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Fluid Phase Equilibria 489 (2019) 104e110

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Fluid Phase Equilibria

journal homepage: www.elsevier .com/locate/fluid

Thermodynamic properties of amyloid fibrils: A simple model ofpeptide aggregation

Tomaz Urbic a, *, Cristiano L. Dias b, **

a Faculty of Chemistry and Chemical Technology, University of Ljubljana, Vecna Pot 113, 1000, Lubljana, Sloveniab New Jersey Institute of Technology, Physics Department, Newark, NJ, USA

a r t i c l e i n f o

Article history:Received 6 October 2018Received in revised form20 January 2019Accepted 3 February 2019Available online 19 February 2019

Keywords:AggregationFibrilsMonte Carlo

* Corresponding author.** Corresponding author.

E-mail addresses: tomaz.urbic@fkkt.uni-lj.si (T. Ur

https://doi.org/10.1016/j.fluid.2019.02.0020378-3812/© 2019 Elsevier B.V. All rights reserved.

a b s t r a c t

In this manuscript, we develop a two-dimensional coarse-grained model to study equilibrium propertiesof fibril-like structures made of amyloid proteins. The phase space of the model is sampled using MonteCarlo computer simulations. At low densities and high temperatures proteins are mostly present asmonomers while at low temperatures and high densities particles self-assemble into fibril-like struc-tures. The phase space of the model is explored and divided into different regions based on the structurespresent. We also estimate free-energies to dissociate proteins from fibrils based on the residual con-centration of dissolved proteins. Consistent with experiments, the concentration of proteins in solutiondoes not affects their equilibrium state. Also, we study the temperature dependence of the equilibriumstate to estimate thermodynamic quantities, e.g., heat capacity and entropy, of amyloid fibrils.

© 2019 Elsevier B.V. All rights reserved.

1. Introduction

Native and unfolded states of proteins coexist in an equilibriumthat depends on temperature and solvent condition [1e4]. Thisequilibrium is disturbed at high protein concentrations whereinthese molecules aggregates forming ordered and elongated struc-tures known as amyloid fibrils [5,6]. At the molecular level, seg-ments of proteins that are incorporated into these fibrils adoptextended conformations, i.e., b-strands, that are hydrogen bondedto neighboring proteins forming a b-sheet [7,8]. Stacking of b-sheets through side chain interactions accounts for the commonpattern of amyloid fibrils, also known as cross-b structure [9,10]esee Fig. 1a. Proteins that form fibrils at physiological conditions canbe either functional, e.g., silk fibrils that are used by spiders tocapture their prey, or related to plaque formation in amyloid dis-eases that include Alzheimer's and Parkinson's [11,12]. The forma-tion of amyloid fibril is a non-equilibrium process and its kineticshas been extensively studied. In particular, fibril formation has beenshown to proceed through a nucleation phase followed by growth[13e15]. Recently, fibril growth has been shown to evolve towardsan equilibrium state where mature fibrils co-exist with monomers

bic), cld@njit.edu (C.L. Dias).

in the solution [16]. Understanding this equilibrium is of great in-terest as it could enable the development of a thermodynamicframework to study these structures. In analogy with proteinfolding [17], thermodynamics may provide insights into the sta-bility of fibrils and their underlying molecular mechanisms [18].

Equilibrium thermodynamic properties of amyloid fibrils arechallenging to measure using conventional experimental methods.For example, heat capacities measured as a function of temperaturein calorimetric experiments were found to be irreversible for mostfibrils precluding the use of analytic frameworks to compute free-energies [19,20]. Accordingly, fibril formation is commonlythought as an irreversible process in which proteins are beingincorporated to it without dissociating [21]. Only recently have“amyloid assembly equilibriummeasurements” shown that maturefibrils can exist in equilibrium with dissolved proteins, i.e., mono-mers, at least for some amino acid sequences [16,22e25]ealthoughthis idea had been proposed previously [26]. Equilibrium betweenfibrils and monomers may also be inferred from self-assembly oflipids inmembrane bilayer [27]. This equilibrium has been exploredto measure free-energies required to dissociate a protein from afibril, i.e., DG. Studies of the temperature dependence of thisequilibrium can also provide insights of other thermodynamicquantities, i.e., change in enthalpy (DH), entropy (DS), and heatcapacity (DCp). These quantities played a key role in determiningthemolecular driving forces of protein folding [1,3] but they remainmostly unknown for amyloid fibrils. Only few experiments have

Fig. 1. (a) Structure of a fibril made from residues 16e21 of the amyloid-b protein (PDB ID: 3OW9). Backbone and side chains of proteins are depicted using cartoon-like and beadrepresentations, respectively. (b) Schematic representation of a protein in the model. Side chain and backbone beads are highlighted as well as Hbond arms. The representation of adimer depicts the vectors (blue arrow) used in interaction potential to account for Hbonds. The representation of an amyloid fibril shows two b-sheets made of three peptides eachstack on top of each other. The fibril axis is also shown.

T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110 105

estimated DCp for amyloid fibrils finding both positive and negativevalues [20,28e30]. Currently, it is unknown how the magnitudeand sign of DCp relate to the peptide sequence, fibril structure, and/or condition of the solvent. Notice that unfolding of proteins ac-counts for a large and positive (never negative) change in heatcapacity [3,4,31e37]. Timescales to equilibrate fibrils and proteinsin a solution are of the order of hours and days which is beyondreach of all-atom computer simulations [22,25,38]. Thus, models tostudy these equilibrium reaction require a coarse-grained approach[39e41].

Here, we develop a coarse-grained model of peptides that canhydrogen bond to each other forming extended aggregate struc-tures mimicking b-sheets. Two b-sheets in the model can stack ontop of each through side chain interactions mimicking the structureof amyloid fibrils. We explore phase space of themodel for differenttemperatures and densities. In agreement with experiments, weshow that under certain conditions of temperature and pressure,amyloid fibrils grow towards an equilibrium state where fibrils andmonomers coexist in equilibrium [42]. We determine the free-energies DF to add or dissociate a protein from a fibril at thisequilibrium and we show that conditions of density accounting fordifferent fibril growth rates can lead to the same fibril-monomerequilibrium state. Moreover, DH, DS, and DCp can be obtained bystudying the temperature dependence of this equilibrium. Wediscuss how the greater stability of fibrils with respect to temper-ature when compared to native protein structures implies a smallDCp (close to zero). Despite its small value, we show that DCp can bemeasured in our model.

2. Model details

Monomers of fibrils are represented as twoedimensional (2D)fused LennardeJones (LJ) disks at a fixed separation l ¼ 1:0. Firstdisk has two arms separated by angle 180+ which can associatewith arm of another molecule and form hydrogen bond. Both armsform 90+ angle with second particle of dimer (See Fig. 1b). Theinteraction potential between two particles is a sum of aLennardeJones terms between the centers of Lennard-Jones disksand an associative termwhich mimic formation of hydrogen bonds

U�Xi!; Xj!�

¼X2a;b¼1

UabLJ

����ria � rjb����þ Ua

�Xi!; Xj!�

: (1)

Xi!

and Xj!

denotes the vector representing the coordinates and theorientation of the ith and jth particle, ria is the position vector ofsite a (1 represent backbone which can form hydrogen bonds and 2side-chains which can only interact by means of Lennard-Jones

interaction) in molecule i. The form of the associative part of theinteraction potential is similar as used in Mercedez-Benz model ofwater [43e47].

Ua

�Xi!; Xj!�

¼X2k;l¼1

εaG���ri1 � rj1

��� ra�G�i!

uij!� 1

�G�j!

uij!

þ 1�

(2)

where G(x) is an unnormalized Gaussian function

GðxÞ ¼ exp�� x2

2s2

�: (3)

Further, εa ¼ �1 is an associative energy parameter andra ¼ 1:2l is a characteristic length at which hydrogen bonds areformed. uij

! is the unit vector along rj1 � ri1, i!

and j!

are the unitvectors representing the arm of the ith and jth particle. Thestrongest association occurs when the arm of one particle iscollinear with the arm of another particle and centers with arms areon distance ra from each other. Thewidth of Gaussian (s¼ 0.085l) issmall enough that it is not possible to havemore than one hydrogenbond per arm. The Lennard-Jones interactions take the standardform

UabLJ ðrÞ ¼ 4εab

"�sabr

�12

��sabr

�6#; (4)

where r is the distance between two beads. Interaction betweenside-chains (ε22 ¼ 0:2) is stronger than between backbones (ε11 ¼0:1) to favor formaiton of fibrils instead of stacking of chains in anyorder. LJ sizes of all particles is equal and it is equal to 0.9 (sab¼0.9l).The parameters for dissimilar segments are obtained from theLorentz-Berthelot combining rules [48]. Distances and energies inthis work are given in units of l and εHB, respectively, and theBoltzmann constant kb is defined as one.

3. Monte Carlo simulations

Here, we performMonte Carlo (MC) simulations of the 2D fibrilsmodel in the canonical (NVT) ensemble [47,49]. All simulationswere performedwith N ¼ 120 or N ¼ 250molecules. Increasing thenumber of particles had no significant effect on the calculatedquantities. The simulation box was chosen to be a square of size L2

where L is given in units of l. To account for different densities,simulations were performed using different box sizes. The density

T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110106

of monomers r* was calculated as ratio between number ofmonomers and the square of size L2. Periodic boundary conditionsand the minimum image convention are used to mimic an infinitesystem of particles. Starting configurations are selected randomly.At each time-step, we try to translate, rotate or switch (rotated for180+) a randomly chosen protein, each with same probability.Probabilities for translation, rotation and switch were the same. AMC cycle which corresponds to N moves of particles (translations,rotations and switches) is used as our unit of time. In one cycle eachparticle was on average translated, rotated and switched. Averagequantities are computed from 106 MC cycle simulations that areperformed after an equilibration period of 105-107 cycles

Fig. 2. Density dependence of ratio of molecules with 0 (black solid line), 1 (red dashed line)simulations and lines are guide for the eyes. Results are plotted for different temperatures

Fig. 3. Temperature dependence of ratio of molecules with 0 (black solid line), 1 (red dashcomputer simulations and lines are guide for the eyes. Results are plotted for different den

depending on temperature and density. Thermodynamic quantitiessuch as energy were calculated as statistical averages over thecourse of the simulations [49]. Cut off of the potential was half-length of the simulation box.

4. Results and discussion

4.1. Bond formation: density and temperature dependence

In Figs. 2 and 3we show the average fraction of peptides that arehydrogen bonded to just one neighboring peptide n1, two peptidesn2, and no-peptides n0. These quantities are depicted in Fig. 2 as a

and 2 hydrogen bonds (blue dotted line). Points are results from Monte Carlo computer(a) T� ¼ 0.1, (b) T� ¼ 0.15, (c) T� ¼ 0.2 and (d) T� ¼ 0.25.

ed line) and 2 hydrogen bonds (blue dotted line). Points are results from Monte Carlosities (a) r� ¼ 0.025, (b) r� ¼ 0.1, (c) r� ¼ 0.2 and (d) r� ¼ 0.25.

Fig. 5. Probability pðNcÞ of finding molecule in cluster of size Nc for temperature T� ¼0.15 and density (a) r� ¼ 0.1 (b) r� ¼ 0.17 and (c) r� ¼ 0.22.

T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110 107

function of density for four different temperatures. At all temper-atures, n2 and n0 increase and decrease, respectively, withincreasing density. These behavior can be rationalized by theincreased proximity of peptides at high density that makes themmore reactive and accounts for an increase in n2 and a decrease inn0. Conservation of the number of peptides dictates the behavior ofn1. This quantity decreases as a function of density at low tem-perature (Fig. 2a) whereas it increases at high temperature (Fig. 2cand d). At intermediate temperature (Fig. 2b), n1 reaches amaximum at intermediate densities. In Fig. 3, we show the tem-perature dependence of n1, n2, and n0 for four densities. At alldensities, n2 and n0 decrease and increase, respectively, withincreasing temperature. These behaviors emerge because thermalfluctuation accounts for bond rupture. Thus, increasing tempera-ture favors the population of non-interaction peptides, i.e., n0,while it accounts for a reduction in n2. Notice that the range oftemperatures for which n0, n1, or n2 dominates depends on thedensity.

In Fig. 4 we depict how density and temperature affect thebonding of peptides. The dotted line shows densities and temper-atures for which 50% of the peptides are not forming hydrogenbonds. The full line corresponds to densities and temperatures forwhich 50% of the peptides are forming two hydrogen bonds. Thus,above the dashed line, most peptides are not forming hydrogenbonds while below the full line most peptides are forming twohydrogen bonds with neighboring peptides. In between the twolines, peptides are mainly bonded to neighboring molecules by justone hydrogen bonds. Since most peptides in a fibril form twohydrogen bonds with neighboring peptides, Fig. 4 provides insightsinto the conditions at which fibrils are more likely to form. How-ever, conditions at which most peptides form two hydrogen bondsis not enough to guarantee that fibrils are formed, see Section B.Similarly it is possible to form fibrils even if the dominant bondingspecie of peptides form only one hydrogen bond, see Section C.

4.2. Cluster analysis

To better define regions in Fig. 4 where fibrils are the dominantphase of the system, we count the number Nc of proteins in anaggregate or cluster. We use an energy criteria wherein proteins areconsidered bonded when their potential energy is less than �0.05.Small variations of this energy cutoff did not account for significantdifferences in the bonding state of the system. In Fig. 5, we showthe probability of a protein to belong to a cluster of size Nc for threedensities at T� ¼ 0:15. At low density, the system is composed ofmany small clusters (see Fig. 5a) whereas at high density it is madeof one large cluster with only a few proteins belonging to smallclusters (see Fig. 5c). At intermediate density (Fig. 5b), clusters of allsizes are equally populated.

To estimate the propensity of the system to form large clusters

Fig. 4. Phase space with marked regions with regions dominated by particles with nohydrogen bond (0 HB), with one (1 HB) and with 2 hydrogen bonds (2 HB).

that resemble fibrils, we compute the fraction pa of proteins thatbelongs to the largest cluster. In Fig. 6a, this quantity is shown as afunction of the density of the system at four temperatures. Proteinsin the system become more connected to each other withincreasing density and decreasing temperature. The same result isalso observed in Fig. 6b where pa is shown as a function of tem-perature for four different densities. Following how states aredefined in percolation theory, we define region where fibrils areformed when the system is connected, meaning that at least 50% ofthe proteins are bonded to one big cluster. While there is somearbitrariness in this definition of fibrils, in the next section weprovide visual evidence that this definition enables the differenti-ation of different states of the system. The full line in Fig. 6c showsdensities and temperature for which 50% of proteins are part of onebig cluster. For conditions on the right hand side of this line, pro-teins are highly connected to each other forming fibrils. In contrastfor conditions on the left side of the line proteins are either in themonomeric state or in the form of b� sheets, i.e. structures inwhich several proteins are hydrogen bonded to each other.

4.3. Dominant structures in phase space

Fig. 7 combines our bond formation analysis (i.e., lines fromFig. 4) and cluster analysis (i.e., line in Fig. 6c). In this figure, “M”

corresponds to conditions of temperature and density where mostproteins in the system are not bonded to each other, i.e., they are inthe monomeric state (See Fig. 8a). Small clusters, i.e., oligomers, arefavored at “O1” (Fig. 8b) and “O2” (Fig. 8c). At “O1”, proteins form

Fig. 7. Phase space with marked regions with regions dominated by monomers (M),oligomers (O1 and O2) and fibrils (F1 and F2).

Fig. 8. Snapshots of the system for different regions (a) monomer region at T� ¼ 0.25,r� ¼ 0.1; (b) oligomer region O1 at T� ¼ 0.15, r� ¼ 0.1; (c) oligomer region O2 at T� ¼0.1, r� ¼ 0.05; (d) fibrils region F1 at T� ¼ 0.15, r� ¼ 0.22; (e) fibrils region F2 at T� ¼0.05, r� ¼ 0.1 and (f) fibrils region F2 at T� ¼ 0.1, r� ¼ 0.22.

Fig. 6. (a) Density dependence of ratio of molecules belonging to biggest cluster insystem. Points are results from Monte Carlo computer simulations and lines are guidefor the eyes. Results are plotted for different temperatures T� ¼ 0.1 (black solid line),T� ¼ 0.15 (red dashed), T� ¼ 0.2 (blue dotted) and T� ¼ 0.25 (green long dashed-dotted). (b) Temperature dependence of ratio of molecules belonging to biggest clus-ter in system. Points are results from Monte Carlo computer simulations and lines areguide for the eyes. Results are plotted for different densities r� ¼ 0.05 (black solidline), r� ¼ 0.1 (red dashed), r� ¼ 0.15 (blue dotted), r� ¼ 0.2 (green long dashed-dotted) and r� ¼ 0.25 (orange dashed-dotted). (c) Phase space with marked regionswhere system is connected (C) and where not (U).

T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110108

mainly small clusters, e.g., dimers, while at “O2” proteins form b-sheets. Thermal fluctuation which accounts for bond rupture anddestabilizes b-sheets can rationalize the structural difference be-tween “O1” and “O2”. State “F1” in Fig. 8d represents regions inspace where fibrils are distorted and coexist with many smallclusters, e.g., dimers. State “F2” represents ideal fibrils inwhich two

T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110 109

beta-sheets are stacked on top of each other. Characteristic con-figurations of these states are shown in Fig. 8e and f.

Fig. 10. Dependence of the free-energy DF to dissociate a protein from a fibril asfunction of temperature. Points are results from Monte Carlo computer simulationsand line is best fit of the data point to Eq. (7).

4.4. Thermodynamics of fibrils

Experimentally, conditions at which fibrils coexist with mono-mers and small clusters have been explored to study the stability offibrils [50]. In these studies, the addition/dissociation of monomersto/from fibrils was shown to be reversible which enabled mea-surements of the free-energy DF of these processes from theequilibrium dissociation constant Kd:

Kd ¼ ½F�½M�½F� ¼ ½M�; (5)

where ½F� is the equilibrium concentrations of fibrils and ½M� ¼ r�o isthe density of monomers in equilibrium. In our simulations, coex-istence of monomers and fibrils occur in region F1 of the phasespaceesee Fig. 8d. This region of the phase space can, therefore, beexplored to compute thermodynamic properties of the system.

In Fig. 9 we show the density of monomers (i.e., non-interactingproteins) as a function of the density of proteins in the system atfour temperatures. The density of monomers is computed bydividing the number of monomers No by the area of the simulationbox L2, i.e., ro ¼ No=L2. ro increases and reaches a plateau at alltemperatures. The existent of this plateau emerges from theassumption that the number of fibrils in the system is constant andmonomeric proteins can be added/dissociate to/from them:

dNo

dt¼ �kof No þ kfoN; (6)

where kof and kfo are rate constants for adding and dissociatingmonomers to/from fibrils, respectively. In equilibrium (i.e., dNo

dt ¼ 0),we obtain that r�o ¼ No=N ¼ kfo=kof which shows that r�o is inde-pendent of the density of the system. Notice that r�o can be used tocompute DF ¼ � kbTlnKd ¼ � kbTlnr�o=rs, where rs is the stan-dard concentration rs ¼ 1. In Fig. 10, we show the dependence ofDF on temperature which, according to thermodynamics, can bedescribed by Refs. [4,17,51,52]:

DF ¼ DUo � TDSo þ DCo

ðT � ToÞ � T log

�TTo

��(7)

where DUo, DSo, and DCo are the change in energy, entropy, andheat capacity related to the dissociation of a protein from a fibrilcomputed at temperature To. The temperature dependence of DF

Fig. 9. Dependence of density of free monomers in system as function of totalmonomer density. Points are results from Monte Carlo computer simulations and linesare guide for the eyes. Results are plotted for different temperatures T� ¼ 0.12 (blacksolid line), T� ¼ 0.15 (red dashed), T� ¼ 0.18 (blue dotted) and T� ¼ 0.2 (green longdashed-dotted).

from our simulations is shown in Fig. 10 and the best fit of thisdependence to Eq. (7) using To ¼ 0:15 gives DUo ¼ 0:740±0:006,DSo ¼ 1:22±0:03, and DCo ¼ � 4:29±0:14. One should note thatthis is a simple model which does not have enough detail to ac-count for the magnitude and sign of these quantities in real pro-teins. As next it is interesting to see that the free energy isdominated by DU0 close to T0. This is expected since fibrils arehighly stable and internal energy (or enthalpy) favors the fibrillarstate while entropy opposes it.

5. Conclusion

In this work we developed a two-dimensional coarse-grainedmodel which was used to study equilibrium properties of amyloidfibrils. We checked properties by Monte Carlo computer simula-tions for different state points in the phase space. We analyzedstructures of the model and determined characteristic structuresfor different parts of phase space. At low densities and high tem-peratures molecules are mostly present as monomers while at lowtemperatures and higher densities fibrils start to form and thanfibrils associate to form higher structures like stack of fibrils. Weprovided computational proof of principle that fibrils can exist inequilibrium with proteins dissolved in solution and this equilib-rium can be used to compute free-energies DF to dissociate aprotein. These quantities remain mostly unknown but they havethe potential to provide important insights into molecular mech-anisms underlying fibril stability.

Acknowledgements

T. U. are grateful for the support of the NIH (GM063592) andSlovenian Research Agency (P1 0103-0201, N1-0042) and the Na-tional Research, Development and Innovation Office of Hungary(SNN 116198).

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