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Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers
Modeling of the inhibitory effect of nanoparticles on amyloid # fibrillation Nirmal Kumar Ramesh, Swathi Sudhakar, and Ethayaraja Mani Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00388 • Publication Date (Web): 19 Mar 2018 Downloaded from http://pubs.acs.org on March 20, 2018
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Modeling of the inhibitory effect of nanoparticles on amyloid β fibrillation Nirmal Kumar Ramesh, Swathi Sudhakar, Ethayaraja Mani* Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai – 600036, India.
KEYWORDS: kinetics, homogeneous aggregation, nucleation, growth, inhibition.
ABSTRACT: Experiments have shown that charged nanoparticles (NP) inhibit, partially or completely, the aggregation of Aβ protein monomers into fibrils. The equilibrium fibril content is found to be inversely proportional to the concentration of NP. In this work, we report a kinetic model for the fibrillation of Aβ protein in the presence of NP. In the model, apart from nucleation, elongation and fragmentation processes, the effect of NP is considered to cause a conformational change to the protein monomer, making the latter incompatible for aggregation. The simulated results explain the growth kinetics of pure Aβ (1-40) protein, and the kinetics in the presence of NP. The NP-monomer interaction considered in the model captures the significant effect of NP on the fibrillation process at a very molar ratio (NP to Aβ monomer) as low as 10-4. The model predictions are compared with two different NP systems namely gold and silica NP. The model can be applied to explain the inhibitory effect of other additives such as small molecules, NP, lipids and surfactants that show a similar inhibition trend for fibril formation of Aβ and other proteins.
1. INTRODUCTION Protein fibrillation has been extensively studied for many decades because of its association with many disorders such as neurodegenerative Alzheimer’s, type-II diabetes, Parkinson’s, Huntington’s, Prion diseases and sickle cell anaemia gelation, to mention a few.1–5 Fibrillation is a generic property of many proteins and polypeptide molecules, where the molecules form fibril-like structures when the surrounding conditions and requirements are met.6,7 However, the existence of functional fibrils has been known in many biological species. The role of these fibrils are diverse and some of them include the formation of bio-filaments8 including actin and tubulin, growth and transformation of cytoskeletal structures, regulation of melanin synthesis9,10 and encryption of long term memory.11 Apart from living systems, aggregation of proteins is also a major challenge in industries and applications where the process can interfere with the production of therapeutic drugs,12 and decreases the shelf life of bottled proteins.13 Aggregates of proteins in-vitro are characterized by their high content of β-sheet rich structures where the β-strands are aligned perpendicular to the axis of fibrils. Many experimental techniques are used to study and analyse the properties of protein fibrils.14 Aggregation can be monitored by Thioflavin T or Congo red dye which selectively bind to the β-sheets of the fibrils15,16 and the fluorescence intensity plots can be used to study the kinetics of fibril formation. With more sophisticated instruments like transmission electron microscopy (TEM) and atomic force microscopy (AFM), the microstructure of protein aggregates can be imaged in real space.17,18 Typically, a sigmoidal growth profile is observed in the experimental plots for the mass of aggregates for a wide range of
proteins. It has a lag phase associated with a lag time ‘τ’, which can be reduced or eliminated by the addition of preformed fibrils also called as seeding.19 The lag time shows an inverse dependency on initial protein monomer concentration. Following the lag phase, a rapid growth phase ensues, characterized by an exponential profile that prolongs until a final plateau phase is reached.20 The existing mechanisms on fibrillation kinetics of protein aggregation are mainly nucleation dependent polymerization models (NDP)21–23 with the inclusion of secondary processes such as fragmentation24 and secondary nucleation25 in later contributions. In primary nucleation models, a critical nucleus is formed from the self-assembly of protein monomers, which then grows into fibrils by subsequent monomer addition.26 Some variations of NDP models include the conformational conversion of nucleus8 and the formation of nucleus from micelles.27 The NDP models predict the kinetics of actin, sickle cell haemoglobin and other protein fibrils, but they are not able to explain some of the features such as the more pronounced lag phase and a rapid exponential growth phase that could be observed in other in-vitro experiments.28,29 Subsequently, Wegner et al24 included the fragmentation mechanism, where a larger fibril disintegrates into two smaller fibrils. The initial observations to support fragmentation mechanism came from the studies reporting that sample agitation and sonication of seed material accelerate the rate of fibrillation.30,31 Fragmentation was experimentally observed in the fibrillation of prion proteins.32 Secondary nucleation is another mechanism of formation of critical nucleus, where the size of nucleus may be smaller than the primary nucleus.33 The energy barrier for the formation of nucleus by secondary pathway can be orders of magnitude lower than primary nu-
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cleation pathways.25 The review papers by Morris et al34 and Gillam et al35 may be referred to for more details of models and mechanisms of protein fibrillation. These fibrils are known to be toxic in many cell lines, which supports the argument that the fibrils cause several neurodegenerative disorders mentioned previously. Several recent reports has demonstrated the complete or partial inhibition of fibril formation due to the interaction with nanoparticles (NP), small molecules, functionalized polymers, peptides, lipids and surfactants.36–38 A large volume of experimental work are now available on the inhibition kinetics of protein fibrillation using different kinds of additives. We have recently showed that oppositely charged inhibitors (with respect to the charge on the protein residues that are responsible for β sheet) can intercept and inhibit the fibrillation process.39 One of the striking features of NP-based inhibitor is that only very small quantity of the order of 10-3-10-4 per monomer is required. Significant reduction in the fibril content is achieved with NP at very low concentrations. Such low levels of NP offer potential advantage to develop NP-based therapeutics for protein fibrillation. The kinetic curves obtained in the presence of NP show increased lag phase and reduced amount of fibril content. However, the available models in literature for pure protein fibrillation can not explain the kinetics in the presence of inhibitors. To the best of our knowledge, there are no models available to capture the effect of NP on the inhibition kinetics of aggregation. A kinetic model that is applicable for protein fibrillation in the presence of NP is desired. The model should be able to explain the macroscopic experimental data namely the effect of NP concentration on the lag time, equilibrium fibril content and the overall kinetic data, under the condition that the NP to protein molar ratio in the range of 10-4:1. One of the key challenge in developing such a model will be the role of NP on the protein monomers, oligomers and fibrils. The interaction between gold nanoparticles and Aβ protein was extensively studied in experiments and molecular dynamics simulation and was reported in one of our previous works.39 Using ATRFTIR spectorscopy, bond vibrations that are characteristics of α-helix and β-sheet structures of Aβ protein are used to quantify the secondary structural information of the protein. For instance, the protien sample treated with 15 nm gold NP of 10 nM concentration showed a decrease in the β structure content from 68% to 25%, and correspondingly an increase in the αhelical structure content from 22% to 45% after 21 h of incubation time, when compared with the control sample.39 Molecualr Dynamics simulation results showed that the Lys16 and Lys28 residues of Aβ(1-40), responsible for the fibrillation, adsorbed strongly on gold NP surface, thus causing secondary structural changes in protein, thereby inhibiting their tendency for fibrillation. Further, the possibility of salt-bridge formation between Asp23 and Lys28 is hindered as the latter is confined onto the NP surface. The salt-bridge between these two residues is required to form β sheet structure. The main objective of this paper is the development of a kinetic model with minimal elementary steps involved in the fibrillation process in the presence of NP. In this work, experiments are carried out to glean the role of nanoparticles in the protein fibriallation, and the inferred mechanisms are included in the
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model based on population balance equations. We show that the proposed model is able to capature the kinetics of fibrillation for two different NP sytems namely negatively charged gold NP and silica NP. The model will be useful in predicting the required dosage of NP required to effect reduction in the fibril content within a tolerated limit. 2. EXPERIMENTAL SECTION 2.1 Synthesis of gold nanoparticles Detailed procedure for the synthesis of negatively charged gold nanoparticles is described elsewhere.39 Briefly, a 50 mL sample of aqueous HAuCl4.3H2O (1mM) was prepared in a 250 mL flask. The solution was brought to boil under reflux while being stirred and 5 mL of 38.8 mM aqueous solution of sodium citrate was added. Then the reaction was continued until the solution reached a wine-red color, indicating the formation of gold seed nanoparticles of about 15 nm diameter. For the synthesis of 40 nm gold nanoparticles, 1 mL of 15 nm gold nanoparticles solution and 450 µL of 42 mM sodium citrate are added to 100 mL of water, and the mixture was stirred and heated to reflux. Then a 50 ml of 0.1 M HAuCl4.3H2O was added. The heating was continued for 15 min, and then the suspension was removed from heat and stirred until it was cooled to room termperature. UV-VIS spectrum and TEM images (S1 in the Supplementary Information) of the gold nanoparticles showed the size of the nanoparticles is 40 nm. 2.2 Nanoparticle- protein interaction Experimental details for the study the kinetics of Aβ fibrillation in the presence of nanoparticles is reported earlier (also see Figure S2-S3 in the Supplementary Information).39 Here, it is shown that it was possible to measure concentration of protiens that are in the fibrils (Cf), attached to nanoparticles (CX) and in inactive forms (CA*). To measure concentrations of these populations, nanoparticles were treated with the protein for 21 h with initial concentration of 10 µM and then the nanoparticles were separated by using millex-vv pvdf syringe filters (0.02 µM). The concentration of the proteins adsorbed in the nanoparticles were quantified using standard brandford assay.40,41 This assay is usually used to measure the concentration of protein in a sample. The principle is that protein molecules binds to Coomassie dye resulting in a color change from brown to blue where the absorbance can be measured at 595 nm under UV-spectroscopy technique. To 800 µl of each protein sample 200 µl Bradford reagent were added and incubated at room temperature for 20 minute. Protein solutions were normally assayed in triplicate manner. The standard solutions with concentration range from 1 to 30 µM were prepared for reference. The supernatant were collected and the total active proteins in the form of fibrils in the supernatant were measured using congo red assay.42 The congo red is the dye which binds to the beta sheet structure (fibrillar proteins) and its absorbance is proportional to the mass of protein only in the fibrillar form. The deactived proteins (conformationally altered by nanoparticles) in the supernatant were obtained from the difference between initial protein concentration (CA0) and the sum of protein adsorbed on the nanoparticle and proteins in fibrillar form (CA* = CA0 – CX - Cf). The congored assay is used to avoid inner filter effects, if any, caused by gold nanoparticles in ThT Fluorescent assay (in our experiments
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there is no significant innerfilter effects by nanoparticles as the concentration used is in nanomolar range). Both the ThT and congo assay yielded same results.
3. MATHEMATICAL MODEL A population balance based model is derived to describe the fibrillation of protein monomers in the presence of physically interacting NP. The fibrillation process of protein alone has been described by many models.21–23,34,35 They consist of mainly three steps: (i) nucleation, (ii) elongation and (iii) fragmentation. The nucleation step is described within the framework of classical nucleation theory, wherein n* number of monomers form a critical nucleus and elongation/growth ensues. Any cluster of size less than the critical nucleus size (n*) dissociates into monomers or smaller clusters, as the former is thermodynamically unfavored. Models for elongation of a fibril involve either addition of monomers or another fibril. In the fragmentation step, a fibril randomly undegoes breakage into two smaller fibrils. These elementary processes are shown as a schematic in Figure 2.
(A)
(B) Figure 1. Concentrations of the fibrils, deactived and proteins adsorbed onto the nanoparticle in (A) 21 h and (B) 40 h for different concentration of 15 nm gold nanoparticles. Figure 1A shows the proportions of proteins – deactived (A*), adsorbed on nanoparticles (X) and fibrils (f) – after 21 h of incubation with different concentrations of gold nanoparticles. It can be observed that the amounts of protein adsorbed on the nanoparticle (X) and deactivated protein (A*) increase with the concentration of nanoparticles. As shown in Figure 1B, when measured at 40 h, the concentration of the deactivated proteins (CA*) increased with a concomittent decrease in the protein attached to nanoparticles (CX). These data suggests that the adsorbed proteins on nanoparticles are detached from the surface into the solution in deactiveed form. Therefore, adsorption and desorption processes are characterized by a respective timescale or kinetic rate constant. However, the protein fibrillatin is nearly complete at 21 h, and the changes in the concentration of CA* and CX between 21 – 40 h do not affect the fibril content. Were the desorbed protein be in active form, there should have been an increase in the fibril content at 40 h compared to 21 h. Hence, nanoparticle-protein interaction leads to conformational changes on the protein and subsequent desorption. Thus the nanoparticles act like a catalyst to deactivate fresh monomer proteins. These observations are invoked in the model developed subsequently.
None of the existing models include the effect of an additive such as nanoparticle, surfactant or polymer that inhibit the fibrillation process, although there are ample experimental data to this effect.43–45 Inspired by the experimental observation as detailed in the Experimental section, the nanoparticleprotein interaction is modelled in the model such that a protein monomer binds to a nanoparticle and undergoes a change in its conformation (secondary structures), which makes the monomer unable to aggregate further. One of the possibilities is that the nanoparticle converts some of the beta-sheet forming residues into alpha-helix structures, thereby the hydrophobic interactions causing the fibrillation may be hindered.39,46–48 In the Introduction, it was highlighted that the alpha-helix content was increased in the presence of nanoparticles with a concomittent decrease in the beta-sheet structure.39 Short-time scale structural changes in the presence of nanoparticles is further shown in Figure S4 (Supplementary Information). However, these observations are based on macroscopic measurements from spectroscopic techniques, whereas the molecular level structural changes are unknown. Even the molecular dynamics simulation showed the adsorption and structural changes on the protein while adsorbed but due to the long time scales involved in the desorption, MD simulations could not offer structural information of the desorbed proteins. The nanoparticle-protein interaction is effectively included in the present model such that the nanoparticle acts like a catalyst to induce structural changes to the protein monomer. The cascade of elementary processes can be described by a set of reactions. In these reactions “A1” refers to protein monomer, “Ai” refers to a fibril containing ‘i’ number of monomers. Aggregation of monomers reversibly form dimer, trimer and by subsequent monomer addition a nucleus of size ‘n*’, also known as the smallest fibril. These pre-nucleation steps are considered to be reversible with forward and backward rate constant as k1, as shown in Eq. 1. The elongation process takes place irreversibly with the addition of monomers to fibrils. The rate constant involved in elongation of fibrils is k2 (Eq. 2) Thus a thermodynamic favorability is imposed on the elongation of fibrils over the nucleation steps. Unlike many other models, which have a single step nucleation, we consider a multiple-step nucleation model. Recent experiments have sug-
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gested that the oligomers of size smaller than the nucleus may be more toxic than the mature fibrils.49,50 Therefore, it may be desired to track the concentration of oligomers in the system.
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protein that is incapable of participating in protein aggregation. This process is governed by two rate constants k3 and k4. 3 4 A1 + B → X → B + A1*
k
k
…….. (4)
In writing kinetic equation to describe the nanoparticle-protein interaction, we note that i) it can not be an elementary reaction as the nanoparticle to protein concentration ratios are of the order of 10-4 (as will be discussed in the Results section), and if accounted so, the rate equation is effectivly a psuedo-first order which can not predict the experimentlaly observed effect of nanoparticles, and ii) the there are multiple events in the whole process of protein adsorption, conformation changes and desorption. Therefore, there should be some complex reactions and intermediates involved in the overall interaction between protein and nanoparticles. Therefore, we write the rate equation of species ‘X’ with a power law (partial order) dependence on concentration of nanoparticle as:
dX = k3 A1 B p − k4 X dt
...….. (5)
The rate of change of free NP concentration is given by the following equation:
dB = −k3 A1 B p + k4 X dt Figure 2. A schematic representation of microscopic mechanisms involved in the formation of protein fibrils incubated with NP. The first three steps (a-c) are the nucleation, elongation and fragmentation mechanisms, respectively. The last step (d) describes the interaction between negatively charged NP and positively charged protein monomers leading to the conformational changes to the protein monomer.
Eq. 3 represents the fragmentation of a fibril into two smaller fibrils. Any fibril of size greater than the nucleus size ‘n*’ is allowed to fragment into smaller species which can even be the size of a monomer. The rate of fragmentation kf is considered to be independent of the length of the fibril to keep the model simple. k1 k1
Ai + A1 ←→ Ai+1
for 1 ≤ i < n *
…..….. (1)
…… (6)
Then the differential equations for protein species of all sizes are : i) monomer ∞ dA1 dA = − ∑ i i − k3 A1B p dt dt i=2
…..….. (7)
This equation is obtained from the mass (or number) balance of protein and from the reaction scheme given in Eq. 4. ii) dimer ∞ dA2 = k1 (0.5A12 − A2 − A1 A2 + A3 ) + 2k f ∑ Ai dt i=n
………(8)
iii) aggregates of size 2 < i < n *
∞ dAi = k1 (A1 Ai−1 − Ai − A1 Ai + Ai+1 ) + 2k f ∑ Aj dt j=n
……….(9)
iv) critical nucleus n* k
2 Ai + A1 → Ai+1
k
for i ≥ n *
f Ai → A j + Ai− j where i ≥ n*; j ≥ 1
…..….. (2)
.….….. (3)
To describe the catalyst-like mechanism of NP as discussed earlier, Eq. 4 describes the interaction between NP and protein monomer. Here, B refers to the NP and X is the nanoparticle with monomer in a bound state, while A1* is the deactivated
∞ dAn* = k1 (A1 An*−1 − An* ) − k 2 A1 An* − k f (n * −1)An* + 2k f ∑ Ai dt i=n*+1 ………..... (10) v) aggregates of size i > n *
dAi = k2 (A1 Ai−1 − A1 Ai ) − k f (i − 1)Ai + 2k f dt
∞
∑A
i
j=i+1
..... (11)
The first two terms in Eq. 11 represents the formation of Ai from Ai-1 and its disappearance by elongation process to form
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Ai+1. The third term is the fragmentation of fibril Ai which can break at ‘i-1’ points and the fourth term accounts for the appearance of Ai due to the fragmentation of larger sized fibrils. Moments of the distribution of fibril sizes {Ai} can be used to calculate statistical quantities such as total mass of fibrils, mean and variance of size as a function of time. The kth moment is defined as ∞
()
M k (t) = ∑ i k Ai t
………… (12)
i=n* ∞ dA dM k (t) = ∑ i k i dt dt i=n*
M 1 (t) A1 0
()
………… (14) where A1(0) is the concentration of the monomer at time t = 0. The mean size of the fibrillar population or the mean number of monomers per fibril is calculated from the zeroth and first moments as:
n (t) =
M 1 (t) M 0 (t)
………… (15)
The variance of size of the fibrils is obtained from the first 3 moments as:
σ 2 (t) =
2 M 2 (t) − n(t) 0 M (t)
………… (16)
Using the definition of moments and through equations (7-11), we derive the following differential moment equations:
dM 0 = k1 ( A1 An*−1 − An* ) + k f ( M 1 − (2n * −1) M 0 ) dt dM 1 = k1 ( A1 An*−1 − An* )n * + ( k2 A1 − n( n − 1) k f ) M 0 dt n−1 dA1 dA dM 1 = −∑ i i − − k3 A1 B p dt dt dt i=2
dA2 = k1 (0.5A12 − A2 − A1 A2 + A3 ) + 2k f M 0 dt
……. (17)
…. (18)
It is noted that to obtain standard deviation of size (σ) viz. Eq. 16, the M2(t) is required. However, the differential equation for M2(t) could not be derived in a closed form for this kinetics. Therefore, in cases where σ is to be calcualted, M2(t) is calculated vis Eq. 12 after solving all differential equations for all the species via Eqs: 5, 6, 7 – 11. 4. RESULTS The model described through Eqs. 5, 6, 17–22 contains 7 unknowns namely the rate constants k1-k4, kf, and parameters n* and p. Of these, the parameters k1, k2, and kf characterizing the aggregation of proteins are obtained by fitting the model (in the absence of nanoparticles) with the experimental data39 corresponding to an initial protein monomer concentration of 30 μM. The fitted parameters are given in Table 1. For strongly aggregating proteins like Aβ, the critical nucleus size is very small and is in the range of 2-3. We have fixed n* = 3. The orders of magnitude of the rate constants are comparable to other proteins that undergo similar fibrillation such as insulin, beta-lactoglobulin and so on.51 The comparison of the experimental data and the model fit on the temporal evolution of mass fraction of Aβ is shown in Figure 3. The figure also shows a comparison of the simulated data with experimental data for a different initial concentration (20 μM) with the same set of rate constants given in Table 1. It can be seen that the model is able to predict the experimental data on protein fibrillation in the absence of nanoparticles reasonably well. As the initial concentration of the monomer is increased, the lag time decreased as there is a higher probability of nucleation at higher monomer concentration. This trend is captured well by the model. The time required to reach complete conversion of monomers to fibril decreases with increasing the protein monomer concentration, and this feature is also explained by the model.
………… (19)
……...… (20)
For 2 < i < n *
dAi = k1 (A1 Ai−1 − Ai − A1 Ai + Ai+1 ) + 2k f M 0 dt …...…… (21) and,
…………. (22) Eqs: 5, 6, 17 – 22 are solved simultaneously with the initial conditions such that A1 (t=0) = A1(0) and B(t=0) = B(0), while the rest of the species concentrations being zero. The simulations are performed in Matlab with ode45 solver. The typical cpu time taken is about 3 minutes in an Intel i7 2.3 GHz Processor to run a simulation for a 30 h kinetics with 30 µM protein incubated with NP. The number of equations to be solved depends on the choice of n*: 7 for n* = 3, for instance.
………… (13)
The summations over ‘i’ starts from ‘n*’ as only species of size larger than or equal to n* are required to account for characterizing the fibril population. The fraction of monomers converted into fibrils is obtained from the first moment as:
f (t) =
dAn* = k1 (A1 An*−1 − An* ) − k2 A1 An* − k f (n * +1)An* + 2k f M 0 dt
Next we test the validity of the model to explain the fibrillation kinetics in the presence of NP. In this regard, Sudhakar et al39 reported the effect of nanoparticle (gold and silica) concentration on the fibrillation kinetics of Aβ protein. Figures 4 and 5 show the experimental data for 15 nm size gold and the same sized silica NP, respectively. It may be noted that in these set of experiments the initial protein concentration is fixed at 30 μM, while the concentration of NP is varied in the range of 1.36 – 8 nM. The molar ratio between nanoparticle to protein monomer to is of the order of 10-4:1. To use the model to predict the experimental data, we still have 3 unknowns namely k3, k4 and p. Assuming that the final mass of monomers that are not utilized for fibril formation is in the form of
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1
A=30µM A=20µM
0.8
1
Fibril mass fraction
A1*, and the species ‘X’ is too short lived and converts instantaneously to A1*, we estimated the value of ‘p’ in equation (5) to be in the range of 0.25 to 0.55 for gold and silica NP. The values of ‘p’ used in the simulations for gold and silica NP are given in Table 1.
Control 1.36nM 3.2nM 6.4nM 8nM
0.8 0.6 0.4 0.2
0.6 0 0
0.4
10
20
30
Time (h)
0.2 0 0
10
20
30
40
Time (h) Figure 3. Comparison between predictions from the model and experimental data39 on the fraction of monomers in the fibrillar form. The experimental data corresponding to 30 µM is used to fit the rate constants k1, k2 and kf, and these constants are used to predict the data for an initial concentration of 20 µM (colour online).. The fitted parameters k1, k2, and kf from pure protein fibrillation (as in Figure 3) are used in the simulations with NP. The rate constants k3 and k4 are chosen to fit the experimental data corresponding to NP concentration of 1.36 nM. With all the parameters now known, simulations are carried out for other NP concentrations. Figure 4 shows the comparison of simulation and experimental data for the control, and samples incubated with different concentration of gold NP. It can be seen that the power law model is able to predict the experimental data reasonably well although several assumptions have been made in the interaction between protein and NP. The model is able to capture the final fibril content for all NP concentrations quantitatively. Significant differences are seen between experiment and model in the elongation phase, which is governed by the fragmentation rate constant kf. The assumption of fragmentation rate independent of NP may cause this discrepancy. As kf for pure protein aggregation itself has not been measured explicitly, relaxing the above assumption will only increase the number of
Figure 4. Comparison of model and experiments on the inhibition of fibrillation by gold NP at various NP concentrations. The points are from experiments and the lines are simulations. Initial monomer concentration is 30 µM39 (colour online). Table 1. Parameters used in model Parameter
Value
n* k1 k2 kfrag
3 8x10-3 m3/(mol.s) 1.3x104 m3/(mol.s) 1.5x10-8 s-1
p k3 k4
1
Fibril mass fraction
Fibril mass fraction
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Gold
Silica
0.35 11.52x101 0.2
0.53 18.9x101 (m3/mol)p.s-1 0.2 s-1
Control 1.36nM 3.2nM 6.4nM 10nM
0.8 0.6 0.4 0.2 0 0
10
20
30
Time (h) Figure 5. Comparison of model and experiments on the inhibition of fibrillation by silica NP at various NP concentrations. The points are from experiments and the lines are simulations. Initial monomer concentration is 30 µM39 (colour online). unknown parameters further. Since the model includes the most important elementary processes in the inhibition kinetics
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of NP, the simulated kinetic curves are able to show the overall qualitative features of the process.To check the generality of model with other NP, we also performed simulations for silica NP of different concentrations and plotted the growth of fibrils in Figure 5, along with the experimental values. In this case also, the model prediction and experimental data are reasonably comparable to each other. If we take a closer look at the parameter values in Table 1, it reveals that the ‘p’ value is different for gold (0.35) and silica (0.53) NP. This suggests that the gold NP is able to deactivate protein monomer at a slightly faster rate compared to silica NP. Because of this, we observe a large reduction in the fibril content in the sample incubated with gold NP compared to silica NP at the same initial concentrations. The other difference is in the k3 value which can be considered as a measure of the size, shape and material composition of NP that will determine the rate of reaction that converts A1 to X.
The lag time that characterizes the waiting time before stable nuclei form, and it is calculated by drawing a tangent at the point of inflection in the kinetic growth curve and its corresponding intercept at the abscissa.20 While this method works well for homogeneous fibrillation, which exhibits a smooth sigmoidal curve, the presence nanoparticle makes the inflection point difficult to detect. Alternative definition of lag time has been used in such scenarios. For example, Arosio et al53 defined the lag time as the time required to attain a 10% fibril formation of the total initial mass of protein. We adopted this definition of lag time in this work. Experimental data suggest that the lag time increased when nanoparticle concentration is increased. The comparison of lag times from experiments and simulation for different NP concentrations is shown in Figure 7. A good agreement between the lag time of experiments and simulations suggests that the nucleation phase of fibrils is well predicted by the model.
Average Length of Fibril
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Figure 6. Average fibril length plotted using simulation results from model (color online) for various concentrations of gold NP incubated in 30 µM protein solution. The standard deviations are shown only for the control (0 nM) and 10 nM concentration of NPs. In Figure 6, the average size of protein fibrils at any given instant of time is plotted for different gold NP concentrations. This data is obtained from the simulations. The size of fibrils increases steadily by elongation and finally decreases because of the reduction in monomer concentration and increased rate of fragmentation. Due to these two processes, the average size reaches a maximum size. It is interesting to observe that the average fibril sizes are comparable in the samples incubated with various NP concentrations, even though Figure 3 shows a stark difference in the amount of fibrils formed. The standard deviation of the size towards the end point i.e. after 25 h for the control case and the system incubated with 10 nM gold NP are similar. This suggests that NP have an effect only on the quantity of fibrils formed, but not on their average size and its standard deviation. The data on temporal change of average fibril size could not be compared with experimental TEM images52 where the protein fibrils are seldom seen, probably because their number is very low number in short timescales.
Figure 7. Comparison of lag time between model and experiment for the amyloid protein solution incubated with gold NPs (color online). 5. DISCUSSION We had initially considered simple mechanisms to explain the experimental data, yet it was not possible to predict the data for different concentrion of nanoparticles. In our first model, it was assumed that nanoparticles instantaneously adsorb “q” number of protein monomers, thereby reducing the protein monomer concentration to CA = CA0-qCB0, where CA0 and CB0 are the concentration of the monomer and nanoparticles, respecitvley. Simulating the kinetic equations with the effective monomer concentration with the constaint that “q” should be independent of CB0 yielded results which disagreed with experimental data. The next model we had considered was to relax the instantaneous adsorption of monomers on the nanoparticle surface, and to consider it to occur at finite rate as: A+B→X With a rate equation
dCA = −kqC ACB dt
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This type of equation is used to descirbe protein corona formation on nanoparticles. Solving this equation along with protien fibrillation kinetic equations (nucleation, growth and fragmentation) yielded results which are in disagreement with experiments (Figures S5 in Supplementary Information). Sensitivity analysis of the parameter “q” and “k” did not improve the prediction for all nanoparticle concentrations. Only when we introduced a mechanim where there is a continuous adsorption of protein monomers on the nanoparticle surface, changes in the conformation leading to inactive structures (A*) and desorption from nanoparticle surface, we could explain the experimental data satisfactorily for nanoparticles of different types (gold, silica) and at varying nanoparticle concentrations. In many heterogeneous catalyst reaction mechanism there are steps such as external mass transfer, adsoption on the surface of catalyst, reaction on catalyst surface, desorption from catalyst surface and leaving the boundary of catalyst. Often, these steps can be lumped into a single reaction mechanism where we get reaction order in fractions less than 1. The use of fractional order of reaction can be found in Gowda et al54, Qamruzzaman et al55,56 ,Ilyas et al57 and Akram et al58 in oxidative degradation of organic compounds. Therefore, to account for the inhibition of fibrillation by different concentration of NP at very low concentrations, the exponent p
