Kinetic Model for Two-Step Nucleation of Peptide Assembly - The

Article pubs.acs.org/JPCB

Kinetic Model for Two-Step Nucleation of Peptide Assembly Ming-Chien Hsieh,† David G. Lynn,‡ and Martha A. Grover*,† †

School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, United States Departments of Chemistry and Biology, Emory University, Atlanta, Georgia 30322, United States

‡

S Supporting Information *

ABSTRACT: Intermediate dynamic assemblies are increasingly seen as necessary for the initial desolvation and organization of biomaterials to achieve their final crystalline order. Here we present a general peptide assembly model for two-step nucleation. The model predicts the phase transitions and equilibria between different phases by employing a combination of the Flory−Huggins parameter, the particle growth constant, and the binding energy to assemblies. Monte Carlo simulations are used to demonstrate how the system evolves from pure solution phases to the final thermodynamic assembly phase via an intermediate metastable particle phase. The final state of the system is determined by the solubility of the particle and assembly phases, where the phase with the lower solubility accumulates. A rare three-phase equilibrium exists when the solubilities of the particles and assemblies are similar. Experimental support for this model is achieved with assembly of the amyloid peptide Ac-KLVFFAE-NH2 (Aβ(16−22)) in mixed acetonitrile/water systems. Increasing the acetonitrile concentration decreases the number of particles, increases the particle size, and accelerates the assembly rate, all consistent with acetonitrile increasing the Aβ(16−22) peptide’s solubility of particles but with little influence on the stability of the assemblies. Taken together, our model captures the transition from the metastable particle phase to the higher order peptide assembly through two-step nucleation.

1. INTRODUCTION A detailed understanding of the assembly pathway for the formation and maturation of peptide assemblies becomes increasingly important both for the construction of functional biomaterials1−6 and, more acutely, for the development of therapeutic approaches to control the devastating protein misfolding diseases.7−9 The range of protein sequences involved in amyloid assembly has complicated defining the assembly pathways, particularly the critical nucleation step.10 Single-step nucleation (1SN)11−14 is the most direct model in assuming that the free peptide monomers nucleate directly into ordered structures in solution. However, strong desolvation energy barriers can slow or prevent this direct nucleation.15,16 According to Ostwald’s rule of stages,17 the peptides form a metastable phase prior to organizing as fibers.18 This two-step nucleation (2SN) has been observed widely from peptide amyloid assemblies19−23 to crystallization of proteins,16,24−29 colloids,30−32 minerals,33,34 and polymers.35 2SN has been observed for the peptide assemblies in the dynamic combinatorial network as well;36 the formation of the metastable particle phase is triggered with the accumulation of the peptide oligomers, and the assemblies of the oligomers nucleate inside the particles. The morphologies of the metastable phases range from molecular clusters to macroscopic dense solution phases,37 which may either (a) be incorporated as the assembly phases develop,15,25,38 (b) coexist with the assembled phases,24,38,39 or (c) persist if assembly nucleation is slow or thermodynamically disfavored.9,18,38,40 © 2017 American Chemical Society

The kinetic 2SN process has been mathematically modeled to reveal the key factors impacting assembly formation. Pan and co-workers have simulated 2SN of proteins with a phenomenological kinetic model as a function of temperature.41 Here the reversible formation of the metastable phase and irreversible nucleation of the assembly phase are assumed to be Arrhenius-type. They conclude that the rate-determining step of the assembly phase is assembly nucleation, which is controlled by both viscosity and monomer concentration of the metastable phase. Since the amount of the metastable phase significantly impacts the maturation of the assembly phase,42 Kashchiev and co-workers simulated the assembly nucleation rate as a function of the individual metastable particle size.43 Later Auer et al. compared the difference between 1SN and 2SN18 using classical nucleation theory to simulate the formation of the metastable phases. However, these efforts focus on assembly nucleation rate without including assembly maturation or monomer depletion. To explain protein crystallization data, Sauter and co-workers proposed in 2015 a 2SN kinetic model37 that includes formation of a metastable phase, monomer depletion, and the transition to an assembly phase. The model used the total peptide concentration to describe the metastable phase because of its poorly defined morphology. Received: March 31, 2017 Revised: June 28, 2017 Published: July 19, 2017 7401

DOI: 10.1021/acs.jpcb.7b03085 J. Phys. Chem. B 2017, 121, 7401−7411

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The Journal of Physical Chemistry B

Figure 1. Peptide assembly mechanism under two-step nucleation. (a) Initially, peptides are dissolved in the solution, and (b) the metastable peptide particles nucleate if the peptides are not completely soluble. (c) The particles grow when the solution is supersaturated for particles. (d) Later, the assemblies nucleate inside the particles and (e) extend into the solution after growing into a critical size by consuming the peptides inside the particles. (f) The assemblies propagate by consuming the free peptides in the solution phase, which decreases the free peptide concentration. (g) The particles start to dissolve when the solution becomes undersaturated for particles. (h) If assembly dissolution is negligible, after all free peptides are depleted, the assemblies become the only species remaining in the solution.

2. METHODS 2.1. Model Development. The peptide self-assembly under two-step nucleation starts with the formation of the metastable particle phase (Figure 1a,b). This particle nucleation is assumed to be analogous to nucleation of droplets in vapor or solution, which follows the classical nucleation theory.18,52,53 The particle nucleation rate (R1) is calculated as p ⎞ ⎛ R1 = p1 S exp⎜ − 22 ⎟ ⎝ ln S ⎠ (1)

Significant experimental evidence now suggests that AcKLVFFAE-NH2 (Aβ(16−22)), the nucleation core of the Aβ peptide implicating Alzheimers’ disease,5,44,45 assembles via 2SN.38 The metastable particles and the final assemblies of Aβ(16−22) are structurally characterized, allowing the assembly pathway to be morphologically defined,38,46−48 similar to other peptide systems.23,38,39 Here, the size of the metastable particles is critically tied to assembly nucleation rate; the assembly forming inside the particle does not extend into the solution phase until its length exceeds the particle diameter.38 Liang and co-workers observed Aβ(16−22) fibers emerging from the metastable particles using fluorescent probes, but they did not report the concentration of assembled peptides during assembly.49 Nilsson and co-workers did measure the concentrations of assembled Aβ(16−22) and several congeners over time using HPLC, but they did not achieve sufficient kinetic resolution to define the nucleation pathway of the assemblies.50 Yang and co-workers provided further support that the Aβ(16− 22) fibers arise from metastable oligomers in phosphatebuffered saline,23 but the presence of metastable flake-like side products complicated their analyses. As the intermediate flakes also bind thioflavin T, the dye used to probe fibrillation, the signal arising from fibers and flakes are indistinguishable, and fiber formation kinetics cannot be described independently. Here we propose a kinetic peptide assembly model through 2SN. The metastable phase is assumed to be spherical particles with dynamically changing sizes, while the assembly phase is composed of unidirectional fibers. This sequence of morphologies, experimentally observed for peptide systems,23,38,39 leads to a model executed with Monte Carlo simulations51 that predicts the transitions from solution (free monomer) to metastable particle to paracrystalline assembly phases. We demonstrate the utility of this 2SN model by defining the kinetics of Aβ(16−22) self-assembly in various acetonitrile− water solvent mixtures.

in which p1 and p2 are peptide concentration-independent constants. S is the supersaturation for particles, defined as the ratio of peptide concentration (C) over the solubility of particles (C1*) such that S = C/C1*. The solubility of particles is estimated with the Flory−Huggins solution theory:53−55 C1 * =

ρmon e−χ Wmon(1 − e−χ )

(2)

where ρmon is the peptide density, Wmon is the peptide molecular weight, and χ is the Flory−Huggins parameter. The typical density of protein and molecular weight of peptide AcKLVFFAE-NH2 are used for ρmon and Wmon, respectively (Table S1). A stable particle nucleates with a critical number of peptides to overcome the interfacial energy with the bulk free energy change. This critical number (n*) is then the particle nucleus size determined by the Gibbs−Thomson equation:56 ⎛ W ⎞2 32πγ 3 mon ⎜⎜ ⎟⎟ n* = 3(kbT )3 ln 3 S ⎝ NAρmon ⎠

(3)

where γ is the interfacial tension between peptide and solvent, kB is Boltzmann’s constant, NA is Avogadro’s number, and T is the temperature, which is set as a constant in this study. Whenever a particle is formed, n* peptides are removed from the solution phase and form a new particle. Equation 3 suggests 7402


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The Journal of Physical Chemistry B that, under a supersaturation S, a particle remains stable against dissolution as long as its size is at least n*; hence, the stability of a particle depends on its size and so does the peptide solubility of that particle.57−59 However, here the peptide solubility of all particles is assumed to be size-independent60−62 for simplicity. The particles may grow or dissolve, which depends on the solubility of particles (C1*; Figure 1c). When particle growth is bulk diffusion-limited, the net particle growth rate for an individual particle with x peptides may be expressed as58,61 r2(x) = kg0x1/3(C − C1 *)

Before the sizes of the assemblies reach the critical value LP, the assemblies stay in the particles. Once an assembly with y peptides emerges into the solution, that assembly then may grow in the solution. The following growth of assembly in the solution phase (Figure 1f-1h) is formulated as a function of the remaining peptides in the solution phase, and the corresponding growth rate (R5) for assemblies with y peptides is R 5(y) = 2kgA(y)C

where kg is the assembly growth constant in solution phase and A(y) is the concentration of assemblies with y peptides in the solution phase. The leading “2” in R5 stands for the two active ends per assembly. The peptide dissociation may occur at the assembly ends due to instability. The assembly dissolution rate (R6) is formulated as

(4a)

and the overall particle growth rate for all the particles with x peptides is R 2(x) = kg0x1/3(C − C1 *)P(x)

(4b)

where kg0 is the particle growth constant and P(x) is the total number of particle with x peptides. This net expression results in either particle growth or dissolution based on the peptide concentration in the solution phase, C. When C > C1*, the solution is supersaturated for particles, and particles grow according to R2 until C = C1*. On the other hand, when C < C1*, the solution is undersaturated and the particles dissolve until C = C1*. Thus, for all particles, both their growth and dissolution stop when C = C1* and never happen simultaneously. Assemblies nucleate inside the metastable particles (Figure 1d). As the assembly nucleation rate (R3) is a function of particle mass,43,63 R3 for the particles with x peptides is formulated as R3(x) = knP(x)x

⎛ −ΔG ⎞ R 6(y) = k bpexp⎜ ⎟B(y)A(y) ⎝ kBT ⎠

(5)

( )

kbp exp C2 * =

kg

−ΔG kBT

(10)

The model is executed with a kinetic Monte Carlo simulation algorithm,51 with 200 parameter sets constructed based on the parameter ranges in Table S1 and a Latin hypercube sampling. These 200 calculated results are classified into pure single phases, multiphase equilibria, or slow kinetics, based on their phase distributions. To account for stochastic fluctuations, a phase is considered to exist after the reaction time only if its concentration is greater than 1% of the total concentration. If there is only one phase remaining, the result is classified as a pure single phase; the equilibrium between different phases is achieved when multiple phases exist at the end and the phase concentrations do not change significantly near the end of the reaction time. If the concentration distribution between the phases is still changing at the end, the kinetics are considered to be “slow”. Further details about the parameter values, the Monte Carlo simulation algorithm, and criteria to classify the results are provided in the Supporting Information (SI). 2.2. Aβ(16−22) Solution Preparation. Aβ(16−22) peptides are synthesized using solid phase peptide synthesis, and the synthetic products are purified with high-performance liquid chromatography (HPLC), using a water/acetonitrile gradient with 0.1% trifluoroacetic acid in a C-18 reverse phase column.38 After lyophilization, the purified peptide is stored at

(6)

where kAG is the growth constant and Ap(x) is the total number of assemblies in particles with x peptides. With R3 and R4, multiple nuclei may nucleate and grow within the same particle, as observed previously.64 This makes the model more flexible compared to other studies,18,43,63 which are limited to one nucleus per particle at most. Once the assemblies grow long enough, their sizes (LA) become greater than the particle diameter (LP), and thus they are exposed to the solution phase (Figure 1e). By assuming that the assembly propagates at the two ends, LA for an assembly containing y peptides is then calculated as 0.5(y − 1) nm, where 0.5 nm is the typical spacing between peptides in the assemblies.65 The particle diameter is calculated by assuming that the particle density (ρpar) is similar to the typical protein density (Table S1). The diameter for a particle with x peptides is calculated as ⎛ xW ⎞1/3 mon 3 ⎟ ⎜ L P(x) = 2⎜ ⎟ ⎝ NAρpar 4π ⎠

(9)

where kbp is the prefactor of the rate constant, ΔG is the binding energy66 between the peptides, and B(y) is the number of breakable bonds at the assembly ends. If y = 2, there is only one bond breakable and thus B(y) = 1; if y > 2, both the bonds at the ends are breakable and B(y) = 2. The dissolution rate constant is expressed in an Arrhenius-type instead of a single constant to keep it flexible enough to predict an assembly system with peptide oligomerization,67 whose binding energy is proportional to the oligomer length.66 When the assembly nucleation and growth inside the particle are slow and negligible, the assembly phase reaches a steady state if the assembly growth rate (R5) and the assembly dissolution rate (R6) are equal; given R5 = R6, the solubility of assemblies (C2*) may be calculated with the assumption that for all assemblies, y > 2:

where kn is the assembly nucleation constant. The assembly nucleus size is assumed to be two, the minimal number to form assemblies with an intermolecular bonding.11 As the peptide concentration in the particles is fixed, it is not explicitly represented in eq 5. Once the reaction happens, two of the peptides in the particle become an assembly, and the assembly grows by consuming the other peptides in the same particle. The net assembly growth rate inside particles with x free peptides (R4) is expressed as37 R 4(x) = kAGA p(x)

(8)

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Figure 2. Peptide solubility for particles as a function of the Flory−Huggins constant (χ). (a) The peptide becomes less soluble as χ increases. Simulation of the particle phase development with (b) χ = 6, (c) 8, and (d) 10. The parameters used for these simulation are taken from Table S1 with kn = 0, kg0 = 10 mol−1 s−1, and C0 = 1 mM.

−20 °C for later use. To disassemble the preexisting assemblies, the peptide powder is dissolved in hexafluoroisopropanol (HFIP) to a concentration of 1 mg/mL, and the mixture is sonicated for 2 h in a bath sonicator. HFIP is removed with a Labconco CentriVap Concentrator-7970010, and the resulting peptide film is placed in a desiccator overnight to remove the remaining HFIP. Immediately prior to conducting an assembly experiment, the peptide film is suspended with HFIP and sonicated for 15 min. The sonicated solution is diluted with 40%, 60%, or 80% (v:v) acetonitrile in water, to obtain a final HFIP concentration of 1% by volume with the desired peptide concentration. The peptide solution is incubated at 22 °C and aliquots of the solution are removed at predetermined times for measurement of assembly kinetics. 2.3. Kinetics and Structure of Aβ(16−22) SelfAssembly Measured with CD, TEM, and FTIR. Circular dichroism (CD) is used extensively to determine β-sheet content of the peptide solutions. Here a volume of 23 μL of peptide solution is loaded into a demountable window cell with a 0.1 mm path length. The ellipticity is obtained at wavelengths from 185−260 nm under a resolution of 0.2 nm with a bandwidth of 2 nm at a scanning rate of 200 nm/min using a Jasco J-810 spectropolarimeter. After background correction, the spectrum from the average of three scans is saved and the resulting ellipticity at 217 nm is used as the indicator of β-sheet content. The morphologies of the particles and assemblies are observed with transmission electron microscopy (TEM). To

deposit the assemblies and particles onto the grid, the peptide solution is loaded onto the copper grid for 1 min, and the excess peptide solution is wicked away with filter paper. The resulting grid is negatively stained with 1.5 wt. % methylamine tungstate solution for 3 min before the stain solution is wicked away with filter paper. The grid is then stored in a vacuum desiccator overnight to remove the remaining liquid. TEM images are recorded with a Hitachi H-7500 transmission electron microscope. The particle size distribution is obtained by analyzing the TEM images with ImageJ 1.48v (National Institutes of Health, MD). Once the Aβ(16−22) assemblies are matured and the CD signatures of the solution do not change significantly with time, FTIR is used to probe the final peptide strand arrangement. Peptide solutions with a volume of 8 μL are loaded onto the diamond chip of a JASCO FT/IR-4100. The absorbance spectra are recorded against a background spectrum at a wavenumber range of 1000−3600 cm−1 at room temperature for 250 scans.

3. RESULTS AND DISCUSSION 3.1. Construction of Parameter Sets and Categories for Final Outcomes. A model is built to simulate the process of peptide assembly under 2SN, with the particles as the intermediate phase in which nucleation of the stable assemblies is initiated. The behavior of the particle phase is investigated with different particle growth constants (kg0) and Flory− Huggins parameters (χ). The miscibility of the peptide and the 7404


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Figure 3. Peptide solubility for assembly as a function of the assembly stacking energy (ΔG). (a) The peptide becomes less soluble as (ΔG) increases, and the simulations of the assembly phase development with (b) ΔG = 10−20, (c) 1.14 × 10−19, and (d) 10−17 J. The parameters used for these simulation are taken from Table S1 with χ = 8, kg0 = 10 mol−1 s−1, and C0 = 1 mM.

The upper and lower limits for ΔG are selected to span the range from very soluble to extremely insoluble, under a 1 mM peptide concentration (Figure 3). By changing kg0, the contribution of the net particle growth rate may be investigated without altering the thermodynamic constants, χ and ΔG, given the reaction time of 100 h. A total of 200 parameter sets are generated based on these three variables with a Latin hypercube sampling,68 and the behaviors of the model under these parameters are classified in Table 1. For some parameter sets, the simulated results show no particle formation within the reaction time (Figures 1a and 2b). There are seven parameter sets (3.5%) that remain in this pure solution phase (case 1). For all parameter sets, at the initial

solvent is critically tied to the Flory−Huggins parameter (χ). As χ increases, the peptide solubility for particles decreases, and the peptides then have a stronger propensity to nucleate particles. Figure 2a shows the simulated Aβ(16−22) peptide solubility as a function of the Flory−Huggins constant (χ), and Figure 2b−d show the development of the particle phase given different χ. The peptides remain soluble with a small χ (Figure 2b), while the particles nucleate and the soluble peptides decrease as χ increases (Figure 2c,d). The stability of the assemblies is varied as a function of stacking energy of the peptides (ΔG; Figure 3), which impacts the assembly dissolution rate. Figure 3 shows the solubility for assembly as a function of the stacking energy between the assembled peptides (ΔG). As ΔG increases, the peptide solubility for assemblies decreases significantly (Figure 3a); the assemblies become stable and start to accumulate by consuming the other phases (Figure 3b−d). These three parameters are used to construct 200 different parameter sets, with parameter values and ranges given in Table S1. The particle growth constants (kg0) and Flory−Huggins parameters (χ) are varied to investigate the transition of the metastable particle phase, while the stability of assemblies and the assembly dissolution rate are varied as a function of stacking energy of the peptides (ΔG). To verify the threshold supersaturation for particle nucleation, the lower limit of χ is selected to make the peptide solubility close to the initial peptide concentration of 1 mM, while the upper limit is selected to make the peptide extremely insoluble (Figure 2).

Table 1. Distribution of Simulation Results from the General Model with 200 Parameter Setsa case 1. 2. 3. 4. 5. 6.

Pure solution phase Pure assembly phase Solution-assembly equilibrium Solution-particle equilibrium Three-phase equilibrium Slow kinetics

no. of cases

percentage (%)

7 52 16 103 5 17

3.5 26 8 51.5 2.5 8.5

The results are classified into six cases based on their final states, including cases that have reached equilibrium and those that have not because of slow rates.

a

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3 in Table 1. The final peptide concentration, determined by the solubility of assemblies, is not sufficient to sustain the particles. The metastable particle phase dissolves as in case 2, and the difference between cases 2 and 3 is the reversibility of assembly growth. In case 2 the growth of assemblies is nearly irreversible, while case 3 is obtained when the assemblies exhibit significant reversibility. When the stacking energy of assemblies is not strong enough, the assemblies cannot grow in the solution. The remaining solution and particle phases reach equilibrium with each other, as shown in Figures 1c,d and S4. This solution-particle equilibrium occurs in over half of the parameter sets (case 4 in Table 1) and is characterized by the absence of the assembly phase. As shown in Figure S2, on average case 4 has the weakest stacking energy (ΔG) compared to the other cases with an assembly phase, and this solution-particle equilibrium has been experimentally observed.38 Kinetically limiting the assembly nucleation rate (kn) may also result in achieving a stationary state between the solution and particle phases.18 Under this condition, the particle phase develops and coexists with the solution phase because the assembly nucleation rate is extremely slow, and the system remains kinetically trapped. When both the assembly and particle phases are observed and the distribution between phases reaches a stationary state, the relatively rare three-phase equilibrium can be observed (Figure 1f and Figure S5). Both particle and assembly supersaturations are crucial to achieving the coexistence of all three phases (Figure 5). When the particle phase has the higher

time the solution is supersaturated since the free peptide concentration C is greater than the solubility of particles C1*. Thus, the supersaturation for particles, defined as S = C/C1*, is greater than one. However, the peptides will not form particles within the reaction time unless the solubility C1* is below a threshold value,37 and this threshold value controls the particle nucleation kinetics. As shown in Figure 4, within the reaction

Figure 4. Threshold supersaturation to trigger particle nucleation. The particles nucleate within the simulation time only when the initial supersaturation is greater than 1.35; the corresponding threshold value for particle nucleation is 0.74 mM. The maximum particle mass concentration for each parameter set is defined as the maximum peptide concentration in the particle phase within the reaction time.

time of 100 h, the particles do not nucleate unless the initial supersaturation is greater than 1.35. The threshold value for particle nucleation is then around 0.74 mM, based on this critical supersaturation of 1.35 and the initial peptide concentration of 1 mM. A supersaturation between 1.0 and 1.35 is then defined as the metastable zone for particle formation. Particles nucleate within the reaction time only when the supersaturation for particles is greater than this threshold value; otherwise no particles are observed within the reaction time. Under two-step nucleation, the metastable particles transition as the assembly phase matures. As illustrated in Figure S1 for a single parameter set, all peptides are consumed by the assembly phase, and this result is categorized as the pure assembly phase (case 2 in Table 1, Figure 1h). The range of ΔG considered is 0.794 × 10−19 to 1.58 × 10−19 (Table S1), while the assembly dissolution is negligible when ΔG > 1.26 × 10−19; this value corresponds to having only 1% of peptide soluble for assemblies at the 1 mM peptide concentration (Table S2). As shown in Figure S2, case 2 on average has the highest ΔG compared to the other cases. The assemblies in the solution phase do not consume the peptides inside the particle phase directly. Rather, once the solution is undersaturated for particles, the particles start to dissolve and release the peptides into the solution. The assemblies then consume the released peptides and become the dominating phase at equilibrium. If the free peptides are soluble for assemblies due to a weaker stacking energy, ΔG, the free peptides may coexist and equilibrate with the assemblies, as illustrated in Figures 1g and S3. This solution-assembly equilibrium is classified as case

Figure 5. Average final supersaturations for particles and assemblies determine the final phase distribution. The numbers in the parentheses are the numbers of results in each case.

solubility, it is eliminated and the result is classified as either case 2 or 3. When the assembly phase has the higher solubility, the result is classified as the solution-particle equilibrium (case 4). To obtain the rare three-phase equilibrium (case 5), the solubilities for particles and assemblies need to be high enough to maintain the solution phase and close enough to each other to stabilize both phases. Figure 5 suggests that the final equilibrium states can be rationally predicted as a function of the solubilities for particles and assemblies, as they are 7406


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The Journal of Physical Chemistry B influenced by the Flory−Huggins parameter (χ) and the stacking energy of the assemblies (ΔG), respectively. The final states of a 1.0 mM peptide solution are simulated with different solubilities for particles and assemblies (Table S3). For a 1 mM peptide solution, both solubilities are in the range from 0.005 mM (least soluble) to 1.1 mM (totally soluble). Figure 6 shows the prediction of the solubility

Figure 7. Final states predicted by the model for 1.0 mM peptide solution as a function of solubilities for particles and assemblies. The parameter sets are listed in Tables S1 and S3 with kg0 = 104 mol−1 s−1 and C0 = 1 mM, except kn = 0.5 × 10−7 s−1. kn is decreased from 10−7 s−1 in Table S1 to ensure that the solubility of the assemblies is not affected by the assembly nucleation in the particles, and thus the threephase equilibrium may be achieved within the reaction time.

Figure 6. Solubility difference between the particles and the assemblies as a function of the Flory−Huggins constant (χ) and the stacking energy (ΔG). The ranges of the χ and ΔG values in Table S3 and the parameter values in Table S1 are used. The difference is calculated as the solubility difference between particles and assemblies: C1* − C2*.

particle phase dissolves slowly even though the free peptide concentration is close to zero. On average, the parameter sets contributing to slow kinetics have the lowest particle growth constant (kg0), as shown in Figure S7. Particle dissolution, which shares the same rate equation with particle growth (eqs 4a and 4b), delays the entire process, and thus the process remains unfinished after 100 h. To verify the importance of this kinetic constant without changing the final thermodynamic distribution, the simulation in Figure S6 is repeated with kg0 increased to its upper limit indicated in Table S1. With fast particle dissolution, the peptide self-assembly process is finished within the reaction time (Figure S8). Taken together, these results demonstrate that the final states in peptide assembly via 2SN are controlled principally by the Flory−Huggins parameter and the peptide stacking energy. The system may be slowed by a small particle growth constant controlling both the particle growth and dissolution rates, such that assembly growth is delayed due to the scarce peptide resource in solution. 3.2. Model Predicts Aβ(16−22) Assembly. The structurally defined Aβ(16−22) assemblies provide an opportunity to validate the model’s utility, and accordingly we incubated the purified Aβ(16−22) peptide in different aqueous acetonitrile (ACN) mixtures to follow assembly maturation. We initially predicted the kinetics of Aβ(16−22) assembly would be slower in more hydrophobic environments, where the assemblies are more soluble.38 Here ACN is used to increase the solvent hydrophobicity and stabilize the free Aβ(16−22) peptide. A series of topography images of Aβ(16−22) solutions in 40% ACN are measured by transmission electron microscopy (TEM) over time. As shown in Figure 8a after 1 h of incubation, 0.5 mM of Aβ(16−22) is above the threshold value for particle nucleation in 40% ACN. The particles are evenly distributed on the EM copper grid with homogeneous particle sizes. In the TEM image of the sample incubated for 5 h

difference between the particle and assembly phases based on Table S3, with eqs 2 and 10 used to derive the particle (C1*) and assembly (C2*) solubilities, respectively. The diagonal of Figure 6 represents the equal solubility between the two phases; the region above the diagonal represents the assembly rich results as the particles are more soluble, while the region below the diagonal represents the particle-rich results, as the assemblies are more soluble. Although Figure 6 predicts the final thermodynamic states, it does not take any kinetics into consideration. Figure 7 shows the kinetic simulation with the entire twostep nucleation model, from eqs 1−9, as a function of the Flory−Huggins constant (χ) and the stacking energy (ΔG). As shown in Figure 7, no particle forms when the solubility of particles is equal to or greater than 0.8 mM, consistent with the threshold value (0.74 mM) obtained from Figure 4. When the solubility of particles is equal to or less than 0.7 mM, different phase equilibria are observed. The assemblies survive when they are less soluble than the particles, and vice versa. The rare three-phase equilibrium is reproduced with equal solubilities for particles and assemblies, as predicted. However, when both the solubilities of particles and assemblies are as low as 0.1 mM, the transition to the three-phase equilibrium becomes slow and does not reach a steady state within the reaction time. Such slow kinetics, which are classified as case 6, will be further discussed below. Not all parameter sets achieve their final thermodynamic equilibria by the end of the reaction time. Although most of the parameter sets do predict a final distribution within the reaction time of 100 h, 17 out of 200 parameter sets exhibit slow kinetics (case 6), and the unfinished reactions are attributed to slow particle dissolution (Figure S6). Although the assembly phase is already growing outside of the particles, the relatively stable 7407


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Incubation of Aβ(16−22) in 60% and 80% ACN exhibits particle and assembly phases that are kinetically distinct from 40% ACN. Counter to our original hypothesis, the fiber maturation kinetics are now faster; fibers appear in 60% (Figure 8d) and 80% (Figure 8g) conditions after 1 h of incubation, in contrast to the pure particle phase in 40% ACN (Figure 8a). The observed particle size at 1 h also increases with increasing ACN concentration (Figure 8a, d, and g). Through these EM images, the properties of the dynamic particle phase, the particle size, and the particle number are certainly tunable with different solvent compositions. The final fiber assemblies remain unchanged in all three solvents (Figure 8c, f, and i) and mature through the same intermediate particle transitions (Figure 8b, e, and h).49 These results are consistent with the morphological evolution of Aβ(16−22) assemblies occurring under a 2SN process. In addition to the TEM images, circular dichroism (CD) analyses provide time dependent evaluation of the assembly progress. While Thioflavin T (ThT) and Congo red (CR) are typically used to probe peptide assemblies and protein aggregation, they do not bind to Aβ(16−22) assemblies in organic solvent,71,72 and CD provides a valuable option. As shown in Figures 9a and S9, the ellipticity at 217 nm73 indicates that β-sheets form immediately. By this analysis, Aβ(16−22) assembles faster in 80% ACN than in 40% ACN (Figure 9a), and all assemblies reach equilibrium by 48 h regardless of the solvent composition, consistent with the TEM analyses (Figure 8e, f, and i). FT-IR analyses of the final products verify amide stretches at 1625 and 1690 cm−1 (Figure S10), consistent with antiparallel β-sheet amyloid fibers under all conditions. Both TEM images and IR spectra then confirm that the structure of Aβ(16−22) fibers are not ACN concentration-dependent, allowing the Aβ(16−22) assembly kinetics under different ACN concentrations to be compared. The experimentally observed kinetics contradict our original hypothesis that the assembly rate should be slower in more hydrophobic surroundings due to a reduced driving force. While the number of particles decreases and the particle size increases with increasing ACN concentrations (Figures 8a, d, and g and S11), the rate of assembly increases (Figure 9a), but

Figure 8. TEM images of 0.5 mM Aβ(16−22) solution in (a−c) 40%, (d−f) 60%, and (g−i) 80% acetonitrile in water. Images are taken after incubation for (a, d, and g) 1 h, (b, e, and h) 5 h, and (e, f, and (i) 48 h. Scale bar = 100 nm.

(Figure 8b), the fiber phase appears together with larger particles, while the density of particles decreases significantly. Such dynamic particle size variation shows the metastable phase to be distinct from amphiphilic micelles, whose sizes are determined by the monomer structure.69,70 Although Figure 8a−c shows that the particle number decreases and particle size increases, these images on dried grids are not sufficient to accurately quantify the overall mass change of the particle phase. After 48 h of incubation, the metastable particle phase is depleted and the stable fiber phase dominates the system (Figure 8c). This series of images indicates that Aβ(16−22) assembles into fibers with the formation and the subsequent dissolution of the metastable particle phase under these conditions.

Figure 9. Kinetics of Aβ(16−22) self-assembly (a) measured experimentally as a function of acetonitrile concentration and (b) simulated as a function of the Flory−Huggins parameter. (a) The CD intensity at 217 nm is normalized to the average value of the last three time points, when the signatures are stable and the kinetics reaches equilibrium. (b) As χ decreases, Aβ(16−22) assembles faster. Table S3 is used for the parameter set with kg0 = 200 mol−1 s−1, kg = 5 mol−1 s−1, ΔG = 10−18.5 J, kn = 5 × 10−7 s−1, and C0 = 0.5 mM. 7408


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The Journal of Physical Chemistry B the final equilibrium position is not impacted (Figure S12). These results are consistent with two key model parameters describing the effect of varying ACN concentration. The model shows that the particle nucleation and particle growth rates are related to supersaturation C1* and thus the Flory−Huggins parameter (χ), while the assembly growth rate depends directly on the rate constant (kg). The assembly growth constant kg is selected instead of the stacking energy (ΔG) because the constant final equilibria suggest that stacking energy remains sufficiently large with different ACN concentrations (Figure S12). Accordingly, simulations with varied Flory−Huggins parameters (χ) and assembly growth constants (kg) are specifically compared with the experimental data. The trends of the Aβ(16−22) kinetics are simulated as a function of the Flory−Huggins parameter (χ), and the peptides assemble more quickly as χ decreases (Figure 9b). As defined in the model and shown in Figure S13, when χ increases, the peptide solubility of particles (C1*) decreases, resulting in fewer free peptides in solution available for assembly. By comparing Figures 9a and 9b, χ and the ACN concentration appear correlated inversely, such that increasing the ACN concentration decreases χ in the system. This observation is consistent with our original hypothesis, since higher solubility is predicted with lower χ. However, the higher solubility of particles has the effect of speeding up rather than slowing down final assembly. The observed inverse relationship between ACN concentration and Flory−Huggins parameter χ, as seen in the assembly fraction kinetics, can be further investigated by comparing particle size. The TEM images at 1 h show that the particle size increases with increasing ACN concentration (see Figures 8a, d, and g and S11), while the observed particle density decreases. The corresponding model predictions show that particle size increases as χ decreases (Figure S14a), with a corresponding decrease in particle density (Figure S14b). Thus, the inverse relationship between ACN concentration and Flory−Huggins parameter χ is further supported by the comparison of model predictions and experimental data for particle size, as well as assembly fraction. The Flory−Huggins parameter is not the only parameter that may affect the assembly kinetics. The assembly growth constant (kg) increases the rate directly (Figure S15), and the simulated results are consistent with the experimental kinetics (Figure 9a). However, kg changes the assembly growth rate only and has no effect on early particle phase evolution, as shown in Figures S16. Neither the number of particles nor the average particle size changes with kg after 1 h of incubation. Hence, although kg may increase with increasing ACN concentrations and accelerate assembly kinetics, it alone is not sufficient to explain the changes in the particle phase in different solvents, as shown in Figures 8 and S11. Originally ACN was expected to slow the assembly kinetics of Aβ(16−22) by increasing the hydrophobicity of the solvent, increasing the solubility of the free peptides, and decreasing the driving force for assembly. However, via two-step nucleation, ACN appears to decrease the driving force for particles instead but does not significantly affect the more stable assemblies. As the solubility of particles increases with the hydrophobicity of the solvent, more peptides become available in the solution phase for assembly growth, which makes the overall kinetics faster.

4. CONCLUSION A peptide assembly model using two-step nucleation is constructed to describe the maturation of assemblies via a metastable particle phase. The model predicts different phase transitions and equilibria are accessible by varying the Flory− Huggins parameter (χ), the particle growth constant (kg0), and the peptide stacking energy in the assemblies (ΔG). The model shows that the solubilities for particles and assemblies, which are significantly influenced by χ and ΔG, determine the final thermodynamic states of the system. When the solubility of the particles is higher than that of assemblies, the particle phase dissolves while the assembly phase either dominates the system or exists in equilibrium with free peptides. When the solubility of the assemblies is higher, the particle phase equilibrates with the solution phase, and no assemblies are apparent. The final state of the system can therefore be rationally predicted given the values of the solubility of the particles and assemblies, but slow particle dissolution can delay maturation of the assembly phase. This model is validated with Aβ(16−22), the nucleation core of amyloid β peptide of Alzheimer’s disease, by evaluating assembly kinetics in solvents with different acetonitrile concentrations. TEM analyses show that the particle number increases and particle size decreases with decreasing acetonitrile concentrations. Both TEM and CD data confirm that the kinetics of Aβ(16−22) assembly are faster with increasing acetonitrile, while the solvent composition does not significantly impact the final equilibrium. The model predicts the more hydrophobic acetonitrile stabilizes the free Aβ(16−22) peptides, leaves more free peptides for assembly, and thus drives the final peptide assembly. These results can now be used to evaluate the assembly of more complex systems, such as recently published dynamic chemical networks36 that juxtapose chemical polymerization with physical phase transitions to create functional assemblies.6

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.7b03085. Ranges of parameters and constants for the model, the Monte Carlo simulation algorithm, the criteria to classify the simulation results, representative results from simulation, and data from CD and IR. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Ming-Chien Hsieh: 0000-0001-9786-8145 Martha A. Grover: 0000-0002-7036-776X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank McDonnell Foundation 21st Century Science Initiative Grant on Studying Complex Systems No. 220020271 and NSF CHE-1507932 for financial support. The authors also thank Robert P. Apkarian Integrated Electron Microscopy Core at Emory University for the assistance in TEM measurements. 7409


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