- Email: [email protected]
Thermodynamic properties of amyloid fibrils: A simple model of peptide aggregation
Fluid Phase Equilibria 489 (2019) 104e110
Contents lists available at ScienceDirect
Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Thermodynamic properties of amyloid fibrils: A simple model of peptide aggregation Tomaz Urbic a, *, Cristiano L. Dias b, ** a b
Faculty of Chemistry and Chemical Technology, University of Ljubljana, Vecna Pot 113, 1000, Lubljana, Slovenia New Jersey Institute of Technology, Physics Department, Newark, NJ, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 October 2018 Received in revised form 20 January 2019 Accepted 3 February 2019 Available online 19 February 2019
In this manuscript, we develop a two-dimensional coarse-grained model to study equilibrium properties of fibril-like structures made of amyloid proteins. The phase space of the model is sampled using Monte Carlo computer simulations. At low densities and high temperatures proteins are mostly present as monomers while at low temperatures and high densities particles self-assemble into fibril-like structures. The phase space of the model is explored and divided into different regions based on the structures present. We also estimate free-energies to dissociate proteins from fibrils based on the residual concentration of dissolved proteins. Consistent with experiments, the concentration of proteins in solution does not affects their equilibrium state. Also, we study the temperature dependence of the equilibrium state to estimate thermodynamic quantities, e.g., heat capacity and entropy, of amyloid fibrils. © 2019 Elsevier B.V. All rights reserved.
Keywords: Aggregation Fibrils Monte Carlo
1. Introduction Native and unfolded states of proteins coexist in an equilibrium that depends on temperature and solvent condition [1e4]. This equilibrium is disturbed at high protein concentrations wherein these molecules aggregates forming ordered and elongated structures known as amyloid fibrils [5,6]. At the molecular level, segments of proteins that are incorporated into these fibrils adopt extended conformations, i.e., b-strands, that are hydrogen bonded to neighboring proteins forming a b-sheet [7,8]. Stacking of bsheets through side chain interactions accounts for the common pattern of amyloid fibrils, also known as cross-b structure [9,10]e see Fig. 1a. Proteins that form fibrils at physiological conditions can be either functional, e.g., silk fibrils that are used by spiders to capture their prey, or related to plaque formation in amyloid diseases that include Alzheimer's and Parkinson's [11,12]. The formation of amyloid fibril is a non-equilibrium process and its kinetics has been extensively studied. In particular, fibril formation has been shown to proceed through a nucleation phase followed by growth [13e15]. Recently, fibril growth has been shown to evolve towards an equilibrium state where mature fibrils co-exist with monomers
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (T. Urbic), [email protected] (C.L. Dias). https://doi.org/10.1016/j.fluid.2019.02.002 0378-3812/© 2019 Elsevier B.V. All rights reserved.
in the solution [16]. Understanding this equilibrium is of great interest as it could enable the development of a thermodynamic framework to study these structures. In analogy with protein folding [17], thermodynamics may provide insights into the stability of fibrils and their underlying molecular mechanisms [18]. Equilibrium thermodynamic properties of amyloid fibrils are challenging to measure using conventional experimental methods. For example, heat capacities measured as a function of temperature in calorimetric experiments were found to be irreversible for most fibrils precluding the use of analytic frameworks to compute freeenergies [19,20]. Accordingly, fibril formation is commonly thought as an irreversible process in which proteins are being incorporated to it without dissociating [21]. Only recently have “amyloid assembly equilibrium measurements” shown that mature fibrils can exist in equilibrium with dissolved proteins, i.e., monomers, at least for some amino acid sequences [16,22e25]ealthough this idea had been proposed previously [26]. Equilibrium between fibrils and monomers may also be inferred from self-assembly of lipids in membrane bilayer [27]. This equilibrium has been explored to measure free-energies required to dissociate a protein from a fibril, i.e., DG. Studies of the temperature dependence of this equilibrium can also provide insights of other thermodynamic quantities, i.e., change in enthalpy (DH), entropy (DS), and heat capacity (DCp ). These quantities played a key role in determining the molecular driving forces of protein folding [1,3] but they remain mostly unknown for amyloid fibrils. Only few experiments have
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
105
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 bead representations, 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 a dimer 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 each stack on top of each other. The fibril axis is also shown.
estimated DCp for amyloid fibrils finding both positive and negative values [20,28e30]. Currently, it is unknown how the magnitude and sign of DCp relate to the peptide sequence, fibril structure, and/ or condition of the solvent. Notice that unfolding of proteins accounts for a large and positive (never negative) change in heat capacity [3,4,31e37]. Timescales to equilibrate fibrils and proteins in a solution are of the order of hours and days which is beyond reach of all-atom computer simulations [22,25,38]. Thus, models to study these equilibrium reaction require a coarse-grained approach [39e41]. Here, we develop a coarse-grained model of peptides that can hydrogen bond to each other forming extended aggregate structures mimicking b-sheets. Two b-sheets in the model can stack on top of each through side chain interactions mimicking the structure of amyloid fibrils. We explore phase space of the model for different temperatures and densities. In agreement with experiments, we show that under certain conditions of temperature and pressure, amyloid fibrils grow towards an equilibrium state where fibrils and monomers coexist in equilibrium [42]. We determine the freeenergies DF to add or dissociate a protein from a fibril at this equilibrium and we show that conditions of density accounting for different fibril growth rates can lead to the same fibril-monomer equilibrium state. Moreover, DH, DS, and DCp can be obtained by studying the temperature dependence of this equilibrium. We discuss how the greater stability of fibrils with respect to temperature when compared to native protein structures implies a small DCp (close to zero). Despite its small value, we show that DCp can be measured 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. First disk has two arms separated by angle 180+ which can associate with arm of another molecule and form hydrogen bond. Both arms form 90+ angle with second particle of dimer (See Fig. 1b). The interaction potential between two particles is a sum of a LennardeJones terms between the centers of Lennard-Jones disks and an associative term which mimic formation of hydrogen bonds
interaction) in molecule i. The form of the associative part of the interaction potential is similar as used in Mercedez-Benz model of water [43e47].
2 ! X ! ! ! ! ! Ua Xi ; Xj ¼ εa G ri1 rj1 ra G i uij 1 G j uij k;l¼1
þ1
(2)
where G(x) is an unnormalized Gaussian function
x2 GðxÞ ¼ exp 2 : 2s
(3)
Further, εa ¼ 1 is an associative energy parameter and ra ¼ 1:2l is a characteristic length at which hydrogen bonds are ! ! ! formed. uij is the unit vector along rj1 ri1 , i and j are the unit vectors representing the arm of the ith and jth particle. The strongest association occurs when the arm of one particle is collinear with the arm of another particle and centers with arms are on distance ra from each other. The width of Gaussian (s ¼ 0.085l) is small enough that it is not possible to have more than one hydrogen bond per arm. The Lennard-Jones interactions take the standard form
U ab LJ ðrÞ ¼ 4εab
"
sab r
12
sab r
6 # ;
(4)
where r is the distance between two beads. Interaction between side-chains (ε22 ¼ 0:2) is stronger than between backbones (ε11 ¼ 0:1) to favor formaiton of fibrils instead of stacking of chains in any order. LJ sizes of all particles is equal and it is equal to 0.9 (sab ¼0.9l). The parameters for dissimilar segments are obtained from the Lorentz-Berthelot combining rules [48]. Distances and energies in this work are given in units of l and εHB , respectively, and the Boltzmann constant kb is defined as one. 3. Monte Carlo simulations
2 X ! ! ! ! U Xi ; Xj ¼ U ab LJ ria rjb þ Ua Xi ; Xj :
(1)
a;b¼1
! ! Xi and Xj denotes the vector representing the coordinates and the orientation of the ith and jth particle, ria is the position vector of site a (1 represent backbone which can form hydrogen bonds and 2 side-chains which can only interact by means of Lennard-Jones
Here, we perform Monte Carlo (MC) simulations of the 2D fibrils model in the canonical (NVT) ensemble [47,49]. All simulations were performed with N ¼ 120 or N ¼ 250 molecules. Increasing the number of particles had no significant effect on the calculated quantities. 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
106
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
of monomers r* was calculated as ratio between number of monomers and the square of size L2 . Periodic boundary conditions and the minimum image convention are used to mimic an infinite system of particles. Starting configurations are selected randomly. At each time-step, we try to translate, rotate or switch (rotated for 180+ ) a randomly chosen protein, each with same probability. Probabilities for translation, rotation and switch were the same. A MC cycle which corresponds to N moves of particles (translations, rotations and switches) is used as our unit of time. In one cycle each particle was on average translated, rotated and switched. Average quantities are computed from 106 MC cycle simulations that are performed after an equilibration period of 105 -107 cycles
depending on temperature and density. Thermodynamic quantities such as energy were calculated as statistical averages over the course of the simulations [49]. Cut off of the potential was halflength of the simulation box. 4. Results and discussion 4.1. Bond formation: density and temperature dependence In Figs. 2 and 3 we show the average fraction of peptides that are hydrogen bonded to just one neighboring peptide n1 , two peptides n2 , and no-peptides n0 . These quantities are depicted in Fig. 2 as a
Fig. 2. Density dependence of ratio of molecules with 0 (black solid line), 1 (red dashed line) and 2 hydrogen bonds (blue dotted line). Points are results from Monte Carlo computer simulations and lines are guide for the eyes. Results are plotted for different temperatures (a) T ¼ 0.1, (b) T ¼ 0.15, (c) T ¼ 0.2 and (d) T ¼ 0.25.
Fig. 3. Temperature dependence of ratio of molecules with 0 (black solid line), 1 (red dashed line) and 2 hydrogen bonds (blue dotted line). Points are results from Monte Carlo computer simulations and lines are guide for the eyes. Results are plotted for different densities (a) r ¼ 0.025, (b) r ¼ 0.1, (c) r ¼ 0.2 and (d) r ¼ 0.25.
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
function of density for four different temperatures. At all temperatures, n2 and n0 increase and decrease, respectively, with increasing density. These behavior can be rationalized by the increased proximity of peptides at high density that makes them more reactive and accounts for an increase in n2 and a decrease in n0 . Conservation of the number of peptides dictates the behavior of n1 . This quantity decreases as a function of density at low temperature (Fig. 2a) whereas it increases at high temperature (Fig. 2c and d). At intermediate temperature (Fig. 2b), n1 reaches a maximum at intermediate densities. In Fig. 3, we show the temperature dependence of n1 , n2 , and n0 for four densities. At all densities, n2 and n0 decrease and increase, respectively, with increasing temperature. These behaviors emerge because thermal fluctuation accounts for bond rupture. Thus, increasing temperature favors the population of non-interaction peptides, i.e., n0 , while it accounts for a reduction in n2 . Notice that the range of temperatures for which n0 , n1 , or n2 dominates depends on the density. In Fig. 4 we depict how density and temperature affect the bonding of peptides. The dotted line shows densities and temperatures for which 50% of the peptides are not forming hydrogen bonds. The full line corresponds to densities and temperatures for which 50% of the peptides are forming two hydrogen bonds. Thus, above the dashed line, most peptides are not forming hydrogen bonds while below the full line most peptides are forming two hydrogen bonds with neighboring peptides. In between the two lines, peptides are mainly bonded to neighboring molecules by just one hydrogen bonds. Since most peptides in a fibril form two hydrogen bonds with neighboring peptides, Fig. 4 provides insights into the conditions at which fibrils are more likely to form. However, conditions at which most peptides form two hydrogen bonds is not enough to guarantee that fibrils are formed, see Section B. Similarly it is possible to form fibrils even if the dominant bonding specie of peptides form only one hydrogen bond, see Section C.
107
Fig. 5. Probability pðNc Þ of finding molecule in cluster of size Nc for temperature T ¼ 0.15 and density (a) r ¼ 0.1 (b) r ¼ 0.17 and (c) r ¼ 0.22.
4.2. Cluster analysis To better define regions in Fig. 4 where fibrils are the dominant phase of the system, we count the number Nc of proteins in an aggregate or cluster. We use an energy criteria wherein proteins are considered bonded when their potential energy is less than 0.05. Small variations of this energy cutoff did not account for significant differences in the bonding state of the system. In Fig. 5, we show the probability of a protein to belong to a cluster of size Nc for three densities at T ¼ 0:15. At low density, the system is composed of many small clusters (see Fig. 5a) whereas at high density it is made of one large cluster with only a few proteins belonging to small clusters (see Fig. 5c). At intermediate density (Fig. 5b), clusters of all sizes are equally populated. To estimate the propensity of the system to form large clusters
that resemble fibrils, we compute the fraction pa of proteins that belongs to the largest cluster. In Fig. 6a, this quantity is shown as a function of the density of the system at four temperatures. Proteins in the system become more connected to each other with increasing density and decreasing temperature. The same result is also observed in Fig. 6b where pa is shown as a function of temperature for four different densities. Following how states are defined in percolation theory, we define region where fibrils are formed when the system is connected, meaning that at least 50% of the proteins are bonded to one big cluster. While there is some arbitrariness in this definition of fibrils, in the next section we provide visual evidence that this definition enables the differentiation of different states of the system. The full line in Fig. 6c shows densities and temperature for which 50% of proteins are part of one big cluster. For conditions on the right hand side of this line, proteins are highly connected to each other forming fibrils. In contrast for conditions on the left side of the line proteins are either in the monomeric state or in the form of b sheets, i.e. structures in which several proteins are hydrogen bonded to each other.
4.3. Dominant structures in phase space
Fig. 4. Phase space with marked regions with regions dominated by particles with no hydrogen bond (0 HB), with one (1 HB) and with 2 hydrogen bonds (2 HB).
Fig. 7 combines our bond formation analysis (i.e., lines from Fig. 4) and cluster analysis (i.e., line in Fig. 6c). In this figure, “M” corresponds to conditions of temperature and density where most proteins in the system are not bonded to each other, i.e., they are in the monomeric state (See Fig. 8a). Small clusters, i.e., oligomers, are favored at “O1” (Fig. 8b) and “O2” (Fig. 8c). At “O1”, proteins form
108
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
Fig. 6. (a) Density dependence of ratio of molecules belonging to biggest cluster in system. Points are results from Monte Carlo computer simulations and lines are guide for 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 dasheddotted). (b) Temperature dependence of ratio of molecules belonging to biggest cluster in system. Points are results from Monte Carlo computer simulations and lines are guide for the eyes. Results are plotted for different densities r ¼ 0.05 (black solid line), r ¼ 0.1 (red dashed), r ¼ 0.15 (blue dotted), r ¼ 0.2 (green long dasheddotted) and r ¼ 0.25 (orange dashed-dotted). (c) Phase space with marked regions where system is connected (C) and where not (U).
Fig. 7. Phase space with marked regions with regions dominated by monomers (M), oligomers (O1 and O2) and fibrils (F1 and F2).
mainly small clusters, e.g., dimers, while at “O2” proteins form bsheets. Thermal fluctuation which accounts for bond rupture and destabilizes b-sheets can rationalize the structural difference between “O1” and “O2”. State “F1” in Fig. 8d represents regions in space where fibrils are distorted and coexist with many small clusters, e.g., dimers. State “F2” represents ideal fibrils in which two
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 ¼ 0.1, r ¼ 0.05; (d) fibrils region F1 at T ¼ 0.15, r ¼ 0.22; (e) fibrils region F2 at T ¼ 0.05, r ¼ 0.1 and (f) fibrils region F2 at T ¼ 0.1, r ¼ 0.22.
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
109
beta-sheets are stacked on top of each other. Characteristic configurations of these states are shown in Fig. 8e and f. 4.4. Thermodynamics of fibrils Experimentally, conditions at which fibrils coexist with monomers and small clusters have been explored to study the stability of fibrils [50]. In these studies, the addition/dissociation of monomers to/from fibrils was shown to be reversible which enabled measurements of the free-energy DF of these processes from the equilibrium dissociation constant Kd :
Kd ¼
½F½M ¼ ½M; ½F
(5)
where ½F is the equilibrium concentrations of fibrils and ½M ¼ ro is the density of monomers in equilibrium. In our simulations, coexistence of monomers and fibrils occur in region F1 of the phase spaceesee Fig. 8d. This region of the phase space can, therefore, be explored to compute thermodynamic properties of the system. In Fig. 9 we show the density of monomers (i.e., non-interacting proteins) as a function of the density of proteins in the system at four temperatures. The density of monomers is computed by dividing the number of monomers No by the area of the simulation box L2 , i.e., ro ¼ No =L2 . ro increases and reaches a plateau at all temperatures. The existent of this plateau emerges from the assumption that the number of fibrils in the system is constant and monomeric proteins can be added/dissociate to/from them:
dNo ¼ kof No þ kfo N; dt
(6)
where kof and kfo are rate constants for adding and dissociating o monomers to/from fibrils, respectively. In equilibrium (i.e., dN ¼ 0), dt we obtain that ro ¼ No =N ¼ kfo =kof which shows that ro is independent of the density of the system. Notice that ro can be used to compute DF ¼ kb TlnKd ¼ kb Tlnro =rs , where rs is the standard concentration rs ¼ 1. In Fig. 10, we show the dependence of DF on temperature which, according to thermodynamics, can be described by Refs. [4,17,51,52]:
DF ¼ DUo T DSo þ DCo ðT To Þ T log
T To
(7)
where DUo , DSo , and DCo are the change in energy, entropy, and heat capacity related to the dissociation of a protein from a fibril computed at temperature To . The temperature dependence of DF
Fig. 10. Dependence of the free-energy DF to dissociate a protein from a fibril as function of temperature. Points are results from Monte Carlo computer simulations and line is best fit of the data point to Eq. (7).
from our simulations is shown in Fig. 10 and the best fit of this dependence 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 that this is a simple model which does not have enough detail to account for the magnitude and sign of these quantities in real proteins. As next it is interesting to see that the free energy is dominated by DU0 close to T0 . This is expected since fibrils are highly stable and internal energy (or enthalpy) favors the fibrillar state while entropy opposes it. 5. Conclusion In this work we developed a two-dimensional coarse-grained model which was used to study equilibrium properties of amyloid fibrils. We checked properties by Monte Carlo computer simulations for different state points in the phase space. We analyzed structures of the model and determined characteristic structures for different parts of phase space. At low densities and high temperatures molecules are mostly present as monomers while at low temperatures and higher densities fibrils start to form and than fibrils associate to form higher structures like stack of fibrils. We provided computational proof of principle that fibrils can exist in equilibrium with proteins dissolved in solution and this equilibrium can be used to compute free-energies DF to dissociate a protein. These quantities remain mostly unknown but they have the potential to provide important insights into molecular mechanisms underlying fibril stability. Acknowledgements T. U. are grateful for the support of the NIH (GM063592) and Slovenian Research Agency (P1 0103-0201, N1-0042) and the National Research, Development and Innovation Office of Hungary (SNN 116198). References [1] [2] [3] [4]
Fig. 9. Dependence of density of free monomers in system as function of total monomer density. Points are results from Monte Carlo computer simulations and lines are guide for the eyes. Results are plotted for different temperatures T ¼ 0.12 (black solid line), T ¼ 0.15 (red dashed), T ¼ 0.18 (blue dotted) and T ¼ 0.2 (green long dashed-dotted).
[5] [6] [7] [8] [9] [10] [11] [12] [13]
W. Kauzmann, Adv. Protein Chem. 14 (1959) 1. P.L. Privalov, Crit. Rev. Biochem. Mol. Biol. 25 (1990) 281. K. Dill, Biochemistry 29 (1990) 7133. C.L. Dias, T. Ala-Nissila, J. Wong-ekkabut, I. Vattulainen, M. Grant, M. Karttunen, Cryobiology 60 (2010) 91. €ndrich, M.A. Fletcher, C.M. Dobson, Nature 410 (2001) 165. M. Fa €ndrich, M. Dobson, Christopher, EMBO J. 21 (2002) 5682. M. Fa Z. Su, C.L. Dias, J. Phys. Chem. B 118 (2014) 10830. C. Narayanan, C.L. Dias, J. Chem. Phys. 139 (2013) 115103. E.D. Eanes, G.G. Glenner, J. Histochem. Cytochem. 16 (1968) 673. A. Geddes, K. Parker, E. Atkins, E. Beighton, J. Mol. Biol. 32 (1968) 343. F. Chiti, C.M. Dobson, Annu. Rev. Biochem. 75 (2006) 333. C.M. Dobson, Nature 426 (2003) 884. s, A. Laio, Phys. Rev. Lett. 110 (2013) F. Baftizadeh, F. Pietrucci, X. Biarne 168103.
110
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
[14] S. Auer, P. Ricchiuto, D. Kashchiev, J. Mol. Biol. 422 (2012) 723. [15] T.C.T. Michaels, A.J. Dear, J.B. Kirkegaard, K.L. Saar, D.A. Weitz, T.P.J. Knowles, Phys. Rev. Lett. 116 (2016) 258103. [16] R. Wetzel, Acc. Chem. Res. 39 (2006) 671. [17] A.D. Robertson, K.P. Murphy, Chem. Rev. 97 (1997) 1251. [18] B.P. Roland, R. Kodali, R. Mishra, R. Wetzel, Peptide Science 100 (2013) 780. [19] C.M. Doyle, J.A. Rumfeldt, H.R. Broom, A. Broom, P.B. Stathopulos, K.A. Vassall, J.J. Almey, E.M. Meiering, Arch. Biochem. Biophys. 531 (2013) 44. [20] B. Morel, L. Varela, F. Conejero-Lara, J. Phys. Chem. B 114 (2010) 4010. [21] S.I. Cohen, S. Linse, L.M. Luheshi, E. Hellstrand, D.A. White, L. Rajah, D.E. Otzen, M. Vendruscolo, C.M. Dobson, T.P. Knowles, Proc. Natl. Acad. Sci. Unit. States Am. 110 (2013) 9758. [22] B. O'Nuallain, S. Shivaprasad, I. Kheterpal, R. Wetzel, Biochemistry 44 (2005) 12709. [23] T.M. Doran, A.J. Kamens, N.K. Byrnes, B.L. Nilsson, Proteins: Structure, Function, and Bioinformatics 80 (2012) 1053. [24] A.D. Williams, S. Shivaprasad, R. Wetzel, J. Mol. Biol. 357 (2006) 1283. ISSN 0022-2836. [25] A.M. Bhattacharyya, A.K. Thakur, R. Wetzel, Proc. Natl. Acad. Sci. Unit. States Am. 102 (2005) 15400. [26] J.D. Harper, P.T. Lansbury Jr., Annu. Rev. Biochem. 66 (1997) 385. [27] J.N. Israelachvili, Intermolecular and Surface Forces, Elsevier Science, 2010. [28] T. Ikenoue, Y.-H. Lee, J. Kardos, M. Saiki, H. Yagi, Y. Kawata, Y. Goto, Angew. Chem. Int. Ed. 53 (2014) 7799. [29] J. Kardos, K. Yamamoto, K. Hasegawa, H. Naiki, Y. Goto, J. Biol. Chem. 279 (2004) 55308. [30] M.D. Jeppesen, K. Hein, P. Nissen, P. Westh, D.E. Otzen, Biophys. Chem. 149 (2010) 40. [31] R.L. Baldwin, J. Mol. Biol. 371 (2007) 283. [32] J.T. Edsall, J. Am. Chem. Soc. 54 (1935) 1506. [33] P. Bruscolini, L. Casetti, Phys. Rev. E 61 (2000) R2208. http://prola.aps.org/
abstract/PRE/v61/i3/pR2208_1. [34] P.D.L. Rios, G. Caldarelli, Phys. Rev. E 63 (2001) 031802. http://prola.aps.org/ abstract/PRE/v63/i3/e031802. [35] C.L. Dias, T. Ala-Nissila, M. Karttunen, I. Vattulainen, M. Grant, Phys. Rev. Lett. 100 (2008) 118101. [36] C.L. Dias, Phys. Rev. Lett. 109 (2012) 048104. [37] V. Bianco, G. Franzese, Phys. Rev. Lett. 115 (2015) 108101. [38] J. Nasica-Labouze, P.H. Nguyen, F. Sterpone, O. Berthoumieu, N.-V. Buchete, , A. De Simone, A.J. Doig, P. Faller, A. Garcia, Chem. Rev. 115 (2015) S. Cote 3518. [39] A. Irb€ ack, N. Linnemann, B. Linse, S. Wallin, et al., Phys. Rev. Lett. 110 (2013) 058101. [40] J. Schmit, K. Ghosh, K.A. Dill, Biophys. J. 100 (2011) 450. [41] P. Ricchiuto, A.V. Brukhno, S. Auer, J. Phys. Chem. B 116 (2012) 5384. [42] JS Espinoza Ortiz, L. Cristiano, Dias, Phys. Stat. Mech. Appl. 11 (2018) 154. [43] T. Urbi c, V. Vlachy, Y.V. Kalyuzhnyi, N.T. Southall, K.A. Dill, J. Chem. Phys. 112 (2000) 2843. [44] T. Urbic, V. Vlachy, Y.V. Kalyuzhnyi, N.T. Southall, K.A. Dill, J. Chem. Phys. 116 (2002) 723. [45] T. Urbi c, V. Vlachy, Y.V. Kalyuzhnyi, K.A. Dill, J. Chem. Phys. 127 (2007) 174511. [46] T. Urbic, M. Holovko, J. Chem. Phys. 135 (2011) 134706. [47] T. Urbic, T. Mohoric, J. Chem. Phys. 146 (2017) 094505. [48] D. Frenkel, B. Smit, Computational sciences series 1 (1986) 1. [49] D. Frenkel, B. Smit, Intermolecular and Surface Forces, vol. 1, Elsevier Science, 2010, p. 1. [50] J.N. Israelachvili, Computational sciences series 1 (2002) 1. [51] L. Smeller, Biochim. Biophys. Acta Protein Struct. Mol. Enzymol. 1595 (2002) 11. [52] S.R. Jampani, F. Mahmoudinobar, Z. Su, C. Dias, Proteins: Structure, Function, and Bioinformatics 83 (2015) 19.
Contents lists available at ScienceDirect
Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Thermodynamic properties of amyloid fibrils: A simple model of peptide aggregation Tomaz Urbic a, *, Cristiano L. Dias b, ** a b
Faculty of Chemistry and Chemical Technology, University of Ljubljana, Vecna Pot 113, 1000, Lubljana, Slovenia New Jersey Institute of Technology, Physics Department, Newark, NJ, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 October 2018 Received in revised form 20 January 2019 Accepted 3 February 2019 Available online 19 February 2019
In this manuscript, we develop a two-dimensional coarse-grained model to study equilibrium properties of fibril-like structures made of amyloid proteins. The phase space of the model is sampled using Monte Carlo computer simulations. At low densities and high temperatures proteins are mostly present as monomers while at low temperatures and high densities particles self-assemble into fibril-like structures. The phase space of the model is explored and divided into different regions based on the structures present. We also estimate free-energies to dissociate proteins from fibrils based on the residual concentration of dissolved proteins. Consistent with experiments, the concentration of proteins in solution does not affects their equilibrium state. Also, we study the temperature dependence of the equilibrium state to estimate thermodynamic quantities, e.g., heat capacity and entropy, of amyloid fibrils. © 2019 Elsevier B.V. All rights reserved.
Keywords: Aggregation Fibrils Monte Carlo
1. Introduction Native and unfolded states of proteins coexist in an equilibrium that depends on temperature and solvent condition [1e4]. This equilibrium is disturbed at high protein concentrations wherein these molecules aggregates forming ordered and elongated structures known as amyloid fibrils [5,6]. At the molecular level, segments of proteins that are incorporated into these fibrils adopt extended conformations, i.e., b-strands, that are hydrogen bonded to neighboring proteins forming a b-sheet [7,8]. Stacking of bsheets through side chain interactions accounts for the common pattern of amyloid fibrils, also known as cross-b structure [9,10]e see Fig. 1a. Proteins that form fibrils at physiological conditions can be either functional, e.g., silk fibrils that are used by spiders to capture their prey, or related to plaque formation in amyloid diseases that include Alzheimer's and Parkinson's [11,12]. The formation of amyloid fibril is a non-equilibrium process and its kinetics has been extensively studied. In particular, fibril formation has been shown to proceed through a nucleation phase followed by growth [13e15]. Recently, fibril growth has been shown to evolve towards an equilibrium state where mature fibrils co-exist with monomers
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (T. Urbic), [email protected] (C.L. Dias). https://doi.org/10.1016/j.fluid.2019.02.002 0378-3812/© 2019 Elsevier B.V. All rights reserved.
in the solution [16]. Understanding this equilibrium is of great interest as it could enable the development of a thermodynamic framework to study these structures. In analogy with protein folding [17], thermodynamics may provide insights into the stability of fibrils and their underlying molecular mechanisms [18]. Equilibrium thermodynamic properties of amyloid fibrils are challenging to measure using conventional experimental methods. For example, heat capacities measured as a function of temperature in calorimetric experiments were found to be irreversible for most fibrils precluding the use of analytic frameworks to compute freeenergies [19,20]. Accordingly, fibril formation is commonly thought as an irreversible process in which proteins are being incorporated to it without dissociating [21]. Only recently have “amyloid assembly equilibrium measurements” shown that mature fibrils can exist in equilibrium with dissolved proteins, i.e., monomers, at least for some amino acid sequences [16,22e25]ealthough this idea had been proposed previously [26]. Equilibrium between fibrils and monomers may also be inferred from self-assembly of lipids in membrane bilayer [27]. This equilibrium has been explored to measure free-energies required to dissociate a protein from a fibril, i.e., DG. Studies of the temperature dependence of this equilibrium can also provide insights of other thermodynamic quantities, i.e., change in enthalpy (DH), entropy (DS), and heat capacity (DCp ). These quantities played a key role in determining the molecular driving forces of protein folding [1,3] but they remain mostly unknown for amyloid fibrils. Only few experiments have
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
105
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 bead representations, 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 a dimer 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 each stack on top of each other. The fibril axis is also shown.
estimated DCp for amyloid fibrils finding both positive and negative values [20,28e30]. Currently, it is unknown how the magnitude and sign of DCp relate to the peptide sequence, fibril structure, and/ or condition of the solvent. Notice that unfolding of proteins accounts for a large and positive (never negative) change in heat capacity [3,4,31e37]. Timescales to equilibrate fibrils and proteins in a solution are of the order of hours and days which is beyond reach of all-atom computer simulations [22,25,38]. Thus, models to study these equilibrium reaction require a coarse-grained approach [39e41]. Here, we develop a coarse-grained model of peptides that can hydrogen bond to each other forming extended aggregate structures mimicking b-sheets. Two b-sheets in the model can stack on top of each through side chain interactions mimicking the structure of amyloid fibrils. We explore phase space of the model for different temperatures and densities. In agreement with experiments, we show that under certain conditions of temperature and pressure, amyloid fibrils grow towards an equilibrium state where fibrils and monomers coexist in equilibrium [42]. We determine the freeenergies DF to add or dissociate a protein from a fibril at this equilibrium and we show that conditions of density accounting for different fibril growth rates can lead to the same fibril-monomer equilibrium state. Moreover, DH, DS, and DCp can be obtained by studying the temperature dependence of this equilibrium. We discuss how the greater stability of fibrils with respect to temperature when compared to native protein structures implies a small DCp (close to zero). Despite its small value, we show that DCp can be measured 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. First disk has two arms separated by angle 180+ which can associate with arm of another molecule and form hydrogen bond. Both arms form 90+ angle with second particle of dimer (See Fig. 1b). The interaction potential between two particles is a sum of a LennardeJones terms between the centers of Lennard-Jones disks and an associative term which mimic formation of hydrogen bonds
interaction) in molecule i. The form of the associative part of the interaction potential is similar as used in Mercedez-Benz model of water [43e47].
2 ! X ! ! ! ! ! Ua Xi ; Xj ¼ εa G ri1 rj1 ra G i uij 1 G j uij k;l¼1
þ1
(2)
where G(x) is an unnormalized Gaussian function
x2 GðxÞ ¼ exp 2 : 2s
(3)
Further, εa ¼ 1 is an associative energy parameter and ra ¼ 1:2l is a characteristic length at which hydrogen bonds are ! ! ! formed. uij is the unit vector along rj1 ri1 , i and j are the unit vectors representing the arm of the ith and jth particle. The strongest association occurs when the arm of one particle is collinear with the arm of another particle and centers with arms are on distance ra from each other. The width of Gaussian (s ¼ 0.085l) is small enough that it is not possible to have more than one hydrogen bond per arm. The Lennard-Jones interactions take the standard form
U ab LJ ðrÞ ¼ 4εab
"
sab r
12
sab r
6 # ;
(4)
where r is the distance between two beads. Interaction between side-chains (ε22 ¼ 0:2) is stronger than between backbones (ε11 ¼ 0:1) to favor formaiton of fibrils instead of stacking of chains in any order. LJ sizes of all particles is equal and it is equal to 0.9 (sab ¼0.9l). The parameters for dissimilar segments are obtained from the Lorentz-Berthelot combining rules [48]. Distances and energies in this work are given in units of l and εHB , respectively, and the Boltzmann constant kb is defined as one. 3. Monte Carlo simulations
2 X ! ! ! ! U Xi ; Xj ¼ U ab LJ ria rjb þ Ua Xi ; Xj :
(1)
a;b¼1
! ! Xi and Xj denotes the vector representing the coordinates and the orientation of the ith and jth particle, ria is the position vector of site a (1 represent backbone which can form hydrogen bonds and 2 side-chains which can only interact by means of Lennard-Jones
Here, we perform Monte Carlo (MC) simulations of the 2D fibrils model in the canonical (NVT) ensemble [47,49]. All simulations were performed with N ¼ 120 or N ¼ 250 molecules. Increasing the number of particles had no significant effect on the calculated quantities. 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
106
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
of monomers r* was calculated as ratio between number of monomers and the square of size L2 . Periodic boundary conditions and the minimum image convention are used to mimic an infinite system of particles. Starting configurations are selected randomly. At each time-step, we try to translate, rotate or switch (rotated for 180+ ) a randomly chosen protein, each with same probability. Probabilities for translation, rotation and switch were the same. A MC cycle which corresponds to N moves of particles (translations, rotations and switches) is used as our unit of time. In one cycle each particle was on average translated, rotated and switched. Average quantities are computed from 106 MC cycle simulations that are performed after an equilibration period of 105 -107 cycles
depending on temperature and density. Thermodynamic quantities such as energy were calculated as statistical averages over the course of the simulations [49]. Cut off of the potential was halflength of the simulation box. 4. Results and discussion 4.1. Bond formation: density and temperature dependence In Figs. 2 and 3 we show the average fraction of peptides that are hydrogen bonded to just one neighboring peptide n1 , two peptides n2 , and no-peptides n0 . These quantities are depicted in Fig. 2 as a
Fig. 2. Density dependence of ratio of molecules with 0 (black solid line), 1 (red dashed line) and 2 hydrogen bonds (blue dotted line). Points are results from Monte Carlo computer simulations and lines are guide for the eyes. Results are plotted for different temperatures (a) T ¼ 0.1, (b) T ¼ 0.15, (c) T ¼ 0.2 and (d) T ¼ 0.25.
Fig. 3. Temperature dependence of ratio of molecules with 0 (black solid line), 1 (red dashed line) and 2 hydrogen bonds (blue dotted line). Points are results from Monte Carlo computer simulations and lines are guide for the eyes. Results are plotted for different densities (a) r ¼ 0.025, (b) r ¼ 0.1, (c) r ¼ 0.2 and (d) r ¼ 0.25.
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
function of density for four different temperatures. At all temperatures, n2 and n0 increase and decrease, respectively, with increasing density. These behavior can be rationalized by the increased proximity of peptides at high density that makes them more reactive and accounts for an increase in n2 and a decrease in n0 . Conservation of the number of peptides dictates the behavior of n1 . This quantity decreases as a function of density at low temperature (Fig. 2a) whereas it increases at high temperature (Fig. 2c and d). At intermediate temperature (Fig. 2b), n1 reaches a maximum at intermediate densities. In Fig. 3, we show the temperature dependence of n1 , n2 , and n0 for four densities. At all densities, n2 and n0 decrease and increase, respectively, with increasing temperature. These behaviors emerge because thermal fluctuation accounts for bond rupture. Thus, increasing temperature favors the population of non-interaction peptides, i.e., n0 , while it accounts for a reduction in n2 . Notice that the range of temperatures for which n0 , n1 , or n2 dominates depends on the density. In Fig. 4 we depict how density and temperature affect the bonding of peptides. The dotted line shows densities and temperatures for which 50% of the peptides are not forming hydrogen bonds. The full line corresponds to densities and temperatures for which 50% of the peptides are forming two hydrogen bonds. Thus, above the dashed line, most peptides are not forming hydrogen bonds while below the full line most peptides are forming two hydrogen bonds with neighboring peptides. In between the two lines, peptides are mainly bonded to neighboring molecules by just one hydrogen bonds. Since most peptides in a fibril form two hydrogen bonds with neighboring peptides, Fig. 4 provides insights into the conditions at which fibrils are more likely to form. However, conditions at which most peptides form two hydrogen bonds is not enough to guarantee that fibrils are formed, see Section B. Similarly it is possible to form fibrils even if the dominant bonding specie of peptides form only one hydrogen bond, see Section C.
107
Fig. 5. Probability pðNc Þ of finding molecule in cluster of size Nc for temperature T ¼ 0.15 and density (a) r ¼ 0.1 (b) r ¼ 0.17 and (c) r ¼ 0.22.
4.2. Cluster analysis To better define regions in Fig. 4 where fibrils are the dominant phase of the system, we count the number Nc of proteins in an aggregate or cluster. We use an energy criteria wherein proteins are considered bonded when their potential energy is less than 0.05. Small variations of this energy cutoff did not account for significant differences in the bonding state of the system. In Fig. 5, we show the probability of a protein to belong to a cluster of size Nc for three densities at T ¼ 0:15. At low density, the system is composed of many small clusters (see Fig. 5a) whereas at high density it is made of one large cluster with only a few proteins belonging to small clusters (see Fig. 5c). At intermediate density (Fig. 5b), clusters of all sizes are equally populated. To estimate the propensity of the system to form large clusters
that resemble fibrils, we compute the fraction pa of proteins that belongs to the largest cluster. In Fig. 6a, this quantity is shown as a function of the density of the system at four temperatures. Proteins in the system become more connected to each other with increasing density and decreasing temperature. The same result is also observed in Fig. 6b where pa is shown as a function of temperature for four different densities. Following how states are defined in percolation theory, we define region where fibrils are formed when the system is connected, meaning that at least 50% of the proteins are bonded to one big cluster. While there is some arbitrariness in this definition of fibrils, in the next section we provide visual evidence that this definition enables the differentiation of different states of the system. The full line in Fig. 6c shows densities and temperature for which 50% of proteins are part of one big cluster. For conditions on the right hand side of this line, proteins are highly connected to each other forming fibrils. In contrast for conditions on the left side of the line proteins are either in the monomeric state or in the form of b sheets, i.e. structures in which several proteins are hydrogen bonded to each other.
4.3. Dominant structures in phase space
Fig. 4. Phase space with marked regions with regions dominated by particles with no hydrogen bond (0 HB), with one (1 HB) and with 2 hydrogen bonds (2 HB).
Fig. 7 combines our bond formation analysis (i.e., lines from Fig. 4) and cluster analysis (i.e., line in Fig. 6c). In this figure, “M” corresponds to conditions of temperature and density where most proteins in the system are not bonded to each other, i.e., they are in the monomeric state (See Fig. 8a). Small clusters, i.e., oligomers, are favored at “O1” (Fig. 8b) and “O2” (Fig. 8c). At “O1”, proteins form
108
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
Fig. 6. (a) Density dependence of ratio of molecules belonging to biggest cluster in system. Points are results from Monte Carlo computer simulations and lines are guide for 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 dasheddotted). (b) Temperature dependence of ratio of molecules belonging to biggest cluster in system. Points are results from Monte Carlo computer simulations and lines are guide for the eyes. Results are plotted for different densities r ¼ 0.05 (black solid line), r ¼ 0.1 (red dashed), r ¼ 0.15 (blue dotted), r ¼ 0.2 (green long dasheddotted) and r ¼ 0.25 (orange dashed-dotted). (c) Phase space with marked regions where system is connected (C) and where not (U).
Fig. 7. Phase space with marked regions with regions dominated by monomers (M), oligomers (O1 and O2) and fibrils (F1 and F2).
mainly small clusters, e.g., dimers, while at “O2” proteins form bsheets. Thermal fluctuation which accounts for bond rupture and destabilizes b-sheets can rationalize the structural difference between “O1” and “O2”. State “F1” in Fig. 8d represents regions in space where fibrils are distorted and coexist with many small clusters, e.g., dimers. State “F2” represents ideal fibrils in which two
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 ¼ 0.1, r ¼ 0.05; (d) fibrils region F1 at T ¼ 0.15, r ¼ 0.22; (e) fibrils region F2 at T ¼ 0.05, r ¼ 0.1 and (f) fibrils region F2 at T ¼ 0.1, r ¼ 0.22.
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
109
beta-sheets are stacked on top of each other. Characteristic configurations of these states are shown in Fig. 8e and f. 4.4. Thermodynamics of fibrils Experimentally, conditions at which fibrils coexist with monomers and small clusters have been explored to study the stability of fibrils [50]. In these studies, the addition/dissociation of monomers to/from fibrils was shown to be reversible which enabled measurements of the free-energy DF of these processes from the equilibrium dissociation constant Kd :
Kd ¼
½F½M ¼ ½M; ½F
(5)
where ½F is the equilibrium concentrations of fibrils and ½M ¼ ro is the density of monomers in equilibrium. In our simulations, coexistence of monomers and fibrils occur in region F1 of the phase spaceesee Fig. 8d. This region of the phase space can, therefore, be explored to compute thermodynamic properties of the system. In Fig. 9 we show the density of monomers (i.e., non-interacting proteins) as a function of the density of proteins in the system at four temperatures. The density of monomers is computed by dividing the number of monomers No by the area of the simulation box L2 , i.e., ro ¼ No =L2 . ro increases and reaches a plateau at all temperatures. The existent of this plateau emerges from the assumption that the number of fibrils in the system is constant and monomeric proteins can be added/dissociate to/from them:
dNo ¼ kof No þ kfo N; dt
(6)
where kof and kfo are rate constants for adding and dissociating o monomers to/from fibrils, respectively. In equilibrium (i.e., dN ¼ 0), dt we obtain that ro ¼ No =N ¼ kfo =kof which shows that ro is independent of the density of the system. Notice that ro can be used to compute DF ¼ kb TlnKd ¼ kb Tlnro =rs , where rs is the standard concentration rs ¼ 1. In Fig. 10, we show the dependence of DF on temperature which, according to thermodynamics, can be described by Refs. [4,17,51,52]:
DF ¼ DUo T DSo þ DCo ðT To Þ T log
T To
(7)
where DUo , DSo , and DCo are the change in energy, entropy, and heat capacity related to the dissociation of a protein from a fibril computed at temperature To . The temperature dependence of DF
Fig. 10. Dependence of the free-energy DF to dissociate a protein from a fibril as function of temperature. Points are results from Monte Carlo computer simulations and line is best fit of the data point to Eq. (7).
from our simulations is shown in Fig. 10 and the best fit of this dependence 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 that this is a simple model which does not have enough detail to account for the magnitude and sign of these quantities in real proteins. As next it is interesting to see that the free energy is dominated by DU0 close to T0 . This is expected since fibrils are highly stable and internal energy (or enthalpy) favors the fibrillar state while entropy opposes it. 5. Conclusion In this work we developed a two-dimensional coarse-grained model which was used to study equilibrium properties of amyloid fibrils. We checked properties by Monte Carlo computer simulations for different state points in the phase space. We analyzed structures of the model and determined characteristic structures for different parts of phase space. At low densities and high temperatures molecules are mostly present as monomers while at low temperatures and higher densities fibrils start to form and than fibrils associate to form higher structures like stack of fibrils. We provided computational proof of principle that fibrils can exist in equilibrium with proteins dissolved in solution and this equilibrium can be used to compute free-energies DF to dissociate a protein. These quantities remain mostly unknown but they have the potential to provide important insights into molecular mechanisms underlying fibril stability. Acknowledgements T. U. are grateful for the support of the NIH (GM063592) and Slovenian Research Agency (P1 0103-0201, N1-0042) and the National Research, Development and Innovation Office of Hungary (SNN 116198). References [1] [2] [3] [4]
Fig. 9. Dependence of density of free monomers in system as function of total monomer density. Points are results from Monte Carlo computer simulations and lines are guide for the eyes. Results are plotted for different temperatures T ¼ 0.12 (black solid line), T ¼ 0.15 (red dashed), T ¼ 0.18 (blue dotted) and T ¼ 0.2 (green long dashed-dotted).
[5] [6] [7] [8] [9] [10] [11] [12] [13]
W. Kauzmann, Adv. Protein Chem. 14 (1959) 1. P.L. Privalov, Crit. Rev. Biochem. Mol. Biol. 25 (1990) 281. K. Dill, Biochemistry 29 (1990) 7133. C.L. Dias, T. Ala-Nissila, J. Wong-ekkabut, I. Vattulainen, M. Grant, M. Karttunen, Cryobiology 60 (2010) 91. €ndrich, M.A. Fletcher, C.M. Dobson, Nature 410 (2001) 165. M. Fa €ndrich, M. Dobson, Christopher, EMBO J. 21 (2002) 5682. M. Fa Z. Su, C.L. Dias, J. Phys. Chem. B 118 (2014) 10830. C. Narayanan, C.L. Dias, J. Chem. Phys. 139 (2013) 115103. E.D. Eanes, G.G. Glenner, J. Histochem. Cytochem. 16 (1968) 673. A. Geddes, K. Parker, E. Atkins, E. Beighton, J. Mol. Biol. 32 (1968) 343. F. Chiti, C.M. Dobson, Annu. Rev. Biochem. 75 (2006) 333. C.M. Dobson, Nature 426 (2003) 884. s, A. Laio, Phys. Rev. Lett. 110 (2013) F. Baftizadeh, F. Pietrucci, X. Biarne 168103.
110
T. Urbic, C.L. Dias / Fluid Phase Equilibria 489 (2019) 104e110
[14] S. Auer, P. Ricchiuto, D. Kashchiev, J. Mol. Biol. 422 (2012) 723. [15] T.C.T. Michaels, A.J. Dear, J.B. Kirkegaard, K.L. Saar, D.A. Weitz, T.P.J. Knowles, Phys. Rev. Lett. 116 (2016) 258103. [16] R. Wetzel, Acc. Chem. Res. 39 (2006) 671. [17] A.D. Robertson, K.P. Murphy, Chem. Rev. 97 (1997) 1251. [18] B.P. Roland, R. Kodali, R. Mishra, R. Wetzel, Peptide Science 100 (2013) 780. [19] C.M. Doyle, J.A. Rumfeldt, H.R. Broom, A. Broom, P.B. Stathopulos, K.A. Vassall, J.J. Almey, E.M. Meiering, Arch. Biochem. Biophys. 531 (2013) 44. [20] B. Morel, L. Varela, F. Conejero-Lara, J. Phys. Chem. B 114 (2010) 4010. [21] S.I. Cohen, S. Linse, L.M. Luheshi, E. Hellstrand, D.A. White, L. Rajah, D.E. Otzen, M. Vendruscolo, C.M. Dobson, T.P. Knowles, Proc. Natl. Acad. Sci. Unit. States Am. 110 (2013) 9758. [22] B. O'Nuallain, S. Shivaprasad, I. Kheterpal, R. Wetzel, Biochemistry 44 (2005) 12709. [23] T.M. Doran, A.J. Kamens, N.K. Byrnes, B.L. Nilsson, Proteins: Structure, Function, and Bioinformatics 80 (2012) 1053. [24] A.D. Williams, S. Shivaprasad, R. Wetzel, J. Mol. Biol. 357 (2006) 1283. ISSN 0022-2836. [25] A.M. Bhattacharyya, A.K. Thakur, R. Wetzel, Proc. Natl. Acad. Sci. Unit. States Am. 102 (2005) 15400. [26] J.D. Harper, P.T. Lansbury Jr., Annu. Rev. Biochem. 66 (1997) 385. [27] J.N. Israelachvili, Intermolecular and Surface Forces, Elsevier Science, 2010. [28] T. Ikenoue, Y.-H. Lee, J. Kardos, M. Saiki, H. Yagi, Y. Kawata, Y. Goto, Angew. Chem. Int. Ed. 53 (2014) 7799. [29] J. Kardos, K. Yamamoto, K. Hasegawa, H. Naiki, Y. Goto, J. Biol. Chem. 279 (2004) 55308. [30] M.D. Jeppesen, K. Hein, P. Nissen, P. Westh, D.E. Otzen, Biophys. Chem. 149 (2010) 40. [31] R.L. Baldwin, J. Mol. Biol. 371 (2007) 283. [32] J.T. Edsall, J. Am. Chem. Soc. 54 (1935) 1506. [33] P. Bruscolini, L. Casetti, Phys. Rev. E 61 (2000) R2208. http://prola.aps.org/
abstract/PRE/v61/i3/pR2208_1. [34] P.D.L. Rios, G. Caldarelli, Phys. Rev. E 63 (2001) 031802. http://prola.aps.org/ abstract/PRE/v63/i3/e031802. [35] C.L. Dias, T. Ala-Nissila, M. Karttunen, I. Vattulainen, M. Grant, Phys. Rev. Lett. 100 (2008) 118101. [36] C.L. Dias, Phys. Rev. Lett. 109 (2012) 048104. [37] V. Bianco, G. Franzese, Phys. Rev. Lett. 115 (2015) 108101. [38] J. Nasica-Labouze, P.H. Nguyen, F. Sterpone, O. Berthoumieu, N.-V. Buchete, , A. De Simone, A.J. Doig, P. Faller, A. Garcia, Chem. Rev. 115 (2015) S. Cote 3518. [39] A. Irb€ ack, N. Linnemann, B. Linse, S. Wallin, et al., Phys. Rev. Lett. 110 (2013) 058101. [40] J. Schmit, K. Ghosh, K.A. Dill, Biophys. J. 100 (2011) 450. [41] P. Ricchiuto, A.V. Brukhno, S. Auer, J. Phys. Chem. B 116 (2012) 5384. [42] JS Espinoza Ortiz, L. Cristiano, Dias, Phys. Stat. Mech. Appl. 11 (2018) 154. [43] T. Urbi c, V. Vlachy, Y.V. Kalyuzhnyi, N.T. Southall, K.A. Dill, J. Chem. Phys. 112 (2000) 2843. [44] T. Urbic, V. Vlachy, Y.V. Kalyuzhnyi, N.T. Southall, K.A. Dill, J. Chem. Phys. 116 (2002) 723. [45] T. Urbi c, V. Vlachy, Y.V. Kalyuzhnyi, K.A. Dill, J. Chem. Phys. 127 (2007) 174511. [46] T. Urbic, M. Holovko, J. Chem. Phys. 135 (2011) 134706. [47] T. Urbic, T. Mohoric, J. Chem. Phys. 146 (2017) 094505. [48] D. Frenkel, B. Smit, Computational sciences series 1 (1986) 1. [49] D. Frenkel, B. Smit, Intermolecular and Surface Forces, vol. 1, Elsevier Science, 2010, p. 1. [50] J.N. Israelachvili, Computational sciences series 1 (2002) 1. [51] L. Smeller, Biochim. Biophys. Acta Protein Struct. Mol. Enzymol. 1595 (2002) 11. [52] S.R. Jampani, F. Mahmoudinobar, Z. Su, C. Dias, Proteins: Structure, Function, and Bioinformatics 83 (2015) 19.