Thermodynamic phase diagram of amyloid-β(16–22) peptideYiming Wanga, Samuel J. Bunceb,c, Sheena E. Radfordb,d, Andrew J. Wilsonb,c, Stefan Auerc, and Carol K. Halla,1

aDepartment of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, NC 27695-7905; bAstbury Centre for Structural MolecularBiology, University of Leeds, LS2 9JT Leeds, United Kingdom; cSchool of Chemistry, University of Leeds, LS2 9JT Leeds, United Kingdom; and dSchool ofMolecular and Cellular Biology, Faculty of Biological Sciences, University of Leeds, LS2 9JT Leeds, United Kingdom

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved December 20, 2018 (received for review November 26, 2018)

The aggregation of monomeric amyloid β protein (Aβ) peptide intooligomers and amyloid fibrils in the mammalian brain is associatedwith Alzheimer’s disease. Insight into the thermodynamic stability ofthe Aβ peptide in different polymeric states is fundamental to de-fining and predicting the aggregation process. Experimental deter-mination of Aβ thermodynamic behavior is challenging due to thetransient nature of Aβ oligomers and the low peptide solubility.Furthermore, quantitative calculation of a thermodynamic phase di-agram for a specific peptide requires extremely long computationaltimes. Here, using a coarse-grained protein model, molecular dynam-ics (MD) simulations are performed to determine an equilibrium con-centration and temperature phase diagram for the amyloidogenicpeptide fragment Aβ16–22. Our results reveal that the only thermo-dynamically stable phases are the solution phase and the macro-scopic fibrillar phase, and that there also exists a hierarchy ofmetastable phases. The boundary line between the solution phaseand fibril phase is found by calculating the temperature-dependentsolubility of a macroscopic Aβ16–22 fibril consisting of an infinite num-ber of β-sheet layers. This in silico determination of an equilibrium(solubility) phase diagram for a real amyloid-forming peptide,Aβ16–22, over the temperature range of 277–330 K agrees well withfibrillation experiments and transmission electron microscopy(TEM) measurements of the fibril morphologies formed. This insilico approach of predicting peptide solubility is also potentially usefulfor optimizing biopharmaceutical production andmanufacturing nano-fiber scaffolds for tissue engineering.

phase diagram | solubility | amyloid | protein aggregation |coarse-grained simulation

Amyloid β protein (16–22) (Aβ16–22) is a seven-residue amy-loidogenic peptide (Ac-K-L-V-F-F-A-E-NH2) comprising

the central, fibril-forming core of the full-length Aβ peptide, amajor constituent of the extracellular plaques associated withAlzheimer’s disease (1, 2). Aβ16–22 has been widely studied due toits relative ease of synthesis and its ability to form well-characterized antiparallel β-sheet fibril structures at concentra-tions above 55 μM (3, 4). At neutral pH, monomeric Aβ16–22 ispredicted to adopt a random coil configuration that then oligo-merizes, passing through an intermediate out-of-register statebefore the final in-register antiparallel alignment (5–8). Atomisticsimulations of a small number of peptides on the nanosecond timescale are possible and have been used to examine the structuralstability of a variety of Aβ16–22 oligomers, including β-barrels (9),β-sheet–rich dimers (10), trimers (11, 12), and hexamers (13, 14).Coarse-grained MD simulations have succeeded in predictingspontaneous formation of octamer (15), trilayer, tetralayer (6),and heptalayer (16) twisted Aβ16–22 nanofibrils involving up to 192peptides on the microsecond time scale (17). A prerequisite tounderstanding which structures Aβ16–22 peptides form, and underwhat conditions, is knowledge of the phase diagram, which char-acterizes the thermodynamic stability of the peptide phases as afunction of relevant parameters, most importantly temperatureand peptide concentration. However, despite being studied undera wide range of conditions, a quantitative thermodynamic phase

diagram has never been generated for Aβ16–22, or any otherpeptide using a realistic protein model.In general, experiments (18–20) determining the coexistence lines

(solubilities) between oligomeric or fibrillar aggregates and theprotein solution are difficult to perform because of the metastablenature of the oligomers and the low solubilities of fibrils. Mostexperiments are performed by preparing a solution at a givenconcentration and temperature, and are left for up to weeks todetermine whether oligomeric or fibrillar aggregates have formed(21, 22). The so-observed concentrations and temperatures at whichfibrils form can often be far from the true equilibrium concentra-tions (solubilities), as the observed structures can be kineticallytrapped, and thus not representative of the most stable state ther-modynamically (23). Experiments in which preformed fibrils are putin solution and the concentration at which they elongate or dissolveis measured can also provide estimates of solubility (24, 25).The challenge in determining a peptide phase diagram for

realistic protein models in silico is the long computational timeneeded to calculate the solubility. Molecular simulations aimedat constructing phase diagrams of peptides (26, 27) are allnonequilibrium and only simulate the self-assembly of peptidesfrom the monomeric state to the aggregate state. There is noguarantee, however, that structures so obtained will coincidewith the thermodynamically stable phases. Furthermore, pre-vious studies do not consider the existence of metastable phases.It has only recently become possible to perform Monte Carlosimulations that are capable of constructing a concentration andtemperature phase diagram at thermodynamic equilibrium forsimple generic peptide models (28, 29). Given the complexity of

Significance

Phase diagrams of atomic systems are calculated routinely bycomputer simulations, but such calculations are absent for eventhe simplest peptides. Previous simulations are mainly non-equilibrium and focus on the assembly of peptides from themonomeric state to the aggregated state. To obtain accurateequilibrium solubilities, it is necessary to simulate many assem-bly and disassembly events of fibrillar aggregates, which is no-toriously difficult as it requires breaking many hydrogen bonds.We overcome these challenges and calculate the equilibriumphase diagram of amyloid β protein (16–22) (Aβ16–22), the ar-chetypal amyloid former, using a realistic protein model. Im-portantly, our prediction of Aβ16–22 solubility over temperaturesfrom 277 to 330 K agrees well with experimental measurements.

Author contributions: Y.W., S.J.B., S.A., and C.K.H. designed research; Y.W. and S.J.B.performed research; Y.W., S.J.B., and S.A. analyzed data; and Y.W., S.E.R., A.J.W., S.A.,and C.K.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1To whom correspondence should be addressed. Email: hall@ncsu.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1819592116/-/DCSupplemental.

Published online January 23, 2019.

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biomolecules, the relevance of this finding to any real peptidesystem is not clear.Here, we construct a temperature concentration phase diagram

for the Aβ16–22 peptide by determining the solubility of oligomericaggregates and fibrils using coarse-grained MD and classical nu-cleation theory. The phase diagram reveals that the solution phaseand fibril phase are the only two thermodynamic stable phases, andthat there also exists a hierarchy of metastable aggregate phases.The thermodynamic boundary between the solution phase and thefibril phase is taken to be the solubility line of an infinite layer (∞)β-sheet fibril predicted using discontinuous MD simulation andclassical nucleation theory. The strength and soundness of this ap-proach lie in the fact that our force field, PRIME20, is knowledge-based and sequence-specific, and accurately predicts peptide orien-tation and strand–strand and sheet–sheet distances of Aβ16–22 fibrilstructures (6). To validate the phase diagram predicted in silico, weconduct Aβ16–22 fibrillation experiments at peptide concentrationsvarying from 10 to 200 μMat 277–330 K. Judging from TEM images,the experimental conditions under which the Aβ16–22 solution formsfibrils agree well with the predicted solution/fibril phase boundary.

Results and DiscussionProtein Model, Structure of Aβ16–22 Oligomer, and Fibril. The Aβ16–22aggregates are modeled using a coarse-grained protein model,called PRIME20 (30, 31), in which each amino acid residue isrepresented by three backbone spheres, one each for N-H, C-H,and C = O, and a single side-chain sphere R (Fig. 1A). Thenondirectional interaction between side-chain spheres and thedirectional hydrogen bond between the N-H and C = O spheresare modeled as square-well potentials (Fig. 1 B and C). Whilethe strength of the directional backbone hydrogen bonding in-teraction is the same for all residues, the strength of the non-directional side-chain–side-chain interaction is side-chain–specific. Details are provided in Methods.Here, we focus on five types of aggregates formed by Aβ16–22:

non–hydrogen-bonded (non-HB) oligomers (Fig. 1D); hydrogen-bonded (HB) oligomers (Fig. 1E); and fibrils composed of two,three, and four β-sheet layers (Fig. 1 F–H). The non-HB oligomeris defined to be an aggregate formed solely by interpeptide hy-drophobic (HP) side-chain–side-chain interactions with no inter-peptide hydrogen bonds; however, intrapeptide hydrogen bondsare allowed. The HB oligomer is defined to be an aggregate thathas, at most, one interpeptide hydrogen bond between any twoneighboring peptides and any number of intrapeptide hydrogenbonds. We consider the non-HB and HB oligomers to be two

distinct states because the peptides in these two states have dif-ferent numbers of hydrogen bonds and HP side-chain contactswith neighboring peptides (SI Appendix, Fig. S1). The β-sheetaggregates are stabilized by having side-chain HP contacts be-tween tightly packed antiparallel β-sheets and a maximum numberof backbone HBs between fully extended peptides within eachβ-sheet, consistent with X-ray diffraction and solid-state NMRmeasurements (2, 6). Our simulations show that the probabilitydistributions for the number of HB and HP side-chain contacts forthe five aggregates do not change significantly with temperatureover the studied temperature range.

Measurement of Solubility for Aβ16–22 Oligomers and Fibrils. To de-termine the solubility of aggregates (oligomers and fibrils) formedby the Aβ16–22 peptides, we perform MD simulations in the ca-nonical ensemble. The solubility is defined to be the peptideconcentration above which the peptide will join with others toform an aggregate and below which the peptide will remain insolution. In our simulations, the solubilities for the different Aβ16–22aggregates (oligomers and fibrils) at a given temperature aremeasured as the equilibrium monomer concentration at which theaggregate neither grows nor shrinks (details are provided inMethods). This approach was used by Bai and Li (32) to measuresolid–liquid phase equilibrium for the Lennard–Jones fluid. Theircalculation of the solid–liquid interfacial energy (surface tension)using classical nucleation theory was in qualitative agreement withresults calculated using other theoretical approaches [e.g., in-terface fluctuation method (33), reversible work integration (34)].Later, Auer and Kashchiev (28) and Auer (29) used this method,along with Monte Carlo simulation, to calculate a phase diagramfor a simple homopolypeptide model, where each amino acid isrepresented by one Cα bead. The predicted values for the surfacetension were in agreement with theoretical predictions for proteincrystals using a simple lattice model (35).We first measured the solubility for the non-HB and HB

oligomers. Fig. 2A shows that the solubilities for the non-HB andHB oligomers increase with increasing temperature. This isexpected, as at higher temperatures, the thermal energy makes iteasier for peptides to detach from the oligomer so that higher

Fig. 1. (A) Representation of the Aβ16–22 peptide in the PRIME20 model. (Band C) Schematic representations of the square-well interaction between twoside chains and the directional square-well interaction (hydrogen bonding)between backbone NH and C = O spheres. (The spheres are not drawn to scalefor ease of viewing.) (D–H) Simulation snapshots of non-HB oligomer; HBoligomer; and two, three, and four β-sheet fibrils, respectively, generated byVisual Molecular Dynamics (VMD) software. The peptides in the five aggre-gates and in solution are shown in green and magenta, respectively.

180 210 240 270 300 330 3600

4

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4β

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C,ytilibuloSe

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e)

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020406080

100120 Binding energy

Latent heat

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HB olig.

non-HB olig.

lom/Jk,ygrenE

C

330K342K353K

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D

Fig. 2. (A) Plot of the solubility, Ce, for non-HB oligomer (olig.); HB oligomer;and two, three, and four β-sheet fibrils as a function of the temperature. (B)Solubility data for non-HB oligomer; HB oligomer; and two, three, and fourβ-sheet fibrils at four to six different temperatures. Lines are fitted to Eq. 1. Thecolor scheme is the same as used in the plot in A. Ln(Ce), natural logarithm of Ce.(C) Comparison of the average binding energy per peptide within the aggregateand the corresponding latent heat of peptide aggregation into the oligomer andfibril. (D) Dependence of the fibril solubility on thickness (i= 2, i = 3, and i= 4) fortwo, three, and four β-sheet fibrils, respectively. Lines are fitted to Eq. 2.

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concentrations are required to stabilize these aggregates. Thelatent heat of aggregation for monomeric peptides into oligo-mers can be obtained by fitting the van’t Hoff equation,

Ce =Crexp�−

LkBT

�, [1]

to the simulation data (Fig. 2B), where Ce is the equilibrium fibrilsolubility, Cr is a temperature-independent reference concentra-tion, kB is the Boltzmann constant, T is the simulation tempera-ture, and L is the latent heat of monomer peptide aggregation intothe oligomer or fibril. From the simulation, we also calculated theaverage peptide binding energy (the sum of the HP and HB en-ergies per peptide) within the non-HB and HB oligomers, includ-ing all of the peptides on the surface and inside the aggregate, andfound that the latent heat is about half of the average peptidebinding energy, EB, (Fig. 2C) consistent with earlier work (29).Although the solubility of the oligomers should depend on the

oligomer size as described by the Ostwald formula (28, 29, 36) (whichpredicts that solubility decreases with increasing oligomer size), wedo not consider the effect of oligomer size here. This is because largeoligomers do not form for this Aβ16–22 system. The simulations showthat the non-HB and HB Aβ16–22 oligomers are relatively small(containing around 10 peptides) and easily break apart into smalleraggregates when they contain more than 15 peptides.Next, we determined the solubilities for two, three, four, and

infinite β-sheet Aβ16–22 fibrils. The simulation results (Fig. 2A)show that at a given temperature, the solubilities for the two,three, and four β-sheets decrease with increasing thickness(number of β-sheet layers) of the fibril. As the number of β-sheetsin the fibril increases, the average number of HP contacts perpeptide increases, as shown in SI Appendix, Fig. S1C, leading toincreased stability of the fibril. A quantitative description of thedependence of the solubility of a fibril with i layers (i = 1, 2, 3, etc.)of a β-sheet, Ciβ, on thickness at fixed temperature can be derivedfrom classical nucleation theory and is given by (37)

Ciβ =C∞βexp�2ψh

kBT·1i

� , [2]

where C∞β is the solubility of an infinite thick fibril, ψh = ahσh isthe fibril surface energy parallel to the fibril thickening axis, σh isthe specific surface energy of the face perpendicular to the fibrilaxis, and ah is the lateral surface area occupied by each peptidewithin one β-sheet as defined by Kashchiev and Auer (37). Alinear fit of our simulation data (lnCiβ vs. 1/i) to Eq. 2 (Fig. 2D)yields an estimate for the solubility C∞β as a function of temper-ature for the infinitely thick fibril. As before, the latent heats ofpeptide aggregation into fibril phases from solution can beobtained by a fit of our data to Eq. 1 (Fig. 2B), and the obtainedvalues are shown in Fig. 2C. From Fig. 2C, we find that for two,three, and four β-sheet fibrils, the values of the latent heat arecomparable to the average peptide binding energy. The relationbetween latent heat and protein–protein interaction energy for aprotein crystal was first derived by Haas and Drenth using a simplelattice model (35). They found that the latent heat should be half ofthe average protein–protein interaction energy; this was derived byusing the zero-temperature approximation, which does not con-sider the entropic effects. We speculate that it might be the neglectof the entropic effect that causes the relation between latent heatand peptide binding energy to hold for oligomers but not for two,three, and four β-sheet fibrils in this work. In addition, the fit of oursimulation data to Eq. 2 yields values for ψh = ahσh, and knowledgeof ah enables us to estimate the average surface tension, σh, of theAβ16–22 fibril (SI Appendix, Fig. S2). The lateral surface area perpeptide, ah = 106.1Å, is the product of the length of a peptidewithin a β-sheet (22.1 Å) and the interpeptide distance in oneβ-sheet (4.8 Å). Hence, depending on temperature, the average

surface tension, σh, for the Aβ16–22 fibril ranges from 24 to 30 mJ/m2

(SI Appendix, Fig. S2), which is in the range of 0.1–30 mJ/m2 reportedfor protein crystals in aqueous solution (35, 38–40).

Aβ16–22 Phase Diagram. The major finding of this study is ourquantitative calculation of the thermodynamic phase diagram forthe Aβ16–22 peptide fragment (Fig. 3A). The peptide solutionphase (Fig. 3A, cyan region) and the macroscopic fibrillar phase(Fig. 3A, yellow region) are the only two thermodynamicallystable phases. The thermodynamic boundary between thesephases is taken to be the solubility line for a fibril with an infinitenumber of β-sheets. [Error bars are not shown in Figs. 3 and 4Abecause the R2 value for the linear fit of C∞β vs. T data to obtainC∞β(T) is close to 1.] Within the fibril phase region of the phasediagram, there exists a series of metastable aggregate phasesseparated by the metastable solubility lines for the non-HBoligomer; HB oligomer; and two, three, and four β-sheet fi-brils. A hierarchy of metastable phases, with the ultimate stableboundary being between solution and infinitely thick fibril, wasfirst observed by Auer and Kashchiev (28).To justify the infinite thickness approximation in calculating

fibril solubility, we calculated Aβ16–22 fibril solubility versus fibrilthickness (SI Appendix, Fig. S3) and found that fibril solubilityremains unchanged after the thickness exceeds 30 layers. Wethink that an actual Aβ16–22 fibril may contain more than 30layers of β-sheets at neutral pH, because Aβ1–42 (41), Aβ10–35(42), and Aβ18–28 (43) form fibrils in vitro with 2, 6, and 24 layersof β-sheets, respectively, at neutral pH. This is consistent with ahypothesis by Lu et al. (44) that shorter amyloidogenic peptides

Fig. 3. (A) Phase diagram for Aβ16–22 peptide. Solubility, Ce, versus tem-perature data are presented for the non-HB oligomer (orange dashed line),HB oligomer (purple dashed line), two β-sheet (2β) fibril (red dashed line), 3βfibril (green dashed line), 4β fibril (blue dashed line), and infinite layerβ-sheet (∞β) fibril (black line). The fibril and solution phases are coloredyellow and cyan, respectively. (B) DMD/PRIME20 simulation snapshots at theconcentration and temperature of phase points A and B are taken at dif-ferent time points, respectively.

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tend to adopt more planar β-sheets and maintain more layers inthe fibril than longer peptides.To better illustrate why the solubility line for the fibril is

regarded as the coexistence line, and why the oligomer andvarious multisheet fibril phases are metastable with respect tothe fibril phase, we put our calculations in the context of morefamiliar thermodynamic arguments in which the compositions oftwo phases in equilibrium are found by minimizing their freeenergies. SI Appendix, Fig. S4 plots the Helmholtz free energiesof the solution, oligomer, and fibril phases as a function of thepeptide mole fraction, xp (which is simply related to the peptideconcentration for a dilute solution), in a peptide and watermixture at fixed temperature and volume. A double-tangent(also called convex-envelop) construction on the free ener-gies of the solution and fibril phases determines the peptidemole fractions, xsolnð1Þp and xfibp , at which the solution and fibrilphases are in equilibrium. The construction locates the peptidemole fractions at which the peptide chemical potentials (thexp = 1 intercepts of lines tangent to the free energy curves) are

equal [i.e., μsolnð1Þp

�xsolnð1Þp

�= μfibp

�xfibp

�]. The composition, xsolnð1Þp ,

is equivalent to the peptide concentration above which themonomer goes through a phase transition to another phase,the solubility line determined in our simulations. Similarly

a double-tangent construction on the solution and oligomerphases yields peptide mole fractions xsolnð2Þp and xoligp , with equal peptide

chemical potentials, μsolnð2Þp

�xsolnð2Þp

�= μoligp ðxoligp Þ, signifying equilib-

rium between solution and oligomer phases in this concentrationrange. This is, however, a metastable equilibrium compared with theequilibrium between solution at xsolnð1Þp and fibril at xfibp , becausethe peptide chemical potential in solution is higher than that whenthe solution is in equilibrium with the fibril. Note also that havingμsolnð2Þp

�xsolnð2Þp

�> μsolnð1Þp

�xsolnð1Þp

�means that xsolnð2Þp > xsolnð1Þp , consistent

with what is measured in simulations. Notice also that in SI Appendix,Fig. S4, there are no metastable phases when xsolnð1Þp < xp < xsolnð2Þp ;the oligomer phase only becomes metastable with respect to thefibril phase when xp > xsolnð2Þp . This is also reflected in the solubilityversus temperature phase diagram (Fig. 3A), which shows a regionof concentrations where the HB oligomer is unstable (below thedashed purple line and above the black solubility line) and a re-gion where the oligomer is metastable (above the dashed purpleline). Similar arguments can be made for the other metastablephases, such as the two, three, and four β-sheet fibril phases. Weemphasize that we did not calculate free energies as we were ableto calculate the solubilities directly; the discussion here is mainlypresented for pedagogical purposes.Knowledge of the thermodynamically stable and metastable

phases in the phase diagram allows us to determine the condi-tions under which the oligomer or fibril can form. A previousdiscontinuous molecular dynamics (DMD) study of Aβ16–22 ki-netic aggregation by Cheon et al. (6) found that at 20 mM andT* = 0.2, corresponding to 342 K, a system of peptides in arandom configuration first forms a fibril nucleus, which then growsinto fibrils. At the same concentration but at a lower temperatureT* = 0.17, corresponding to T = 273 K, the peptides first form HBoligomers that later merge and rearrange to form fibrils. Snapshotsfrom new simulations at T = 342 K and T = 273 K are shown in Fig.3B. Our thermodynamic phase diagram helps explain why Aβ16–22aggregates via a “one-step” pathway at high temperature and a“two-step” pathway at low temperature. From Fig. 3A, at point A(20 mM, T = 342 K), the peptide concentration is greater than thesolubility of fibril, C > Ce,fibril (T = 342 K) but smaller than thesolubility of the HB oligomer, C < Ce,HB oligomer (T = 342 K). Aspoint A satisfies the thermodynamic criteria for stable fibril for-mation but not for HB oligomer formation, it is not surprising thatthe aggregation kinetics are characterized by nucleus formationand subsequent fibril growth. In comparison, point B (20 mM, T =273 K) satisfies the thermodynamic condition for formation of astable fibril and a metastable HB oligomer C > Ce,fibril (T = 273 K)and C > Ce,HB oligomer (T = 273 K); this is conducive to two-stepaggregation kinetics in which several oligomers form and latermerge and rearrange to form fibrils. These arguments about theconnection between the fibril formation kinetics and aggregatestability and metastability are meant here only to be suggestive.The best way to predict Aβ16–22 aggregation pathways at dif-ferent temperatures and concentrations would be to constructaggregation free energy landscapes along the appropriate reactioncoordinates.To validate the in silico prediction of Aβ16–22 solubility at

biophysically relevant temperatures, TEM was used to determinewhether fibrils had formed after a predetermined time (2-wkincubation). Fig. 4 is a summary of our experimental results nearthe predicted solubility boundary line at temperatures rangingfrom 277 to 330 K. At T = 330 K, fibrils form at both 200 μM and300 μM, but no fibrils are observed when the peptide concen-tration is at or below 100 μM, which is consistent with the pre-dicted solubility (177 μM) at 330 K. Furthermore, at T = 277 K,T = 296 K, and T = 310 K, fibrils form when the concentration isequal to or above 20 μM, which also agrees with our predictedsolubility at T = 277 K, T = 296 K, and T = 310 K of 0.2 μM,2.6 μM, and 16.5 μM, respectively. It is important to note thateven though fibrils were not observed in the TEM images at

Fig. 4. (A) Summary of both the simulation-predicted temperature-dependent solubility line, Ce(T), for Aβ16–22 peptide (black curve) and the fi-brillation experiments performed under given conditions. Red dots and redcircles indicate conditions at which fibrils have been found to form and not toform, respectively, via TEM (at T = 277 K, T = 296 K, T = 310 K, and T = 330 K).Blue dots indicate that fibrils have been reported in the literature to formunder these conditions (SI Appendix, Fig. S5). (B) Six selected TEM images (i–vi)showing that Aβ16–22 forms fibrils under the conditions that correspond to thesix red dots labeled i–vi in A. The buffer is 100 mM ammonium bicarbonate (pH7), with a final concentration of 1% DMSO (vol/vol). (Scale bar: 200 nm.)

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10 μM at all temperatures tested, this does not rule out thepresence of low populations of fibrils under these conditionsthat are below the detection limits of this method (i.e., lowpossible surface adsorption of the fibrils). Thus, we concludethat the boundaries predicted by the phase diagram quantita-tively agree with the experimental findings, demonstrating thepower of this approach to understand the thermodynamicstability of peptide assemblies.The concentration and temperature conditions at which Aβ16–22

forms fibrils reported from a number of other in vitro studies alsoagree with our simulation prediction (2, 3, 8, 45) (Fig. 4 and SIAppendix, Fig. S5). It should be noted that the only studies in-cluded in this comparison use Aβ16–22 with capped N and C ter-mini, since uncapping the termini can have a substantial twistingeffect on the fibril supramolecular structure (46). Most of thesestudies were performed at high concentrations (>200 μM), placingthem in regions of the phase diagram that would predict fibrilformation, even at the highest temperature studied (55 °C, 328 K)(7). Only one study by Senguen et al. (3) demonstrated that fibrilsare formed at a concentration of 55 μM at 37 °C (310 K), again inagreement with our phase diagram.Knowledge of the peptide solution/fibril phase boundary could

allow us to modulate the quality of nanofiber structures formed.This can be accomplished by carefully controlling the drivingforce for fibril formation (e.g., highly ordered fibrils can beobtained at solution conditions close to the phase boundary,while less-ordered entangled fibrils can be obtained at condi-tions away from the phase boundary). This suggests that ourcomputation-based approach could inform future fabrication ofamyloid nanofiber substrates/scaffolds with desired microstruc-ture properties for various tissue engineering and biomedicalapplications (47, 48). In addition, controlling protein aggrega-tion is a key problem for the purification and storage of ther-apeutic proteins (49, 50). Thus, knowledge of the solubilityof specific peptides and proteins over a wide range of tem-peratures could be used to effectively prevent their unwantedagglomeration.Our approach can be applied to investigate the effect of mu-

tations on peptide solubility. For example, in preliminary simu-lations, we show that the F19A mutant of Aβ16–22 remainssoluble at C = 20 mM and T = 342 K in contrast to the fibril-forming behavior of the wild type at the same conditions. In fact,our predicted solubility of the F19A mutant is about 600 mM at307 K (SI Appendix, Fig. S6). This result is consistent with ex-perimental findings by Senguen et al. (3) that the F19A mutantof Aβ16–22 is much more soluble (at least 350 μM) than its wildtype at body temperature. One of our next steps will be to cal-culate the solubility of Aβ16–22 mutants at positions 19 and 20and to compare our results with experimental measurements.

ConclusionIn this work, we demonstrated that by using a relatively realisticcoarse-grained protein model, it is possible to calculate viasimulation an equilibrium concentration and temperature phasediagram for a short amyloidogenic peptide, Aβ16–22. The pre-dicted phase diagram provides insight into the thermodynamicstability of the different types of aggregates formed by Aβ16–22under biophysically relevant conditions. This helps us to un-derstand the conditions under which they can form and how toprevent them from forming. The solubilities predicted in silicoare in the range expected from experiments and, as such, alsoprovide a rigorous test of the validity of the model and meth-odology used. As is shown in SI Appendix, Fig. S6, the solubilityof Aβ16–22 peptide calculated in this work is two to three ordersof magnitude higher than that of the generic polypeptide cal-culated by Auer and Kashchiev (28) and Auer (29). Our work isan in silico determination of an equilibrium (solubility) ther-modynamic phase diagram for a nontrivial peptide. Althoughpeptide phase diagrams have been calculated via simulations (26,27), they are usually based on the kinetics of phase formation asopposed to a direct solubility measurement. Furthermore, the in

silico prediction of Aβ16–22 solubilities over the temperaturerange of 4–57 °C agrees well with fibrillation experiments andTEM measurements of the fibril morphologies formed. This insilico approach of predicting peptide solubility is also potentiallyuseful for engineering nanofiber scaffolds for biomedical appli-cations as well as manufacturing biopharmaceuticals.

MethodsIn Silico Peptide Model. The molecular model that we use is a four sphere perresidue model in which each amino acid residue is represented by threebackbone spheres, one each for N-H, C-H, and C = O, and one for the side-chain sphere, R (30, 31). The PRIME20 model has unique geometric andenergetic parameters for each of the 20 amino acids. Specifically, each side-chain sphere of the 20 amino acids has a distinct hard sphere diameter(effective van der Waals radius) and side chain-to-backbone distances (R-Cα,R-NH, and R-CO). The interactions between the atoms are described bydiscontinuous potentials, including the hard sphere, square-well, andsquare-shoulder potentials. The strength of the directional hydrogen bond isdescribed by the well depth, eHB, and is the same for all residues, and thestrength of the nondirectional HP interaction between side chains i and j isdescribed by a side-chain–specific well depth (eHP-ij). The potential energyparameters between the 20 different amino acids include 210 independentsquare-well widths and 19 independent square-well depths derived by usinga perceptron learning algorithm that optimizes the energy gap between 711known native states from the Protein Data Bank and decoy structures (31).For example, the interaction strength, eHP-AA, between two alanine sidechains is 0.084eHB, where the hydrogen bonding strength, eHB, is chosen tobe 12.47 kJ/mol as was used in previous work (17). The details of the as-signment of the values of the interaction parameters of the PRIME20 modelare described in earlier work (17, 30, 31). The reduced temperature is de-fined as T* = kBT/eHB. It can be related to a real temperature by T [K] = 2,288.46T*− 115.79; this equation was obtained by matching the folding temperature ofalanine-rich polypeptides in our previous DMD simulations (17) to the ex-perimental values (51). The reduced time unit is Δtreduced = 0.96 ns,which has been obtained by matching the self-diffusion coefficient of Aβ16–22 obtained from DMD simulations to that calculated from atomistic MDsimulation (17).

DMD simulations with the PRIME20 model are ideally suited for simulatingprotein aggregation in three respects:

i) They allow simulation of the complete aggregation process from a sys-tem of random coil peptides to a fibrillar structure (up to 200 peptides)at time scales up to 100 μs (16, 17). For illustrative purposes, SI Appendix,Fig. S7 shows a simulation snapshot of the structure of a two-layerβ-sheet fibril formed by Aβ16–22 peptides.

ii) The PRIME20 model is realistic enough to ensure that the predicted fibrilstructures agree well with X-ray diffraction and solid-state NMR mea-surements (6). Its adequacy and efficiency have been demonstrated byits application to short peptide systems, including Aβ16–22 (6); thedesigned hexapeptide sequences of López de la Paz et al. (52) andWagoner et al. (53); the tau fragment (54); and longer peptide systems,including Aβ17–36 (55) and Aβ17–42 (56).

iii) The side-chain spheres of the 20 possible amino acids each have distinctgeometric and energetic parameters, ensuring that the peptide is mod-eled in a sequence-specific manner.

Simulation Method. The solubilities for the different Aβ16–22 aggregates(oligomers and fibrils) were measured using an approach first proposed byBai and Li (32) and later adopted by Auer (29). The method is based on thedefinition of the solubility of an aggregate at a given temperature as theequilibrium monomer concentration at which the aggregate neither growsnor shrinks. The preformed aggregate is initially placed in the center of acubic box surrounded by monomer peptides. Periodic boundary conditionsare imposed. At constant temperature, we constrain the fibril to avoid newβ-sheet creation, while allowing it to elongate or shrink; the concentrationof monomeric peptides is monitored until it reaches a plateau and remainsconstant thereafter for a long time (SI Appendix, Fig. S8). The solubility ofthe aggregate at any given temperature is defined to be the monomericpeptide concentration in that final equilibrium state. Although this methodis simple in principle, it is very costly in terms of simulation time. In fact, toobserve whether a preformed aggregate grows or shrinks requires simula-tion of hundreds of attachment and detachment events to and from theaggregate. Detachment events of peptides from fibrils are particularly rareas they require breakage of several strong hydrogen bonds. Due to the

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infrequency of detachment events, the total number of peptides used in oursimulations is relatively small (16–80 peptides).

Here, we measured the solubilities of five types of Aβ16–22 aggregates,including a non-HB oligomer; an HB oligomer; and fibrils containing two,three, and four β-sheets. We choose to focus our attention on the non-HBand HB oligomers because they represent two limiting examples of theforces that can stabilize oligomers formed early in the fibril formationprocess. The solubility profiles of HB oligomers with more than one hydro-gen bond formed between any peptide pair are expected to be similar tothe ones we study here, and thus are omitted. The two, three, and fourβ-sheet fibrils are defined to be aggregates that contain two, three, and fourstacked β-sheets, respectively. The system size and simulation details aredescribed in SI Appendix. To measure the solubility of potentially metastableaggregates, constraints are required to avoid structural conversion of onetype of metastable aggregate into another (i.e., the growth of a non-HBoligomer into an HB oligomer, the growth of a two β-sheet fibril into a threeβ-sheet fibril). Methods used to accomplish this are described in SI Appendix.

Aβ16–22 Solid Phase Peptide Synthesis, Aggregation Conditions, and TEM. Theexperimental details of Aβ16–22 solid phase peptide synthesis, aggregationconditions, and TEM are described in SI Appendix.

ACKNOWLEDGMENTS. We thank Dr. Matt Iadanza for help with processingthe EM images. Y.W. and C.K.H. acknowledge the support of a CheneyVisiting Scholar Fellowship from the University of Leeds. This work wassupported by NIH Grant R01 EB006006 and, in part, by National ScienceFoundation (NSF) Research Triangle Materials Research Science and Engi-neering Centers Grant DMR-1121107. It was also supported by NSF Divisionof Chemical, Bioengineering, Environmental, and Transport Systems Grants1743432 and 1512059, and by Engineering and Physical Sciences ResearchCouncil Grants EP/N035267/1, EP/N013573/1, and EP/KO39292/1. S.J.B. wassupported by a Biotechnology and Biological Sciences Research CouncilPhD Studentship (Grant BB/J014443/1). The JEM-1400 (Jeol) was purchasedwith funding from the Wellcome Trust (Grant 094232/Z/10/Z) and fitted witha CCD camera funded by Grant 090932/Z/09/Z.

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