Abstract
Aggregation of amyloid β (Aβ) peptides is a significant event that underpins Alzheimer disease (AD) pathology. Aβ aggregates, especially the lowmolecular weight oligomers, are the primary toxic agents in AD and hence, there is increasing interest in understanding their formation and behavior. Aggregation is a nucleationdependent process in which the prenucleation events are dominated by Aβ homotypic interactions. Dynamic flux and stochasticity during prenucleation renders the reactions susceptible to perturbations by other molecules. In this context, we investigate the heterotypic interactions between Aβ and fatty acids (FAs) by two independent toolsets such as reduced order modelling (ROM) and ensemble kinetic simulation (EKS). We observe that FAs influence Aβ dynamics distinctively in three broadlydefined FA concentration regimes containing nonmicellar, pseudomicellar or micellar phases. While the nonmicellar phase promotes onpathway fibrils, pseudomicellar and micellar phases promote predominantly offpathway oligomers, albeit via subtly different mechanisms. Importantly offpathway oligomers saturate within a limited molecular size, and likely with a different overall conformation than those formed along the onpathway, suggesting the generation of distinct conformeric strains of Aβ, which may have profound phenotypic outcomes. Our results validate previous experimental observations and provide insights into potential influence of biological interfaces in modulating Aβ aggregation pathways.
Introduction
Aggregation of Aβ constitutes the central process in Alzheimer disease (AD) pathology. Brains of AD patients contain large amounts of proteinacious plaques mainly comprised of insoluble Aβ fibril deposits. Monomeric Aβ peptides (Aβ40 and Aβ42) spontaneously undergo aggregation towards large fibrils in a nucleation dependent manner. The effects of nucleation on aggregation dynamics have been extensively studied^{1,2,3,4,5,6,7,8} that point to a key ratelimiting step for the formation of nucleus/nuclei^{9,10,11,12,13}. Therefore, the dynamics associated with reactions leading up to nucleation is critical for aggregation. Energetically, the prenucleation phase is nebulous with monomers and oligomers (dimers, trimers etc.) involved in a dynamic flux dominated by stochastic interactions^{14,15,16}. Furthermore, intrinsic disorder and amphipathic nature of Aβ facilitate multiple phase transitions and heterogeneous interactions during nucleation, making the process particularly sensitive to environmental factors such as pH, ionic strength, temperature and other interacting partners^{17,18,19,20,21,22}. This is significant because smaller, soluble aggregates have emerged as the primary neurotoxic agents responsible for memory loss in AD^{23,24,25,26}. Furthermore, it is clear that oligomers may not be the obligatory intermediates to fibril formation, and that the oligomers can also be populated along alternate pathways of aggregation (offpathways)^{22, 27,28,29,30,31,32}. Offpathway oligomers differ from those formed along the onpathway resulting in multiple conformational variants with distinct biochemical and cellular properties. Given the conformational diversity among oligomers, it is imperative to determine the factors that affect dynamics involved in such oligomer formation to establish a framework of molecular mechanisms that better defines amyloid progression.
One class of biologically important interacting partners that affect Aβ prenucleation dynamics are the anionic surfactants such as fatty acids and lipids^{33,34,35,36,37,38,39}. Lipids and fatty acids are profoundly important in physiological contexts as they are abundant in both brain tissues and CSF^{40, 41}. The amphipathic Aβ peptide is known to have strong affinity for membranes and hence, these interactions may affect the prenucleation dynamics. Several reports also demonstrate the effects of phospholipids on Aβ aggregation^{42,43,44}. Similarly, both polyunsaturated (PUFAs) and saturated fatty acids are also known to have significant effects on the AD brain^{45, 46}. Kumar and coworkers have previously reported the generation of 4–5 mers and 12–24 mers from lauric acid along a nonfibril formation pathway^{22}. More importantly, using carbon chain lengths of C9C12 fatty acids (FAs), they established that below (nonmicellar), near (pseudomicellar) and above (micellar) their respective critical micelle concentrations (CMCs), FAs generate Aβ oligomers or fibrils via distinct pathways^{22}. A schematic of the switching dynamics from on to the offpathway with increasing concentration of FAs is shown in Fig. 1. Here the bold horizontal arrow signifies increasing FA concentration while the vertical arrows conceptually depict the dynamical aggregation regimes of the system with the inclusion of monomeric Aβ. It can be observed that low FA concentration involves the nonmicellar phase where only the onpathway reactions are active; while high FA concentration (beyond the CMC) involves the micellar phase where only the offpathway reactions are active (with onpathway being completely switched off). The middle phase of the FA concentration arrow denotes the nearCMC (i.e., pseudomicellar) phase and is the most interesting one; here, both on and offpathway reactions are active and the monomeric Aβ may form aggregated oligomers from both pathways simultaneously (as denoted by the arrows).
In this report, we sought to understand the dynamics of heterotypic interactions between Aβ and FAs and how phase transitions of FAs modulate Aβ aggregation. We first use a reduced order model (ROM) and a corresponding mathematical analysis rooted in a linear stability argument to identify the switching behavior from on to offpathway in the presence of micellar interfaces. The ROM can be thought of as a toyversion of the far more complex biophysical problem. However, despite considering only 4species in this model, the ROM still deals with a highly nonlinear problem which captures the overall aggregation dynamics of the Aβ system and allows for rigorous examination of features such as dynamic stability which would be intractable in the case of a very fine grained model. Mathematical biologists consistently simplify complex biological problems to gain initial insight into problems, to understand the nature of equilibria, stability, bifurcations in the larger system, before venturing to speak of specific details which is the objective of the ROM. This analysis was used as the basis to further develop a more detailed model, which was built around the discoveries of the ROM and experimental data.
Secondly, using an ensemble kinetic simulation (EKS) model, we demonstrate that in three distinct phasetransition regimes, FAs significantly and consistently modulate Aβ aggregation by altering the pathways. These results are in agreement with the reported in vitro experimental observations, and provides detailed molecular insights into the heterotypic interactions between Aβ and FAs. These insights also reveal that generation of conformationallydistinct oligomeric strains (produced by the oligomers involved in the on and offpathways) are highly likely in physiological environments containing lipid and FA interfaces due to the dynamics involved in the heterotypic FAAβ interactions.
Results
Stability analysis for a reduced order model (ROM)
We examined the sets of on and offpathway reactions shown below. The model system considered is as follows:
where the reaction 2 is referred to the onpathway, while reaction 3 is the offpathway. In these reactions, A_{1} denotes the Aβ monomers while ${A}_{1}^{\prime}$ denotes the offpathway monomers created in the presence of fatty acids, FA (considering critical micelle concentration, CMC). A_{ n } are the onpathway oligomers while ${A}_{n}^{\prime}$ are the offpathway oligomers made of n monomeric units. Finally, ${k}_{i}^{+}$ and ${k}_{i}^{}$ are the forward and backward rate constants of the corresponding reactions (i = 1, 2, 3). The details of the modeling methodology are elaborated in the Methods section. We are aware that the onpathway fibrils are considerably larger than the terminal species of the offpathway, i.e. size of (A_{ n }) ≫ size of (A^{′}_{ n }) or n ≫ n′. In the ROM analysis however, both pathways terminate at the same value of n since we believe this to be sufficient to elucidate the competition between the two pathways in our model. Since the only “bridge” from the on to offpathway is between the monomeric species, once the reaction chooses the path from ${A}_{1}\to {A}_{n}$, the onpathway dominates in the steady state (or viceversa) and higher order oligomer dynamics do not reveal anything more about the process. The asymmetric nature of the two pathways is however, considered in our followup analysis to understand detailed mechanistic phenomena.
Based on the results of our analysis presented in the Methods section, Fig. 2 shows the eigenvalues λ_{ i } (i = 1–4), of the Jacobian matrix for the mass action kinetic equations (18–21). These eigenvalues appear as a function of α_{3}, a normalized parameter which depend on the CMC and rate constants. The eigenvalues depict the rate of decay of any perturbation to the stable, nonzero equilibria of the system of equations (18–21). It so happens that the critical parameter α_{3} is key to determining whether the onpathway or offpathway is dominant. The Fig. 2 shows some significant events in the dynamics of this system. As α_{3} is increased from zero, the real part of the eigenvalues λ_{ i } are all negative, indicating that the equilibria are stable. When α_{3} < 1.0, we observe that λ_{2} < λ_{1}. However, when α_{3} > 1.0, it follows that λ_{2} > λ_{1} indicating an exchange in the magnitude of the stability. For α_{3} > 1.0, the species B_{1} (dimensionless version of A_{1}) dominate in magnitude and also become more stable than ${B}_{1}^{\prime}$ (dimensionless version of ${A}_{1}^{\prime}$), indicating a transcriticallike bifurcation at α_{3} = 1.0.
Note that in the dynamical systems literature, transcritical bifurcation refers to a stable equilibrium and an unstable equilibrium which exchange stabilities with respect to some parameter^{47}. In our model system, the switching of stabilities is revealed by α_{3}, which, by its definition, contains the ratio of the forward to backward rate constants for this reaction, as well as the fatty acid concentration which enhances the formation of the offpathway species. Figure 3(b) shows the rate of reformation of the species B_{ n } and ${B}_{n}^{\prime}$, which is slower than the monomeric species.
Another interesting phenomenon that occurs in this system is a discontinuity in the eigenvalues or second switching which occurs at around α_{3} = 1.87 (see Fig. 3(a)). The Table 1 in the Supplementary Information captures the variety of switching behaviors and changes in behavior of the ROM system when different parameters (rate constants) are varied. At this stage, the biophysical implication of this second critical point is unclear to us although it is most certainly a symptom of the high nonlinearity of the system.
We explore the effect of n on our system, for 1 ≤ n ≤ 20. Our analysis reveals significant changes to the stability of the system as n varies; in particular, the eigenvalues λ_{1} and λ_{2} show some dramatic changes with n. For 1 ≤ n < 12, multiple switching points (at least two) were observed with λ_{2} eventually being more negative than λ_{1}, i.e. offpathway eventually becomes more stable for sufficiently large α_{3}. However, for 12 ≤ n ≤ 20, the onpathway eventually becomes more stable. The physical significance of the multiple switching, in particular the one beyond α_{3} = 1 still eludes us and it remains to be verified if this an indication of some underlying bifurcation in the system or is simply an artifact of the nonlinear system or an indication of some nucleation type event observed in our previous work using similar methods^{48}.
The results of our stability analysis with support from the Nash equilibrium for this system (see Methods section), allows us to draw the following conclusions: (i) The simplified, though highly nonlinear system of ROM equations with 4species captures the essential dynamics of the Aβ system and can help explain the switching dynamics considering only the CMC; (ii) The nondimensional parameter α_{3} is seen to be the key control parameter in this analysis indicating that FA concentration, along with the ratio of the rate constants along this “bridge” reaction pathway determine the ultimate fate of the system. (iii) The presence of a second, discontinuous switching of eigenvalues at α_{3} ≥ 1.87 potentially indicates a different biophysical phenomenon occurring along the offpathway implicating the involvement of additional species that may play an important role in the FA dynamics. To explore the dynamics between on and offpathways of Aβ aggregation, we then conducted a more detailed ensemble simulation using chemical kinetics under varying concentration of FAs.
Ensemble kinetic simulation (EKSs) on AβFA dynamics
In order to further investigate the dynamics between the on and offpathways in the presence of FAs, we used the EKS approach. EKS approach involves numerical ordinary differential equation (ODE)based simulations that encompasses ensembles of the dynamic species involved in the reaction flux. Fundamental aspects of EKSs were used in the onpathway model that we had previously reported^{10, 48}. In this report, we have expanded the EKS model to simulate various phasetransition regimes of FAs and Aβ, especially along offpathway (elaborated in the Methods section). Before modeling on and offpathway dynamics, it is imperative to simulate the phasetransitions of FAs during micelle formation. We specifically chose three concentration regimes at which FAs show distinct phase changes: low concentrations where the FAs are nonmicellar (FA_{ n }), near their CMCs where they are pseudomicellar (FA_{ pm }), and high concentration where they are fully micellar (FA_{ m }). We conjecture the nomenclature, pseudomicelles to define the dynamic state between a nonmicelle and a micelle, which seemed to affect Aβ in a distinct way^{22}. The hydrodynamic radii (Rh) estimates of FA_{ pm } and FA_{ m } indicated that the former is ~6–7 times larger than the latter, confirming the looselyheld, amorphous state of FA_{ pm } (Table 1).
Dynamics in nonmicellar phase  Aβ–FA _{ n } interactions
First, for the control experiment in the absence of FAs (onpathway reactions, exclusively), we find the forward rate constants during the nucleation stage (k_{ nuon }) is considerably lower than the forward rate constant (k_{ fbon }) of elongation stage, as expected (Fig. 4(a)). The simulated data showed good correlation with the experimental data with an R^{2} value of 96.2, and an equally good agreement in the lag times (Fig. 4(a)) for C12, as well as C9C11 FA data (Fig. 4(b)). The estimated rate constants for control, i.e., onpathway, are shown in Supplementary Table 2 in AppendixB under Supplementary Information. Secondly, in the presence of FA_{ n }, onpathway reactions still dominate due to the absence of micelles (or pseudomicelles) in the solution. We hypothesize that FA_{ n }s interact with Aβ and catalyze aggregation by altering the rate constants along the onpathway by a factor of K, resulting in a change in the final fibril concentration. The corresponding set of reactions for this model are shown in AppendixC under Supplementary Information. We obtain the value of K < 1 for the best fit with the experimental data pointing to a slowdown in the elongation stage of the onpathway (Supplementary Table 3 in AppendixC under Supplementary Information).
The simulations of Aβ interactions with C12 FA_{ n } when compared to the onpathway fibril growth characteristics, provided two important observations. First, the overall aggregation was augmented with the introduction of FA_{ n }, consistent with the overall ThT intensity observed in experiments (Fig. 4(a); red). Second, the third phase of fibril growth curve (i.e., the slower growth phase towards saturation subsequent to the exponential growth phase) was delayed for a considerable amount of time with C12 FA_{ n } (as opposed to the onpathway), despite no change in the lag time (Fig. 4(a)). We hypothesize these effects are caused by the introduction of FA_{ n }, which modulate the onpathway rate constants. To test this hypothesis, we systematically analyzed the effects that FA_{ n } may have on the rates of onpathway reactions at critical times of the aggregation process. First, the forward rate constants during the prenucleation stage was altered (increased/decreased) by a factor of K, which did not fit the fibril growth curve for the C12 FA_{ n } case and concomitantly could not validate this behavior (data not shown). Next, upon varying the forward elongation rate constant (k_{ fbon }), a small decrease in the forward elongation rate constant render higher ThT intensity with delayed saturation time, and without changing the lag time. This suggests that the addition of FA_{ n } molecules effectively decrease the forward elongation rate constants. According to the Lee’s and our own onpathway models^{48, 49}, the elongation stage (ie. ${A}_{12}+{A}_{1}\leftrightarrows {A}_{12}$; ${A}_{12}+{A}_{2}\leftrightarrows {A}_{12}$; … and ${A}_{12}+{A}_{11}\leftrightarrows {A}_{12}$) does not lead to the increase in ThT intensity directly as these reactions may not increase the number of ThT binding sites significantly to augment fluorescence intensities. Hence, a decrease in the elongation rate constant essentially extends the duration of the prenucleation stage, although without increasing the lag time. This subsequently could increase the formation of prenucleated onpathway aggregates (specifically 12 mers, i.e., A_{12}s) resulting in higher ThT intensity/fibril growth. The increased number of A_{12}s resulting from these reactions then react with one another as well as the onpathway oligomers during the prenucleation stage (monomers to 11 mers, i.e., A_{1}–A_{11}) to generate higher order aggregates with concomitant higher ThT intensities. The predicted K value for different initial FA_{ n } concentrations is shown in Supplementary Table 3 in AppendixC under Supplementary Information. Similar corresponding plots for ThT vs simulations for C9C11 FA_{ n } are shown in Fig. 4(b). The high R^{2} values obtained for the fits (Supplementary Table 3) suggest that FA_{ n }s promote onpathway fibrils by a slightly different mechanism as compared to the Aβ aggregation in the absence of FAs. Hence, as shown in Fig. 4(c), the ratio of ${A}_{4}^{\prime}$/A_{4} is effectively zero.
We further consider two hypothetical but physically relevant scenarios in which varying rates of entry of Aβ monomers and FAs were considered as opposed to a fixed concentration^{50}. The reaction models for this scenario were then minimally modified to consider these additional entry rates as variables to evaluate the oligomer concentration dynamics. As expected in Fig. 4(c), the interaction of FA_{ n } with Aβ generated no offpathway oligomers due to the absence of micelles or pseudomicelles. Only onpathway aggregates were formed exclusively, although by a slightly different mechanism. Based on these results, we determine that FA_{ n }s decrease the rates during the fibril elongation stage along the onpathway, which seems to hold good for each type of FA used (C9C12).
In (Fig. 4(d)), we report the oligomer ratios (of the same size) between the control and FA_{ n } setups. We considered four oligomers for this test: A_{1}, A_{4}, A_{8}, A_{12}; in each case, these ratios rise above the value of 1, showing that the presence of more belowCMC oligomers than onpathway oligomers in the system as time increases. Note that these were generated from two standalone simulations, one for the control and the other one for FA_{ n } as in both cases only onpathway oligomers are produced. The results clearly suggest that the FA_{ n } scenario produces more oligomers (and possibly more fibrils) with time which supports the experimental observation of increased ThT intensity for FA_{ n }. Additionally, as expected the ratios are smaller for larger oligomers; this is simply because of the reaction order as fewer oligomers of larger size are formed in the system. Thus our reaction hypothesis of a slowdown of the elongation stage in the presence of FA_{ n }s, although counterintuitive, accurately validates the experimental observations of increased ThT intensity.
Dynamics in pseudomicellar phase  Aβ/FA _{ pm } interactions
FA concentration around the CMC increases the concentration of FA_{ pm } in the bulk solution, the presence of which shifts the reaction flux towards predominantly offpathway as established experimentally^{22, 51} (Fig. 1). As the forward elongation rate constant becomes crucial only at the later phase of the onpathway reactions, the effect of K was neglected here. This allowed us to assume that effectively all FAs convert to FA_{ pm }s and thus making the effects of free FAs during the onpathway elongation stage negligible. The predicted rate constants from the control experimental data were used here for the onpathway set of reactions, and the parameter space was iterated to only estimate the rate constants involving offpathway reactions. Thus, our combined on and offpathway model was able to explain all the experimental observations in the presence of FA_{ pm } (corresponding R^{2} values are reported in Table 2 for C9C12 FAs). Furthermore, previous reports identified that 12–24 mers (LFAOs) are the predominant species generated in the presence of C12 FA_{ pm }^{22, 51}. Our simulations predict the presence of multiple conformers within the oligomer of a specific size (strains) as observed before^{52}. The 12–24 mer LFAOs show conformational dynamics (represented as ${F}_{1}^{\prime}$ and ${F}_{1}^{\u2033}$ here), wherein the ${F}_{1}^{\prime}$, and not ${F}_{1}^{\u2033}$, is able to associate with itself. Note that both species (${F}_{1}^{\prime}$ and ${F}_{1}^{\u2033}$) correspond to 12–24 mer LFAOs and we considered up to fourfold lateral associations between them to form ${F}_{4}^{\prime}$ which are essentially 48–96 mers as previously established^{52}. Such ${F}_{4}^{\prime}$ LFAOs then undergo a secondary fragmentation to produce four ${F}_{1}^{\u2033}$ which are not allowed to laterally associate thereby making the 12–24 mers as the predominant species. This is due to subtle yet distinct structural differences between the two 12–24 mer LFAOs^{53}. In our model, we identified that lateral association and fragmentation are both critically important in Aβ/FA_{ pm } reaction dynamics, which ultimately explains the formation of stable oligomers of size 12–24 i.e., ${F}_{1}^{\u2033}$. The predicted concentration of pseudomicelles are shown in Table 2 while the predicted rate constants and shown in Supplementary Table 4 in AppendixD under Supplementary Information.
Overall simulation of the ThT intensity data for FA_{ pm } regime yielded good correspondence with the experimental results for C12 (Fig. 5(a); green). We obtained a high R^{2} value and nearly similar correspondence with the experimental lag time and saturation time for C12 as well for C9C11 FAs (Fig. 5(b)). The comparative concentration dynamics of offpathway and onpathway aggregates in the presence of FA_{ pm } are shown in Fig. 5(c–f) for C12 fatty acid. Figure 5(e) is the data for constant FA_{ pm } and varying Aβ monomer entry rates while Fig. 5(f) is the data obtained for constant monomers and varying FA_{ pm } entry rates. In our model, for the sake of simplicity and comparison, we considered aggregates of similar sizes from both on and offpathway. We observe that the onpathway oligomers A_{4}, A_{7} and A_{12} show delayed emergence (Fig. 5(c)). It is noteworthy that A_{4} emerges rapidly, and saturates before decreasing in concentration. This behavior is due to the elongation of oligomers by A_{12}s that occurs along the onpathway, which form larger fibrils at the cost of A_{4} consumption. Along the offpathway, aggregates ${A}_{4}^{\prime}$, ${A}_{8}^{\prime}$ and ${A}_{12}^{\prime}$ show expected characteristics of being formed rapidly initially before saturating (Fig. 5(d)). It is important to note that the elongation in the offpathway leads to ${F}_{1}^{\prime}$ consuming ${A}_{4}^{\prime}$ and ${A}_{12}^{\prime}$ oligomers to form the 12–24 mers as discussed previously. Such ${F}_{1}^{\prime}\mathrm{s}$ then associate to form ${F}_{4}^{\prime}$ which are seen in greater quantities towards the beginning with their concentration decreasing with time, which is likely due to fragmentation of high molecular weight aggregates. Finally, most of the FA_{ pm } dynamics leads to the formation of the ${F}_{1}^{\u2033}$ with time, which becomes the predominant ThT positive species in the system. This dynamics is further corroborated by incorporation of physiologically relevant monomer entry rates into the system (Fig. 5(d)). Here, the dynamics show more conclusively the dominance of the ${F}_{1}^{\u2033}$ oligomer in the system with increasing time followed by ${A}_{8}^{\prime}$ and ${A}_{4}^{\prime}$. In order to verify the potential fragmentation identified by our simulation, we investigated this possibility experimentally by incubating Aβ and C12 FA near its CMC (Fig. 6). The results indicate that incubation rapidly results in the formation of aggregates corresponding to 50–60 nm diameter within one hour of incubation (Fig. 6(a) and (b)). However, after two hours of incubation, the size of the aggregates decreased to 10 nm diameter. No more change in the aggregate size was observed even after 48 h (data not shown). This phenomenon was also observed in the aggregate size monitored by immunoblotting (Fig. 6(c)), and supports the possibility of fragmentation of aggregates as predicted by the EKS model predictions.
Figure 5(e) and (f) show the ratio of the oligomer distributions for C12 fatty acids considering constant entry rates of Aβ monomers with fixed initial FA_{ pm } concentration (Fig. 5(e)) and FA_{ pm } into the system for fixed initial monomer concentration (Fig. 5(f)), respectively, as described before for FA_{ n }. These plots illustrate the switching behavior between the on and offpathways as predicted by the ROM stability analysis (Figs 2 and 3). For example, we can observe that for negligible monomer or FA_{ pm } entry rates, the onpathway is active, but the switching occurs quickly to offpathway when the monomer entry rate is about 0.001 × 100 μM h^{−1} (Fig. 5(e)). This switching effect is however faster when the FA_{ pm } entry rate is considered as shown in Fig. 5(f) (the amplitude of the ${F}_{1}^{\u2033}/{A}_{12}$ ratio is higher in the latter case). Also, the ratios are higher after 50 h of incubation as compared to after 100 or 200 h (Fig. 5(e) and (f)). This suggests that either the ${F}_{1}^{\u2033}$ concentration decreases with time or the A_{12} increases with time or both cases happen simultaneously. As shown in Fig. 5(d), the increase in A_{12} is more pronounced that that in ${F}_{1}^{\u2033}$ concentration with time (note that both concentration increase with time) resulting in a decrease in the recorded ratio with time. Nevertheless, we can conclude that generally the switch to offpathway is extremely fast with higher FA_{ pm } entry rates being a more crucial factor than an increase in the monomer entry rates for the observed effect.
Comparing the FA_{ pm } ThT intensity plots (Fig. 5(b)), we additionally observe that the concentration of FA_{ pm } significantly affects the behavior of aggregation. For example, higher the FA_{ pm } concentration present in the system (correlates with smaller fatty acid chain length), more the saturation is delayed for the ThT growth curve. As expected, our predicted FA_{ pm } concentration is highest near the CMC, and the concentration starts to decrease when total FA concentration is either increased or decreased. The predicted FA_{ pm } concentration for each fatty acid chain length at the near CMC zone experiments is shown in Table 2. Note that the prediction for the C9 case shows a small deviation primarily because the experiment was done at a 100 mM concentration while the actual CMC was 85 mM, i.e., a deviation of 17.6%.
Dynamics in micellar phase  Aβ − FA _{ m } interactions
The aboveCMC regime (FA_{ m }) only produces ${A}_{4}^{\prime}$ along the offpathway that do not seem to aggregate any further due to the stabilization by the micelles^{22} (Fig. 7(a)). Therefore, the corresponding reaction model only considers the prenucleation stage in the offpathway. Furthermore, the ${A}_{4}^{\prime}$ oligomers do not bind to ThT and hence, are invisible to this assay. This complicates the precise predictions of ${A}_{4}^{\prime}$ concentrations and direct correlation with the experimental data. Therefore, we could not estimate the accurate rate constants for FA_{ m }. It is noteworthy that we assumed the offpathway reactions in the presence of FA_{ pm } also undergoes similar prenucleation stage reactions besides others. The rate constants of the prenucleation stage from FA_{ pm } data was used for this simulation to determine the dynamics of ${A}_{4}^{\prime}$ production. As shown in Fig. 7(b) and (c), one could observe that ${A}_{4}^{\prime}$ is generated rapidly accompanying a switching to the offpathway, even with negligible monomer or micelle entry rates. As the ${A}_{4}^{\prime}$ formation presumably involves a concerted single step reaction, the concentration profile is different from its onpathway counterpart, A_{4}. Specifically, as micelle entry rate increases, number of micelles in the system also increases such that the ${A}_{4}^{\prime}$ concentration increases much faster than that of the A_{4} concentration as shown in Fig. 7(c). However, Fig. 7(c) suggests that for a fixed initial micelle concentration, and as the Aβ monomer entry increases, the ratio of ${A}_{4}^{\prime}/{A}_{4}$ goes towards saturation. This is potentially due to reduction in the number of micelles available for accommodating Aβ on the micellar surface.
Discussion
The results reported here by two independent methods (ROM and EKS) reaffirm the experimental observations that FAs modulate Aβ aggregation. Specifically, the following significant conclusions can be drawn from this work: (a) interactions of Aβ with nonmicellar FAs (FA_{ n }) generate predominantly high molecular weight, onpathway aggregates. (b) Interactions of Aβ with FAs near the latter’s CMC (FA_{ pm }) generate predominantly low molecular weight oligomers (12–24 mers) along offpathway along with some minor quantities of onpathway fibrils. (c) Interactions of Aβ with high concentrations of FAs (FA_{ m }; above their CMCs) exclusively generate offpathway 4–5 mer oligomers without any fibrils suggesting that the onpathway was completely turned off.
Perhaps most significantly, simulations from both ROM and EKS, as well as the previously demonstrated experimental evidence have identified that both concentration and phasetransitions of FAs dictate switching of Aβ aggregation pathways. Furthermore, the report demonstrates that under three distinct phases, FA_{ n }, FA_{ pm }, and FA_{ m }, the FAs not only modulate Aβ aggregation pathways but also promote low molecular weight oligomers, which are increasingly known to be the main pathogenic species in AD and other related neurodegenerative disorders. Specifically under pseudomicellar conditions, where FA_{ pm } undergoes a distinct phase transition, they preferentially promote and stabilize lowmolecular weight oligomers. The ability of certain phases of FAs, or in general any biological surfactants, to promote the formation of low molecular weight, offpathway presents high significance in AD pathology. This is because oligomers generated in this manner could lead to the formation of various toxic ‘conformeric strains’ of Aβ aggregates, which could manifest in distinct phenotypic manifestations observed in AD. Indeed, Dean et al., have shown that LFAOs, which are generated in the presence of C12 FAs, are able to propagate morphologically distinct fibrils and caused excessive cerebral amyloid angiopathy (CAA) in transgenic mice brains, which was absent in similar reactions seeded with onpathway fibrils^{52}.
Mechanistically, simulations reported here have revealed a deeper understanding of AβFA interactions. Some of the salient dynamics involved in the modulation of Aβ aggregation by FAs in distinct phases include: (i) in the presence of FA_{ n }, the free FAs seem to slow the kinetics of elongation (or postnucleation) stage along the onpathway, (ii) in the presence of FA_{ m }, aggregation does not proceed beyond ${A}_{4}^{\prime}$ (or ${A}_{5}^{\prime}$) likely due to the stabilization of Aβ by FA_{ m } as AβFA_{ mm } complex. Further investigation of this complex by molecular dynamics based studies to infer why such complexes cannot aggregate is ongoing and will be reported at a later date, and (iii) in the presence of FA_{ pm }, Aβ forms A_{4} during the prenucleation phase followed by a slower progression to ${A}_{12}^{\prime}$, and elongation to form 12–24 mers denoted by ${F}_{1}^{\prime}$. Association of ${F}_{1}^{\prime}$ to form ${F}_{4}^{\prime}$, which is thermodynamically unstable, leads to a fragmentation to form ${F}_{1}^{\u2033}$ as supported by experimental evidence (Fig. 6). The most likely possibility is that the ${F}_{1}^{\u2033}$ are structurally distinct and different from ${F}_{1}^{\prime}$ that are trapped in some kinetic minimum along the offpathway as thought to be^{22, 51}, which renders them incapable of aggregating further.
In addition to these insights, EKS model also considered a physiological scenario in which instead of constant amount or Aβ or FAs, a constant influx of Aβ monomers as well as pseudomicellar and micellar FAs were considered. Incorporating specific entry rates for Aβ and FAs, we observe that the presence of pseudomicellar phase influences the switching of pathways more than that induced by the monomers. However, we observed that even a slight increase in the entry rates of either FA or Aβ is enough to switch the dynamics preferentially towards the offpathway. This is evident in Fig. 8, which shows a surface plot which considers the monomer and FA_{ pm }/FA_{ m } rates. The corresponding oligomer concentration ratios along off and onpathway show that the onpathway is active only when FA_{ pm } entry rates are very low (Fig. 8(a) and (b)). After 50 h of incubation, offpathway is predominantly activated for the highest FA_{ pm } entry rate and low monomer entry rates. In other words, all monomers entering the system are consumed by FA_{ pm } leaving little scope for onpathway reaction to occur. However, after 200 h, offpathway is most active when both FA_{ pm } and monomer entry rates are high resulting in most monomers being consumed by the FA_{ pm }. The scenario is bit different for FA_{ m }, where the offpathway is most active when both monomer and FA_{ m } entry rates are high both after 50 and 200 h (Fig. 8(c) and (d)). The dynamics seems to be more stable after 200 h than after 50 h, as oligomer concentrations saturate by then. However after 50 h, a switch to offpathway happens only with sufficiently high Aβ monomer and FA_{ m } entry rates, and there may be a critical point in this entry rate for both that would dictate whether a switch to the offpathway occurs or not. In later times though (200 h, for example), there is a higher propensity of the system to switch to the offpathway even with lower monomer or micelle entry rates.
Overall, this report has brought forth several interesting observations on the dynamics of AβFA interactions, which seems to depend critically on the concentration and phase of FAs. Furthermore, the switching dynamics in conjunction with the constant infusion of Aβ monomers and FAs in their distinct phases illustrate the potential of biological interfaces to influence Aβ into forming several different conformationally distinct aggregates. Such conformeric strains could then influence phenotypic outcomes in AD pathology.
Methods
Experimental
Preparation of large fatty acidderived oligomers (LFAOs)
C12:0 fatty acid (5 mM) with 50 mM NaCl was added to freshly purified Aβ monomer (50 μM) buffered in 20 mM Tris, pH 8.0, and kept at 37°C in quiescent conditions. Separate reactions were started in a staggered fashion such that at the time of data collection, the sample would reach the desired time point.
Dynamic light scattering (DLS)
DLS data was collected using a Zetasizer Nano S instrument (Malvern, Inc., Worcestershire, UK). At each time point, DLS was collected by averaging 24 runs of 5 s each with a pre equilibration time of 30 s. This average was used to determine the diameter (nm) using the volume (%) function. For determining micelle size without Aβ, stock concentrations of C9:0, C10:0, C11:0, and C12:0 were diluted to the appropriate concentration in 50 mM NaCl buffered in 20 mM Tris, pH 8.0.
SDSPAGE with immunoblotting
LFAO preparations were staggered in start such that at the time of initiating SDSPAGE experiments, each sample was at the desired time point. Samples were diluted into 1X Laemmli loading buffer with 1% SDS, and loaded onto 4–20% BioRad TGX gels without boiling. Prestained MW markers (Novex Sharp Protein Standard, Life Technologies) were run in parallel for MW determination. Proteins were transferred to 0.2 μm nitrocellulose membrane (BioRad) and boiled for 1 min in a microwave oven in 1X PBS, followed by blocking for 1.5 h in 1X PBS containing 5% nonfat dry milk with 1% tween 20. Blots were then probed overnight at 4 °C with a 1:6000 dilution of Ab5 monoclonal antibody, which detects amino acids 1–16 of Aβ. Blots were then incubated with a 1:6000 dilution of antimouse, horseradish peroxidase conjugated secondary antibody and developed with ECL reagent (Thermo Scientific).
Reduced Order model (ROM)
The system of differential equations corresponding to Eqns 1 determines the evolution of the 4 species (${A}_{1},{A}_{1}^{\prime},{A}_{n}^{\prime},{A}_{n}^{\prime}$) and are given by
The above equations are better suited for analysis in nondimensional form. If we chose the characteristic concentration and time as A_{0} (the equilibrium concentration of monomers) and $\frac{1}{{k}_{1}^{}}$, respectively, then the dimensionless variables can be defined as follows:
The corresponding dimensionless differential equations, in terms of dimensionless time ‘s’, now can be written as
Equilibrium points
In the equations (4–7), if we consider the limit $({k}_{1}^{+},{k}_{1}^{},FA)\to (0,0,0)$, then the offpathway dynamics is turned off, leaving only the reaction
$$n\cdot {A}_{1}\underset{{k}_{2}^{}}{\overset{{k}_{2}^{+}}{\rightleftharpoons}}{A}_{n}.$$However, the onpathway cannot be switched off and always persists. In this case the equilibrium solution is given by the pair $\left({A}_{0},{\left(\frac{{k}_{2}^{+}}{{k}_{2}^{}}{A}_{0}\right)}^{\mathrm{1/}n}\right)$.
Turning our attention now to the more general case of the nondimensional equations (9–12), the equilibrium points, ${\mathbf{B}}_{e}=({B}_{\mathrm{1,}e},{B}_{\mathrm{1,}e}^{\prime},{B}_{n,e},{B}_{n,e}^{\prime})$ can be obtained by the vanishing of the equations (9–12). Solving for these four simultaneous equations, we obtain the following relations, in terms of the equilibrium concentration B_{1,e }
The subscript e indicates that these species are in equilibrium.
It should also be noted that if one treats the on and offpathway dynamics as a “competition” in accordance with the term used in gametheory^{54, 55}, then the equilibrium ${\mathbf{B}}_{e}{}_{{\alpha}_{3}=1}={\mathbf{B}}_{Nash,e}$ is nothing but the Nash equilibrium of the system given by
$${\mathbf{B}}_{Nash,e}:=\left({B}_{\mathrm{1,}e},{B}_{\mathrm{1,}e},\frac{{\alpha}_{2}}{{\alpha}_{1}}{B}_{\mathrm{1,}e}^{n},\frac{{\beta}_{2}}{{\beta}_{1}}{B}_{\mathrm{1,}e}^{n}\right)$$(see Fig. 3 for an example of such an equilibrium). This equilibrium exists only at α_{3} = 1 which also will be seen to have special significance in terms of the stability of the system. It follows that when α_{3} > 1, then the offpathway species dominate creating more equilibrium concentrations of ${B}_{\mathrm{1,}e}^{\prime}$ and for α_{3} < 1, onpathway dominates resulting in greater production of ${B}_{\mathrm{1,}e}$.
We define ${k}^{+}={k}_{1}^{+}FA$ and ${k}^{}={k}_{1}^{}$ so that the former leads to offpathway and the latter to onpathway. Table 3 below points to a sample case of the various strategies in the competition between the two pathways with the payoff for each “strategy” given by the equilibrium concentrations.
Time evolution of the different species
The system of equations (4–7) were solved using the Matlab ode45 function. A sample solution is shown in Fig. 2 corresponding to initial conditions: ${B}_{1}(0)=2$, ${B}_{1}^{\prime}(0)={B}_{n}(0)={B}_{n}^{\prime}(0)=0$, ${\alpha}_{1}={\beta}_{1}=1$, ${\alpha}_{2}={\beta}_{2}=1$ and n = 4. Also, here ${\alpha}_{3}=1$ which results in ${B}_{1,e}={B}_{1,e}^{\prime}$.
Linear stability analysis
To study the stability of the four species, we linearly perturb the system which is mathematically represented by
then the linearized set of differential equations for the perturbations, at o(ε) becomes
which can be expressed in operator form with the perturbation matrix, B given by
$$\mathbf{B}=\left(\begin{array}{cccc}{n}^{2}{\alpha}_{2}{B}_{\mathrm{1,}e}^{n1}{\alpha}_{3}& 1& n{\alpha}_{1}& 0\\ {\alpha}_{3}& {n}^{2}{\beta}_{2}{\alpha}_{3}^{n1}{B}_{\mathrm{1,}e}^{n1}1& 0& n{\beta}_{1}\\ n{\alpha}_{2}{B}_{\mathrm{1,}e}^{n1}& 0& {\alpha}_{1}& 0\\ 0& {\beta}_{2}{\alpha}_{3}^{n1}{B}_{\mathrm{1,}e}^{n1}& 0& {\beta}_{1}\end{array}\right)$$The stability of the equilibrium is found from computing the eigenvalues, λ_{ i } (i = 1, 2, 3, 4) of the matrix B. This is best done numerically. In the figures below we discuss the results of the computations. In particular, the term α_{3} is varied throughout the analysis since it captures a significant dynamical feature of this problem; the switching from onpathway to offpathway. Over all, we choose the following ranges for our parameters: ${10}^{10}\le {\alpha}_{i},{\beta}_{i}\le {10}^{10}$ for i = 1, 2. Also, $1\le n\le 20$ and $1\le {B}_{1,e}\le 20$. We begin with the base case defined by the choice ${\alpha}_{1}={\beta}_{1}=0.1$, ${\alpha}_{2}={\beta}_{2}=100$, n = 4 and B_{1,e } = 1 and then study the sensitivity of the system and our results to each of the parameters.
Ensemble Kinetic Simulations (EKS) of on and offpathways
Experimental observations and model assumptions
The detailed simulations and subsequent analysis are based on prior work where the invitro experiment using the nonesterified fatty acids of chain length C9 to C12 revealed a few crucial characteristics of Aβ42 aggregation^{22}:

1.
At FA concentration less than CMC range (FA_{ n }), only onpathway aggregates were produced. Furthermore an increase in ThT intensity than the control experiment was detected.

2.
At near CMC (FA_{ pm }), offpathway fibrils of length 12–24 mers were noticed with very few fibrils being produced leading to the conclusion that the onpathway was almost switched off.

3.
At FA concentration higher than CMC (FA_{ m }), offpathway oligomers of length 4–5 mers were observed while no fibrils were produced, i.e., the onpathway was switched off completely.
Before establishing a proper modeling of the offpathway, we wish to explore the formation of micelles first. Ideally, CMC is the concentration of the surfactants (here, FAs) in the solution from where the micelle formation starts. Therefore, if high concentration of surfactants are introduced into the solution (FA_{ m }), it directly forms micelles. At FA_{ n } and FA_{ pm } concentrations, ideally there should not be any micelles present in the solution whereas at FA_{ m } range, almost all of the FAs should convert into micelles. As discussed earlier, at FA_{ pm }, the FAs loosely form pseudomicelles that are sixseven times larger than regular micelles. It can hence be hypothesized that as the FA concentration is increased beyond the CMC, extra FA molecules bind to the pseudomicelles, which may then undergo structural change to form more stable and compact structured micelles (i.e., L) that were considered in the ROM model.
Based on the experimental observations, we make the following considerations to formulate the underlying reaction mechanism.

1.
At FA_{ n } range, only onpathway reactions take place, as in this zone no micelles are present in the solution. We assume that the presence of FAs have catalytic effects in altering the rate constants of the onpathway with a factor of K resulting in a change in the final fibril concentration; the corresponding set of reactions are shown in AppendixC under Supplementary Information.

2.
At FA_{ m } concentrations, both onpathway and offpathway occur simultaneously; however, as seen in the ROM model, the existence of micelles can switch the Aβ monomers more towards the offpathway aggregates. We further assume that one micelle binds four Aβ monomers at once to form the offpathway species ${A}_{4}^{\prime}$; such ${A}_{4}^{\prime}$ oligomers cannot aggregate any further. This is because our experimental observations at FA_{ m } range point to the formation of ${A}_{4}^{\prime}\mathrm{s}$ which do not aggregate to form fibrils.

3.
Similarly at FA_{ pm } range, both on and offpathway reactions occur simultaneously. The reaction mechanism for ${A}_{4}^{\prime}$ oligomer formation is similar as in the previous step. However, in this case the oligomer ${A}_{4}^{\prime}$ further aggregates to form 12–24 mers.
Onpathway reaction model
The onpathway reaction model is motivated by our previous works in refs 9, 10, 47, 52, 56. Here, two different sets of reactions are considered. Firstly, the Aβ monomers form higher order aggregates (A_{12}) through monomer addition and eventually form fibrils; the modeling abstraction considers A_{12} as fibrils, F. These reactions mainly depend on the monomer concentration and are slow being termed as prenucleation. Next, the higher order oligomers (A_{12}) react with the monomers and other onpathway aggregates (A_{2}, …, A_{11}) and get elongated. The rates of these reactions depend on both monomer and fibril concentration. The prenucleation stage is slower, whereas the elongation phase is rapid, thereby causing the sigmoidal growth of the fibril concentration over time. The nucleation number of onpathway reactions is taken as 12 as reported in ref. 48; these reactions are shown in AppendixB under Supplementary Information.
FA _{ n } reaction model
At the FA_{ n } range, we assume the reactions to be exactly similar to the onpathway model; however the rate constants for each phase of the sigmoidal ThT growth curve were varied by a factor of K. The corresponding reaction model is shown in AppendixC under Supplementary Information.
FA _{ m } reaction model
The proposed offpathway reaction set is quite different from the onpathway model and motivated by the experimental observations. In this model, at FA_{ m } range, the four Aβ monomers directly convert into the offpathway oligomer (${A}_{4}^{\prime}$) by reacting with micelles, L. These ${A}_{4}^{\prime}\mathrm{s}$ can not able elongated any further. These reactions are shown in AppendixE under Supplementary Information.
FA _{ pm } reaction model
The modeling of offpathway reactions at FA_{ pm } zone is more challenging however; there are no existing models for offpathway reactions at this zone. Furthermore, the behavior of aggregation, although sigmoidal, is much dissimilar than the onpathway reaction in terms of the faster timescale involved; the formation of 12–24 mers in this zone exhibit considerably less lag time and saturation time. We consider the following reactions at this range. In the first stage denoted as primary nucleation, the ${A}_{4}^{\prime}$ form higher order oligomers (${A}_{12}^{\prime}$) by monomer addition (i.e., adding A_{1}). In the next stage, ${A}_{12}^{\prime}$ further elongates with the intermediate oligomers (${A}_{4}^{\prime}$–${A}_{11}^{\prime}$) to form higher order oligomers (${F}_{1}^{\prime}$); this stage is termed as the elongation stage. Note that a ${F}_{1}^{\prime}$ is a modeling abstraction and is of variable length; it’s length ranges from 16 mers to 23 mers (considering the addition of ${A}_{4}^{\prime},\mathrm{...},{A}_{11}^{\prime}$ each to A_{12}). This results in a 6fold increase in the oligomer size hosted by the pseudomicelles as compared to the micelles (which can only host upto ${A}_{4}^{\prime}$, whereas pseudomicelles can host 23 mers) and is consistent with the experimental observations on the diameters of pseudomicelles and micelles. We next consider such ${F}_{1}^{\prime}$ oligomers to laterally associate and create bigger oligomers (${F}_{4}^{\prime}$); we term this stage as lateral association. As experimentally validated in Fig. 6, there is approximately a fourfold increase in oligomer size towards the beginning of the FA_{ pm } dynamics after which the oligomer size goes down to that of ${A}_{12}^{\prime}$ and ${F}_{1}^{\prime}$; hence it is highly likely that the laterally associated ${F}_{4}^{\prime}$ undergoes a secondary fragmentation into the lower order oligomer ${F}_{1}^{\u2033}$.
It is noteworthy that ${F}_{1}^{\u2033}$ is a structurally different species than ${F}_{1}^{\prime}$; the latter can laterally associate while the former cannot. This assumption was necessary to fit with the experimentally observed ThT kinetics data. Considering the secondary fragmentation to produce ${F}_{1}^{\prime}$ instead of ${F}_{1}^{\u2033}$ could not correlate the simulation plots with the experimentally observed ThT dynamics. The set of nearCMC reactions are shown in AppendixD under Supplementary Information.
Thus, the model at the nearCMC range deals with several rate constants and parameters. There are four rate constants for onpathway reactions: (i) forward and backward rate constants of prenucleation reaction (k_{ nuon }, k_{ nuon_}) from the onpathway; (ii) forward and backward rate constants of elongation reactions (k_{ fbon }, k_{ fbon_}) from the onpathway; (iii) forward and backward rate constants of prenucleation in the offpathway (k_{ con } and k_{ con_}); (iv) forward and backward rate constants of nucleation reactions (k_{ nouff }, k_{ nouff_}); (v) forward and backward rate constants of elongation reactions (k_{ fboff }, k_{ fboff_ }) (vi) forward and backward rate constants of lateral association reactions (k_{ eloff }, k_{ eloff_ }); (vii) forward and backward rate constants of secondary fragmentation (k_{ fagoff }, k_{ fagoff_ }); (viii) the fatty acid effect on the onpathway rate constant parameter, K; and finally (ix) the pseudomicelle concentration: p. Note that, the estimation of the pseudomicelle concentration from the CMC values of fatty acids poses a different problem and needs controlled experimentation; however, it is biophysically not possible to exactly measure the pseudomicelle concentration. We could only measure the diameter of pseudomicelles. Hence we considered p as a free parameter along with the other rate constants to estimate the pseudomicelle concentration.
Parameter estimation
It is difficult to estimate the proper rate constant values of all these parameters at once. Hence, we follow our divide and conquer strategy from ref. 10 to determine all the rate constants step by step. First, we matched the experimental data of the control experiment with the simulation considering the onpathway reaction setup only and estimated the four rate constants involved in the onpathway reactions. Subsequently, we used these values in the combined onoffpathway model to estimate all the additional rate constants. More precisely, our simulation involves the steps below:

1.
First, determine the rate constants for the onpathway reactions from control experiment data.

2.
Estimate the parameter K for the FA_{ n } experimental data by using the estimated onpathway rate constants.

3.
Estimate the rate constants of offpathway aggregation with FA_{ pm } experimental data utilizing the estimated onpathway rate constants above.

4.
Validate the formation of offpathway oligomers (${A}_{4}^{\prime}$) in FA_{ m } concentration using all the estimated rate constants.
We first calculated the reaction flux for all reactions at a particular stage; then using these reaction fluxes, the differential equations for the rate of change of concentration is formulated for each oligomer. Next, these differential equations were solved using MATLAB’s ode solver and the R^{2} value between the simulated curve and experimental data was calculated for the different rate constant combinations. We solved these differential equations for various rate constant combinations with each rate constant ranging from 10^{−5} to 10^{5} and the pseudomicelle concentration p from 1 to 100 μM with multiples of 5. After that, these rate constants were manually finetuned to better match the experimental data and obtain better rate constant estimates. The bestfitted simulation parameters are taken as estimated rate constants of these reactions; each of these parameters are reported in the Appendices under the corresponding reaction models. Moreover, all the rate constants estimated here by fitting with experimental ThT intensity plots required a mapping of the cumulative effects of the concentration of higher order ThT positive oligomers to the ThT values; such mapping is explained in AppendixF in the Supplementary Information.
Additional Information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.
Murphy, R. M. Peptide aggregation in neurodegenerative disease. Annu Rev Biomed Eng 4, 155–174 (2002).
 2.
Thirumalai, D., Klimov, D. K. & Dima, R. I. Emerging ideas on the molecular basis of protein and peptide aggregation. Curr Opin Struct Biol 13, 146–159 (2003).
 3.
Ross, C. A. & Poirier, M. A. Protein aggregation and neurodegenerative disease. Nat. Med. 10, Suppl, S10–17 (2004).
 4.
Harper, J. D. & Lansbury, P. T. Jr. Models of amyloid seeding in Alzheimer’s disease and scrapie: mechanistic truths and physiological consequences of the timedependent solubility of amyloid proteins. Annu Rev Biochem 66, 385–407 (1997).
 5.
Lomakin, A., Teplow, D. B., Kirschner, D. A. & Benedek, G. B. Kinetic theory of fibrillogenesis of amyloid betaprotein. Proc Natl Acad Sci USA 94, 7942–7947 (1997).
 6.
Murphy, R. M. & Pallitto, M. M. Probing the kinetics of betaamyloid selfassociation. J Struct Biol 130, 109–122 (2000).
 7.
Nichols, M. R. et al. Growth of betaamyloid(140) protofibrils by monomer elongation and lateral association. Characterization of distinct products by light scattering and atomic force microscopy. Biochemistry 41, 6115–6127 (2002).
 8.
Morris, A. M., Watzky, M. A. & Finke, R. G. Protein aggregation kinetics, mechanism, and curvefitting: a review of the literature. Biochim Biophys Acta 1794, 375–397 (2009).
 9.
Ghosh, P., Datta B. & Rangachari V. Computational predictions for the lagtimes and nucleation mass involved in Aβ42 peptide aggregation. Proceedings of the International Conference on Bioinformatics Models, Methods and Algorithms (BIOINFORMATCIS) 312–316 (2012).
 10.
Ghosh, P., Kumar, A., Datta, B. & Rangachari, V. Dynamics of protofibril elongation and association involved in Aβ42 peptide aggregation in Alzheimer’s disease. BMC Bioinformatics 11(Suppl 6), S24 (2010).
 11.
Saric, A., Chebaro, Y. C., Knowles, T. P. & Frenkel, D. Crucial role of nonspecific interactions in amyloid nucleation. Proceedings of the National Academy of Sciences of the United States of America 111, 17869–17874 (2014).
 12.
Buell, A. K. et al. Solution conditions determine the relative importance of nucleation and growth processes in αsynuclein aggregation. Proceedings of the National Academy of Sciences 111, 7671–7676 (2014).
 13.
Cohen, S. I. et al. Proliferation of amyloidbeta42 aggregates occurs through a secondary nucleation mechanism. Proceedings of the National Academy of Sciences of the United States of America 110, 9758–9763 (2013).
 14.
Arosio, P., Knowles, T. P. & Linse, S. On the lag phase in amyloid fibril formation. Physical chemistry chemical physics: PCCP 17, 7606–7618 (2015).
 15.
Powers, E. T. & Powers, D. L. Mechanisms of protein fibril formation: nucleated polymerization with competing offpathway aggregation. Biophysical journal 94, 379–391 (2008).
 16.
Powers, E. T. & Powers, D. L. The kinetics of nucleated polymerizations at high concentrations: amyloid fibril formation near and above the ”supercritical concentration. Biophysical journal 91, 122–132 (2006).
 17.
Kirkitadze, M. D., Condron, M. M. & Teplow, D. B. Identification and characterization of key kinetic intermediates in amyloid betaprotein fibrillogenesis. J Mol Biol 312, 1103–1119 (2001).
 18.
O’Nuallain, B., Williams, A. D., Westermark, P. & Wetzel, R. Seeding specificity in amyloid growth induced by heterologous fibrils. J Biol Chem 279, 17490–17499 (2004).
 19.
Wood, S. J., Maleeff, B., Hart, T. & Wetzel, R. Physical, morphological and functional differences between ph 5.8 and 7.4 aggregates of the Alzheimer’s amyloid peptide Abeta. J Mol Biol 256, 870–877 (1996).
 20.
Fraser, P. E., Darabie, A. A. & McLaurin, J. A. Amyloidbeta interactions with chondroitin sulfatederived monosaccharides and disaccharides: implications for drug development. J Biol Chem 276, 6412–6419 (2001).
 21.
Nichols, M. R., Moss, M. A., Reed, D. K., Hoh, J. H. & Rosenberry, T. L. Rapid assembly of amyloidbeta peptide at a liquid/liquid interface produces unstable betasheet fibers. Biochemistry 44, 165–173 (2005).
 22.
Kumar, A. et al. Nonesterified Fatty Acids Generate Distinct Lowmolecular Weight Amyloidβ(Aβ42) Oligomers along pathway Different from Fibril Formation. PLoS One 6, e18759 (2011).
 23.
Smith, A. M., Jahn, T. R., Ashcroft, A. E. & Radford, S. E. Direct observation of oligomeric species formed in the early stages of amyloid fibril formation using electrospray ionisation mass spectrometry. J Mol Biol 364, 9–19 (2006).
 24.
Cleary, J. P. et al. Natural oligomers of the amyloidbeta protein specifically disrupt cognitive function. Nat Neurosci 8, 79–84 (2005).
 25.
Chromy, B. A. et al. Selfassembly of Abeta(142) into globular neurotoxins. Biochemistry 42, 12749–12760 (2003).
 26.
Walsh, D. M., Klyubin, I., Fadeeva, J. V., Rowan, M. J. & Selkoe, D. J. Amyloidbeta oligomers: their production, toxicity and therapeutic inhibition. Biochem Soc Trans 30, 552–557 (2002).
 27.
Liu, C. et al. Outofregister betasheets suggest a pathway to toxic amyloid aggregates. Proc Natl Acad Sci USA 109, 20913–20918 (2012).
 28.
Gellermann, G. P. et al. Abetaglobulomers are formed independently of the fibril pathway. Neurobiol Dis 30, 212–220 (2008).
 29.
Rangachari, V. et al. Amyloidβ (142) Rapidly Forms Protofibrils and Oligomers by Distinct Pathways in Low Concentrations of Sodium Dodecylsulfate. Biochemistry 46, 12451–12462 (2007).
 30.
Necula, M., Kayed, R., Milton, S. & Glabe, C. G. Small molecule inhibitors of aggregation indicate that amyloid beta oligomerization and fibrillization pathways are independent and distinct. J Biol Chem 282, 10311–10324 (2007).
 31.
Goldsbury, C., Frey, P., Olivieri, V., Aebi, U. & Muller, S. A. Multiple assembly pathways underlie amyloidbeta fibril polymorphisms. J Mol Biol 352, 282–298 (2005).
 32.
Bitan, G. et al. Amyloid beta protein (Abeta) assembly: Abeta 40 and Abeta 42 oligomerize through distinct pathways. Proc Natl Acad Sci USA 100, 330–335 (2003).
 33.
Williams, T. L., Day, I. J. & Serpell, L. C. The effect of Alzheimer’s Abeta aggregation state on the permeation of biomimetic lipid vesicles. Langmuir 26, 17260–17268 (2010).
 34.
Kim, S. I., Yi, J. S. & Ko, Y. G. Amyloid beta oligomerization is induced by brain lipid rafts. J Cell Biochem 99, 878–889 (2006).
 35.
Mandal, P. K. & Pettegrew, J. W. Alzheimer’s disease: soluble oligomeric Abeta(140) peptide in membrane mimic environment from solution NMR and circular dichroism studies. Neurochem Res 29, 2267–2272 (2004).
 36.
Yong, W. et al. Structure determination of micellelike intermediates in amyloid betaprotein fibril assembly by using small angle neutron scattering. Proc Natl Acad Sci USA 99, 150–154 (2002).
 37.
Kakio, A., Nishimoto, S., Yanagisawa, K., Kozutsumi, Y. & Matsuzaki, K. Interactions of amyloid betaprotein with various gangliosides in raftlike membranes: importance of GM1 gangliosidebound form as an endogenous seed for Alzheimer amyloid. Biochemistry 41, 7385–7390 (2002).
 38.
ChooSmith, L. P. & Surewicz, W. K. The interaction between Alzheimer amyloid beta(140) peptide and ganglioside GM1containing membranes. FEBS Lett 402, 95–98 (1997).
 39.
ChooSmith, L. P., GarzonRodriguez, W., Glabe, C. G. & Surewicz, W. K. Acceleration of amyloid fibril formation by specific binding of Abeta(140) peptide to gangliosidecontaining membrane vesicles. J Biol Chem 272, 22987–22990 (1997).
 40.
Carpentier, Y. A. & Hacquebard, M. Intravenous lipid emulsions to deliver omega 3 fatty acids. Prostaglandins Leukot Essent Fatty Acids 75, 145–148 (2006).
 41.
Schlame, M., Haupt, R., Wiswedel, I., Kox, W. J. & Rustow, B. Identification of shortchain oxidized phosphatidylcholine in human plasma. J Lipid Res 37, 2608–2615 (1996).
 42.
Zhao, H., Tuominen, E. K. & Kinnunen, P. K. Formation of amyloid fibers triggered by phosphatidylserinecontaining membranes. Biochemistry 43, 10302–10307 (2004).
 43.
Mandal, P. K., McClure, R. J. & Pettegrew, J. W. Interactions of Abeta(140) with glycerophosphocholine and intact erythrocyte membranes: fluorescence and circular dichroism studies. Neurochem Res 29, 2273–2279 (2004).
 44.
Fletcher, T. G. & Keire, D. A. The interaction of betaamyloid protein fragment (1228) with lipid environments. Protein Sci 6, 666–675 (1997).
 45.
Solfrizzi, V. et al. Dietary Fatty Acids in Dementia and Predementia Syndromes: Epidemiological Evidence and Possible Underlying Mechanisms. Ageing Res Rev 9(2), 184–99 (2009).
 46.
Lukiw, W. J. & Bazan, N. G. Docosahexaenoic acid and the aging brain. J Nutr 138, 2510–2514 (2008).
 47.
Strogatz, S. H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (Westview press, 2014).
 48.
Ghosh, P., Vaidya, A., Kumar, A. & Rangachari, V. Determination of Critical Nucleation Number for a Single Nucleation Amyloidβ Aggregation Model. J. Mathematical Biosciences 273, 70–79 (2016).
 49.
Lee, C.C., Nayak, A., Sethuraman, A., Belfort, G. & McRae, G. J. A threestage kinetic model of amyloid fibrillation. Biophysical Journal 92, 3448–3458 (2007).
 50.
Ghosh, P., Ghosh, S., Basu, K., Das, S.K. & Daefler, S. An analytical model to estimate the time taken for cytoplasmic reactions for stochastic simulation of complex biological systems. Granular Computing, 2006 IEEE International Conference on 79–84 (2006).
 51.
Kumar, A. et al. Specific soluble oligomers of amyloidβ peptide undergo replication and form nonfibrillar aggregates in interfacial environments. J Biol Chem 287, 21253–21264 (2012).
 52.
Dean, D. N. et al. Strainspecific Fibril Propagation by an Abeta Dodecamer. Scientific reports 7, 40787 (2017).
 53.
Dean, D. N., Pate, K. M., Moss, M. A. & Rangachari, V. Conformational Dynamics of Specific Abeta Oligomers Govern Their Ability To Replicate and Induce Neuronal Apoptosis. Biochemistry 55, 2238–2250 (2016).
 54.
Hines, W. G. S. Evolutionary stable strategies: a review of basic theory. Theoretical Population Biology 31(2), 195–272 (1987).
 55.
Smith, J. M. Evolution and the Theory of Games (Cambridge University press, 1982).
 56.
Ghag, G., Ghosh, P., Mauro, A., Rangachari, V. & Vaidya, A. Stability analysis of 4species Aβ aggregation model: A novel approach to obtaining physically meaningful rate constants. Applied Mathematics and Computation 224, 205–215 (2013).
Acknowledgements
The authors wish to thank the following agencies for financial support: National Center for Research Resources (5P20RR01647611) and the National Institute of General Medical Sciences (8 P20 GM10347611) from the National Institutes of Health for funding through INBRE (to VR), National Institute of Aging (R15AG046915) (to VR) and NSF Graduate Research Fellowship Program (NSF 1445151) and Gk12 Program at USM (NSF 0947944) (to DND) and NSF 1351786 to PG.
Author information
Affiliations
Department of Computer Science, Virginia Commonwealth University, Richmond, VA, 23284, USA
 Pratip Rana
 & Preetam Ghosh
Department of Chemistry & Biochemistry, University of Southern Mississippi, Hattiesburg, MS, 39406, USA
 Dexter N. Dean
 & Vijayaraghavan Rangachari
Department of Mathematical Science, Montclair State University, Montclair, NJ, 07043, USA
 Edward D. Steen
 & Ashwin Vaidya
Authors
Search for Pratip Rana in:
Search for Dexter N. Dean in:
Search for Edward D. Steen in:
Search for Ashwin Vaidya in:
Search for Vijayaraghavan Rangachari in:
Search for Preetam Ghosh in:
Contributions
P.G., A.V. and V.R. conceived the work. P.R. and P.G. formulated and implemented the EKS simulations. E.D.S. and A.V. formulated and implemented the ROM model. D.N.D. and V.R. conducted the in vitro experiments and corresponding data analysis. All authors contributed in writing the paper.
Competing Interests
The authors declare that they have no competing interests.
Corresponding author
Correspondence to Preetam Ghosh.
Electronic supplementary material
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.