicate the presence of asmaller subpopulation of fibrils ascribed to closedloops. To further explore the equilibrium status ofpreformed apoC-II fibrils, we developed proceduresto purify and characterize the closed-loop subpop-ulation. Preformed apoC-II fibrils were subjected tocycles of freeze–thawing to enrich for closed loops,and preparative ultracentrifugation was used tosediment the linear fibrils leaving closed-loop fibrilsenriched in the supernatant. The results in Fig. 3show representative electron micrographs of apoC-

(a)c(

s)

0.000

0.002

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0.006

0.008

0.010

(b)

Sedimentation coefficient (S)0 50 100 150 200

c(s)

0.000

0.002

0.004

0.006

0.008

Fig. 2. ApoC-II fibril size distributions followingexposure to shear. (a) Sedimentation coefficient distribu-tions for preformed apoC-II fibrils (0.3 mg/ml; continuousline) and for fibrils subjected to rapid vortexing for 20 min(dotted line) and bath sonication for 15 min (broken line).(b) Sedimentation coefficient distributions for preformedapoC-II amyloid fibrils (0.3 mg/ml) before (continuousline) and after (broken line) freeze–thaw treatment.Distributions for the untreated sample (dotted line) andfreeze–thawed samples (dot/dot/dash line), followingincubation at 25 °C for 24 h.


II fibrils in the supernatant fraction, revealing a highproportion of closed-loop fibrils. Analysis of thecontour lengths of the closed loops (Fig. 3) yieldedan average length of approximately 292 nm, corre-sponding to approximately 621 subunits and com-parable to the results from previous studies.18 Basedon a worm-like chain model, this value for theaverage length of the fibrils gives an estimate of45.0 S for the average sedimentation coefficient ofclosed loops, comparable with the size of the smallerpopulation of fibrils observed in Fig. 1.

Subunit exchange experiments

An isodesmic self-association model for fibrilformation proposes a final equilibrium statecomposed of a mixture of active monomers andfibrils with a single equilibrium constant describ-ing both the addition of an active monomer tothe ends of fibrils and the reversible breaking andjoining of fibrils. A prediction of this model isthat, in the equilibrium state, the addition oflabeled monomers to preformed fibrils will lead

to subunit exchange of the labeled monomerswith the fibrils. To provide additional support foran isodesmic self-association model, we per-formed subunit exchange experiments. The resultsin Fig. 4 show that the addition of Alexa488-labeled apoC-II to preformed apoC-II fibrils orpurified closed-loop fibrils leads to the rapidincorporation of Alexa488-apoC-II into fibrils asindicated by the decrease in the amount offluorescence detected in the supernatant fractionfollowing centrifugation. A dynamic exchange ofsubunits has also been observed for otheramyloid assemblies.27,28

Subunit exchange was also monitored by fluores-cence decay experiments. The self-assembly ofAlexa488-labeled apoC-II into fibrils has previouslybeen shown to reduce Alexa488 emission.29 Theaddition of Alexa488-labeled apoC-II to preformedapoC-II fibrils or purified closed loops led to adecrease in Alexa488 fluorescence, indicating a time-dependent subunit exchange with both the pre-formed fibrils and the purified closed-loop samples(Fig. 4). Control experiments using free Alexa488-maleimide showed no change in fluorescence in thepresence of preformed apoC-II fibrils.The incorporation of Alexa488-labeled apoC-II

into the closed loops suggested an equilibriumbetween closed loops and linear fibrils thatpermits subunit exchange with closed loops.Alternatively, the observed exchange could arisefrom contaminating linear fibrils within the puri-fied closed-loop sample. Size distribution analysisof Alexa488-labeled apoC-II incubated for 4 h withpreformed or purified closed-loop fibrils is alsoshown in Fig. 4. Monitoring of the sedimentationvelocity behavior at 280 nm to detect total proteinshows that the average sedimentation coefficientof the purified closed loops is approximately 40 Sand is similar to the values observed for thesmaller subpopulation present in preformed fibrils(Fig. 1). Monitoring the sedimentation behavior at495 nm shows that the distribution of Alexa488absorbance is similar to the distribution of totalprotein, indicating substantial equilibration ofAlexa488-labeled apoC-II with both preformed orpurified closed-loop apoC-II fibrils. This behavioris consistent with a dynamic isodesmic self-association model for fibrils at equilibrium with asubpopulation of closed loops.

Dilution of preformed apoC-II amyloid fibrils

A further test of the reversible isodesmic self-association model was to examine whether dilutionof preformed apoC-II fibrils reduced the averagesize of the fibrils. The results in Fig. 5 show that a 1:3dilution of mature apoC-II fibrils formed at 0.3 mg/ml reduced the modal sedimentation coefficient ofthe fibrils. This reduction was also apparent in

length of closed-loops (nm)0 100 200 300 400 500

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Fig. 3. Electron microscopy ofpurified apoC-II closed loops.ApoC-II closed loops were puri-fied from preformed apoC-II fibrilsby preparative ultracentrifugation.(a–d) Negative staining transmis-sion electron micrographs showingrepresentative images. Scale barsrepresenting 50 nm are shown. (e)Distribution of the contour lengthsof the closed loops.


calculated values of the weight-average sedimenta-tion coefficient and was maintained followingincubation of the fibrils over a period of 7 days(Table 2). Experiments employing higher dilutionfactors were performed using Alexa488-labeledapoC-II fibrils and fluorescence detection in theanalytical ultracentrifuge. The results in Fig. 5 andTable 2 show that dilution of the fibrils by factors of1:10 and 1:100 decreased the modal sedimentationcoefficient of the main population of fibrils and the

weight-average sedimentation coefficients of theentire fibril population. These changes were main-tained after incubation of the samples for 14 days.The results in Fig. 5 also show that high dilutiongenerated a smaller population of fibrils withsedimentation coefficients in the range 10–30 S.The overall reductions in average fibril size follow-ing dilution attest to the reversibility of apoC-II fibrilformation and are consistent with an equilibriummodel for fibril formation.


An equilibrium model to simulate the sizedistribution of apoC-II amyloid fibrils

A model to describe our experimental results ispresented in Fig. 6. Themodel proposes an ensembleof misfolded apoC-II monomers in equilibrium withan active conformer, governed by an isomerizationconstant, Ki. The model further proposes that thisactive conformer self-assembles via an isodesmicself-association process governed by an equilibriumconstant, Keq. An intrinsic feature of an isodesmicself-association model is that monomer addition andreversible fibril breaking and joining are character-ized by the same equilibrium constant. The revers-ible formation of closed loops is accounted for by anequilibrium constant, KL, which describes fibrillength-dependent closed-loop opening and closing.

(c)

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The model shown in Fig. 6 was used to fittheoretical curves to the size distribution data forapoC-II fibrils formed at different starting con-centrations. The parameters of the model used tofit the data in Fig. 7 are summarized in Table 3.The results in Fig. 7 show good agreementbetween the modal sedimentation coefficient forthe two main populations of fibrils formed atinitial concentrations of 0.1, 0.3 and 0.5 mg/ml.The fitted values obtained for the concentration ofthe active conformer over the concentration range0.1–0.5 mg/ml varied over a very narrow rangefrom 3.05578 to 3.05690 pM. These valuescorrespond to values of the product nKeq rangingfrom 0.9942 to 0.9978. For an isodesmic self-association, values of nKeq close to 1 lead to theformation of large fibrils [see Eq. (1)].In general, the shapes of the theoretical size

distributions in Fig. 7 are in reasonable agreementwith the experimental results. The main divergencearises from the width of the theoretical fits com-pared to the experimental distribution. While therecould be several explanations, one possibilityconsidered was that the equilibrium constant forfibril self-association is size dependent. The resultsin Fig. 7 also show fits to the experimental sizedistributions assuming an empirical relationshipbetween the self-association constant Keq and thefibril length (see Materials and Methods section).The fitted value for exponent “a” of −0.001 [Eq. (3)]corresponds to a size-dependent fractional decreaseof 0.8% in Keq for fibrils composed of 3000 subunitscompared to those composed of 2 subunits. The

Fig. 4. Subunit exchange with apoC-II fibrils. (a)Alexa488-labeled apoC-II monomer was added to pre-formed apoC-II fibril samples (filled circles) or purifiedclosed loops (open circles). Subunit exchange was mon-itored by centrifugation of the samples and determinationof the Alexa-labeled apoC-II fluorescence in the superna-tant fraction. (b) Alexa488 fluorescence emission mea-sured as a function of time for monomeric Alexa488-labeled apoC-II added to linear fibrils (filled circles),purified closed loops (open circles), linear fibrils alone(filled triangles) and closed loops alone (open triangles).The fluorescence of free Alexa488-maleimide (0.76 μM),following the addition of apoC-II fibrils (15.7 μM),expressed relative to the fluorescence of Alexa488-mal-eimide alone (filled squares). (c) Sedimentation velocitysubunit exchange analysis of preformed and closed-loopapoC-II fibrils. Monomeric Alexa488-labeled apoC-II(0.016 mg/ml) was incubated with preformed fibrils orpurified closed loops (0.14 mg/ml) for 4 h at 25 °C.Sedimentation coefficient distribution analysis was per-formed at 280 nm to detect total protein for preformedfibrils or purified closed loops (continuous line and brokenline, respectively) and at 495 nm to monitor thedistribution of Alexa488-labeled apoC-II in preformedfibrils or purified closed loops (dotted line and dot/dot/dash line, respectively).

Table 2. Effect of dilution on the weight-averagesedimentation coefficient of preformed apoC-II fibrils

DilutionDetectionmethoda Initiallya

After 7 or14 daysa

Undiluted OD 98.4 98.21:3 OD 86.1 84.2Undiluted Fluorescence 96.2 98.51:10 Fluorescence 76.3 62.41:100 Fluorescence 65.0 65.3

Studies were performed using preformed Alexa488-labeled apoC-II fibrils.

a Weight-average sedimentation coefficients in Svedberg units(S) were measured using SEDFIT.44 Results are presented forinitial samples and for samples incubated at 25 °C for 7 days foroptical density at 280 nm measurements or 14 days for samplesmonitored by Alexa488 fluorescence.

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Fig. 5. Effect of dilution on apoC-II amyloid fibril sizedistributions. (a) Sedimentation coefficient distributionsfor preformed apoC-II fibrils (0.3 mg/ml) initially(continuous line) and after incubation at 25 °C for 7 days(dotted line). Data are also presented for a 1:3 dilution ofpreformed fibrils, initially (broken line) and after incuba-tion at 25 °C for 7 days (dot/dot/dash line). Sedimenta-tion velocity was monitored at 280 nm. (b and c)Sedimentation coefficient distributions for preformedapoC-II fibrils (0.3 ml/ml), incubated with Alexa488-labeled apoC-II and DTT (1 mM) for 4 h prior to analysis.Sedimentation velocity was monitored by fluorescencedetection. Data are presented for samples diluted 1:10 (b)and 1:100 (c), initially (broken lines) and after incubationat 25 °C for 7 days (dot/dot/dash line). The continuouslines in (b) and (c) are scaled distributions obtained for theundiluted sample after incubation for 7 days.


results show that, while the theoretical size distri-butions are sensitive to the inclusion of a size-dependent self-association constant, both modelsprovide a satisfactory description of the average sizeof the fibrils observed at equilibrium and over therange of concentrations studied.

Application of the model to fit the weight-averagesedimentation coefficient of fibrils

The fitted parameters in Table 3, assuming aconstant value of Keq, were used to calculateequilibrium values for the weight-average sedimen-tation coefficients offibrils formed at different apoC-IIconcentrations (Fig. 1). The results in Fig. 8 showgoodagreement between these calculated values andexperimental values in the concentration range 0.1–0.5 mg/ml. In particular, the model accuratelydescribes the concentration-dependent increase inthe average sedimentation coefficient for the largerpopulation of fibrils and the concentration indepen-dence of the average sedimentation coefficient of the“closed-loop” subpopulation. At higher total apoC-IIconcentrations, the weight-average sedimentationcoefficients of the fibrils deviate from the theoreticcurve. This divergence is evenmore evident in studiesat higher concentrations where values in the range100–1000 S have been observed.30 These additionaleffects appear to be irreversible and are attributed tofibril tangling and interactions between fibrils.30

Analysis of kinetics of fibril formation

The model in Fig. 6 was extended to examine thekinetics of apoC-II fibrils formed at different startingconcentrations.17 The kinetic description was possi-ble with the simplifying treatment of all fibrils as afibril of average length, thereby excluding directconsideration of closed-loop formation (see Mate-rials andMethods). Given this approximation,whichis justified on the basis of the small proportion of theclosed-loop population, the results in Fig. 9 showreasonable agreement between the lines of best fit forthe model and experimental data on the rate of fibrilformation and changes in the weight-average sedi-mentation coefficient during fibril formation. Thebest-fit lines were constrained to the values of Ki andKeq obtained from an analysis of the data in Fig. 7,

Misfoldedmonomers

Amyloid fibrilsActive conformer

AxisKi Keq Keq

Keq

Linear fibrilsClosed loops

KL

Keq

Fig. 6. Equilibrium model forapoC-II fibril assembly. The modelproposes an initial isomerization ofmisfolded apoC-II monomers toform an active conformer that self-assembles into fibrils according toan isodesmic self-association pro-cess [Eq. (1)]. An integral part of themodel is fibril breaking and joiningcoupled to size-dependent closed-loop formation. The equilibrium

constants for the various steps are shown. The structural model for apoC-II fibrils shows a linear assembly of monomersin a “letter-G-like” conformation.19

(a)

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assuming a size-independent value of Keq, assummarized in Table 3. The fitting process involvedfitting three additional parameters, namely, valuesfor kon (the rate constant for formation of the activefolded conformer), ke (the rate constant for fibrilelongation by conformer addition) and kj (the rate forfibril joining) (Table 3). While a more elaboratetreatment, including closed-loop formation, mayimprove the fits shown in Fig. 9, the presenttreatment indicates that the equilibrium modelshown in Fig. 6 satisfactorily accounts for the majorfeatures of the kinetics of apoC-II fibril formation.

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Fig. 7. Fit of experimental fibril size distributions withtheoretical model. Experiment data for fibrils formed atdifferent apoC-II concentrations (Fig. 1b) were fitted to themodel presented in Fig. 6. Parameters used to fit the dataare summarized in Table 3. (a–c) Experimental sizedistributions at 0.1, 0.3 and 0.5 mg/ml, respectively(continuous lines), and the corresponding theoretical fitsassuming size-independent (dotted lines) or size-dependent(broken lines) isodesmic self-association constants.

Discussion

It is pertinent to summarize the evidence support-ing the isodesmic self-association model for amyloidfibril formation by apoC-II. The model is consistentwith the stacking of identical subunits within apoC-II fibrils to form a twisted-ribbon architecture witheach subunit interface stabilized by identicalinteractions.19 The reversible isodesmic self-associ-ation model in Fig. 6 is also consistent with theconcentration dependence of the equilibrium sizedistributions, the rapid exchange of labeled subunitswith mature fibrils and the recovery toward theequilibrium position following sonication or dilu-tion of preformed fibrils (Figs. 1–5). However, itshould be noted that, while the model fits theexperimental data in the 0.1- to 0.5-mg/ml concen-tration range, a limitation of the model applies athigher apoC-II concentrations (N0.6 mg/ml) whereadditional irreversible processes occur to generatelarge fibrillar aggregates attributed to fibril aggre-gation or tangling.30 A further consideration is thatwhile the model assumes that the equilibriumconstant for the formation of each subunit–subunitinterface is identical, there exists the possibility thatthe initial dimerization constant differs from theequilibrium constant describing subsequent mono-mer addition steps. Such a possibility appears to be

Table 3. Parameters used to fit apoC-II fibril sizedistributions and the kinetics of fibril formation to anisodesmic self-association model

Isodesmicself-association

modela

Size-dependentisodesmic

self-associationmodela

Kineticmodelb

Keq (M−1)×10−11 3.27059 37.2750c 3.27

Ki ×106 1.51 0.134 1.5

KL 1 10 —kon (h−1)×10−3 — — 3ke (M

−1 h−1)×10−12 1.8kj (M

−1 h−1)×10−8 — — 3.2a Procedures used to fit the size distribution data (Figs. 7 and 8)

and kinetic data (Fig. 9) are presented in Materials and Methods.The values of μ and σ [Eq. (1)] used to fit the data in Figs. 7 and8 were 210 and 100, respectively.

b The kinetic data on the rate of fibril formation and change inweight-average sedimentation coefficient (Fig. 9) were fittedassuming fixed values for Keq, and Ki was obtained from theanalysis of the size distribution data (Fig. 7). The kinetic data werefitted assuming the absence of closed loops.

c The fitted value for Keq refers to the value obtained for Keq(2)assuming a size-dependent isodesmic self-association model andwas calculated according to Eq. (3). The fitting procedure yieldeda fitted value for the exponent “a” of −0.001.

time (h)0 48 96 144 192 240

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time (h)0 48 96 144 192 240

Fig. 9. Application of the equilibrium model to fit thekinetics of apoC-II fibril formation. Data on the rate offibril formation (a) and on the change in weight-averagesedimentation coefficient with time (b) are from Ref. 17.The experimental data were analyzed according to theequilibrium model presented in Fig. 6 and the parameterssummarized in Table 3, extended to include rate constantsfor the various equilibria (see Theory section). Experimen-tal data (symbols) and theoretical fits (lines) are shown forfibrils formed at 0.2 mg/ml (diamonds, dot/dot/dashline), 0.3 mg/ml (squares, broken line), 0.4 mg/ml


the case for apoC-II fibril formation in the presenceof amphipathic activators or inhibitors where thereis good evidence for the formation of discretetetramers or dimers, respectively.31 The slowformation of discrete oligomeric species coupledwith an isodesmic self-association model may alsoaccount for the appearance of a distinct concentra-tion-dependent lag phase for some amyloid fibrilforming systems.A useful property of an isodesmic self-association

model is the ability to define the concentration of allfibrillar species based on the assignment of a single

Total apoC-II concentration (mg/mL)0.0 0.1 0.2 0.3 0.4 0.5 0.6

Wei

ght a

vera

ge s

(S

)

0

20

40

60

80

100

120

140

160

Fig. 8. Fit of model shown in Fig. 6 to the dependence ofapoC-II fibril size on total apoC-II concentration. Weight-average sedimentation coefficient values for the fastermoving and slower moving population of fibrils (Fig. 1c)are shown together with theoretical data (long broken andshort broken lines, respectively) calculated using the fittedparameters shown in Table 3, assuming a constant valuefor Keq.

(triangles, dotted line) and 0.5 mg/ml (circles, continuousline).

equilibrium constant. This simplifies the extensionof the model to include fibril size-dependentprocesses such as closed-loop formation and lateralinteractions allowing simulation and analysis offibril-derived oligomeric intermediates. A furtherproperty of the model arises from a consideration ofEq. (1), which shows that the system converges to anequilibrium fibril distribution when the productKeq×n is less than 1. For apoC-II fibrils, thiscondition occurs when the concentration of activemonomer decreases to below 1/Keq, correspondingto a concentration of approximately 3×10−12 M or3 pM. Fibril formation by apoC-II can be comparedwith other systems where it is difficult to define sizedistributions due to the very large fibrillar aggre-gates formed, and in some cases, stirring is requiredto obtain fibrils in a reasonable timescale.8,9,32,33 A


distinction between the behavior of different amy-loid fibril forming systems can be made, based onthe value of the isodesmic self-association constantKeq. For apoC-II fibril formation, the value of Keq issuch that, during fibril formation, the concentrationof the free active conformer n decreases to below 1/Keq and that a discrete and well-defined fibril sizedistribution is formed. For systems characterized byvery large values of Keq, the product Keq×n remainsgreater than 1, and the system does not reachequilibrium. In these cases, the concentration of freeactive conformer reaches very low values, and fibrilformation comes to a kinetic rather than equilibriumend point. According to this classification, distin-guishing between equilibrium and nonequilibriumself-association schemes becomes an importantconsideration in the analysis of amyloid fibrilforming pathways.The best-fit values reported in Table 3 for Keq, ke

and kj allow the relative rates of monomer dissoci-ation from fibrils (keoff) and the rate of fibril breakage(kb) to be calculated, where Keq=ke/keoff=kj/kb. Thecalculated values for keoff and kb (5.4 h−1 and9.7×10−4 h−1, respectively) indicate that monomerdissociation from fibrils is a rapid process comparedto fibril breakage, at least during the early stageswhen the size of the fibrils is small. It should benoted, however, that keoff describes the rate constantfor breakage at each interface in the fibril, as opposedto the rate constant of fibril breakage anywhere. Atlater stages, the size of the fibrils increases such that,for fibrils composed of 3000 subunits, the rate ofbreakage per fibril is approximately 3 h−1.The equilibrium model proposed in Fig. 6 differs

from the nucleation–elongation models commonlyapplied to the analysis of amyloid fibril formation. Bythe consideration of the equilibrium condition, thenumber of parameters required to describe the finalsize distribution is reduced considerably compared toa nucleation–elongation model based on the assign-ment of kinetic rate constants. An additional advan-tage of the isodesmic self-association model is theability to distinguish the energy contributions associ-ated with monomer folding from the energy ofsubunit interactions. For apoC-II, fitting the fibrilsize distribution data provided an estimate of Keq andthe equilibrium concentration of free active conformern. Knowledge of n and the size of the total freemonomer pool then allows the concentration of theunfolded monomer m to be calculated and hence theisomerization constantKi. The values obtained for theisomerization constant Ki (1.5×10

−6) and the isodes-mic self-association constant Keq (3.3×1011 M−1) forapoC-II fibril formation (Table 3) indicate an unfa-vorable positive free-energy value of 32.7 kJ/mol foractive conformer formation that is coupled with alarge negative free energy of −64.6 kJ/mol forsubunit–subunit interactions within fibrils. It shouldbe noted that, although the free-energy change for

conformer formation is positive, the rate of isomeri-zation is fast with values for kon and koff (Table 3),ensuring rapid interconversion between an ensembleof unfolded monomers and the active monomericconformer over the millisecond-to-second timescale.The kinetics of the isomerization–isodesmic self-association model may be contrasted with calcula-tions based on the nucleation–elongation model. Afree pool of monomer, indicative of a reversibleprocess, has been reported for fibril formation byAβ7

and used as a measure of the equilibrium constant forfibril elongation to estimate the effects ofmutations onfibril stability.34 An uncertainty introduced with thenucleation–elongation model is that it is not possibleto distinguish the effects of mutations on monomerfolding or monomer addition to fibrils.The equilibrium model proposed for apoC-II

fibrils has implications for in vivo amyloid fibrilformation where deposits accumulate over a longtimescale such that a steady-state equilibriumcondition is achieved, governed by the relativerates of protein synthesis and breakdown. Studiesaimed at controlling amyloid fibril formation anddepositions have mainly focused on procedures tolimit the formation of amyloid fibrils. The presentstudy suggests procedures targeted at reversingfibril formation or modulating fibril breaking, andjoining may also be effective. Another aspect of theequilibrium model, relevant to intracellular amy-loid fibril formation, is the potential for spatialresolution of amyloid fibril forming conditionswithin the cell. A recent study of Huntingtonaggregation within cells identified three majorpopulations corresponding to monomers, oligo-mers (modal sedimentation coefficient of 140 S)and intracellular inclusion bodies.35 The appear-ance of soluble oligomers is consistent with theformation of a discrete size distribution predictedfor an isodesmic self-association. Variations in thetotal concentration of amyloid fibril forming pro-teins in different regions of the cell could determinewhether equilibrium or nonequilibrium self-associ-ation mechanisms apply and the nature of thefibrils and oligomeric intermediates formed. ForapoC-II fibril formation, a variety of physiologicalfactors including macromolecular crowding, 36

mutations,37 molecular chaperones,38,39 apolipo-protein E,40 shear forces,41 lipids42 and oxidation43

modulate fibril formation. The development of asimple isodesmic self-association model for apoC-IIfibril formation will facilitate kinetic and thermo-dynamic studies of the mechanism these modula-tors exert on amyloid fibril formation.

Materials and Methods

ApoC-II was expressed, purified and stored as a stocksolution in 5 M guanidine hydrochloride and 10 mM


Tris–HCl (pH 8.0) at a concentration of approximately45 mg/ml.29 A cysteine-containing derivative, apoC-IIS61C, provided by Dr. Chi Pham (University ofMelbourne), was conjugated with Alexa488 C5 malei-mide (Invitrogen–Molecular Probes, Eugene, OR).29

ApoC-II fibril formation was initiated by dilution ofthe stock apoC-II solution into refolding buffer [100 mMsodium phosphate and 0.05% sodium azide (pH 7.4)]and incubation at 25 °C. Preformed apoC-II fibrils wereprepared by incubation of apoC-II at 0.3 mg/ml for7 days at 25 °C.

Sedimentation analysis

Sedimentation experiments were conducted using anXL-I analytical ultracentrifuge (Beckman Coulter Instru-ments, Inc., Fullerton, CA), an An-60Ti rotor and double-sector 12-mm-pathlength cells containing quartz windowsand charcoal-filled epon centerpieces. Sedimentationvelocity data were collected at 8000 rpm, using opticaldensity measurements at 280 nm and 8-min time intervals.A fluorescence detection system (Aviv Biomedical) wasused to monitor the fluorescence of Alexa-labeled apoC-II.44 Fluorescence data were collected at 1-min intervalsfrom 6.0 to 7.25 cm with the excitation laser focused to aspot 20 μm in diameter, 31 μm below the surface of thewindow. Sedimentation velocity data were analyzedusing the c(s) model in SEDFIT9.4.45 Second derivativeTikhonov–Phillips regularization was used to convert theexperimental data into continuous size distributions.46

The hydrodynamic scaling law derived on the basis of aworm-like chain model for noninteracting fibrils was usedto relate the sedimentation coefficient to fibril molecularweight as described previously.30

Shear experiments

Preformed apoC-II fibrils were subjected to shear usingrapid vortexing, bath sonication or freeze–thaw treatment.Vortexing was carried out for 20 min using a benchtopvortex instrument at a speed setting of 600 rpm. Bathsonication was performed using a bath sonicator (Ultra-sonic FXP 10) for 15 min. Freeze–thaw treatment involvedfour cycles of freezing the sample (1 ml aliquots) in liquidnitrogen and thawing it in warm water of 42 °C.

Fluorescence spectroscopy

All fluorescence measurements were recorded in a CaryEclipse fluorescence spectrometer (Varian, Inc.). Thefluorescence emission of Alexa488-labeled apoC-II(0.00675 mg/ml) was determined using an excitationwavelength of 495 nm and collecting the emission over therange 495–700 nm using a 495-nm long-pass filter.

Purification of apoC-II closed-loop fibrils

ApoC-II fibrils were formed at 0.3 mg/ml by incubationin refolding buffer at 25 °C for 7 days. Preformed fibrils(1 ml aliquots) were freeze–thawed 10 times by submerg-ing them in liquid nitrogen for 25 s followed by rapidthawing in water at 42 °C. After annealing at 25 °C for24 h, the samples were centrifuged at 50,000 rpm for 8 min

in 50,000 rpm (47,500g) using an OptimaMax centrifugeand a TL-100.1 rotor (Beckman Coulter Instruments, Inc.)to separate fibrils into pellet and supernatant fractions forfurther analysis.

Transmission electron microscopy

Carbon-coated Formvar 300 mesh copper grids wererendered hydrophilic by glow discharge under reducedatmosphere for 10 s. ApoC-II closed-loopfibril sampleswerediluted to a concentration of 0.1 mg/ml with deionizedwater. The diluted samples were applied to the grids for1 min within 15 min of glow discharging and stained twicewith 2% potassium phosphotungstate at pH 6.8. The gridswere air-dried and inspected using a Tecnai TF30 transmis-sion electronmicroscope (FEI, TheNetherlands) operating at200 kV. Images were acquired digitally using a GatanUS1000 2k×2k CCD Camera (Pleasanton, CA).

Subunit exchange and fibril dilution experiments

A centrifugation assay was used to monitor apoC-IIsubunit exchange. Monomeric Alexa488-labeled apoC-II(6.75 μg/ml) was added to preformed apoC-II fibrils orpurified closed-loop samples (0.14 mg/ml). Followingincubation at 25 °C, for specified times, the samples(100 μl) were centrifuged at 100,000 rpm for 30 min, andAlexa488 fluorescence in the supernatant was measured.Subunit exchange was also monitored by the quenching ofAlexa488-labeled apoC-II fluorescence that occurs on fibrilformation.19 Monomeric Alexa488-labeled apoC-II(6.75 μg/ml) was added to preformed apoC-II fibrils orclosed-loop samples (0.14 mg/ml), and Alexa488 fluores-cence was measured continuously during incubation at25 °C. Fluorescence emission was normalized to datacollected for Alexa488-labeled apoC-II (6.75 μg/ml) alonethat showed only minor variation over the same timecourse (b5%). Fibril dilution experiments and sizedistribution analysis using fluorescence detection wereperformed using preformed apoC-II fibrils (0.3 mg/ml)incubated with Alexa488-labeled apoC-II (6.75 μg/ml)and DTT (1 mM) for 4 h.

Computer simulations

The equilibrium constants shown in Fig. 6 were used toobtain expressions for the various species as a function oftotal apoC-II concentration. With the molar concentrationsof unfolded monomer, active folded conformer, linearfibrils of length i and closed loops of length i denoted asm,n, fi and Li, respectively, the relevant expressions are:

m ¼ n=Ki

f i ¼ Ki−1eq ni

Li ¼ PiKLf i

Pi ¼ e− i−μð Þ2= 2σ2ð Þ

ð1Þ

P is the length-dependent probability of closed-loopformation, assuming a Gaussian distribution with μ andσ as the mean length and standard deviation of closed


loops, respectively. With this formulation, the total apoC-II concentration ct is given by:

ct ¼ m þ n þXi¼l

i¼2

if i þXi¼l

i¼2

iLi ð2Þ

Combination of Eqs. (1) and (2) and assigning values for n,the concentration of active folded conformer, allowedfibril size distributions to be calculated as a function oftotal concentration for specified values of Ki, Keq, KL, μ andσ. The summations in Eq. (2) were truncated for largerspecies (iN35,000) since the concentration of these speciesbecame vanishingly small. Fibril size distributions wereconverted into sedimentation coefficient distributions thatassumed a worm-like chain model.30 Fitted values for Keqwere obtained by fitting theoretical fibril size distributionsto experimental data using a sequential process. Initially,the data obtained at a total apoC-II concentration of0.3 mg/ml were analyzed, and the total concentration offibrils was calculated by subtracting the free pool of non-sedimenting monomer (m+n). Values were then assignedto n to fit the total fibril concentration for assigned valuesof Keq, neglecting closed-loop formation (KL=0). Thisprocedure was then repeated for different values of Keq toobtain fibril distributions that fitted the modal sedimen-tation coefficient of fibrils observed experimentally. Thefitted value for Keq was then fixed, and the process wasrepeated iteratively with the inclusion of closed-loopformation to obtain a fibril size distribution that ade-quately described the experimental profile for manuallyassigned values of KL, μ and σ. This procedure yieldedfitted values for Keq, KL, μ and σ, which were then fixed toobtain simulated size distributions to compare withexperimental distributions obtained for apoC-II fibrilsformed over the concentration range 0.1–0.6 mg/ml. Thisprotocol also yielded fitted values for n, which, togetherwith experimental values for the free pool of apoC-IImonomer (m+n), allowed calculation of the correspond-ing values for Ki. Simulated size distributions were alsocalculated for the case where Keq varied with fibril length i.For these calculations, the following expression was usedfor the length-dependent equilibrium constant K′eq(i):

K eq′ ið Þ ¼ K eq 2ð Þ � ia ð3Þwhere the exponent “a” was used as an additional fittingparameter.The time courses for fibril formation and the changes in

weight-average sedimentation coefficient were computedfor the model shown in Fig. 6 using procedures similar tothose used previously.17 For practical reasons, we used anapproximate description based on the simplifying ap-proximation of replacing all fibrils of different length witha single, average fibril species, f. This approach neglectedthe formation of closed-loop structures, with the approx-imation that the proportion of closed loops relative to totalfibrils is small. The relevant equations are:

dm

dt¼ −konm þ koffn

dn

dt¼ konm−koff n−kenf−2ke n2 þ 2keofff

df

dt¼ ke n

2 þ kb c0−m−n−fð Þ−k jf 2

ð4Þ

where c0 is the total molar concentration of subunits in theexperiment, kon and koff are rate constants for formationand dissociation of active folded conformer, ke and keoff arethe rate constants for fibril elongation and activeconformer dissociation from each end and kb and kj arethe rates of fibril breaking or end-to-end joining persubunit–subunit interface. The average fibril length (inmonomer units) can be calculated as (c0−m−n)/f, which,in turn, can be converted into a weight-average sedimen-tation coefficient expected to be observed in the sedimen-tation experiments.30 Eq. (1) was solved in MATLAB andimplemented for nonlinear regression of the experimentaldata.

Acknowledgements

This research was supported under the AustralianResearch Council's Discovery Projects fundingscheme (project number DP0984565). M.D.W.G. isthe recipient of an Australian Research Council PostDoctoral Fellowship (project number DP110103528).This work was supported in part by the IntramuralResearch Program of the National Institute ofBiomedical Imaging and Bioengineering, NationalInstitutes of Health.

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an equilibrium model for linear and closed-loop amyloid fibril formation

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